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[ "Formation and evaporation of quantum black holes from the decoupling mechanism in quantum gravity", "Formation and evaporation of quantum black holes from the decoupling mechanism in quantum gravity", "Formation and evaporation of quantum black holes from the decoupling mechanism in quantum gravity", "Formation and evaporation of quantum black holes from the decoupling mechanism in quantum gravity" ]
[ "Johanna N Borissova [email protected] \nPerimeter Institute for Theoretical Physics\n31 Caroline Street NorthN2L 2Y5WaterlooONCanada\n\nDepartment of Physics and Astronomy\nUniversity of Waterloo\n200 University Avenue WestN2L 3G1WaterlooONCanada\n", "Alessia Platania [email protected] \nPerimeter Institute for Theoretical Physics\n31 Caroline Street NorthN2L 2Y5WaterlooONCanada\n\nKTH Royal Institute of Technology and Stockholm University\nHannes Alfvéns väg 12SE-106 91Nordita, StockholmSweden\n", "Johanna N Borissova [email protected] \nPerimeter Institute for Theoretical Physics\n31 Caroline Street NorthN2L 2Y5WaterlooONCanada\n\nDepartment of Physics and Astronomy\nUniversity of Waterloo\n200 University Avenue WestN2L 3G1WaterlooONCanada\n", "Alessia Platania [email protected] \nPerimeter Institute for Theoretical Physics\n31 Caroline Street NorthN2L 2Y5WaterlooONCanada\n\nKTH Royal Institute of Technology and Stockholm University\nHannes Alfvéns väg 12SE-106 91Nordita, StockholmSweden\n" ]
[ "Perimeter Institute for Theoretical Physics\n31 Caroline Street NorthN2L 2Y5WaterlooONCanada", "Department of Physics and Astronomy\nUniversity of Waterloo\n200 University Avenue WestN2L 3G1WaterlooONCanada", "Perimeter Institute for Theoretical Physics\n31 Caroline Street NorthN2L 2Y5WaterlooONCanada", "KTH Royal Institute of Technology and Stockholm University\nHannes Alfvéns väg 12SE-106 91Nordita, StockholmSweden", "Perimeter Institute for Theoretical Physics\n31 Caroline Street NorthN2L 2Y5WaterlooONCanada", "Department of Physics and Astronomy\nUniversity of Waterloo\n200 University Avenue WestN2L 3G1WaterlooONCanada", "Perimeter Institute for Theoretical Physics\n31 Caroline Street NorthN2L 2Y5WaterlooONCanada", "KTH Royal Institute of Technology and Stockholm University\nHannes Alfvéns väg 12SE-106 91Nordita, StockholmSweden" ]
[]
We propose a new method to account for quantum-gravitational effects in cosmological and black hole spacetimes. At the core of our construction is the "decoupling mechanism": when a physical infrared scale overcomes the effect of the regulator implementing the Wilsonian integration of fluctuating modes, the renormalization group flow of the scale-dependent effective action freezes out, so that at the decoupling scale the latter approximates the standard quantum effective action. Identifying the decoupling scale allows to access terms in the effective action that were not part of the original truncation and thus to study leading-order quantum corrections to field equations and their solutions. Starting from the Einstein-Hilbert truncation, we exploit for the first time the decoupling mechanism in quantum gravity to investigate the dynamics of quantum-corrected black holes from formation to evaporation. Our findings are in qualitative agreement with previous results in the context of renormalization group improved black holes, but additionally feature novel properties reminiscent of higher-derivative operators with specific non-local form factors.
10.1007/jhep03(2023)046
[ "https://export.arxiv.org/pdf/2210.01138v5.pdf" ]
252,715,684
2210.01138
f7522cd1b09c646d4913c23f4a040e2cc7e9a861
Formation and evaporation of quantum black holes from the decoupling mechanism in quantum gravity 23 May 2023 Johanna N Borissova [email protected] Perimeter Institute for Theoretical Physics 31 Caroline Street NorthN2L 2Y5WaterlooONCanada Department of Physics and Astronomy University of Waterloo 200 University Avenue WestN2L 3G1WaterlooONCanada Alessia Platania [email protected] Perimeter Institute for Theoretical Physics 31 Caroline Street NorthN2L 2Y5WaterlooONCanada KTH Royal Institute of Technology and Stockholm University Hannes Alfvéns väg 12SE-106 91Nordita, StockholmSweden Formation and evaporation of quantum black holes from the decoupling mechanism in quantum gravity 23 May 2023Published in JHEP We propose a new method to account for quantum-gravitational effects in cosmological and black hole spacetimes. At the core of our construction is the "decoupling mechanism": when a physical infrared scale overcomes the effect of the regulator implementing the Wilsonian integration of fluctuating modes, the renormalization group flow of the scale-dependent effective action freezes out, so that at the decoupling scale the latter approximates the standard quantum effective action. Identifying the decoupling scale allows to access terms in the effective action that were not part of the original truncation and thus to study leading-order quantum corrections to field equations and their solutions. Starting from the Einstein-Hilbert truncation, we exploit for the first time the decoupling mechanism in quantum gravity to investigate the dynamics of quantum-corrected black holes from formation to evaporation. Our findings are in qualitative agreement with previous results in the context of renormalization group improved black holes, but additionally feature novel properties reminiscent of higher-derivative operators with specific non-local form factors. Introduction The observation of gravitational waves [1] and the reconstruction of the image of a black hole shadow [2] have provided impressive support to Einstein's General Relativity (GR), and to the existence of astrophysical objects whose properties reflect very closely those of GR's black holes. Yet, it is expected that GR eventually breaks down at Planckian scales, leaving the stage to a more fundamental theory accounting for the quantum nature of gravity. Among the different theories of quantum gravity, string theory [3], loop quantum gravity [4] and spin foams [5], asymptotically safe gravity [6,7], and non-local gravity [8,9] have gained considerable attention. While seemingly diverse, a common feature is that the effective action and field equations stemming from their ultraviolet (UV) completions display additional higher-derivative terms [10][11][12][13][14][15][16][17][18][19] which complement the Einstein-Hilbert dynamics. These corrections are expected to play an important role in determining the quantum spacetimes allowed by a principle of least action [20] and their dynamics. In this respect, black holes and their alternatives are particularly important avenues: on the one hand, quantum gravity is expected to yield non-singular solutions or spacetimes with integrable singularities [60,61]; on the other hand, the quantum dynamics could shed light on how these objects are formed in a gravitational collapse, what the final stages of the evaporation process could be, and how singularity resolution can be achieved in the gravitational path integral [62]. Moreover, accounting for the quantum dynamics is key to establish whether the linear instabilities that potentially affect the inner horizon of regular or rotating black holes [63,64] are damped or enhanced by quantum effects. Finally, the number of derivatives in the effective action is crucially related to the type of allowed solutions: truncating the full effective action to quartic order in a derivative expansion, the phase space of all possible solutions is dominated by wormholes and singular black holes [65][66][67][68][69]. Adding terms with six or more derivatives, the field equations instead admit spherically symmetric regular solutions [20,70]. Determining the shape and properties of quantum black holes from first principles is highly challenging: it requires resumming quantum-gravitational fluctuations, deriving an effective action or a similar mathematical object parametrized by finitely many free parameters, and finally determining the spacetime solutions to the corresponding field equations. In turn, computing the effective action requires solving either the path integral or its integrodifferential re-writing in terms of functional renormalization group (FRG) [71] equations. To avoid these complications, the so-called renormalization group (RG) improvement has been used extensively in the framework of asymptotically safe gravity to investigate how quantum-gravitational effects could impact the short-distance behavior of gravity beyond GR and its solutions. This approach has emerged in the context of gauge theories [72][73][74][75] as a way to access leading-order quantum effects while avoiding the computation of quantum loops or a full solution of the beta functions. It consists of promoting the classical constants to running couplings and subsequently replacing the RG scale with a characteristic energy scale of the system. At a qualitative level, the application of the RG improvement to gravity [41-44, 49, 50, 76-98] has pointed to the following tentative conclusions (see [99] for a review). First, classical static black holes are replaced by regular black holes [22,44,49,77,78,83,92,[100][101][102][103] or by compact objects [50,104]. Secondly, accounting for the formation of black holes from the gravitational collapse of a massive star makes singularity resolution less straightforward and typically results in a weaker condition: black hole singularities are not fully resolved, but are rather replaced by so-called integrable singularities [41-43, 80, 105]. Thirdly, singularity resolution in cosmology leads to either bouncing cosmologies or to cyclic universes [85,87]. Finally, in a cosmological context, the spectrum of temperature fluctuations in the cosmic microwave background radiation is intuitively understood in terms of fundamental scale invariance [106,107] in the UV-a key requirement for a theory to be UV-complete at a fixed point of the RG flow, cf. [82,84,88,93,95,[108][109][110][111][112][113][114][115][116][117][118]. Yet, the connection of these results with asymptotic safety and the FRG seems vague, as the application of the RG improvement to gravity is subject to ambiguities. In particular, in the context of gravity it is not obvious how to identify the RG scale consistently, as several characteristic physical scales may compete in a given process or phenomenon. This has led to a plethora of applications of the RG improvement in gravity, where the scale is identified based on physical intuition. In addition to this ambiguity, it is not clear whether the RG improvement should be implemented at the level of the action, field equations, or solutions. While these details typically do not affect the qualitative conclusions obtained via the RG improvement (at least when the scale is reasonably motivated and not manifestly inconsistent, e.g., with diffeomorphism invariance [119]), a more rigorous approach might allow to determine the connection of these results with first-principle computations in quantum gravity, and in particular with the form factors program [17,51,[120][121][122][123][124][125]. The importance of the latter lies in the possibility to compute (via FRG calculations) the effective action in a curvature expansion, including infinitely many higher-derivative terms, and thus to determine formal properties of the theory [122,123,[126][127][128][129][130][131][132] and of its solutions [51]. The scope of this work is to fill the gap between such FRG calculations and the current practise of the RG improvement. We do so by exploiting the so-called decoupling mechanism [133] 1 : if below a certain critical RG scale-dubbed the decoupling scale-there are infrared (IR) scales dominating over the regulator which implements the Wilsonian integration of quantum gravitational fluctuations, then the RG flow freezes out and the scale-dependent effective action at the decoupling scale provides a good approximation to the effective action. In particular, identifying the decoupling scale typically grants access to some higher-derivative interaction terms which were not taken into account in the original truncation. For instance, this is the case in scalar electrodynamics, where the decoupling mechanism allows to derive the logarithmic interaction term in the Coleman-Weinberg effective potential (see [118,133] for details). In this paper we investigate the first application of the decoupling mechanism in gravity. In particular, we will use it to determine qualitative features of the dynamics of black holes beyond GR, from formation to evaporation. As a first attempt in this direction, we will start from the Einstein-Hilbert truncation and use a simple model for the gravitational collapse where the mass function is linear in the advanced time. Our key results can be summarized as follows. The dynamics of quantum-corrected black holes is governed by an effective Newton coupling which decreases both in time (down to a certain non-zero value), and along the radial direction. In particular, its radial dependence smoothly interpolates between the observed value at large distances and zero at the would-be singularity. As a consequence, the curvature of the quantum-corrected spacetime is weakened compared to 1 The decoupling of UV modes from IR physics was originally studied in the context of renormalizable field theories and led to the decoupling theorem by Appelquist and Carazzone [134]. The decoupling theorem famously applies to the Euler-Heisenberg Lagrangian [135] and Fermi's theory of weak interactions [136]. In systems where spontaneous symmetry breaking or mixing effects occur, the decoupling does not generically take place. An example is chiral perturbation theory [137]. its classical counterpart. Although we started from the Einstein-Hilbert truncation, the effective Newton coupling also features characteristic damped oscillations reminiscent of black hole solutions in higher-derivative gravity with specific non-local form factors: free oscillations in the lapse function are typical of black holes in local quadratic gravity assuming a specific sign of the Weyl-squared coupling [68,138], whereas their damping requires the presence of non-local form factors in the quadratic part of the action [139]. This is an expected outcome of the decoupling mechanism and provides evidence that a careful application of the RG improvement, where the scale is not set by physical intuition, but rather by rigorously exploiting the decoupling condition, might provide important insights into quantum gravity phenomenology [140]. Finally, within some approximations, a standard study of the black hole evaporation leads to conclusions in line with the literature [76]: in the evaporation process, quantum black holes get hotter, and after reaching a maximum temperature, they start cooling down, eventually resulting in a cold black hole remnant. The present paper is organized as follows. In Sect. 2 we introduce the FRG and the decoupling mechanism. Next, we show how the decoupling mechanism can be exploited to access some of the higher-derivative terms in the effective action, and thus how to derive corrections to the solutions of GR. We present our setup in Sect. 3, where we also derive the equations governing the dynamics of the quantum-corrected spacetime. We provide numerical and analytical solutions to these equations in Sects. 4, 5, and 6, where we study the dynamics of quantum-corrected black holes in three distinct regimes: formation, static configuration at the end of a collapse, and evaporation, whereby we assume that the evaporation starts only after the collapse is over. We discuss our results in Sect. 7. Functional renormalization group and decoupling mechanism This section introduces the key novel ingredient in our derivation of the dynamics of black holes beyond GR: the decoupling mechanism [133]. To this end, we shall start by briefly summarizing the FRG, its relation to quantum field theory, and its use in quantum gravity. Next, we shall clarify the difference between the RG scale built into the FRG and the physical running appearing in the effective action and in scattering amplitudes (see also [141,142]). Finally, we will review the idea behind the decoupling mechanism and we will explain how it can be exploited to extract qualitative information on quantum spacetimes and their dynamics. Effective actions and the functional renormalization group Schwarzschild black holes and the Friedmann-Lemaître-Robertson-Walker cosmology can be found as solutions to the Einstein field equations δS EH δg µν = 0 , (2.1) S EH being the classical Einstein-Hilbert action. In a quantum theory of gravity these field equations are replaced by their quantum counterpart, δΓ 0 δg µν = 0 ,(2.2) where Γ 0 is the gravitational quantum effective action. The knowledge of the effective action thus paves the way to the investigation of quantum black holes and quantum cosmology. Yet, computing the effective action is extremely challenging. One should solve either the gravitational path integral Dg µν e i S bare [gµν ] ,(2.3) equipped with a suitable regularization, or the FRG equation. Within the FRG, the idea is to first regularize the path integral by introducing an ad hoc regulator term in the bare action, and then transform the integral over field configurations (2.3) into a functional integro-differential equation for a scale-dependent version of the effective action, Γ k , called effective average action. The resulting flow equation for Γ k [143,144] reads k∂ k Γ k = 1 2 STr Γ (2) k + R k −1 k∂ k R k . (2.4) Here Γ (2) k denotes the matrix of second functional derivatives of the effective average action with respect to the quantum fields at fixed background. The function R k is a regulator whose properties guarantee the suppression of IR and UV modes in the flow equation, such that the main contribution to Γ k comes from momentum modes at the scale k. Finally, the supertrace "STr" denotes a sum over discrete indices as well as an integral over momenta. The solution to Eq. (2.4) for a given initial condition identifies a single RG trajectory. The set of all RG trajectories defines the RG flow. A solution Γ k is physically well defined (i.e., the corresponding theory is renormalizable) if its RG trajectory approaches a fixed point in the UV, k → ∞. In this limit Γ k ought to approach the bare action S bare , up to the reconstruction problem, see, e.g., [145][146][147]. The opposite limit, k → 0, corresponds to the case where all quantum fluctuations are integrated out, and yields the standard quantum effective action Γ 0 . First steps towards computing the gravitational effective action have been taken in [17,120,123,129,[148][149][150] in the context of asymptotically safe gravity and in [10,[151][152][153][154][155][156][157][158][159][160][161] within string theory. While deriving the coefficients and form factors in the effective action is highly challenging, one may attempt to find solutions to Eq. (2.2) using alternative strategies. Before describing one of them, that is based on the decoupling mechanism, in the next subsection we shall first clarify a fundamental difference between the momentum scale k in Eq. (2.4) and the physical momentum dependence of Γ k , as this difference is often a source of confusion. Clarifying nomenclature: RG scale dependence versus physical running The effective average action Γ k is constructed as an RG scale dependent action functional, where all couplings or functions are promoted to k-dependent quantities. The action Γ k can thus be parametrized by an infinite-dimensional coordinate vector containing the couplings associated with all possible diffeomorphism-invariant operators. In full generality, the flow equation (2.4) can be associated with infinitely many ordinary coupled differential equations for the couplings. However, in practice the technical complexity requires a truncation of the theory space to a manageable subspace. For instance, at quadratic order in a curvature expansion one has Γ k = d 4 x √ −g 1 16πG k (R − 2Λ k ) + R g R,k (□) R + C µνσρ g C,k (□) C µνσρ , (2.5) where G k = g k k −2 and Λ k = λ k k 2 are the RG scale dependent versions of the Newton and cosmological constants, g k and λ k being their dimensionless counterparts, and g R,k (□) and g C,k (□) are quartic couplings which can generally depend on the d'Alembertian operator. The k-dependence is attached with the Wilsonian integration of fluctuating modes from the UV to the IR. In particular, it is typically used to study the fixed point structure of the action, as the existence of suitable fixed points relates to renormalizability and guarantees that observables computed using the effective action Γ 0 are finite. Provided that such a suitable fixed point exists, one can integrate the flow down to the physical limit k = 0, where the effective average action reduces to the quantum effective action Γ 0 . It is important to remark that the k-dependence is not related to the physical momentum dependence of couplings, which is to be read off from the effective action Γ 0 . Specifically, the structure of the effective average action (2.5) should be contrasted with that of the effective action [17] Γ 0 = d 4 x √ −g 1 16πG N (R − 2Λ) + R F R (□) R + C µνσρ F C (□) C µνσρ , (2.6) where the Newton coupling and the cosmological constant are constants whose values are fixed by observations, while the physical running-encoded in the form factors F i (□) ≡ g i,0 (□)-is attached to the couplings related to the terms at least quadratic in curvature. Note that the dependence on the d'Alembertian is the curved-spacetime generalization of the dependence of couplings on a physical momentum p 2 [17]. The so-called RG improvement was originally devised as a method to obtain an approximation to the effective action (2.6) (or to the solutions to its field equations) by starting from its k-dependent counterpart (2.5) (typically a local version of it) and subsequently replacing the RG scale k with a physical momentum or energy scale. This seems to be a viable strategy in the context of gauge and matter theories [72][73][74][75]. In the framework of quantum field theory the RG improvement originated from the solutions to the Callan-Symanzik equation, while its relation with the FRG is made concrete by the decoupling mechanism, which we review in the following subsection. Effective actions from the decoupling mechanism The flow of the effective average action Γ k from the UV fixed point to the physical IR is governed by the FRG equation (2.4). In particular, the variation of Γ k on the left-hand side of (2.4) is induced by the artificial regulator R k . The latter is an effective mass-square term, R k ∼ k 2 , suppressing fluctuations with momenta p 2 ≲ k 2 . The decoupling mechanism [133], if at work, could provide a short-cut linking (a truncated version of) Γ k to the effective action Γ 0 and relies on the following observation. In the flow towards the IR, R k decreases as ∼ k 2 , and at a certain scale k dec the running couplings and other physical scales in the action, for instance a mass term, may overcome the effect of the cutoff R k . As a result, the flow of the effective average action Γ k would freeze out, so that at the decoupling scale Γ k=k dec approximates the standard effective action Γ 0 (cf. Fig. 1). By identifying the decoupling scale, certain terms appearing in the full effective action can be predicted which were not contained in the original truncation. An emblematic example is scalar electrodynamics, where the RG improvement, combined with the decoupling mechanism, is able to correctly generate the logarithmic corrections in the Coleman-Weinberg effective potential [72], see [118,133] for details and [73][74][75] for other examples in the context of quantum electrodynamics and quantum chromodynamics. It is important to notice that generally k dec will depend on a non-trivial combination of several physical IR scales appearing in the action, e.g., curvature invariants, masses, or field strengths. To make this statement more precise, one has to look at the structure of the regularized inverse propagator. Schematically, i.e., neglecting any tensorial structure, the latter is conveniently written as UV IR k = ∞ k ≈ k dec k = 0 S bare ≈ Γ * Γ k k Γ 0 ≡ Γ Γ k dec ≈ Γ 0Γ (2) k + R k = c (p 2 + A k [Φ] +R k ) ,(2.7) where c is a constant, Φ denotes the set of fields in the theory defined by Γ k , and by definition A k [Φ] ≡ Γ(2) k /c−p 2 and R k ≡ cR k . The regulatorR k efficiently suppresses modes with p 2 ≲ k 2 when it is the largest mass scale in the regularized inverse propagator. By contrast, if A k [Φ] contains physical IR scales, there might be a critical momentum k dec below whichR k becomes negligible. Grounded on these arguments, the decoupling condition reads R k dec ≈ A k dec [Φ] ,(2.8) and provides an implicit definition of the decoupling scale k dec . We emphasize that whereas the structural form of the inverse propagator (2.7) defines the functional A k for general k, at the decoupling scale k dec , A k dec satisfies the condition (2.8). It is worth noticing that this equation might not have real solutions, in which case the RG improvement would not be applicable. We finally remark that the implementation of the RG improvement at the level of the Callan-Symanzik equation and in the EAA is subtlety different: in the former the decoupling occurs due to a balance of physical scales only, while in the latter the unphysical momentum k is involved. Since the first works independently of the choice of cutoff, the second might inherit this property too. Decoupling mechanism versus the practice of RG improvement The procedure of RG improvement consists of promoting the coupling constants to RG scale dependent couplings, and "identifying" the IR cutoff k with a physical scale in order to capture qualitatively the effect of higher-order and non-local terms in the full effective action (2.6). The scale dependence is governed by the beta functions which can be computed using functional RG methods within a given truncation of the effective action. For instance, if the effective average action for gravity is approximated by monomials up to first order in the curvature, its form reduces to the Einstein-Hilbert action with a scale-dependent Newton coupling and cosmological constant. Neglecting the cosmological constant, the flow equation (2.4) for the running Newton coupling gives rise to the approximate scale dependence [22] 2 G(k) = G 0 1 + ωG 0 k 2 ,(2.9) where ω = 1/g * , g * being the non-Gaussian fixed point value of the dimensionless Newton coupling g(k) = G(k)k 2 . The existence of such a fixed point in the UV, combined with the requirement of a finite number of relevant directions, are key requirements for the definition of asymptotically safe theories. The application of the RG improvement in gravity suffers from the following problems: • The challenge to relate the cutoff k to a physical scale of the system: A priori, the role of k is to provide a way to parametrize the RG flow from the UV towards the IR. In principle, if one ignores the original idea behind the decoupling mechanism, there exists no general prescription of how to perform the scale identification on curved spacetimes and in situations where spacetime symmetries are insufficient to fix k uniquely. Scale-setting procedures relying on diffeomorphism invariance and minimal scale dependence of the action were proposed in [119,133,[162][163][164][165], but these are either not always applicable, or they provide insufficient information to completely fix the function k(x). Moreover, in generic physical situations, there is more than one scale. • Ambiguity in the application at the level of action, field equations, and solutions: Although the RG improvement in gravity is motivated by the decoupling of the RG flow of the action functional Γ k , the sequence of replacements g i → g i (k) → g i [k(x)] can in principle be implemented at the level of the action, field equations, or solutions. Typically, the latter two implementations are more straightforward than that at the level of the action, since in physical applications one could skip the step of deriving the field equations or their solutions, respectively. Nevertheless, the three procedures can yield different results. This can be intuitively understood, as, for instance, the replacement k → k(x) at the level of the action would generate higher-derivative operators which would in turn reflect in additional terms in the field equations. • Backreaction effects in gravity: In the context of quantum field theories other than gravity, the RG improvement can be applied straightforwardly [72][73][74][75]. The reason is that in this case coordinates and momenta are related trivially, p ∼ 1/r, and the background metric is fixed and typically flat. In the context of gravity this procedure is less controlled, since the definition of any diffeomorphism-invariant (i.e., scalar) quantity requires a metric. Classical spacetimes are however singular, and their metric cannot be trusted in the proximity of the would-be singularities. Moreover, the Newton coupling itself, which in a standard RG improvement procedure is supposed to be replaced with its running counterpart, is part of the metric needed for the definition of the map k(x). Finally, when a one-step RG improvement is performed at the level of the solutions, this induces a change in the effective Einstein equations and a change in the spacetime metric, which in turn would lead to a different map k(x). This suggests that the application of the RG improvement in gravity requires taking backreaction effects into account and determining the effective metric self-consistently, e.g., via the iterative procedure devised in [49]. Summarizing, the RG improvement in gravity was originally motivated by the decoupling mechanism, and the scale identification k(x) was meant to act as a short-cut to determine the effective action. Yet, in most works on RG improved spacetimes the function k(x) has been fixed based on physical intuition only, accounting neither for the consistency constraints stemming from Bianchi identities (aside from a few examples, e.g. [119,133,[162][163][164][165]), nor for the decoupling mechanism. In this work we will for the first time exploit the decoupling mechanism to derive the function k(x) from the decoupling condition (2.8) and to determine the dynamics of quantum-corrected black holes. Modified spacetimes from the decoupling mechanism: setup In this section we derive the differential equations describing the evolution of the metric of a spherically-symmetric, asymptotically flat black hole spacetime, including quantum corrections computed by exploiting the gravitational beta functions and the decoupling mechanism. Generalized Vaidya spacetimes One of the key lessons of Einstein's GR is the formation of black holes from the gravitational collapse of matter and radiation. Scenarios for the collapse of a sufficiently massive object have been developed and are discussed controversially in relation to the cosmic censorship conjecture [166]. In its weak form, the conjecture posits that the maximal Cauchy development possesses a complete future null infinity for generic initial data. In other words, an event horizon should exist which prevents an observer at future null infinity from seeing the singularity. The conjecture is however known to be violated in various models for the gravitational collapse. In particular, well-known classical models which violate this conjecture are the Tolman-Bondi spacetime for the spherical collapse of dust clouds [167][168][169] or the imploding Vaidya spacetime describing the spherical collapse of radiation [170,171]. In the latter case the singularity appears when the ingoing radiation hits a chosen spacetime point-typically the origin of the given coordinate system. In this classical model the singularity is initially naked provided that the rate of concentration of the radiation is sufficiently low [172]. In the following we will introduce these Vaidya spacetimes as well as an important generalization of the corresponding class of metrics that will be key in our construction. The classical imploding Vaidya solution in advanced Eddington-Finkelstein coordinates reads [170,171] ds 2 = −f (r, v) dv 2 + 2 dv dr + r 2 dΩ 2 , (3.1) with the lapse function f (r, v) = 1 − 2G 0 m(v) r , (3.2) where G 0 denotes the observed value of the Newton coupling. The mass function m(v) depends on the advanced time coordinate and can be used to model a gravitational collapse or evaporation. The metric (3.1) with lapse function (3.2) is an exact solution to the Einstein equations with vanishing cosmological constant, G µν = 8πG 0 T µν ,(3.3) and an energy-momentum tensor corresponding to a pressureless perfect fluid [170,173], T µν = µ u µ u ν . (3.4) Here u µ is the fluid's four-velocity and µ =ṁ (v) 4πr 2 (3.5) is its energy density. A dot denotes differentiation with respect to the advanced time. More generally, one could consider a generalized mass function depending both on the advanced time v, as well as on the radial coordinate r. The resulting generalized Vaidya spacetime [173] is described by a metric of the form (3.1) with lapse function f (r, v) = 1 − 2M (r, v) r , (3.6) where the Newton constant G 0 is now absorbed in the generalized mass function M (r, v). These spacetimes are solutions to the classical Einstein equations (3.3) with an effective energy momentum tensor T µν = µ l µ l ν + (ρ + p)(l µ n ν + l ν n µ ) + pg µν ,(3.7) where the two null vectors l µ and n µ satisfy l µ n µ = −1. The functions µ and ρ are the two contributions to the energy density associated with the first advanced time and radial derivatives of the generalized mass function M (r, v), while p is the classical pressure computed from its second derivative, µ =Ṁ (r, v) 4πG 0 r 2 , ρ = M ′ (r, v) 4πG 0 r 2 , p = − M ′′ (r, v) 8πG 0 r . (3.8) In the previous definition, dots and primes denote derivatives with respect to v and r, respectively. Generalized Vaidya spacetimes can be used to model deviations from the Schwarzschild solution. In the next subsection we will make use of these dynamical solutions to describe the collapse of black holes in the presence of quantum-gravitational fluctuations. We will take into account both the backreaction generated by the modifications induced on the spacetime by quantum effects [49], as well as the dynamical evolution of the spacetime triggered by a time-varying mass function m(v). To this end, we will combine the techniques developed in [49], which we review below, with the ideas in [41][42][43]76], and with the decoupling mechanism [118,133]. Determining the decoupling scale The scope of this subsection is to determine the decoupling scale k dec at which the RG scale dependent effective action Γ k approximates the full effective action. To this end, one first needs to derive the Hessian Γ (2) k and its regularized version. Within the Einstein-Hilbert truncation the effective average action is given by Γ k = d d x √ g 1 16πG k (2Λ k − R) + L m ,(3.9) where G k and Λ k are the Newton coupling and cosmological constant, d is the number of spacetime dimensions, and L m is a matter Lagrangian. In our case, since one of our main scopes is to describe the quantum-corrected gravitational collapse, we limit ourselves to the Lagrangian of a perfect fluid with energy-momentum tensor (3.4), which reads [174] L m = µ(r, v) , (3.10) µ(r, v) being the energy density of the pressureless fluid as defined in (3.5). The quadratic part of the action is constructed by writing the metric as g µν =ḡ µν + h µν , whereḡ is the background metric and h describes fluctuations about this background, and by expanding the action aboutḡ up to quadratic order in h. The metric fluctuations are split as h µν = h T L µν + d −1ḡ µν ϕ, where ϕ ≡ḡ µν h µν is the trace part of the metric andḡ µν h T L µν = 0 expresses the orthogonality condition of the traceless mode h T L µν . Further restricting the background to a maximally symmetric spacetime 3 and using a harmonic gauge fixing, with Γ gf = 1 2 d d x √ g 1 16πG k ḡ µν D σ h µσ − 1 2D µḡ αβ h αβ D ρ h νρ − 1 2D νḡ αβ h αβ , (3.11) the regularized Hessian becomes diagonal in field space. Its elements in the trace, traceless, and Faddeev-Popov ghost sectors [7] are Γ (2) k + R k h = G −1 k □ + k 2 r k (□/k 2 ) − 2λ k + C TR + µG k , Γ (2) k + R k ϕ = − d − 2 2d G −1 k □ + k 2 r k (□/k 2 ) − 2λ k + C SR + µG k , Γ (2) k + R k gh = G −1 k □ + k 2 r k (□/k 2 ) − 2λ k + C VR ,(3.12) where □ ≡ −D 2 is the d'Alembertian operator built with background covariant derivatives, R is the background Ricci scalar, r k is the dimensionless version of the regulator R k , and C T = d(d − 3) + 4 d(d − 1) , C S = d − 4 d , C V = − 1 d . (3.13) As we are interested in asymptotically flat spacetimes, in the following we shall neglect the contribution from the cosmological constant. At this point, the decoupling condition (2.8) reads G −1 k dec (γR + G k dec µ − k 2 dec r k dec ) = 0 , (3.14) where γ ≡ max{|C T |, |C V |, |C S |}, and γ = 2/3 for d = 4. In particular, to determine the form of the decoupling scale a simple mass-type regulator R k ≃ k 2 suffices. This is tantamount to setting r k = 1, so that the decoupling condition reads k 2 dec ≡ G k dec µ + γR . (3.15) Accounting for the decoupling condition at the level of the action, in combination with the expression (2.9) for the running Newton coupling G k , would thus yield an effective action Γ 0 ≈ Γ k=k dec = d 4 x √ g − 1 + µ ω G 2 0 16πG 0R − γω(1 − µ ω G 2 0 ) 16πR 2 + O(µ 2 ,R 3 ) , (3.16) where we have expanded all terms in a curvature expansion and linearized the final expression with respect to the energy density µ. The resulting effective action contains some of the higher-derivative terms in (2.6), as expected, as well as a non-minimal coupling with matter, encoded in the terms µR and µR 2 . We thus expect that the dynamical spacetimes stemming from the implementation of the decoupling mechanism at the level of the solutions-which are the focus of our work-will reflect the presence of the higher-derivative operators and of the non-minimal coupling µR. As we will see, our findings are consistent with this expectation. Note that this is non-trivial: in past applications of the RG improvement, accounting neither for the backreaction effects of [49] nor for the decoupling employ, the spacetime is initially a flat Minkowski background. Thus, a maximally symmetric background is a consistent choice for the early stages of the gravitational collapse. Thereafter, deviations from a maximally symmetric background ought to be automatically encoded in the dynamical adjustment of all physical energy scales and equations involved. Secondly, we checked that in d = 4 the corrections induced by a generic Vaidya background would only change the numerical prefactors of our expressions-at least to leading order in the radial coordinate, in the two opposite regions r ≪ l P l and r ≫ l P l . condition, the implementation of the replacement k → k(x) at the level of the action or at the level of the solutions would generally yield different results. Before proceeding, we briefly comment on the possibility to further constrain the function k(x) by requiring the validity of the Bianchi identities [119,133,[162][163][164][165]. The simplest starting point is the RG scale dependent version of the Einstein-Hilbert action, Eq. (3.9). If the stress-energy tensor for the matter is separately conserved, the Bianchi identities impose a consistency condition on the function k(x). The specific form of the modified Bianchi identities relies on whether one makes the replacement k → k(x) at the level of the action or at the level of the field equations (see [118] for details). If the scale dependence is first introduced at the level of the action, this condition reads [133] 2G(k)Λ ′ (k) + G ′ (k)(R − 2Λ(k)) = 0 , (3.17) where primes denote the differentiation with respect to k. The requirement (3.17) expresses the fact that the sum of the effective energy momentum tensor introduced by the coordinate dependence of the Newton coupling and the cosmological constant term should be conserved to guarantee consistency with the covariant conservation of the Einstein tensor. Such a requirement turns out to be redundant in our case, since the effective spacetimes are solutions to field equations of the form (3.3) which are found self-consistently. We thus conclude that in our case the consistency conditions [119,133,[162][163][164][165] are automatically satisfied and therefore do not add additional constraints. Effective dynamics from the decoupling mechanism The RG improvement in gravity entails working in a truncated version of the effective average action, where the running of the couplings is determined by their beta functions and the RG scale parameter is to be related to a physical energy scale of the system, e.g. the decoupling scale. Nevertheless, due to the complexity of the resulting modified field equations 4 , the RG improvement is sometimes (in particular in the context of blackhole physics) implemented at the level of classical equations or solutions. In the following, we will exploit an RG improvement at the level of classical Vaidya solutions. In this section we derive the equations governing the effective dynamics of a spherically symmetric black hole by combining the decoupling mechanism with the iterative procedure devised in [49]. The latter replaces the standard RG improvement at the level of the solutions with a self-consistent approach accounting for the backreaction effects generated by the introduction of quantum effects on dynamical spacetimes. We will first review this procedure and will subsequently combine it with the decoupling mechanism to derive quantum-corrected spacetimes of the Vaidya type and to study their dynamics. The starting point is the classical (static) Schwarzschild spacetime with lapse function given by (3.6), m(v) ≡ m being the mass of the black hole as measured by an observer at infinity. While the exterior Schwarzschild metric is a solution to the vacuum Einstein equations, a non-zero effective energy-momentum tensor T µν is expected to be present on the right-hand side of the field equations (3.3). The latter can arise in the presence of (quantum) matter, or via quantum-gravitational effects in the form of higher derivatives in the gravitational effective action, cf. Eq. (2.6). Due to these additional terms, the metric of a static spherically-symmetric black hole is expected to be modified with respect to the classical case. Assuming that the time and radial components of the metric are inversely related, g rr = g −1 tt , as is the case for Schwarzschild black holes, the action of quantum effects can be encoded in the radial dependence of an effective Newton coupling G(r). The radial dependence is introduced via the replacement G 0 → G[k(r)], and leads to an effective metric of the form f (r) = 1 − 2mG[k(r)] r , (3.18) where k(r) is the map between the RG scale k and the radial coordinate r, and is initially constructed by means of the classical metric. The spacetime (3.18) describes an exact solution to the Einstein equations in the presence of a generalized effective energy-momentum tensor T eff µν with energy density ρ eff ∝ ∂ r G and pressure p eff ∝ ∂ 2 r G [49,173]. This effective energy-momentum tensor has the role of mimicking the higher-derivative terms in the full quantum effective action (2.6). Yet, in a gravitational context the simple replacement k → k(x) is not expected to yield a good approximation to the effective field equations [49] since: (i) the scalar quantity k(r) (e.g., the proper distance, or a curvature invariant) is necessarily built on the original Schwarzschild metric which fails to give an accurate description of the spacetime in the region of interest, i.e., where quantum gravity effects are important, (ii) the new metric (3.18) is no longer a solution to the vacuum field equations and this backreaction effect might in turn impact the spacetime metric, and (iii) a new scale k(r) built with (3.18) will not match the function k(r) constructed using the Schwarzschild metric. This points to the conclusion that in gravity backreaction effects induced by the replacement k → k(r) have to be taken into account. This can be done iteratively, until a self-consistent solution is reached. In other words, one should iteratively apply the RG improvement until the scale k n (r) used to define the lapse function f n+1 (r) matches the decoupling scale k n+1 constructed using the metric g (n+1) µν at the step n + 1. The iterative procedure is implemented by defining the lapse function f n (r) at a step n > 0 as f n (r) = 1 − 2mG n [k n−1 (r)] r ,(3.19) i.e., in terms of a scale k n−1 (r) constructed by means of the metric g (n−1) µν at the step n − 1. In general, this will be a function of the first and second derivatives of the effective Newton coupling G n−1 (r). If the sequence {G n } defined in this way converges, the fixed function G ∞ (r) satisfies a differential equation which is fully determined by the functional form of the scale k n (r). Specifically, based on the RG running (2.9), this yields the differential equation G ∞ (r) = G 0 1 + ω G 0 k 2 ∞ (r) ,(3.20) with k 2 ∞ depending on G ∞ and its derivatives. In [49] the scale has been fixed to be k 2 ∝ ρ, giving rise to an analytically solvable first-order ordinary differential equation for G ∞ . Its solution is given by G ∞ (r) = G 0 1 − e − r 3 l 3 , (3.21) where l is a length scale of the order of the Planck length l P . As a key result, the limit of the sequence of metrics is described by a Dymnikova black hole [175] with a regular de Sitter core. We now proceed by generalizing the framework in [49] to Vaidya spacetimes (3.6), including a general dynamical mass m(v) in the lapse function (3.6). This has the ultimate goal to describe the dynamics of quantum black holes from formation to evaporation. In this, the scale k(x) will be derived by an explicit use of the decoupling mechanism, as this is key to connect the RG scale dependent description (2.5) with the physics of the effective action (2.6). Specifically, k(r) should be equated to the decoupling scale k dec in Eq. (3.15) in order for the metric (3.18) to be an approximate solution to the field equations stemming from an effective action of the type (2.6). In the case of dynamical spacetimes, one can set up the iterative procedure by using the lapse function f n (r, v) = 1 − 2M n (r, v) r . (3.22) Here the generalized mass function M n (r, v) = m(v) G n [k n−1 (r, v)] is defined by the classical mass function and the running Newton coupling at the step n of the iteration. The metric defined by the lapse function (3.22) belongs to the class of generalized Vaidya spacetimes [173] introduced in Sect. 3.1. The corresponding metric satisfies the effective Einstein equations G n µν = 8πG n T n µν , where the effective energy-momentum tensor takes the form (3.7) with energy densities and pressure redefined as µ n =Ṁ n (r, v) 4πG n (r, v)r 2 , ρ n = M ′ n (r, v) 4πG n (r, v)r 2 , p n = − M ′′ n (r, v) 8πG n (r, v)r . (3.24) The effective Newton coupling G n will itself depend on the self-adjusting cutoff k n which needs to be determined by the properties of the spacetime at the previous step of the iteration. In particular, the effective Newton coupling will generally depend on both the radial coordinate r and the advanced time v, G n = G n (r, v); in the remainder of this section we will omit this dependence for shortness. Finally, in order to make contact with the FRG and determine solutions which approximate those stemming from the full effective action Γ 0 , we shall fix k to be the decoupling scale k dec . In particular, for a fully consistent implementation of the decoupling mechanism, the decoupling scale has to be built using the iterative procedure detailed above. Setting r k = 1 as before and evaluating the decoupling condition (3.15) on-shell finally yields the definition of the decoupling scale at the step n + 1, k 2 n+1 = G n (µ n + γ16π(ρ n − p n )) ,(3.25) where we have dropped the label "dec" from the decoupling scale and we have written the background Ricci scalar (for metrics of type (3.22)) in terms of the generalized energy density and pressure (3.24), according to R = 4 M ′ n (r, v) r 2 + 2 M ′′ n (r, v) r = 16πG n (ρ n − p n ) . (3.26) As will be important later, we note at this point that the high-energy regime k ≫ m P l , where the flow is close to the UV fixed point g * of the dimensionless Newton coupling, corresponds to the large-curvature regime. For a spherically symmetric black hole spacetime this means that the UV fixed point regime corresponds to the region close to the classical singularity, while the IR corresponds to large radii. Finally, taking the limit n → ∞, the dynamical equation for the effective gravitational coupling becomes G ∞ = G 0 1 + G 0 ωG ∞ µ ∞ + 2 3 16π(ρ ∞ − p ∞ ) , (3.27) where µ ∞ , ρ ∞ and p ∞ are defined by (3.24), with M ∞ (r, v) ≡ G ∞ (r, v) m(v), i.e., µ ∞ =ṁ (v) 4πG ∞ r 2 + m(v)Ġ ∞ 4πG ∞ r 2 , ρ ∞ = m(v) G ′ ∞ 4πG ∞ r 2 , p ∞ = − m(v) G ′′ ∞ 8πG ∞ r . (3.28) Inserting these expressions for the energy densities and pressure in terms of the effective Newton coupling and mass m(v) in Eq. (3.27), we obtain the following second-order nonlinear partial differential equation for G ∞ (r, v) G 0 ωm 16πrG ′′ ∞ + 32πG ′ ∞ + 3Ġ ∞ + 3G 0 ωṁG ∞ + 12πr 2 G ∞ − 12G 0 πr 2 = 0 , (3.29) where the classical Vaidya mass function m(v) is still to be specified. The partial differential equation (3.29) is our first main result and will be used to determine the dynamics underlying the quantum-corrected gravitational collapse and black hole evaporation. Specifically, once a solution to Eq. (3.29) is found, the resulting spacetime metric takes the form of a generalized Vaidya spacetime with lapse function f ∞ (r, v) = 1 − 2 m(v) G ∞ (r, v) r . (3.30) We will use the framework introduced in this section to study the effective metric in different regimes, from formation to evaporation. Dynamics of the collapse process As a result of the decoupling mechanism, the dynamics of the effective Newton coupling G(r, v) is governed by Eq. (3.29) where it remains to specify the Vaidya mass function m(v). In this work we will use one of the simplest models for the gravitational collapse of a massive star, known as Vaidya-Kuroda-Papapetrou (VKP) model [171,172,176]. The same model was considered in [41][42][43] to study the quantum-corrected collapse based on a one-step RG improvement not accounting for the decoupling mechanism. We will present two distinct analytical results showing the expected functional dependence of the effective Newton coupling at early times and for small values of the radial coordinate. Finally, by using these solutions as boundary conditions, together with the requirement of matching the observed value of the Newton constant at large distances and early times, we will provide a complete numerical solution to the partial differential equation (3.29). Non-trivial corrections to the classical black hole spacetime, which describe the outcome of the gravitational collapse, will be the subject of Sect. 5, while the evaporation will be described separately in Sect. 6. Vaidya-Kuroda-Papapetrou collapse model The VKP spacetime [171,172,176] is a simplified model for the gravitational collapse of a massive star. Its geometry is characterized by a linear mass function, m(v) =        0, v ≤ 0 ; λv, 0 < v < v ; m, v ≥ v ,(4.1) as shown in Fig. 2. While for advanced times v ≤ 0 the spacetime is a flat Minkowski vacuum, at v = 0 shells of ingoing radiation originating from the star are infused and concentrated towards the origin, r = 0. The linear increase in mass at the rate λ stops at v = v, when the object settles down to the static classical Schwarzschild spacetime with mass m. Historically, the VKP model was one of the first counterexamples to the cosmic censorship conjecture [172]. Identifying possible boundary conditions In this section we determine the solutions to the dynamical equation (3.29) in two asymptotic regimes, where this equation can be solved analytically. This will provide us with the boundary conditions to solve the full dynamics numerically. Dynamics at early times The radial dependence of the effective Newton coupling at early times v ≪v is dictated by the differential equation (3.29) with the dominant contribution stemming from the energy density associated with µ ∞ . Indeed, for v ≪v the spacetime is approximately Minkowski and thus the radial derivatives of the Newton coupling-defining ρ ∞ and p ∞ -are approximately zero. Moreover, since during the collapse m(v) is modeled as a power of the advanced time, m(v) ∼ v n (with exponent n = 1 in our case), it further suppresses ρ ∞ and p ∞ , cf. Eq. (3.28). In contrast, m(v) enters µ ∞ via its advanced time derivative, and since in our case it is a constant,ṁ = λ, its contribution to µ ∞ will dominate over all terms in ρ ∞ and p ∞ . As a consequence, at early times the effective Newton coupling has only a very weak dependence on the advanced time v, which even drops out if ρ ∞ and p ∞ are neglected. Dropping the ρ ∞ and p ∞ contributions from Eq. (3.27) the effective Newton coupling reduces to a function of the radial coordinate only, G ∞ = G ∞ (r), and obeys the equation 3G 0 ω λ G ∞ (r) + 12πr 2 G ∞ (r) − 12G 0 πr 2 = 0 . (4.2) The positive semi-definite solution to the previous quadratic equation reads G ∞ (r) = 2 G 0 λω −πr 2 + π 2 r 4 + G 0 2 λ ωπr 2 . (4.3) Therefrom, the resulting metric can be determined by inserting the result into the lapse function (3.6). At small radii, the corresponding Kretschmann scalar scales as R µνρσ R µνρσ ∝ 1 r 4 ,(4.4) Compared to the classical curvature singularity ∝ r −6 , the antiscreening of gravity stemming from the fixed point implies a weakening of the curvature singularity already at early times. Moreover, following our later analysis in Sect. 5.1, the divergence of the local curvature at the origin is expected to be further weakened during the collapse. In fact, the curvature for the static solutions at the end of the collapse settles down to a scaling ∝ r −3 . Dynamics close to the classical singularity We shall in the following consider solutions to the field equations in the region of spacetime close to the would-be singularity, i.e., for r ≪ l P l . Our goal is to determine a boundary condition of the form G ∞ (r min , v) = J(v), at a fixed r min ≪ l P l , for the numerical integration of the partial differential equation (3.29). In contrast to the case v ∼ 0 studied in the previous subsection, the asymptotic analysis of Eq. (3.29) for r ∼ 0 is extremely involved, and standard techniques based, e.g., on expansions in power laws, are not effective in this case. Yet, as we are only interested in finding a boundary condition G ∞ (r min , v) = J(v) at a fixed r min ≪ l P l , we will utilize two complementary strategies, in combination with some arguments, which we describe in the following. In our first approach, we neglect the two terms proportional to r 2 in Eq. (3.29), as they are small for r ∼ 0, and dropping them significantly reduces the complexity of the equation. Separating the variables, the ansatz G ∞ (r, v) = G 0 (1 − H(v))F (r) (4.5) further simplifies the remaining differential equation to 3F (r)(−1 + H(v) + vH ′ (v)) + 16πv(−1 + H(v))(2F ′ (r) + rF ′′ (r))) = 0 ,(4.6) which can be rewritten as 2F ′ (r) + rF ′′ (r) F (r) = − 3 16πv 1 − H(v) − vH ′ (v) 1 − H(v) ≡ c 0 . (4.7) Here we have used the fact that the left-hand and right-hand sides of the equation can depend only on r and v, respectively, and thus must be equal to a constant c 0 . As a result we obtain two differential equations determining the functions F and H, 2F ′ (r) + rF ′′ (r) = c 0 F (r) , 1 − v H ′ (v) 1 − H(v) = − 16π 3 c 0 v . (4.8) The solution for the radial function F (r) is F (r) = c 1 + c 2 r . (4.9) for c 0 = 0, while for c 0 ̸ = 0 it is given by F (r) = c 1 I 1 (2 √ c 0 √ r) √ c 0 √ r + c 2 K 1 (2 √ c 0 √ r) √ c 0 √ r ,(4.10) where I n (x) and K n (x) are modified Bessel functions of the first and second kind, respectively, and c 0 > 0 in order for the solution to be real. In both solutions for F (r), c 1 and c 2 are integration constants. The dependence on the advanced time is instead encoded in the function H(v) = 1 + b 1 e − 16π 3 c 0 v v ,(4.11) where b 1 ≡ −v 0 is an integration constant. Expanding the exponential function produces a term ≃ 1/v to first order. At next order, H(v) receives a constant contribution. In general, terms coming with an even power in the series expansion of the exponential give rise to positive odd powers of v with a positive pre-factor in the overall expression for H(v). They would therefore yield positive contributions to the v-dependence of the effective Newton coupling and dominate at late times. As we expect the effective Newton coupling to be dynamically weakened during the collapse process, we set these terms to zero by the choice c 0 ≡ 0 in (4.8), i.e., we proceed by requiring both summands in Eq. (4.6) to vanish simultaneously. In this case the time dependence is encoded in Eq. (4.11), with c 0 = 0, while the function F (r) is given by Eq. (4.9), where the integration constant c 2 must be zero, as otherwise the magnitude of the effective Newton coupling would increase towards r → 0. This requirement follows from the existence of a fixed point of the RG flow, as in this case the RG scale dependent Newton coupling scales as G k ∼ g * k −2 in the UV, and vanishes in the high-energy limit. The anti-screening of the gravitational interaction, making gravity weaker in the UV, is the reason behind the expectation of singularity resolution in asymptotically safe gravity. Finally, the integration constant c 2 for the function F (r) is fixed to c 1 = 1, which guarantees that the ansatz (4.5) is compatible with the observed value of the Newton constant at early times. In summary, for small radii r and times v > 0 the form of the effective Newton coupling can be approximated by G ∞ (r, v) = G 0 v 0 v . (4.12) This result indicates that the injection of radiation into an initially flat Minkowski spacetime, within the quantum-corrected VKP model, describes a highly non-perturbative process at early times and close to the would-be singularity, after which the strength of the coupling rapidly decreases with time. A similar conclusion is also reached by employing an alternative strategy. We shall use once again the ansatz (4.5), and then proceed by replacing it in the full partial differential equation (3.29) (including the last two terms proportional to r 2 ) and by expanding about r = 0 up to linear order in the radial coordinate. This procedure yields a differential equation for the function H(v), whose solution reads H(v) = 1 + b 1 e − 16π 3 c 0 v v ,(4.13) where we have defined c 0 = r F (0)F ′′ (0) + 2 r F ′ (0) 2 + 2F (0)F ′ (0) 3F (0) (2 r F ′ (0) + F (0)) . (4.14) On the one hand, Eq. (4.13) resembles the solution in Eq. (4.11). On the other hand, its explicit dependence on the radial coordinate r-encoded in the expression (4.14) of c 0shows that a simple separation of variables of the form (4.5), while generally successful to study the asymptotics of differential equations, is not effective in our case and leads to contradictions. After all, as already mentioned, the exponent α governing the leading-order scaling of G ∞ ∼ r α for r ∼ 0 is expected to be a function of the advanced time v. The asymptotics (4.12) is thus not expected to be accurate and the absence of an r dependence in Eq. (4.12) should not come as a surprise. Specifically, a weak dependence on the radial coordinate at small radii, making the effective Newton coupling vanishing at r = 0, is expected on physical grounds. Despite these issues, as the two derivations presented above yield the same v dependence at a fixed spatial slice, and since the aim of this subsection is solely to find a second, reasonable input for the numerical integration, we will assume that Eq. (4.12) provides a consistent boundary condition at r = r min ≪ l P l , and we will use it as an input for the numerical integration. Whether this assumption is consistent can then be verified a posteriori, based on the outcome of the numerical integration. In particular, the r dependence ought to be restored in the full solution. We anticipate here that the numerical solution will be compatible with this expectation, and specifically with an effective Newton coupling that vanishes in the limit r → 0. Full numerical solution In this section we combine the previous results and provide a numerical solution to the partial differential equation (3.29) for the VKP model. The numerical integration will be performed in the region (v, r) ∈ [v 0 , v] × [r min , r max ]. Hereby v 0 ≪ 1 and v denote the start and end time, respectively, for the numerical integration of the equations along the advanced time direction. Similarly, r min and r max are the integration boundaries for the radial coordinate. In particular, for the numerical integration we fixed v 0 /t P l = 0.01, v/t P l = 1, r min /l P l = 10 −4 , and r max /l P l = 50. Moreover, we set the parameters λ and ω to one and we chose the mass m of the black hole to be Planckian, m/m P l = 1. In general, in the collapse model introduced in Sect. 4.1, the infusion rate λ and the duration of the collapsev determine the mass m of the configuration at the end of the collapse. Consequently, different choices of m will have an impact on the properties of the final static object, as we shall see in Sect. 5. In Eq. (3.29) the time derivative and second spatial derivative occur with the same sign on the left-hand side of the partial differential equation, resulting in a structure reminiscent of negative diffusion. It is well known that the numerical analysis of this type of differential equations is very involved. We present here a numerical solution stemming from the following initial and boundary conditions. First, we require that the effective Newton coupling reduces to the observed value G 0 both at early times and at large distances. This results in the initial and boundary conditions G ∞ (r, v 0 ) = G ∞ (r max , v) = G 0 . Secondly, we make use of the result (4.12), which describes the dynamics in the proximity of the classical singularity, to fix the remaining boundary condition near the origin, at r min /l P l = 10 −4 . More explicitly, this boundary condition reads G(r min , v) = G 0 v 0 /v. Finally, according to our simplified model for the gravitational collapse, see Sect. 4.1, we evolve the system until a final time v/t P l = 1. Fig. 3 shows the result of the numerical integration. Whereas at early times and large distances the effective Newton coupling is well approximated by its classical value, the situation is drastically different for small radii. As soon as the collapse process has started, the effective Newton coupling at small distances from the radial center becomes weaker, thus providing a direct illustration of the antiscreening effect of gravity in the UV. Its dependence on the advanced time v approximately follows an inverse power, cf. Eq. (4.11). In particular, at the end of the gravitational collapse, the effective Newton coupling interpolates between the classical observed value G 0 , which is recovered at large distances, and zero in the limit r → 0, i.e., where quantum gravity effects become stronger. Importantly, we checked that these qualitative features are insensitive to the initial and boundary conditions. An additional striking feature of the effective Newton coupling lies in its damped oscillations along the radial direction at late times. Such oscillations have been observed in some specific black hole solutions of higher-derivative gravity with specific non-local form factors [139]. This seems to be consistent with the arguments in Sect. 2, and specifically with the insight that the decoupling mechanism might fulfil the Figure 3. Numerical solution to the partial differential equation (3.27) for the effective Newton coupling G ∞ . We use as initial condition at early times and boundary condition in the IR the observed value of the Newton constant, i.e. G ∞ (r, v 0 ) = G ∞ (r max , v) = G 0 . For the remaining boundary condition near the origin at r min we use the result (4.12) associated with the dynamics in the proximity of the would-be singularity. At early times for all r, as well as at large distances for all v, the effective Newton coupling reproduces the observed value of Newton's constant, G 0 . When the collapse process starts, the effective Newton coupling decays as ≃ v −1 until the shell-focusing is over. At the end of the collapse, the effective Newton coupling converges to a function which interpolates between G 0 (large-distance limit) and zero (for small radii). This function features in addition damped oscillations along the radial direction. original scope of RG improvement, granting access to some of the quantum corrections in the effective action. We will come back to this topic in the next section. Finally, Fig. 4 shows the time-evolution of the (0, 0)-component of the resulting metric according to the defining equation (3.6). The collapse drives the formation of a black hole horizon whose location lies initially at a radius smaller than its classical counterpart, the latter being approximately located at r h /l P l = 2m(v)/m P l . In fact, the classical Schwarzschild spacetime at the end of the collapse is reproduced well for sufficiently large masses of the final configuration, and only outside of the Planckian region, r ≫ l P l . In the next section, Sect. 5, we will analyse possible outcomes of our collapse model and provide an analytical explanation for the origin of the oscillations by studying the static limit of the partial differential equation (3.29). Static spacetimes at the end of the collapse A key aspect of classical gravitational collapse models, including the VKP model considered here, is the formation of an event horizon and a spacetime singularity after a finite amount of time. In spherically symmetric settings, the metric at the end of the collapse is the static Schwarzschild geometry and contains a curvature singularity at r = 0. In terms of the Kretschmann scalar the degree of divergence of the final static configuration is R µνρσ R µνρσ ∝ r −6 . On the other hand, for the quantum-corrected VKP spacetime with r, v). The collapse drives the formation of a black hole horizon whose location lies initially at a radius smaller than for the classical VKP model. For sufficiently large masses, the final configuration is approximated well by the Schwarzschild spacetime, at least outside of the Planckian region, i.e., for r ≫ l P l . the effective gravitational coupling determined by (3.20), a weakening of the curvature singularity is expected due to the anti-screening character of gravity encoded in Eq. (2.9). Within the VKP model, the system is static for advanced times v > v, as the mass function reaches the constant value m(v) = m. In such a static limit the effective Newton coupling becomes independent of the advanced time v, G ∞ = G ∞ (r), and the energy density µ ∞ in Eq. (3.25) vanishes. All together, the static limit of the effective Newton coupling is defined by the differential equation (3.27) with all advanced time derivatives set to zero, and it describes the final spacetime configuration at the end of the gravitational collapse. To investigate the properties of the resulting static spacetime, in this section we first study the analytical properties of the effective Newton coupling in two opposite limits: in the small radii regime, close to the classical singularity, and in the large distance limit. In the latter the solution displays the same damped oscillations appearing in the collapse phase. Neglecting such oscillations, we will find a function that interpolates between the small-and large-radii behaviors. This interpolating function will provide us with the starting point to study the evaporation phase, which is the focus of the next section. Analytical solution close to the classical singularity In the following we study the outcome of the quantum-corrected VKP model for small radii. Neglecting for a moment the evaporation effects, at the end of the collapse the effective Newton coupling G ∞ will be a function of the radial coordinate only, governed by the differential equation (3.27) with constant ADM mass m(v) = m. Focusing on the small-r region, corresponding to the UV fixed point regime, the RG scale dependence of the dimensionful Newton coupling is given by G(k) ≃ g * k −2 . Since the fixed point regime is reached for k 2 ≫ m 2 P l /ω and ω = 1/g * ∼ O(1) according to FRG computations, this scaling can be obtained by neglecting the 1 in the denominator of Eq. (2.9). Accordingly, we can study the static spacetime solutions resulting from the collapse, and in the proximity of the classical singularity, by setting m(v) = m (static limit) and by neglecting the 1 in the denominator of Eq. (3.27) (fixed point regime). In these limits the effective Newton coupling G ∞ = G ∞ (r) is completely determined by the ordinary differential equation G 0 ωm 4rG ′′ ∞ + 8G ′ ∞ G ∞ − 3G 0 r 2 = 0 . (5.1) As we are interested in determining the leading-order scaling of G ∞ (r) in the proximity of the would-be singularity, we assume that G ∞ (r) ∼ C r n close to r = 0 and determine the parameters (C, n) by inserting this power law ansatz in Eq. (5.1). Following this approach we find G ∞ (r) = 1 √ 5ωm r 3/2 + O(r 3 ) . (5.2) Let us stress that Eq. (5.2) is expected to approximate the effective gravitational coupling at the end of the collapse and at sufficiently small radial distances, r/r P l ≪ 1. The r 3/2scaling implies that at the origin the effective Newton coupling goes to zero. A positive exponent for the leading power in a series expansion around the origin is consistent with previous results on RG improved black holes in spherical symmetry, e.g. [22]. In particular, the specific exponent 3/2 was also found in [92]. Using the expression (5.2) for the Newton coupling in the lapse function of the classical Schwarzschild spacetime, f ∞ (r) = 1 − 2mG ∞ (r) r ≃ 1 − 2m r r 3/2 √ 5ωm , for r ≪ l P l , (5.3) allows us to investigate properties of the geometry close to the origin. In contrast to the classical solution, the lapse function is regular and takes the value f ∞ = 1 at r = 0, as a consequence of the vanishing effective Newton coupling in the limit r → 0. However, the regularity of the metric at the origin does not imply a curvature singularity-free de Sitter core. Indeed, the Kretschmann scalar of the quantum-corrected VKP model at the end of the collapse becomes R µνρσ R µνρσ ∝ 1 r 3 . (5.4) It diverges due to the divergent metric derivative at the origin. Nonetheless, the degree of divergence is lowered compared to the classical singularity and reproduces the exponent found in [92] using an alternative cutoff scheme, but a similar self-consistent approach. In summary, the anti-screening character of the gravitational force at high energies reduces the strength of the curvature singularity in comparison to the classical Vaidya model. In previous studies it was shown that such an anti-screening effect might in certain cases even lead to singularity resolution [22,76,100,101,103], cf. also [44] for an analysis of necessary conditions. To investigate global properties of the static solutions, an approximate scale dependence of the Newton coupling must be derived from (2.9). This is the focus of the next section. Analytical solution at large distances and interpolating function The scope of this section is to determine static analytic solutions at large distances, complementing the analytic solution (5.3) found in the previous section, which instead describes the endpoint of the VKP collapse for small radii. To this end, we need to solve the differential equation (3.27) with constant mass function m(v) = m and for large radii. Setting m(v) = m, the differential equation (3.27) simplifies to G 0 ωm 4rG ′′ ∞ + 8G ′ ∞ + 3r 2 G ∞ − 3G 0 r 2 = 0 . (5.5) Moreover, at radii r/l P l ≫ 1, we can make the ansatz G ∞ (r) = G 0 1 − F (r) r , (5.6) where |F (r)/r| ≪ 1 at large r and |F (r)/r| → 0 as r → ∞, such that the classical lapse function is recovered at infinity. Inserting the ansatz (5.6) into the differential equation (5.5) leads to 4G 2 0 mω(F (r) − r)F ′′ (r) − 3r 2 F (r) = 0 . (5.7) Using that |F (r)| ≪ r at large distances, the previous equation reduces to a Stokes differential equation F ′′ (r) + 3 4G 2 0 mω rF (r) = 0 . (5.8) Solutions are linear combinations of Airy functions in the form F (r) = Re c 1 Ai(a(m, ω)r) + c 2 Bi(a(m, ω)r) , (5.9) where a(m, ω) = 2 −2/3 3 1/3 (−G 2 0 mω) −1/3 . The two integration constants are taken to be c i ∝ 1/m on dimensional grounds (see also [177]). The left panel of Fig. 5 shows the analytic solution for the effective Newton coupling at large r according to (5.6) with the function F (r) given in Eq. (5.9), together with the analytic solution (5.2) valid at small radii, for a mass parameter corresponding to one Planck mass, m = m P l . The power-law behavior in the UV remains valid up to approximately one Planck length r ≈ l P l away from the origin. Beyond the transition at the Planck scale where no analytic solution to the differential equation (5.5) is available (corresponding to the blue region in Fig. 5), the analytic solution (5.6) characterized by damped Airy functions (5.9) takes over. The presence of Airy functions causes characteristic oscillations around the classical value G 0 = m −2 P l with decaying amplitude and wavelength at increasing radii. In the limit r → ∞ the amplitude of the oscillations goes to zero, such that the classical lapse function is recovered. In particular, since the effective Newton coupling approaches the observed value of Newton's constant in the large-distance limit, the resulting spacetimes are asymptotically flat. In the right panel of Fig. 5 the analytic solution (5.5) at large radii is displayed for different masses. At a given radius r, the amplitudes of the oscillations decrease, whereas their wavelengths increase as the mass parameter m grows. In particular, for astrophysical black holes the mass is m/m P l ≈ m ⊙ /m P l ≈ 10 38 and thus the amplitude of the oscillations becomes tiny and hard to resolve. Accordingly, the energy associated with the inverse wavelength of the oscillations becomes microscopic for large masses. To sum up, the amplitude and wavelength of the oscillations decreases with both the radial coordinate r and with the black hole mass m, making them negligible for astrophysical black holes. We have additionally confirmed these findings, which are based on our analytic results, through different numerical methods, such as a direct integration of the second-order differential equation, a transformation to a first-order system, and a shooting with boundary conditions imposed at the origin and at large radii. Let us now comment on the interpretation of these oscillations of the lapse function. On the one hand, similar oscillation patterns were found in certain models of quadratic gravity [68,138] (where however the amplitude of the oscillations does not decrease by increasing r, and the period does not increase either), in higher-derivative gravity with specific non-local form factors [139], and in the context of corpuscolar gravity [59,178,179]. On the other hand, the RG improvement procedure was originally introduced as a way to explore the leading effects of operators occurring at higher order in the expansion of the effective action. In particular, operators quadratic in the curvature will appear naturally beyond the Einstein-Hilbert truncation. Reproducing solutions to a quadratic action with non-local form factors may therefore be viewed as an indication that the results of the iterative RG improvement coupled with the decoupling mechanism are consistent. Next, we need to determine an analytic approximation to the full static solution. If the oscillations on top of the effective Newton coupling are neglected, we find that the analytic power-law solution (5.2) at the origin and the classical constant Newton coupling at large r are smoothly connected by the approximate interpolating function G int ∞ (r) = G 0 1 − e − r 3/2 √ 5ωr h /2 l P l ,(5.10) with r h = 2mG 0 . This can be seen by computing the next-to-leading order corrections to Eq. (5.2) and comparing them with the expansion of various possible interpolating functions, perhaps inspired by the most commonly studied black holes beyond GR. Specifically, the corrections to Eq. (5.2) read G ∞ (r) = r 3/2 √ 5mω − r 3 21G 0 mω + 25r 9/2 16758 √ 5G 2 0 (mω) 3/2 + O(r 11/2 ) (5.11) and we verified that the above next-to-leading order coefficients do not match those of Bardeen-like or Hayward-inspired solutions. One of the reasons is that they would introduce, e.g., r 4 -or r 5/2 -corrections that are absent in Eq. (5.11). In contrast, the expansion of Eq. (5.10) (inspired by the Dymnikova solution) is G int ∞ (r) = r 3/2 √ 5mω − r 3 10G 0 mω + r 9/2 30 √ 5G 2 0 (mω) 3/2 + O(r 11/2 ) (5.12) and offers a better approximation to the expansion (5.11). The interpolating function (5.10) is shown in the left panel of Fig. 5. In the limit of large masses or large radii the exponential becomes negligible, and deviations from the classical Schwarzschild solution are strongly suppressed. The exponential nature of the interpolating function seems to be a feature of the self-consistent approach, which in the static case leads to a Dymnikova solution, cf. [49], corresponding to an effective Newton coupling of the type (5.10), with characteristic scaling ∼ r 3 close to the origin, in place of ∼ r 3/2 . The Dymnikova scaling is physically more appealing, as it makes curvature invariants finite at r = 0. This is to be contrasted with our case, where the characteristic scaling ∼ r 3/2 of the effective Newton coupling is not strong enough to remove the singularity, although it makes it weaker, cf. (5.4). This result is not surprising: even at the level of a one-step RG improvement, adding quantum corrections to the static Schwarzschild solution leads to singularity resolution [22], while, starting from a dynamical spacetime, the dynamics of the quantum-corrected gravitational collapse typically lead to black holes with gravitationally weak (or integrable [60]) singularities [43]. Replacing the one-step RG improvement with the self-consistent procedure in [49] does not change this intriguing result. Yet, in contrast to the one-step RG improvement, self-consistency favors the appearance of exponential lapse functions. Such an exponential behavior is a highly desirable feature, as it gives hopes that the corresponding spacetime can come from a principle of least action in quantum gravity [20]. In particular, it is conceivable that these spacetimes characterized by exponential lapse functions could stem from an effective action of the type (2.6) with exponential form factors. Notably, this resonates with the findings in [139], where it was shown that quadratic effective actions with exponential form factors lead to damped oscillations resembling those that we have observed. In contrast, the typical polynomial lapse functions obtained from the one-step RG improvement, such as the Bonanno-Reuter metric [22] and the Hayward black hole [26], seem to be incompatible with a principle of least action [20], making their relation with quantum gravity questionable. Dynamics of the evaporation process In the previous section we obtained an approximate analytical result for the effective Newton coupling at the end of the collapse. According to Eq. (5.10), the resulting static metric is characterized by the approximate lapse function f ∞ (r) = 1 − 2mG 0 r 1 − e − r 3/2 √ 5ωr h /2 l P l , (6.1) where we remind the reader that m is the ADM mass measured by an observer at infinity. Although the lapse function (6.1) neglects the oscillations encountered in the previous section, it provides an analytical approximation to the endpoint of the gravitational collapse and sets our starting point to study its evaporation. In the following we will work under the assumption that the black hole radiation remains thermal until the end of the evaporation. While this is likely not realized due to nonperturbative effects, it is not straightforward to describe the complete evaporation in a non-perturbative fashion. Moreover, this assumption will allow us to compare our case to the classical one of Schwarzschild black holes. Causal structure and critical mass The interpolating function (5.10) allows us to study the causal structure of the quantumcorrected static spacetime at the end of the collapse, and to determine the approximate location of its horizon(s). While the classical lapse function has a single horizon at r h = 2G 0 m, the causal structure of the quantum-corrected spacetime is more complicated and, similarly to other proposed alternatives to Schwarzschild black holes, it depends on the ratio m/m P l . At a critical value m = m c there is exactly one horizon. For masses below the critical mass there is no horizon and the curvature singularity is naked and timelike. Above the critical mass instead, as is typical for regular black holes, there are two horizons. We note at this point that our construction does not eliminate the problem of mass inflation characterizing most black holes with two horizons, as the lapse function (6.1) is such that the surface gravity at the inner horizon κ − is non-zero [180]. The critical ratio m c /m P l is expected to be of order one, since no scale other than the Planck mass is included in our physical description. The critical mass parameter can be estimated analytically as follows. First, we start by determining the condition to have a horizon close to the classical singularity, where the Newton coupling is described by the function (5.2). This is done by inserting (5.2) into (5.3) and searching for zeros of the resulting lapse function. There is one zero at r h = 5ω 4m/m P l l P l . . Lapse function f ∞ of the final static configuration as a function of r for different masses, each depicted with a different color. The analytic solution based on the oscillating effective Newton coupling (5.6) with the function F (r) defined by (5.9) (dashed lines) is valid at large radii, r/l P l ≫ 1. In the deep UV instead, for r ≪ l P l , the lapse function is approximated by the analytic solution to equation Eq. (5.2) (solid lines), and takes the value f ∞ (0) = 1 at the origin. The region highlighted in blue is where the transition between these two analytic solutions, which are valid in opposite asymptotic regimes, should occur. Next, we recall that the fixed point scaling of the Newton coupling is valid only at high energies, i.e., at small distances, r/l P l ≪ 1. The previous condition is saturated in Eq. (6.2) at r = r h , if the mass parameter is chosen to be m c /m P l ≈ 5/4 ≃ 1.12. This derivation however is valid only if the horizon lies in the region where the analytic approximation for G ∞ based on the fixed point solution is adequate and we must verify this assumption a posteriori. It turns out that m c /m P l ≈ 5/4 can only provide a rough estimate for the critical mass m c . In fact, the horizon location for this value of the mass parameter would be at r ≈ 2G 0 m, cf. Fig. 6, which is outside the regime of validity of Eq. (5.2) but within the same order of magnitude. A numerical analysis utilizing the analytical approximation (5.6) for the lapse function at large radii shows that the correct value for the critical mass lies slightly above the analytical one derived above, and is m c /m P l ≈ 1.18, cf. Fig. 6. Finally, when neglecting the oscillations, i.e., when considering the interpolating lapse function (6.1) as a starting point, the qualitative causal structure is similar: depending on the value of m, the spacetime exhibits two, one or no horizons, cf. Fig. 7. The location r + of the outer black hole horizon approximates the location of the classical Schwarzschild radius at r h = 2G 0 m as the mass is increased. At the same time, increasing m the inner Cauchy horizon r − moves closer to the origin in units of G 0 m. At the critical value m c both horizons coincide, while spacetimes characterized by smaller masses have no horizon. Finally, if the black hole mass m is below the critical value m c , the spacetime is horizonfree. In this case however the critical mass is m c ≈ 1.55 m P l . The difference with the one previously discussed stems from neglecting the oscillations of the lapse function, as is clear from Fig. 6. As the specific position of the horizon does not impact the qualitative where f ∞ (r) can also be negative or vanish. Specifically, the lapse function is negative between the two horizons, positive outside, and vanishes on the boundary. Thus, for masses m/m P l greater than, equal or less than the critical value m c /m P l , the spacetime has two, one or no horizon(s) respectively. aspects of the evaporation process, and since we do not have a full solution featuring both the oscillations (5.6) at large radii and the correct ∼ r 3/2 scaling at short distances, we will neglect the oscillations and we will use the interpolating lapse function (6.1) as a starting point to study the evaporation process of the corresponding black hole. Evaporation process For m > m c the lapse function f ∞ exhibits a simple zero at r = r + . In particular, its derivative is non-negative for r ≥ r + and its value increases monotonically from zero at r = r + to one at infinity. We may thus associate a temperature to this black hole configuration by following Hawking's analysis of black hole radiance [181] in the language of Euclidean path integrals and thermal Green's functions [182][183][184]. To this end, let us consider a static spherically symmetric spacetime of the form ds 2 = −f (r) dt 2 + f (r) −1 dr 2 + r 2 dΩ 2 . (6.3) A positive definite Euclidean metric can be defined by performing a Wick rotation, i.e., by complexifying the time coordinate, t → iτ . Expanding the lapse function in a Taylor series in the near-horizon region it can be shown that to first order the metric locally describes a Rindler space. A coordinate transformation (τ, r) → (ϕ, ρ), where ϕ = |f ′ (r + )|τ /2 and ρ 2 = 4(r − r + )/f ′ (r + ), allows us to write the metric in the neighborhood of the horizon as ds E 2 = dρ 2 + ρ 2 dϕ 2 + r 2 + dΩ 2 . (6.4) By requiring smoothness of the metric, one is led to identify ϕ with an angle variable having period 2π and to restrict the range of possible values of the radial variable to r > r + . In this case the first two terms in the Euclidean metric correspond to the line element of a 2-dimensional flat plane written in polar coordinates (ϕ, ρ). The resulting manifold is a Euclidean black hole with topology R 2 × S 2 . The periodicity of ϕ translates into one of τ , i.e. τ → τ + β with period β = 4π/f ′ (r + ). If quantized matter fields are considered on the Euclidean black hole background, their Green's functions become thermal with the temperature determined by the inverse of the parameter β, T BH = 1 β = f ′ (r + ) 4π . (6.5) For a Schwarzschild black hole this temperature reproduces the well-known result due to Hawking [181], T Schwarzschild = 1 8πG 0 m . (6.6) In order to apply the previous formula to our case, starting from the configuration with two horizons, one has to identify the location of the outer horizon r + . The latter is given by largest positive zero of the lapse function (6.1), see Fig. 7. We determine this root numerically for varying mass and insert the result into (6.5). Thereby we arrive at the temperature as a function of m, shown in the left panel of Fig. 8. For large masses the spacetime is well approximated by the Schwarzschild solution. As a consequence, in this limit the temperature of the quantum black hole reduces to the Hawking temperature (6.6). Lowering the mass, deviations between the classical and quantum spacetime become significant, with the quantum-corrected temperature always lying below the semi-classical one. While the latter diverges as ∝ 1/m for small m, when lowering the mass of the quantum black hole its temperature reaches a maximum and subsequently falls down to zero. This happens when the two horizons coincide, i.e., when the black hole mass m has reached the critical value m c . Initial configurations characterized by a smaller mass parameter have no horizon and thus the derivation of (6.5) does not apply. Our results are in remarkable agreement with the RG improved spacetimes studied in [22], whereby a cutoff function is constructed from the radial proper distance of an observer to the center: after reaching a maximum temperature, the quantum black hole begins to cool down. The evaporation process comes to an end when its mass is lowered to m = m c = O(m P l ). The critical mass therefore represents a final state of evaporation, leaving behind a Planck-size black hole remnant. We now evaluate how much time is needed for the evaporation of a black hole from an initial mass m i to its final value m f . The mass loss per unit proper time measured by an observer is given by Stefan-Boltzmann's laẇ m = −σA(m)T 4 BH (m) ,(6.7) where a dot denotes differentiation with respect to the proper time t, σ is a constant and A(m) = 4πr 2 + is the area of the outer horizon. For a Schwarzschild black hole the radiation power decreases as ∝ m −2 , which leads to a finite amount of time ∝ m i 3 for the complete evaporation from m i → 0 to happen, as shown in the right panel of Fig. 8. The situation is notably different in the quantum case. Starting from an initial value m i > m c , the critical final value m c is reached only asymptotically, at infinitely late times. This can be explained as follows. As the quantum black hole evaporates it eventually reduces its mass to the value associated with the maximum temperature peak, displayed in Fig. 8. Thereafter the cooling process begins and the temperature gradient becomes negative. When the temperature is close to zero, the mass change per time-which obeys a T 4 -behaviour according to Stefan-Boltzmann's law-becomes tiny. At this stage the black hole cannot radiate away power efficiently anymore. In particular, it is impossible to reach the final stage of evaporation. This result is consistently interpreted in view of the third law of black hole thermodynamics, according to which a zero surface gravity cannot be achieved in a physical process, as has already been observed in [22]. The time dependence of the metric is obtained by plugging the time-dependent mass function m(t) into the Vaidya lapse function (3.6). Fig. 9 shows the evaporation of a Schwarzschild black hole compared to the time evolution of its quantum counterpart. A Schwarzschild black hole evaporates completely within a time t ∝ m 3 i , leaving behind empty Minkowski space, modulo a naked singularity. By contrast, a quantum black hole gradu- ally approaches the critical configuration for which the inner and outer horizon coincide. Following our previous considerations, however, it will take infinitely long to get there. We finally comment on the applicability of the entropic arguments against remnants [185,186] to our case. In principle, if one assumes that only asymptotic states matter and that black holes are uncharged, there are at least two ways to avoid the conclusions of [185,186]: (i) remnants are unstable and only asymptotic states matter; (ii) remnants are stable but there is no degeneracy. Since in our case all remnants have the same mass and in principle the same entropy (provided that information can escape, e.g., via non-thermal processes at the end of the evaporation), this might be enough to remove the dangerous degeneracy typically associated with remnants. This solution might however not work in the case of charged black holes (which should be carefully checked separately), as the presence of a charge would likely not allow having remnants of the same mass. This would likely reintroduce the degeneracy and is one of the strongest arguments supporting the idea that there can be no global symmetries in quantum gravity [187]. While the "no global symmetry" conjecture has not been tested within asymptotic safety 5 , there is currently no evidence supporting a conflict between global symmetries and the asymptotic safety condition. If more refined calculations would confirm this conclusion, alternative explanations for degeneracy avoidance ought to be sought after. A possibility would be that asymptotic safety is a low-energy approximation to a more fundamental theory forbidding global symmetries (e.g., string theory [159,188]). Alternatively, the degeneracy might be avoided if black holes have quantum hair [189], or if the evaporation process becomes highly non-perturbative in its last stages, in a way that allows all information to escape. These are exciting possibilities and intriguing research avenues, to which we hope to contribute in the future. Conclusions A key challenge of quantum gravity is to derive spacetimes whose properties and dynamics are valid at all resolution scales. Such dynamical solutions are expected to emerge from a principle of least action, in which the classical action is replaced by its quantum (or "effective") counterpart. Yet, determining such an effective action as well as finding solutions to the corresponding quantum field equations is technically extremely involved. One should first evaluate the gravitational path integral or, equivalently, solve the RG equations of a scale-dependent version of the effective action [71]. By taking its infrared limit, all quantum fluctuations are integrated out and the scale-dependent effective action reduces to the standard quantum effective action. As a way to circumvent these technical challenges, in the past decades studies of quantum gravity phenomenology in the context of asymptotically safe gravity have strongly relied on the use of "RG improvement" [72][73][74][75]. The latter was originally devised in the context of quantum field theory to provide insights on the quantum dynamics while avoiding the complex procedures of solving RG equations or computing quantum loops in perturbation theory. Its necessary ingredients are an action, the beta functions governing the scale dependence of its couplings, and a functional relation between the RG scale and the characteristic energy of a given phenomenon, e.g., the center of mass energy in a scattering process. Although the use of RG improvement in quantum field theory has been incredibly successful, its application to gravity is subject to several ambiguities (see, e.g., [118] for a summary), making its connection to the asymptotic safety program unclear. In particular, the lack of a clear recipe to relate the RG scale with the variety of competing physical energy scales involved in gravitational phenomena is one of its most severe problems. In this work we put forth a method to address this issue and to determine some of the leading-order quantum corrections to classical spacetimes. Our strategy relies on the socalled decoupling mechanism [133]: when a system is characterized by one or more physical infrared scales, their combination can overcome the regulator term implementing the shellby-shell integration of fast-fluctuating modes in the path integral, thus slowing down the flow of the scale-dependent effective action. At the "decoupling scale"-the critical scale below which the flow freezes out-the scale-dependent effective action approximates the quantum effective action. The decoupling mechanism thus provides a short-cut to the effective action and generally grants access to higher-order terms which were not part of the original truncation. In this work we derived a condition to identify the decoupling scale, given an ansatz for the action, and subsequently exploited this condition to study the dynamics of quantum-corrected black hole spacetimes in asymptotic safety, starting from the Einstein-Hilbert truncation. Our results are remarkably promising. On the one hand, they are in qualitative agreement with previous studies based on the RG improvement. Specifically: (i) Accounting for the dynamics of a gravitational collapse makes full singularity resolution less straightforward than in static settings. Nevertheless, quantum effects make the singularity gravitationally weaker, in agreement with preliminary indications from first-principle computations [51]; (ii) Black holes can have up to two horizons depending on whether their mass is below, equal, or above a critical Planckian mass scale. Astrophysical black holes would thus be characterized by two horizons and their evaporation would resemble closely the one of known black holes in the literature [26,76,175]. On the other hand, in our construction we find additional striking features reminiscent of higher-derivative operators with specific nonlocal form factors. In particular, the lapse function characterizing quantum-corrected black holes decreases exponentially, and displays damped oscillations along the radial direction. Although we started from the Einstein-Hilbert truncation, free oscillations are typical of black holes in local quadratic gravity assuming a specific sign of the Weyl-squared term [68,138]. This result is consistent with the expectation that the decoupling mechanism ought to grant access to higher-derivative terms that were not included in the original truncation, and provides encouraging evidence that our construction could lead to results in qualitative agreement with first-principle calculations in quantum gravity. In addition, the damping of the oscillations indicates the presence of non-local form factors in the quadratic part of the effective action. Specifically, given the exponential nature of the dynamical lapse function we derived, one could speculate that these black holes could stem from an effective action with exponential form factors. In turn, this hypothesis is supported by the findings in [139], where it was shown that exponential form factors in the action yield black holes whose lapse functions oscillate along the radial direction, with a characteristic damped amplitude. Altogether, the decoupling mechanism provides an intriguing novel avenue to systematically compute leading-order corrections to classical spacetime solutions. While in this work we focused on black holes and we started from a simple ansatz for the action, our construction also applies to cosmological frameworks and can be extended to include higherderivative terms. Specifically, it is both interesting and necessary to investigate the stability of our findings against the truncation order. This requires going beyond the Einstein-Hilbert truncation, introducing for instance quartic and cubic curvature invariants, and determining the corresponding black-hole solutions from the decoupling mechanism. We hope to tackle these points in future works. Figure 1 . 1Idea behind the decoupling mechanism. If one or a combination of physical IR scales in the effective average action overcomes the effect of the regulator R k in the flow equation (2.4), the flow freezes out and the effective average action at the critical scale k dec approximates the full effective action Γ 0 . Figure 2 . 2Mass function of the classical VKP spacetime as given in Eq. (4.1). The spacetime is initially flat. At v = 0 the gravitational collapse starts and the mass m(v) increases linearly with an injection rate λ. The collapse lasts until v =v, where the mass function m(v) reaches the plateau m(v) = m, m denoting the final mass of the black hole. Figure 4 . 4Time evolution and radial dependence of the lapse function f ∞ (r, v), i.e. the (0, 0)component, of the VKP spacetime with effective Newton coupling G ∞ ( Figure 5 . 5Effective Newton coupling G ∞ as a function of the radial coordinate in the static limit at the end of the collapse, for ω ≡ 1. The left panel shows static solutions to Eq.(3.27) in different approximations and for m = m P l . Below the Planck scale, r ≲ l P l , the effective Newton coupling is approximated by the solution (5.2) to the equations in the fixed point regime, and scales as ∼ r 3/2 (dotted line). The solid line displays the analytic solution (5.6) for large radii and is characterized by damped Airy functions of the form (5.9) where we set c 1 , c 2 ≡ 1/m. The blue region is where the transition between these two analytical solutions (5.2) and (5.6) should occur. Finally, the dashed line shows the exponential function (5.10) which smoothly interpolates between the analytic solution in the UV and the Newton's constant G 0 = m −2 pl in the IR, and solves Eq. (3.27) in the static limit and in the special case where the amplitude of the oscillations vanishes. The right panel depicts the analytic solution (5.6) with the function F (r) specified by Eq. (5.9) at large radii for different black hole masses. At a given radius, the amplitudes of the oscillations decrease, whereas their wavelengths increase for growing mass parameter m. All solutions are valid at large radii, and are not expected to provide a good approximation in the blue region where the transition to the scaling solution (5.2) occurs. Figure 6 6Figure 6. Lapse function f ∞ of the final static configuration as a function of r for different masses, each depicted with a different color. The analytic solution based on the oscillating effective Newton coupling (5.6) with the function F (r) defined by (5.9) (dashed lines) is valid at large radii, r/l P l ≫ 1. In the deep UV instead, for r ≪ l P l , the lapse function is approximated by the analytic solution to equation Eq. (5.2) (solid lines), and takes the value f ∞ (0) = 1 at the origin. The region highlighted in blue is where the transition between these two analytic solutions, which are valid in opposite asymptotic regimes, should occur. Figure 7 . 7Density plot of the lapse function f ∞ (r), highlighting its positivity for increasing values of the mass parameter m and as a function of the radial coordinates in units of r h = 2G 0 m. In the figure r − and r + denote the inner and outer horizon, respectively. For r > r h or m < m c the lapse function f ∞ (r) is strictly positive. The causal structure is instead non-trivial for m ≥ m c , Figure 8 . 8Temperature and mass of an evaporating classical and quantum black hole. All quantities are in Planck units. The left panel depicts the temperature of the quantum black hole (solid blue line) compared to the classical Hawking temperature of a Schwarzschild black hole (dashed dark blue line). The temperature is displayed as a function of the black hole mass. In the classical evaporation process, the black hole becomes hotter and hotter, leading to a complete evaporation which eventually leaves the classical singularity naked after a finite amount of time. In the quantum version, the temperature at first increases as the mass decreases. However, in contrast to the classical case, it reaches a maximum and then slowly goes to zero. The right panel shows the black hole mass as a function of the proper time measured by an observer at infinity. The function m(t) is determined by solving Eq. (6.7) numerically, with the initial value m i ≡ m(0) set to m i /m P l = 2 > m c and with σ ≡ 1. The dashed dark green line corresponds to the classical case, and shows that the evaporation process occurs in a finite amount of time. By contrast, in the quantum-corrected model (solid green line), a black hole with initial mass m i requires an infinite amount of proper time to convert the mass (m i − m c ) into Hawking radiation, eventually leading to a black hole remnant with mass m c (dotted black line). Figure 9 . 9Schwarzschild (left panel) and quantum-corrected lapse function (right panel) at different times. Classically, the Hawking temperature increases monotonically as the black hole mass is converted into Hawking quanta. The evaporation thus continues for a finite amount of time ∆t = 2048π 3 /3t P l until the mass m vanishes and the classical black hole reduces to a Minkowski spacetime with a naked singularity at r = 0. The corresponding lapse function is one everywhere except at the origin, where it diverges. In the quantum model, evaporation takes an infinite amount of time and a black hole remnant with mass m c ∼ m P l (solid red line) is formed asymptotically. Although the RG scale dependence (2.9) has been first derived in[22] via computations in Euclidean signature, Eq. (2.9) still appears to be a good approximation in Lorentzian signature[130]. This choice is not ideal for a generic black hole background. However, it significantly simplifies the expressions and we do not expect it to impact the qualitative aspects of our results. 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[ "Comparison of Discrete Variable and Continuous Variable Quantum Key Distribution Protocols with Phase Noise in the Thermal-Loss Channel", "Comparison of Discrete Variable and Continuous Variable Quantum Key Distribution Protocols with Phase Noise in the Thermal-Loss Channel" ]
[ "S P Kish \nCentre of Excellence for Quantum Computation and Communication Technology\nDepartment of Quantum Science and Technology\nResearch School of Physics\nThe Australian National University\nCanberraACTAustralia., Science China\n", "P Gleeson \nCentre of Excellence for Quantum Computation and Communication Technology\nDepartment of Quantum Science and Technology\nResearch School of Physics\nThe Australian National University\nCanberraACTAustralia., Science China\n", "P K Lam \nCentre of Excellence for Quantum Computation and Communication Technology\nDepartment of Quantum Science and Technology\nResearch School of Physics\nThe Australian National University\nCanberraACTAustralia., Science China\n", "S M Assad \nCentre of Excellence for Quantum Computation and Communication Technology\nDepartment of Quantum Science and Technology\nResearch School of Physics\nThe Australian National University\nCanberraACTAustralia., Science China\n" ]
[ "Centre of Excellence for Quantum Computation and Communication Technology\nDepartment of Quantum Science and Technology\nResearch School of Physics\nThe Australian National University\nCanberraACTAustralia., Science China", "Centre of Excellence for Quantum Computation and Communication Technology\nDepartment of Quantum Science and Technology\nResearch School of Physics\nThe Australian National University\nCanberraACTAustralia., Science China", "Centre of Excellence for Quantum Computation and Communication Technology\nDepartment of Quantum Science and Technology\nResearch School of Physics\nThe Australian National University\nCanberraACTAustralia., Science China", "Centre of Excellence for Quantum Computation and Communication Technology\nDepartment of Quantum Science and Technology\nResearch School of Physics\nThe Australian National University\nCanberraACTAustralia., Science China" ]
[]
Discrete-variable (DV) quantum key distribution (QKD) based on single-photon detectors and sources have been successfully deployed for long-range secure key distribution. On the other hand, continuous-variable (CV) quantum key distribution (QKD) based on coherent detectors and sources is currently lagging behind in terms of loss and noise tolerance. An important discerning factor between DV-QKD and CV-QKD is the effect of phase noise, which is known to be more relevant in CV-QKD. In this article, we investigate the effect of phase noise on DV-QKD and CV-QKD protocols, including the six-state protocol and squeezed-state protocol, in a thermal-loss channel but with the assumed availability of perfect sources and detectors. We find that in the low phase noise regime but high thermal noise regime, CV-QKD can tolerate more loss compared to DV-QKD. We also compare the secret key rate as an additional metric for the performance of QKD. Requirements for this quantity to be high vastly extend the regions at which CV-QKD performs better than DV-QKD. Our analysis addresses the questions of how phase noise affects DV-QKD and CV-QKD and why the former has historically performed better in a thermal-loss channel. arXiv:2206.13724v2 [quant-ph] 12 May 2023
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[ "https://export.arxiv.org/pdf/2206.13724v2.pdf" ]
258,685,279
2206.13724
bb73b36ac57241474eefaf2632d64af20746ccf7
Comparison of Discrete Variable and Continuous Variable Quantum Key Distribution Protocols with Phase Noise in the Thermal-Loss Channel S P Kish Centre of Excellence for Quantum Computation and Communication Technology Department of Quantum Science and Technology Research School of Physics The Australian National University CanberraACTAustralia., Science China P Gleeson Centre of Excellence for Quantum Computation and Communication Technology Department of Quantum Science and Technology Research School of Physics The Australian National University CanberraACTAustralia., Science China P K Lam Centre of Excellence for Quantum Computation and Communication Technology Department of Quantum Science and Technology Research School of Physics The Australian National University CanberraACTAustralia., Science China S M Assad Centre of Excellence for Quantum Computation and Communication Technology Department of Quantum Science and Technology Research School of Physics The Australian National University CanberraACTAustralia., Science China Comparison of Discrete Variable and Continuous Variable Quantum Key Distribution Protocols with Phase Noise in the Thermal-Loss Channel Discrete-variable (DV) quantum key distribution (QKD) based on single-photon detectors and sources have been successfully deployed for long-range secure key distribution. On the other hand, continuous-variable (CV) quantum key distribution (QKD) based on coherent detectors and sources is currently lagging behind in terms of loss and noise tolerance. An important discerning factor between DV-QKD and CV-QKD is the effect of phase noise, which is known to be more relevant in CV-QKD. In this article, we investigate the effect of phase noise on DV-QKD and CV-QKD protocols, including the six-state protocol and squeezed-state protocol, in a thermal-loss channel but with the assumed availability of perfect sources and detectors. We find that in the low phase noise regime but high thermal noise regime, CV-QKD can tolerate more loss compared to DV-QKD. We also compare the secret key rate as an additional metric for the performance of QKD. Requirements for this quantity to be high vastly extend the regions at which CV-QKD performs better than DV-QKD. Our analysis addresses the questions of how phase noise affects DV-QKD and CV-QKD and why the former has historically performed better in a thermal-loss channel. arXiv:2206.13724v2 [quant-ph] 12 May 2023 INTRODUCTION Quantum key distribution (QKD) enables the sharing of keys between two parties, Alice and Bob. Once a quantum secret key is established, it can later be used by both parties to unlock encrypted communication with total confidentiality. In fact, this form of communication is guaranteed to be secure against an eavesdropper, Eve, by the laws of quantum physics. QKD has become a viable cyber security technology with increasing interest across government agencies and commercial corporations [1]. The first proposed QKD protocol based on discrete-variables (DV) uses two polarization bases, which was named after the authors Bennett & Brassard, is BB84. This protocol and its three polarization bases variant, the six-state protocol, rely on the use of single-photon states and remain robust QKD protocols to this day [2,3]. Fifteen years afterward, QKD was extended to continuous-variables (CV), which was initially based on entangled multi-photon two-mode squeezed states (TMSV) and the use of low-noise coherent detection [4][5][6]. An equivalent scheme-the squeezed-state protocol-only requiring preparation of modulated squeezed states was proposed shortly afterward [7]. Subsequently, the GG02 [8][9][10] with reverse reconciliation-proposed by Grosshans & Grangier-and the SRLL02 [11] protocol based on Gaussian modulation of coherent states eliminated the need for preparing experimentally challenging squeezed states. Although coherent-state protocols are experimentally more accessible [12], the squeezedstate protocol remains relevant due to its ideally better performance and compatibility with certain quantum repeater architectures [13]. A comparison between measurement-device-independent (MDI) DV-QKD and CV-QKD protocols, taking into account * [email protected] † [email protected] experimental imperfections was done by Pirandola et. al [14]. A critical comment argued that this comparison is unfair as it depends on the source and detector technologies used [15]. DV-QKD and CV-QKD protocols in a noisy channel with ideal sources and detectors have been investigated in Ref. [16]. It was shown that CV-QKD protocols are robust against noise when loss is low whereas DV-QKD protocols are superior in strong loss regimes. However, in Ref. [16], the key rates of the QKD protocols were ignored as a metric for the comparison. High key rates are an important requirement for a full QKD network to service many users [17,18]. We hypothesize that one of the factors for the consistent historical performance of DV-QKD protocols is mainly due to their robustness to phase noise, which plagues CV-QKD protocols that rely on encoding information in phase as well as amplitude [5]. We test this hypothesis by introducing a phase noise model consistent with both DV-QKD and CV-QKD. In this article, we compare idealized DV-QKD and CV-QKD protocols, the BB84 protocol, the six-state (6S) protocol, and the squeezed-state protocol, by assuming perfect sources, detectors, and reconciliation efficiency in a thermalloss channel. In doing so, we avoid the dependence on practical implementation and current technological limitations. In the first half of the article, we delve into key-rate comparisons in the thermal-loss channel of QKD protocols. For completeness, we consider the strategy of "fighting noise with noise" for improved performances in both the DV-QKD and the CV-QKD protocols. We also identify gaps, if any, between the ideal performances of these QKD protocols and known bounds on the key capacity in the thermal-loss channel. In the second half of the article, unlike previous works [14][15][16], we address phase noise in both DV-QKD and CV-QKD, which is a discerning factor for the performance of QKD. We make use of the fact that in the DV-QKD protocol, the thermal-loss and phase noise channels are equivalent to the depolarizing and dephasing channels, respectively. Furthermore, we present results in the combined thermal-loss and phase noise channels. Our work addresses an important ques-tion about which QKD protocol performs better by various metrics for a given thermal-loss and phase noise channel. Finally, we discuss and conclude our results in the context of real-world implementations, and possible future directions. I. THERMAL-LOSS IN QKD In this section, we present the security models and secret key rate expressions for the DV-QKD and CV-QKD protocols in the thermal-loss channel. We then present the results of the secret key rate of these calculations. A. Thermal-loss in the BB84 (and six-state) dual-rail protocol We make use of the dual-rail BB84 protocol which is one possible implementation of the original BB84 protocol. In the original BB84 protocol, Alice sends a polarization qubit to Bob with a channel that can support both polarizations. This is equivalent to Alice utilising two quantum channels, each supporting only a single polarization. We present this dualrail BB84 protocol in Fig. 1 a) and 1 b). In the BB84 protocol, Alice prepares a single qubit in either the rectilinear Z-basis {|0 , |1 } or the diagonal X-basis {|+ , |− }. In the rectilinear basis shown in Fig. 1 a), a logical 0 is prepared by Alice sending a single photon state |1 in the top a 1 mode and a vacuum state |0 in the bottom a 2 mode. Similarly, a logical 1 is prepared by sending the vacuum state |0 in the top a 1 mode and a single-photon state |1 in the bottom a 2 mode. The qubits pass through a thermal-loss channel represented by a beamsplitter parameter with transmissivity 0 ≤ η 1,2 ≤ 1 and thermal state ρ Th with N Th thermal photons in the auxiliary port. Bob, after deciding randomly (discussed in detail later) to measure the Z-basis, measures each mode output with singlephoton detectors, only accepting single-photon events at b 1 or b 2 corresponding to logical 0 or 1. Any other detector events are not counted towards the final key. In the diagonal basis (see Fig. 1 b)), Alice interferes with a single-photon with the vacuum using a balanced 50 : 50 beamsplitter to generate the superposition state |+ = 1 √ 2 (|1 a 1 |0 a 2 + |0 a 1 |1 a 2 ) which corresponds to a logical 1 state. A logical 0 corresponds to Alice placing a π-phase shifter after the beamsplitter and generating the state |− = 1 √ 2 (|1 a 1 |0 a 2 − |0 a 1 |1 a 2 ) to send to Bob. Bob, having randomly decided to measure in the X-basis by placing a balanced beamsplitter, measures only single-photon events at b 1 or b 2 corresponding to logical 0 or 1. We assume the modes pass through the thermal-loss channels with η 1 = η 2 = η and thermal noise N Th and no correlations between the two thermal environments. In the final step of the protocol, Bob sends information to Alice about which basis he used. In this reconciliation phase, Alice discards the data that does not match the basis she used to encode her qubits. The key rate (per channel use) for the BB84 protocol with perfect reconciliation efficiency in the asymptotic limit is [19][20][21] K BB84 = P S 2 (1 − h(Q Z ) − h(Q X )),(1) where h(x) = −x log 2 (x) − (1 − x) log 2 (1 − x) is the binary entropy function, P S is the success probability of singlephoton events, Q Z and Q X are the quantum bit error rates (QBERs) of the measurement bases Z and X respectively. Unlike the usual normalization preserving DV channels, the success probability P S is necessary because the thermal environment adds Gaussian noise, and only single-photon events are counted towards the secret key rate. Here, we assume perfect number-resolving detectors as opposed to click detectors that count all non-vacuum n > 0 events. To calculate Q Z , we consider the probability of a bit-flip if Alice sends a logical 0 (i.e. |1 a 1 |0 a 2 ) and Bob detects a logical 1 (i.e. simultaneously detects |0 b 1 and |1 b 2 ) with probability given by (see Appendix A for full calculations): P Z,0→1 = P Z,|0 a 1 →|1 b 1 P Z,|1 a 2 →|0 b 2 = N Th (1 + N Th )(1 − η) 2 γ 4 ,(2) where γ = 1 + N Th − N Th η. Bob only accepts the correct bits and the flipped bits using photon-number resolving detectors. Therefore, we normalize by considering the total probability Bob only detects the logical bits in the Z-basis. Since we assume the channels are symmetric, P Z,1→0 = P Z,0→1 , the QBER is Q Z = P Z,0→1 P Z,0→1 + P Z,0→0 = P Z,1→0 P Z,1→0 + P Z,1→1 ,(3) where P Z,0→0 = P Z,|1 a 1 →|1 b 1 P Z,|0 a 2 →|0 b 2 and P Z,1→1 = P Z,|0 a 1 →|0 b 1 P Z,|1 a 2 →|1 b 2 are the probabilities of Bob detecting the same bits that Alice sent after passing through the channel. The probability of an event (or success) is given by: P S = P Z,0→1 + P Z,0→0 = η + 2 N Th (1 + N Th )(1 − η) 2 γ 4 .(4) To calculate Q X , we consider the bit-flips in the X basis. In this case, the modes a 1 and a 2 are entangled because of the balanced beamsplitter (see Fig. 1 b)). Similar to above, we obtain the QBER, for the X bases as Q X = P X,0→1 P X,0→1 + P X,0→0 . We find due to symmetry that the probabilities for the diagonal basis are the same as for the rectilinear basis and it follows that Q X = Q Z , simplifying the key rate equation. We make use of Eqs. (1), (5), and (4) to calculate the key rate in the asymptotic limit. Conditioned on the outcome with probability P S , it can be shown that the density matrix after the thermal-loss channel ρ B := ρ B P S = η η + 2 N Th (1 + N Th )(1 − η) 2 ρ A + N Th (1 + N Th )(1 − η) 2 η + 2 N Th (1 + N Th )(1 − η) 2 I,(6) where ρ A is Alice's initial density matrix. This represents a depolarizing channel [22] ρ → (1 − λ )ρ + λ 2 I(7) with depolarizing parameter λ = 2 N Th (1 + N Th )(1 − η) 2 η + 2 N Th (1 + N Th )(1 − η) 2 ,(8) which tends to 1 as η → 0 or N Th → ∞, as expected. A property of the depolarizing channel is that the error rate is the same in all bases: Q X,Y,Z = λ 2 = N Th (1 + N Th )(1 − η) 2 η + 2N Th (1 + N Th )(1 − η) 2 ,(9) which can be seen from Eq. (7). In establishing this equivalence between the thermal-loss and depolarizing channel, we extend our analysis to the six-state protocol which makes use of an additional basis Y with QBER Q Y . The key rate for the 6S protocol is given by [21]: K 6S = P S 2 (1 − H(Λ 00 ) − H(Λ 01 ) − H(Λ 10 ) − H(Λ 11 )),(10) where H(x) = −x log 2 x, and Λ 00 = 1 − Q X + Q Y + Q Z 2 , Λ 01 = Q X + Q Y − Q Z 2 , Λ 10 = −Q X + Q Y + Q Z 2 , Λ 11 = Q X − Q Y + Q Z 2 ,(11) where the factor of 1/2 is to normalize the key-rate to per channel use. In the thermal-loss channel, the QBER Q X = Q Y = Q Z is symmetric. However, as we will see when phase noise is introduced, the QBER of the three bases can be asymmetric. Lower bounds of BB84 and 6S protocols in the thermal-loss channel Introducing random bit flips at Alice before the error processing increases the performance of BB84 in a noisy channel and sets a tighter lower bound on the key rate [23]. In this extension of the BB84 protocol which we denote as NBB84, the key rate equation depends on Alice's added bit-flip probability q (or trusted bit-flips). Following Ref. [23], we make use of the QBER for the thermal-loss channel in Eq. (54) and maximize the key rate with respect to q. We note that the six-state protocol (with and without trusted bit-flips) can tolerate higher QBER than the BB84 protocol [23]. Similarly, the lower bound on the secret key rate of the 6S protocol is likewise calculated by introducing bit-flips at Alice which increases the QBER tolerance of the channel [23]. B. Thermal-loss in the squeezed state protocol In the squeezed-state protocol in a prepare-and-measure (PM) scheme presented in Fig. 2 a), Alice introduces a modulation signal in either the X =â+â † or P = −i(â−â † ) quadrature (randomly chosen) a squeezed state with V sq with Gaussian distribution centered at 0 with variance V sig . In the equivalent entanglement-based (EB) scheme presented in Fig. 2 b), Alice performs a homodyne measurement on one mode of a shared two-mode squeezed vacuum state (TMSV) where the other mode passes through the channel E , and Bob performs a homodyne measurement [24]. The parameter transformation V sq , V sig Alice Bob Eve θ X B η ρ Th a) TMSV µ X A Alice Bob Eve between the PM and EB schemes is: V sq = 1/µ, V sig = µ 2 − 1 µ ,(12) where µ is the quadrature variance of X and P of the TMSV source in EB scheme. The following key rate calculations are in the EB scheme. In the asymptotic regime of infinite keys, Eve's most powerful attack is a collective attack. Security proofs in this regime for this protocol are based on reduction of coherent attacks to collective attacks for infinite dimensions and on the optimality of Gaussian attacks [25][26][27]. The secret key rate against collective attacks in the asymptotic regime with reverse reconciliation is given by [28] K RR = β I AB − χ EB ,(13) where β is the reconciliation efficiency, I AB is the mutual information between Alice and Bob, and χ EB is the Holevo information between Bob and Eve. In a Gaussian thermal-loss channel, the quadrature covariance matrix between Alice and Bob is [24]: γ AB = aI cσ z cσ z bI = γ A σ AB σ AB γ B =   (V A + 1)I η(V 2 A + 2V A )σ z η(V 2 A + 2V A )σ z V B I   ,(14) where V A = µ − 1 and V B = η(V A + 1) + (1 − η)(2N Th + 1) are the TMSV variances measured by Alice and Bob (respectively), I = diag(1, 1) is the unity matrix and σ z = diag(1, −1) is the Pauli-Z matrix. We choose homodyne detection (also known as "switching") at Bob, in which Bob switches between X or P quadrature measurements. In the squeezed state protocol with homodyne detection, the mutual information is given by: I hom AB = 1 2 log 2 V B V B|A ,(15) where V B|A = b − c 2 a . The Holevo information between Bob and Eve for the collective attack is given by χ EB = S(E) − S(E|B),(16) where S(E) is Eve's information and S(E|B) is Eve's information conditioned on Bob's measurement. In Eve's collective attack, Eve holds a purification of the state between Alice and Bob with entropy given by S(E) = S(AB) = G[(λ 1 − 1)/2] + G[(λ 2 − 1)/2],(17) where G(x) = (x + 1) log 2 (x + 1) − x log 2 x and λ 1,2 are the symplectic eigenvalues of the covariance matrix γ AB given by λ 2 1,2 = 1 2 [∆ ± √ ∆ 2 − 4D 2 ], where ∆ = Det(γ A ) + Det(γ B ) + 2Det(σ AB ), and D = Det(γ AB ). The conditional covariance matrix of Alice's mode after the homodyne detection by Bob is Γ A|b = µ − η(µ 2 −1) η µ+(1−η)(2N Th +1) 0 0 µ .(18) Therefore, Eve's entropy conditioned on Bob's measurement S(E|B) = S(A|b) is given by G[(λ 3 − 1)/2] where λ 3 is the symplectic eigenvalue of Γ A|b . Introducing trusted noise before Bob's homodyne measurement can help extend a high-noise thermal-loss channel. In this extension of the squeezed-state protocol which we denote NSqz-Hom, trusted Gaussian noise ξ B is added before postprocessing on Bob's homodyne measurement data [24]. The effect is that Eve's information decreases more than the mutual information between Alice and Bob (see Appendix C for calculations), thus increasing the secret key rate of the protocol. Similarly, heterodyne detection at Bob has the same effect of introducing additional noise, thereby extending secure communication distance in a thermal-loss channel [29]. II. PHASE NOISE IN QKD We consider a standard model of bosonic phase noise, known also as dephasing, phase-diffusion, or phase-damping; for an excellent review, see [30]. This channel represented by θ on the right of Fig. 1 applies rotation by a random angle θ to the bosonic state according to a classical distribution f (θ ), giving the transformation ρ −→ π −π dθ f (θ )e iâ †â θ ρe −iâ †â θ .(19) Sinceâ †â is the number operator, a given rotation θ applies a phase e inθ to each Fock state |n , equivalently described by the transformationâ † −→ e iθâ † .(20) The canonical phase distribution is the wrapped normal distribution, which models the random diffusion of an angle and accurately represents the physical process of phase diffusion [30]. Birefringence may produce this behaviour in polarisation-based implementations of the six-state and BB84 protocols or in time-bin implementations, phase drift in between the interferometers at either end [31]. The phase shift θ (assumed here to have mean zero) is normally distributed over the whole real line: f W N (θ ) = 1 σ √ 2π e −θ 2 /2σ 2 : θ ∈ R, which we can 'wrap' into a single 2π interval by summing the contributions from equivalent angles: f W N (θ ) = 1 σ √ 2π ∞ ∑ k=−∞ e −(θ +2πk) 2 /2σ 2 : θ ∈ [−π, π]. The varianceσ 2 of θ over the whole real line is in general not its variance when wrapped; however, the two distributions approach each other in the limit of small variance. The corresponding qubit transformation of the phase noise ignoring the thermal- loss (η = 1) is ρ jk → e iθ j e −iθ k ρ jk , which may be expressed as ρ →Û θ ρÛ † θ whereÛ θ = diag(e iθ i , e iθ j ). If θ is drawn from a distribution f (θ), the qubit channel be- comes ρ → Û θ ρÛ † θ = θ f (θ)Û θ ρÛ † θ dθ where · denotes expected value. If {θ j } are independent, the corresponding transformation of off-diagonal terms (i = j) is ρ jk −→ e iθ j e −iθ k ρ jk = e iθ j e −iθ k ρ jk = r j r * k ρ jk where r j := e iθ j is the so-called 'circular mean' of θ j , given for the wrapped normal distribution: r j = e −σ 2 /2 .(21) Diagonal entries remain unchanged: ρ j j −→ e iθ j e −iθ j ρ j j = ρ j j . If {θ j } are identically distributed then all have the same (real) circular mean r and we obtain a (generalised) dephasing channel [22] ρ −→ ρ dephased = ρ 00 r 2 ρ 01 r 2 ρ 10 ρ 11 (22) which always sends single-photon inputs to single-photon outputs, unlike the thermal-loss channel. By leaving diagonal entries unchanged, Eq. (22) introduces no error in the Z basis. Common DV-QKD protocols such as the (generalised) BB84 and six-state protocols make use of additional bases (X and Y ) which are unbiased with respect to the Z basis. The extension to the combined thermal-loss phase-noise rail presented in Fig. 1 can be obtained by composing the separate depolarization and dephasing channels described in Eqs. (7) and (22), giving ρ −→ (1 − λ ) ρ 00 r 2 ρ 01 r 2 ρ 10 ρ 11 + λ 2 I. The corresponding error rates are thus (1 − λ )ρ dephased + λ 2 I, i.e. Q Z = λ 2 , Q X,Y = 1 2 (1 − λ )(1 − r 2 ) + λ ,(23) with r 2 and λ given by Eqs. (21) and (8) respectively. The probability of success P S remains the same as for the thermalloss channel in Eq. (4), as the subsequent dephasing does not affect which states are discarded. The key rate for the BB84 and 6S protocols are straightforward to calculate from these QBERs and using Eqs. (1) and (10), respectively. For CV-QKD, we make use of the phase noise model shown in Ref. [28,[32][33][34]. Residual phase noise manifests as an added excess noise that Bob measures given by: ε θ = 2V A (1 − e −V θ /2 ),(24) where V θ is the phase noise between the local oscillator and signal. Since the squeezed-state protocol is modulated with equal probability in the X or P quadratures, the excess noise due to phase noise is symmetric [24]. In Appendix D, we show that the phase noise associated with the squeezing angle of φ and the phase noise associated with the coherent state phase of θ can be incorporated into the same phase noise parameter. Application of the rotation operators on squeezed coherent states leads to the same excess noise in Eq. (24). Finally, we assume that the Gaussian phase noise in CV-QKD V θ is equal to the wrapped normal distribution variance σ 2 θ in DV-QKD. We note that in the regime where σ 2 θ is large, the phase diffusion channel becomes non-Gaussian [30]. Since we are considering the squeezed-state protocol with coherent detection, we make use of Eq. (16) to calculate a lower bound on the key rate. It is left for future work to determine optimal protocols in the phase diffusion channel. III. COMPARISON OF QKD PROTOCOLS A. Without phase noise σ 2 θ = 0 In CV-QKD, information is encoded in the X and/or P quadratures in one polarization with access to an infinite Hilbert space. Conversely, in DV-QKD, information is encoded in one or more polarization basis in a 2-dimensional Hilbert space. To make a fair comparison, we assume that Alice uses one polarization basis asymptotically close to 100% of the time (the "computational" basis). The other basis is only measured to characterize channel parameters and the QBER. We make a similar assumption for the squeezed state protocols in the sense that Bob rarely switches the quadrature he measures to characterize the anti-squeezing and determine whether Eve tampered with the shared EPR state, thus removing the usual sifting factor of 1/2 that comes with switching. We also make the following ideal assumptions about the CV-QKD and DV-QKD protocols in the thermal-loss channel: (i) single-photon and laser sources are perfect (ii) detectors that are used are ideal with detector efficiencies η d = 1 and detector noise ξ det = 0 (except for intentionally adding trusted noise in the "fighting noise with noise" protocol) (iii) all channel parameters have been estimated with no statistical error (iv) all channel noise is attributed to Eve (v) reverse reconciliation efficiency is perfect with β = 1 and error correction efficiency is perfect for both CV-QKD and DV-QKD (vi) all security analysis is in the asymptotic limit. Our simplified analysis here is valid in the ideal situation where squeezed and coherent states are only affected by loss, thermal noise and (in the next section) phase noise. Pirandola et. al determined lower bounds (LB) and upper bounds (UB) on the secret key capacity C(η, N Th ) of the thermal loss channel where N Th is the thermal noise and η is the transmissivity of the thermal-loss channel [35,36]. The lower bound is given by the reverse coherent information of C(η, N Th ) ≥ − log 2 [1 − η] − G(N Th ) = K Lower , C(η, N Th ) ≤ − log 2 [(1 − η)η N Th ] − G(N Th ) = K Upper ,(25) given for non-entanglement breaking channels N Th < η/(1 − η) and where G(x) = (x + 1) log 2 (x + 1) − x log 2 x. We present our results in Fig. 3 a)-d) for the secret key rate per polarization channel based on calculations of the BB84 protocol, the 6S protocol, the GG02 protocol (see Appendix E calculations), and the squeezed state with homodyne (Sqz-Hom) protocols in the thermal-loss channel for various thermal noise parameters. We note that since we make use of the dual-rail BB84 protocol which is one possible implementation of the BB84 protocol, the key rate equation for the DV-QKD protocols has been divided by 2 into units of symbols per polarization channel. For the Sqz-Hom protocol, we choose a practically achievable squeezing V sq of 15 dB [37]. We note that adding more squeezing only adds a very small improvement to the key rates (see Appendix F for more details). In the limit of infinite squeezing, the secret key rate of the Sqz-Hom would approach the lower bound (LB) of the secret key capacity in the thermal-loss channel as shown most clearly in Fig. 3 b) and in a pure-loss channel as shown in Fig. 3 a). The BB84 and 6S protocols surpass the lower bound in an intermediate thermal-noise regime as shown in Fig. 3 b). In Fig. 3 e)-h), we present the "fighting noise with noise" versions of the protocols. For the squeezed state protocol with homodyne detection (NSqz-Hom) with 15 dB squeezing we optimized with respect to the trusted noise ξ B . As shown in Fig. 3 g) and h) surpasses the LB for high thermal noise. In addition, the secret key rate of the "fighting noise with noise" versions of the DV-QKD protocols, the NBB84 and N6S protocols are optimized with respect to the added bit-flips by Alice q and a slight advantage is obtained as shown in Fig. 3 c). In Fig. 4, to benchmark the performance of the different protocols, we normalized the key rate to the upper bound of the secret key capacity. We compare the protocols by plotting the parameter: K CV:DV = K Sqz-Hom − K 6S Max[K Sqz-Hom , K 6S ] ,(26) for channel parameters of standard optical fibre of loss 0.2 dB/km with distance D = 10 − η 50 km and N Th in Fig. 5. The key rate (K Sqz-Hom , K 6S ) > K 0 where K 0 is the minimum required key rate. When the squeezed-state protocol is significantly higher in key ratesK = 1, and conversely, when the 6S protocol is best,K = −1. In Fig. 5, from left to right, the protocols are operated at increasingly higher key rates. Given a minimum key rate requirement, we compare the protocols which operate the best in bits per channel use. The main observation here is that the channel parameter space where the 6S protocol dominates shrinks for increasingly higher key rates and CV-QKD is at an advantage. It can also be seen that for higher minimum key-rate requirements, only the Sqz-Hom protocol can operate (see red regions in the middle and right subfigures). However, the 6S protocol can be operated in an intermediate-noise regime at low-key rates where CV-QKD cannot (left and centre subfigure). Our results indicate that common QKD protocols are far from the upper bound secret key capacity in a thermal-loss K =10 (a) Contour plot ofÑ CV:DV Th as a function of the phase noise and distance (or loss) for the Sqz-Hom and 6S protocol. The green line indicates the point at which both protocols tolerate the same amount of thermal noise. In the white regions to the right-hand side, neither one of the protocols tolerate any thermal noise. In the red region, only the Sqz-Hom protocol tolerates thermal noise. We also show current state-of-the-art CV-QKD (red asterisks) and DV-QKD (blue circles) protocols. K =10 K =10 K =10 0 0 0 -9 -6 -3 (b) Contour plot ofL CV:DV orD CV:DV as a function of the phase noise and thermal noise for the Sqz-Hom and 6S protocol. The green line indicates the point at which both protocols tolerate the same amount of loss. In the white regions to the right-hand side, neither one of the protocols tolerate any loss. Figure 6 channel. We also find that the NSqz-Hom protocol has the best excess noise tolerance in very noisy channels in agreement with Ref. [38] but we find that the BB84 and 6S protocols perform better in an intermediate noise regime. B. With phase noise σ 2 θ > 0 In the following section, we quantify the performance of the 6S and Sqz-Hom (with optimized modulation variance V A ) protocols in the combined thermal-loss and phase noise channel. First, consider the maximum tolerable thermal noise given by: N (Max) Th = argN Th 0, if K(0, σ 2 θ , D) < K 0 . K(N Th , σ 2 θ , D) = K 0 , otherwise.(27) In other words, the maximum tolerable noise if the key rate is less than K 0 at N Th = 0 is 0. Otherwise, the maximum tolerable noise is N Th when the key rate falls to K 0 . In Fig. 6 a), we plot the following quantity: N CV:DV Th (σ 2 θ , D) = N (Max) Th,Sqz-Hom − N (Max) Th,6S Max[N (Max) Th,Sqz-Hom , N (Max) Th,6S ] ,(28) which is the difference between the maximum tolerable thermal noise of the Sqz-Hom and the 6S protocols for a given phase noise σ 2 θ and distance D to achieve a key rate K 0 . Highlighted in the figure is the green contour where both protocols tolerate the same amount of thermal noise, i.e.Ñ CV:DV Th = 0. For low key rates, it can be seen that the squeezed-state protocol tolerates more thermal noise than the 6S protocol in short channels and when σ 2 θ < 10 −3 . The Sqz-Hom protocol also tolerates more thermal noise than the 6S protocol at longer distances. In this red region, the 6S protocol tolerates zero thermal noise, whereas the Sqz-Hom protocol tolerates some thermal noise. For higher key-rate requirements, although the region of noise tolerance shrinks for both protocols, the Sqz-Hom tolerates proportionally more thermal noise across the phase noise versus distance parameter space. Next, we consider the maximum distance or maximum tolerable loss given by: D (Max) = argD 0, if K(N Th , σ 2 θ , 0) < K 0 . K(N Th , σ 2 θ , D) = K 0 , otherwise.(29) In other words, the maximum distance if the key rate is less than K 0 at D = 0 is 0. Otherwise, the maximum distance is D when the key rate falls to K 0 . In Fig. 6 b), we plot the following quantity: L CV:DV (σ 2 θ , N Th ) =D CV:DV (σ 2 θ , N Th ) = D (Max) Sqz-Hom − D (Max) 6S Max[D (Max) Sqz-Hom , D (Max) 6S ] ,(30) which is the difference between the maximum distance of the Sqz-Hom and the 6S protocols for a given phase noise σ 2 θ and thermal noise N Th to achieve a key rate K 0 . The Sqz-Hom protocol can tolerate more loss than the 6S protocol at thermal noise between 10 −2 and 0.9, and phase noise σ 2 θ < 10 −3 . As found in [16], at a small region of high thermal noise, the 6S protocol tolerates more loss than the Sqz-Hom protocol. At higher key-rate requirements, the Sqz-Hom protocol can tolerate more loss compared to the 6S protocol. In fact, it can tolerate as much as σ 2 θ = 0.05 for a K > 10 −3 key-rate requirement and N Th < 10 −3 to perform at a longer distance than the 6S protocol. From these results, we can conclude that for low key-rate requirements, the 6S protocol clearly dominates a larger region of parameters. However, for high key-rate requirements, the Sqz-Hom protocol dominates most of the parameter space for phase noise less than a phase noise of σ 2 θ < 10 −3 . [43], the phase noise is upper bounded from the total excess noise. As a comparison, experimental values for the phase noise in CV-QKD protocols are shown in Table. I. These are also shown in Fig. 6 a) along with current state-of-the-art DV-QKD protocols [44] & [45] (converted to distance in standard fibre). For DV-QKD implementations, we convert the time jitter full width at half maximum ∆t FWHM ) to the phase noise using the following equation: σ 2 θ = (2π∆t FWHM ) 2 (2 √ 2 ln 2∆t) 2 ,(31) where ∆t is the timing between pulses. This equation converts FWHM to a Gaussian width [46] and then to a phase noise (in radians squared). The timing between pulses in both experiments is inversely proportional to the repetition rate ∆t = 1/ f . DISCUSSION We discuss our results with less-than-ideal experimental setups of QKD protocols. In optical fibre, the current distance record for DV-QKD is 421 km in ultralowloss (ULL) fibre (0.17 dB/km) corresponding to 71.9 dB loss [44]. A secret key rate of 0.25 bps or equivalently K = 10 −10 bits per channel use was obtained using superconducting single-photon detectors at a repetition rate of 2.5 GHz. Most recently, a high key rate of K = 4.4 × 10 −2 was demonstrated in 10 km of standard optical fiber for DV-QKD [45]. We plot these experimental points normalized to standard optical fiber loss (0.2 dB/km) in Fig. 6 a). Based on these results, for this similar key-rate requirement K 0 = 10 −3 , CV-QKD would, in theory, be able to achieve the same high key rate and tolerate more noise if the same levels of phase noise are maintained as in [39] & [40]. Additionally, CV-QKD can extend up to 150 km as opposed to DV-QKD which cannot tolerate noise beyond 125 km. In the rightmost subfigure in Fig. 6 b), it can be seen that CV-QKD can tolerate more loss than DV-QKD for a large parameter region given a higher key rate requirement. Nonetheless, in terms of distance, DV-QKD is currently leading the benchmark for QKD with the world record for CV-QKD being more than half the distance in ULL fibre at 202.81 km (or 32.4 dB) using the GG02 protocol where a key rate of K ≈ 10 −6 was achieved [43]. On the other hand, the apparent advantage of CV-QKD is in the efficient encoding of keys per symbol and the faster generation and detection of coherent (or squeezed) states with a much larger block size. Post-processing codes at low signal-to-noise ratio were a bottleneck in CV-QKD until it was recently shown that Raptorlike LPDC codes can maintain a high key extraction rate and high reconciliation efficiency, paving the way for practical and deployable CV-QKD [47]. We have also focused mainly on the squeezed-state protocol. Despite renewed interest in the squeezed-state protocol due to its robustness to noise [48][49][50], the difficulty of modulating and generating stable squeezed coherent states remains. However, entanglement-based versions have been demonstrated [51], and sources of highly entangled TMSV states are a promising pathway toward realizing the squeezedstate protocol [52]. Furthermore, one of the limitations of CV-QKD is currently the maintaining of phase reference using a local oscillator (LO), which needs to be practically solved without compromising unconditional security, in a real-world setting outside of the laboratory [43]. It can be seen from our results, that although CV-QKD performs well in a high-thermal noise regime, the introduction of phase noise destroys this advantage. For CV-QKD to maintain this advantage, the amount of phase noise must be less than σ 2 θ < 10 −3 . However, we also find that CV-QKD performs best for high minimum keyrate requirements where it can tolerate more thermal noise at longer distances than DV-QKD. The physical reason behind this is that in CV-QKD, more symbols can be sent that will result in a shared key. Conversely, DV-QKD is limited to single photons. Based on these results, we speculate that the consistent his-torical performance of DV-QKD protocols is mainly due to robustness to phase noise, which plagues CV-QKD protocols that rely on encoding information in phase as well as amplitude. However, with increasingly more robust carrier phase compensation schemes based on machine learning as in Ref. [41,42], phase noise may no longer be a limiting factor in CV-QKD. We note the limitations of our phase noise model for the squeezed-state protocol. We had assumed that the phase noise is Gaussian, whereas the phase diffusion channel is a non-Gaussian channel [30]. It is left for future work to determine the optimal QKD protocol in the phase diffusion channel. Although current upper bounds on the secret key capacity can serve as a benchmark for QKD protocols, no QKD protocol is currently known to saturate these bounds in the thermal-loss channel. We note that energy-constrained upper bounds in the thermal-loss channel have been recently determined, that would be comparable in energy to common DV-QKD protocols [53]. IV. CONCLUSION In this work, we compared DV-QKD and CV-QKD protocols on equal grounds in a thermal-loss channel and we assumed ideal sources and detector performances. We developed analytical formulas for the QBER of the BB84 and six-state protocols in a thermal-loss channel. We introduced the minimum key rate as a metric for QKD performance. We found the squeezed-state protocol dominates most of the channel parameter regimes when there is no phase noise, except for an intermediate-noise regime where the six-state protocol can tolerate more loss and surpasses the lower bound to the secret key capacity. With the addition of phase noise, we find that the overall landscape of the DV-QKD and CV-QKD comparison becomes more complex. Finally, we find DV-QKD is largely unaffected by phase noise, whilst CV-QKD is sensitive but performs better below a threshold phase noise only recently reached in experiments. AUTHOR CONTRIBUTIONS STATEMENT S. P. K. wrote the paper, and produced the calculations and results, P. G. produced some of the analytical calculations, S. M. A. and P. K. L. conceived the main idea and proofread the manuscript. All authors reviewed the manuscript. APPENDICES A. Quantum Bit Error Rate (QBER) To calculate Q Z , we consider the probability of a bit-flip if Alice sends a logical 0 (i.e. |1 a 1 |0 a 2 ) and Bob detects a logical 1 (i.e. simultaneously detects |0 b 1 and |1 b 2 ) given by, P Z,0→1 = P Z,|1 a 1 →|0 b 1 P Z,|0 a 2 →|1 b 2 = Tr(U BS (η)(|1 a 1 1| a 1 ⊗ ρ Th )U † BS (η) |0 b 1 0| b 1 ) × Tr(U BS (η)(|0 a 2 0| a 2 ⊗ ρ Th )U † BS (η) |1 b 2 1| b 2 ),(32) where ρ Th = ∑ ∞ n=0 [N n Th /(N Th + 1) n+1 ] |n n| is the thermal state with average thermal photon number N Th and U BS (η) is the unitary beamsplitter transformation mixing the thermal environment and the state sent by Alice. If Alice prepares a logical 1 and Bob measures 0, the probability is P Z,1→0 = P Z,|0 a 1 →|1 b 1 P Z,|1 a 2 →|0 b 2 = Tr(U BS (η)(|0 a 1 0| a 1 ⊗ ρ Th )U † BS (η) |1 b 1 1| b 1 ) × Tr(U BS (η)(|1 a 2 1| a 2 ⊗ ρ Th )U † BS (η) |0 b 2 0| b 2 ),(33) and since we assume the channels are symmetric, P Z,1→0 = P Z,0→1 . The total un-normalized probability of a bit-flip is 2P Z,0→1 . Bob only accepts the correct bits and the flipped bits. Therefore, we normalize by considering the total probability Bob only detects the logical bits in the Z-basis. Hence Q Z = P Z,0→1 P Z,0→1 + P Z,0→0 = P Z,1→0 P Z,1→0 + P Z,1→1 ,(34) where P Z,0→0 = P Z,|1 a 1 →|1 b 1 P Z,|0 a 2 →|0 b 2 and P Z,1→1 = P Z,|0 a 1 →|0 b 1 P Z,|1 a 2 →|1 b 2 are the probabilities of Bob detecting the same bits that Alice sent after passing through the channel. These probabilities are given by P Z,0→0 = P Z,1→1 = Tr(U BS (η)(|1 a 1 1| a 1 ⊗ ρ Th )U † BS (η) |1 b 1 1| b 1 ) × Tr(U BS (η)(|0 a 2 0| a 2 ⊗ ρ Th )U † BS (η) |0 b 2 0| b 2 ).(35) These probabilities are: P Z,0→1 = P Z,1→0 = (1 − η) 2 (N Th + N 2 Th ) (1 + (1 − η)N Th ) 4(36)P Z,0→0 = P Z,1→1 = η + (1 − η) 2 (N Th + N 2 Th ) (1 + (1 − η)N Th ) 4 .(37) To calculate Q X , we consider the bit-flips in the X basis. In this case, the modes a 1 and a 2 are entangled because of the balanced beamsplitter (see Fig. 1 b)). Therefore, we consider the joint probability given by P X,0→1 = Tr[U 50/50,b 1 b 2 U BS,a 1 (η)U BS,a 2 (η)(|− a 1 ,a 2 −| a 1 ,a 2 ⊗ ρ a 1 ,Th ⊗ ρ a 2 ,Th )U † BS,a 1 (η)U † BS,a 2 (η)U † 50/50,b 1 b 2 M 1 ], P X,1→0 = Tr[U 50/50,b 1 b 2 U BS,a 1 (η)U BS,a 2 (η)(|+ a 1 ,a 2 +| a 1 ,a 2 ⊗ ρ a 1 ,Th ⊗ ρ a 2 ,Th )U † BS,a 1 (η)U † BS,a 2 (η)U † 50/50,b 1 b 2 M 0 ],(38) where U 50/50,b 1 b 2 is the second balanced beamsplitter unitary, |− a 1 ,a 2 −| a 1 ,a 2 = R π,a 1 |+ a 1 ,a 2 +| a 1 ,a 2 R † π,a 1 is obtained by applying a π-phase shifter to the state |+ , M 1 = |0 b 1 0| b 1 ⊗ |1 b 2 1| b 2 is the logical 1 measurement outcome and M 0 = |1 b 1 1| b 1 ⊗ |0 b 2 0| b 2 . Similar to above, we also renormalize to obtain the QBER, Q X = P X,0→1 P X,0→1 + P X,0→0 . We find due to symmetry that the probabilities for the diagonal basis are the same as for the rectilinear basis and it follows that Q X = Q Z , simplifying the key rate equation. B. Thermal-loss to depolarized state Using the model of thermal noise from the previous section we identify Alice's input mode A, Bob's output mode B, and the environmental input and output modes E and F (see Fig. 7), with corresponding creation and annihilation operators (lowercase). A photon-number (Fock) state of a bosonic mode may be expressed as |n = (â † ) n √ n! |0 ; in this representation, the action of the beamsplitter is given entirely by the transformationâ → √ ηb + 1 − ηf e → 1 − ηb − √ ηf .(40) If the beamsplitter receives no photon from Alice and exactly n photons from the environment, then under action (40) the combined input state |0, n AE transforms as Figure 7: A thermal noise rail with modes labeled for Alice, Bob, and the environment. |0, n AE = (ê † ) n √ n! |0, 0 AE −→ ( √ 1 − ηb † − √ ηf † ) n √ n! |0, 0 BF = 1 √ n! n ∑ k=0 n k ( 1 − ηb † ) n−k (− √ ηf † ) k |0, 0 = 1 √ n! n ∑ k=0 n! k!(n − k)! ( 1 − η) n−k (− √ η) k (n − k)! √ k! |n − k, k = n ∑ k=0 n k ( 1 − η) n−k (− √ η) k |n − k, k := |ψ n which is a coherent superposition of Fock states, with n to-tal photons split across rails B and F according to a binomial distribution. If Alice instead sends a single photon we obtain |1, n AE =â † |0, n AE −→ ( √ ηb † + 1 − ηf † ) |ψ n = √ η n ∑ k=0 n k (− √ η) k ( 1 − η) n−k √ n − k + 1 |n − k + 1, k + 1 − η n ∑ k=0 n k (− √ η) k ( 1 − η) n−k √ k + 1 |n − k, k + 1 := |φ n where the first term corresponds with Alice's photon reaching Bob, and the second with it escaping to the environment. It follows that Alice's input can be considered a 2 × 2 density matrix ρ A with terms of form ρ A i j |i j|. The collective input AE to the beamsplitter system is therefore ρ in = ρ A ⊗ ρ E = ∑ i, j,n ρ A i j p n |i, n j, n| . Since quantum channels are linear, the collective output BF is determined by the action of the channel on each |i, n j, n| term (despite |i j| individually representing a nonphysical state whenever i = j). Since |i, n represents an independent input to each beamsplitter, the output is a direct tensor product of the independent single-rail outputs derived above, i.e. |i, n AE −→ |ψ n 0 B 0 F 0 |ψ n 1 B 1 F 1 · · · |φ n 0 B 0 F 0 · · · |ψ n 1 B n 1 F n 1 := |ω i,n BF where only rail i has output |φ n , the symbol ω was chosen for no particular reason. The Hermitian conjugate of Eq. (40) transforms the corresponding bra in the same way, giving j, n| −→ ω j,n | and hence |i, n j, n| → ω i,n ω j,n . Bob's final state is ρ B = Tr F (ρ out ) = Tr F ∑ i, j,n ρ A i j p n ω i,n ω j,n = ∑ i, j ρ A i j ∑ n p n Tr F ω i,n ω j,n(41) obtained by tracing over the environmental modes in the collective output ρ out . We assume that Bob may perform a perfect photon-numberresolving (PNR) measurement in any desired basis, and that like Alice he is interested only in single-photon states |β = ∑ i β i |i and will discard all others. With perfect measurement, Bob's outcome probabilities are given by projection: P(|β ) = β|ρ B |β = Tr |β β| ρ B(42) where only terms of form |i j| in Bob's state ρ B contribute to this expression if |β is a single-photon state. Like ρ A , Bob's state ρ B may therefore be effectively considered a 2×2 matrix ρ B i j , which we now compute. Discarding terms which contain multiple photons in any single one of Bob's rails leaves |ψ n −→ |ψ n = (− √ η) n−1 − √ η |0, n + n(1 − η) |1, n − 1 |φ n −→ |φ n = (− √ η) n−1 [n − ηn − η] |1, n − η(1 − η)(n + 1) |0, n + 1 .(43) Next, to compute Tr F ω i,n ω j,n = ∑ n n ω i,n ω j,n n we need only consider components of ω i,n ω j,n with diago-nal environmental mode |n n|, as all others vanish. Discarding these nondiagonal terms in each of our single-rail outer products gives ψ n ψ n −→ η n−1 η |0, n 0, n| + n(1 − η) |1, n − 1 1, n − 1| (44) φ n φ n −→ η n−1 [n − ηn − η] 2 |1, n 1, n| + η(1 − η)(n + 1) |0, n + 1 0, n + 1| (45) φ n ψ n −→ −η n−1 √ η [n − ηn − η] |1, n 0, n|(46)ψ n φ n −→ −η n−1 √ η [n − ηn − η] |0, n 1, n| .(47) We can decompose ω i,n ω j,n in the collective Fock basis as a sum of terms corresponding with each different combination of photon numbers from Eqs. (44) and/or (45). However, we keep only those terms with a photon in exactly one of Bob's modes; if i = j, terms (46) and (47) provide these photons (albeit in a different rail on each side of the outer product) and hence all other rails must be empty. After simultaneously tracing out the environment, this gives Tr F ω i,n ω j,n = 1 η (η n i [n i − ηn i − η]) (η n j [n j − ηn j − η]) ∏ k =i, j η n k |i j| . If i = j, the photon is received either in the original rail i or an erroneous rail j = i, giving Tr F ω i,n ω i,n = η n i −1 (n i − ηn i − η) ∏ k =i η n k |i i| + ∑ j =i (1 − η) 2 η n i (n i + 1) n j η n j −1 ∏ k =i, j η n k |j j| . Returning to Eq. (41), we now sum over all n. This is done analytically, and can also be done with the aid of Mathematica. The resulting action of the channel is defined by |i j| A −→ η γ 4 |i j| B : i = j, |i i| A −→ η γ 4 |i i| + N Th (1 + N Th )(1 − η) 2 γ 4 I where γ = 1 + N Th − N Th η and I is the identity, i.e. I/2 is the maximally-mixed state. Noting that Tr ρ A = ∑ i ρ A ii = 1, we can thus express this as the qubit transformation ρ A → ρ B (see Eq. (41)): ρ A −→ η γ 2 ρ A + N Th (1 + N Th )(1 − η) 2 γ 2 ∑ i ρ A ii I (48) = η γ 4 ρ A + N Th (1 + N Th )(1 − η) 2 γ 4 I.(49) The trace of this un-normalised output now represents the probability P s of successfully receiving a valid qubit: P S = Tr ρ B = η + 2 N Th (1 + N Th )(1 − η) 2 γ 4 .(50) Conditional on success, we obtain the normalised statẽ ρ B := ρ B P S = η η + 2 N Th (1 + N Th )(1 − η) 2 ρ A + N Th (1 + N Th )(1 − η) 2 η + 2 N Th (1 + N Th )(1 − η) 2 I.(51) This represents a depolarizing channel [22] ρ → (1 − λ )ρ + λ 2 I(52) with depolarizing parameter λ = 2 N Th (1 + N Th )(1 − η) 2 η + 2 N Th (1 + N Th )(1 − η) 2 ,(53) which tends to 1 as η → 0 or N Th → ∞, as expected. A property of the depolarizing channel is that the error rate is the same in all bases: Q = λ 2 = N Th (1 + N Th )(1 − η) 2 η + 2N Th (1 + N Th )(1 − η) 2 ,(54) which can be seen from Eq. (52). In this article, we only focus on the dual-rail case of d = 2. It is left for future work to consider the high-dimensional QKD protocols in-depth. C. Fighting noise with noise squeezed state protocol Introducing trusted Gaussian noise ξ B before Bob's detection modifies the mutual information: I noise AB = 1 2 log 2 V B + ξ B V B|A + ξ B ,(55) The conditional entropy is: S(E|x B ) = S(BC|x B ) = G((λ 3 −1)/2)+G((λ 4 −1)/2),(56) where the symplectic eigenvalues are given by: λ 2 3,4 = 1 2 [A ± A 2 − 4B],(57) where A = 1 V B + ξ B (V B +V A D + ξ B ∆), B = D V B + ξ B (V A + ξ B D),(58) where for ξ B = 0 and ξ B = 1, we obtain the squeezed protocol with homodyne and heterodyne detection, respectively. D. Squeezed-state protocol phase noise In this section, we analyze the squeezed-state protocol more closely. To find the excess noise due to phase noise, we note that a squeezed coherent state has two relevant angles in phase space. These are θ , the angle of the coherent state relative to the X quadrature, and φ , the angle of the squeezing axis. As in [33], we consider the residual phase noise after estimating the angle, but with consideration of this additional φ . The quadratures after homodyne or heterodyne measurements are: x m p m = G 2 cos φ sin φ − sin φ cos φ × cos θ sin θ − sin θ cos θ x A + x 0 p A + p 0 ,(59) where Alice sends a squeezed coherent state with x A ∼ N (0,V x ) and p A ∼ N (0,V p ) centered at x 0 = 0 and p 0 = 0 measured with a coherent detector with gain G. However, we make use of trigonometric identities in Eq. (59) to obtain: x m p m = G 2 cos (θ + φ ) sin (θ + φ ) sin (θ + φ ) cos (θ + φ ) x A + x 0 p A + p 0 . (60) Bob then estimates the phase with the estimatorsθ ∼ N (θ ,V θ ) andφ ∼ N (φ ,V φ ). Bob then sends his phase estimates to Alice who makes corrections and estimates Bob's measurements. The excess noise due to the phase noises would then be: ξ x = var(x m −x m ) ξ p = var(p m −p m ).(61) where var is the variance, andx m andp m are the estimated quadratures as a function of the estimatorsθ andφ . The excess noise depends on the remaining phase noise Θ = θ + φ −θ −φ which we assume is a normally distributed variable Θ ∼ N (0, σ 2 Θ ). Then it is straightforward to calculate the excess noise: ξ x = 2V A (1 − e −σ 2 Θ /2 ) ξ p = 2V A (1 − e −σ 2 Θ /2 ),(62)whereσ 2 Θ = V φ +V θ . E. GG02 protocol with heterodyne detection For heterodyne detection by Bob, the mutual information I AB in a thermal-loss channel is [24] I AB = log 2 V B + 1 V B|A M + 1 = log 2 ηV A + (1 − η)(2N Th + 1) η + (1 − η)(2N Th + 1) ,(63) where V B is Bob's variance and V B|A M = b − c 2 /(a + 1) is Bob's variance conditioned on Alice's heterodyne measurement. S(E|B) = S(A|x B , p D ) is the information obtained by Eve conditioned on Bob's heterodyne measurement result x B and the auxiliary mode p D [24]. The covariance matrix of Alice after a projective measurement by Bob's heterodyne detection is γ out A = γ A − σ AB (γ B + I) −1 σ T AB ,(64) where σ AB = cσ Z . The conditional Von Neumann entropy is S(A|x B , p D ) = G[(λ 3 − 1)/2],(65) where the symplectic eigenvalue λ 3 is λ 3 = a − c 2 /(b + 1).(66) F. Squeezing required for the Sqz-Hom protocol In Fig. 8, we compare the performance of the Sqz-Hom protocol for the amount of squeezing used to the BB84 protocol and GG02 with heterodyne (GG02) protocol. In a pureloss channel (see Fig. 8 a)), Sqz-Hom protocols with more than 10 dB of squeezing are sufficient to be equal to or better than the GG02 protocol for all loss parameters (where the key rate is greater than K = 10 −10 ). However, for an intermediatenoise region (i.e. Fig. 8 b)), the BB84 protocol is robust at higher channel losses. We find that for very noisy thermalloss channels shown in Fig. 8 c) and d), more than 9 dB of squeezing is required to surpass BB84. Figure 8: Regions where QKD protocols give the highest secret key rate greater than K = 10 −10 based on the amount of squeezing V sq prepared by Alice for the squeezed-state protocol with homodyne detection. In the unshaded regions, K is less than 10 −10 for all protocols. Comparison of the squeezed-state protocol with homodyne detection in a pure-loss channel based on the amount of squeezing prepared by Alice. Above 9 dB of squeezing, the Sqz-Hom protocol performs better than GG02 and BB84 protocols. Figure 1 : 1Dual-rail BB84 protocol in the thermal-loss channel. a) and b) show the rectilinear and diagonal polarization bases as the dual-rail equivalent of the BB84 discrete-variable QKD protocol, respectively. Both bases undergo a phase shift θ with phase noise σ 2 θ . In a), based on which mode (top or bottom) Alice chooses to send a single photon determines the logical bit 0 or 1. In b), based on the phase 0 or π of the rotation R (black square), Alice prepares a logical bit 0 or 1, respectively. is, in fact, a depolarized state (see Appendix B): Figure 2 : 2Squeezed-state protocol with homodyne detection in the thermal-loss channel. The phase shifter θ represents the phase noise σ 2 θ . Shown in a) is the equivalent prepare and measure squeezed-state protocol and in b) is the entanglement-based version of the squeezed-state protocol. Figure 3 :Figure 4 : 34Secret key rate per polarization channel in a thermal-loss channel for increasing noise N Th . Figures a)-d) and e)-h) show the QKD versions of the protocols without and with trusted noise, respectively. For comparison, we also include the GG02 in all figures. For the pure-loss channel, the Sqz-Hom and NSqz-Hom essentially overlap with the PLOB bound for the chosen squeezing of 15 dB. Next in b) and f) with some noise in the thermal-loss channel means that the BB84, NBB84, 6S and N6S protocols outperform the CV-QKD protocols. As shown in c), d), g) and h) as more thermal noise is present, the Sqz-Hom and NSqz-Hom outperform BB84, NBB84, 6S, and N6S. In particular, Sqz-Hom saturates the lower bound (LB). Lastly, NSqz-Hom is by far the best protocol in a high noise regime as shown in h) but far from the upper bound (UB). Same asFig. 3but to benchmark the performance of the different protocols, we normalise the key rates by the upper bound K Upper . Figure 5 : 5Comparison ofK CV:DV for protocols for a set of thermal-loss channel parameters. Blue regions indicate where the 6S protocol has a higher key rate than Sqz-Hom and conversely, red regions are where the Sqz-Hom has higher key rates than the 6S protocol. Given a minimum key rate requirement, we compare the protocols which operate the best for single-channel use QKD. The 6S protocol covers a small region of intermediate noise and loss as seen in the first two subfigures. The green line indicates where the QKD protocols can operate up to the minimum key rate. The rest of the parameter space is covered by the squeezed state protocol. For high key rates in the rightmost subfigure, the 6S protocol always performs worse than Sqz-Hom. The purple line is the upper bound (UB) of the key capacity in the thermal-loss channel. The red region in the middle and right subfigures are regions where only the Sqz-Hom can achieve the minimum key rate.the thermal-loss channel and the upper bound by the Gaussian relative entropy of entanglement (of the Choi state in the thermal-loss channel): Table I : IResidual phase noise of locally generated local oscillator Gaussian modulated CV-QKD schemes in the first table and DV-QKD schemes in the second table. With the exception of Ref. 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[]
[ "Implications of Spectra and Polarizations of Fast Radio Bursts: From Perspective of Radiation Mechanisms", "Implications of Spectra and Polarizations of Fast Radio Bursts: From Perspective of Radiation Mechanisms" ]
[ "Yuan-Pei Yang \nSouth-Western Institute for Astronomy Research\nYunnan University\n650504KunmingChina\n\nPurple Mountain Observatory\nChinese Academy of Sciences\n210023NanjingChina\n" ]
[ "South-Western Institute for Astronomy Research\nYunnan University\n650504KunmingChina", "Purple Mountain Observatory\nChinese Academy of Sciences\n210023NanjingChina" ]
[ "MNRAS" ]
The extremely high brightness temperatures of fast radio bursts (FRBs) imply that the radiation process must be coherent, but the radiation mechanism is still unknown. The observed properties of narrow spectra and polarization distributions could be used to constrain the radiation mechanism of FRBs. In this work, we discuss the implications of the spectra and polarizations of FRBs from the perspective of intrinsic radiation mechanisms. We first analyze the observed relative spectral bandwidth of radio bursts from an FRB repeater. Furthermore, we generally discuss the properties of the spectra and polarization of the radiation mechanisms involving the relativistic particle's perpendicular acceleration, which depends on the relation between the particle's deflection angle and the radiation beaming angle 1/ . We find that: (1) If the narrow spectra of FRBs are attributed to the intrinsic radiation mechanism of a single particle, the condition of 1 would be necessary, in which scenario, the observed number fraction between linearly and circularly polarized bursts of some FRB repeaters might be due to the propagation effects;(2) Coherent process by multiple particles with some special distributions can lead to a narrow spectrum even for the scenario with 1;(3) If the observed number fraction between linearly and circularly polarized bursts is attributed to the radiation mechanism with 1, the cumulative distributions of the linear and circular polarization degrees would mainly depend on the particle's beaming distribution.
null
[ "https://export.arxiv.org/pdf/2305.08649v1.pdf" ]
258,685,324
2305.08649
fd5204fd40f89356c6de31cf817eabd150860fa7
Implications of Spectra and Polarizations of Fast Radio Bursts: From Perspective of Radiation Mechanisms 2023 Yuan-Pei Yang South-Western Institute for Astronomy Research Yunnan University 650504KunmingChina Purple Mountain Observatory Chinese Academy of Sciences 210023NanjingChina Implications of Spectra and Polarizations of Fast Radio Bursts: From Perspective of Radiation Mechanisms MNRAS 0002023Accepted XXX. Received YYY; in original form ZZZPreprint 16 May 2023 Compiled using MNRAS L A T E X style file v3.0radiation mechanisms: non-thermal -(transients:) fast radio bursts -polarization The extremely high brightness temperatures of fast radio bursts (FRBs) imply that the radiation process must be coherent, but the radiation mechanism is still unknown. The observed properties of narrow spectra and polarization distributions could be used to constrain the radiation mechanism of FRBs. In this work, we discuss the implications of the spectra and polarizations of FRBs from the perspective of intrinsic radiation mechanisms. We first analyze the observed relative spectral bandwidth of radio bursts from an FRB repeater. Furthermore, we generally discuss the properties of the spectra and polarization of the radiation mechanisms involving the relativistic particle's perpendicular acceleration, which depends on the relation between the particle's deflection angle and the radiation beaming angle 1/ . We find that: (1) If the narrow spectra of FRBs are attributed to the intrinsic radiation mechanism of a single particle, the condition of 1 would be necessary, in which scenario, the observed number fraction between linearly and circularly polarized bursts of some FRB repeaters might be due to the propagation effects;(2) Coherent process by multiple particles with some special distributions can lead to a narrow spectrum even for the scenario with 1;(3) If the observed number fraction between linearly and circularly polarized bursts is attributed to the radiation mechanism with 1, the cumulative distributions of the linear and circular polarization degrees would mainly depend on the particle's beaming distribution. INTRODUCTION Fast radio bursts (FRBs) are millisecond-duration radio bursts with extremely high brightness temperatures of ∼ 10 35 K, which suggests that their radiation mechanisms must be coherent. Some coherent emission mechanisms have been invoked to interpret the emissions of FRBs, including coherent radiation by charged bunches (Katz 2014;Kumar et al. 2017;Yang & Zhang 2018b, 2023Lu et al. 2020;Cooper & Wijers 2021;Wang et al. 2022b;Kumar et al. 2022b;Qu et al. 2023), maser by hydrodynamic instabilities or kinetic instabilities (Lyubarsky 2021;Beloborodov 2017;Waxman 2017;Metzger et al. 2019), coherent plasma radiation (Philippov et al. 2020;Yang & Zhang 2021), etc. However, there is no smoking gun to identify the radiation mechanism of FRBs so far. In addition to the radiation mechanism, the physical origin of FRBs also remains an unsolved puzzle due to the diversity of FRBs (see the recent review of Zhang 2022a). FRB 200428 was detected to be associated with a Galactic core-collapse magnetar SGR J1935+2154 (Bochenek et al. 2020;CHIME/FRB Collaboration et al. 2020;Mereghetti et al. 2020;Ridnaia et al. 2021;Tavani et al. 2021), implying that at least some FRBs originate from the magnetars born from the ★ E-mail: [email protected] (YPY) core collapses of massive stars. However, the association between FRB 20200120E and its host globular cluster with an extremely old age challenges the core-collapse magnetar formation (Bhardwaj et al. 2021;Kirsten et al. 2022), which means that it is more likely produced by an old object or a system associated with a compact binary merger (Wang et al. 2016;Zhang 2020;Kremer et al. 2021;Lu et al. 2022). Up to the present, hundreds of FRB sources have been detected, and dozens of them show repeating behaviors (e.g., CHIME/FRB Collaboration et al. 2021). The increasing number of detected FRBs starts to shed light on the diversity among the phenomena, and the properties of the observed spectra and polarizations provide important information about the radiation mechanism of FRBs. The first CHIME/FRB catalog identified four observed archetypes of burst morphology (Pleunis et al. 2021), including simple broadband, simple narrow band, temporally complex, and downward drifting. Meanwhile, FRB repeaters' bursts have a larger pulse duration, narrower bandwidth, and lower brightness temperature than those of the oneoff FRBs, which might be due to a beaming, propagation effect, or intrinsic populations. Zhou et al. (2022) recently reported over 600 bursts from the repeating FRB 20201124A during an active episode and found that the sub-bursts of FRB 20201124A show narrow emission spectra with a center frequency of 1.09 GHz and a characteristic width of ∼ 277MHz. FRB 20220912A also has many bursts with narrow spectral bandwidth . For the bursts with their spectra within the L band of the Five-hundred-meter Aperture Spherical radio Telescope (FAST), the relative spectral bandwidth of the radio bursts was found to be distributed near Δ / 0 ∼ (0.1−0.2). Furthermore, some FRBs show more extremely narrow bandwidth. One burst of FRB 20190711A has a central frequency of 1.4 GHz and a full-width-at-half-maximum (FWHM) bandwidth of just 65 MHz, and no evidence of any emission in the remaining part of the 3.3 GHz band of the Ultra-wideband Low (UWL) receiver system of the Parkes radio telescope (Kumar et al. 2021), which means that the relative spectral bandwidth is only Δ / 0 ∼ 0.05. The polarization observations of FRBs also show puzzling features, which might originate from the radiation mechanism or propagation effects as proposed by Qu & Zhang (2023). Many sources appear 100% linearly polarized emission (e.g., Michilli et al. 2018), and both constant and varying polarization angle have been observed (e.g., Michilli et al. 2018;Luo et al. 2020). Some FRB repeaters have a certain proportion of circularly polarized bursts. The bursts of FRB 20201124A were found to be highly polarized with the total degree of polarization larger than 90% for most bursts (Jiang et al. 2022), and some of them are circularly polarized with the largest circular polarization degree reaching ∼ 75% (Kumar et al. 2022a;Xu et al. 2022). The polarization distribution of FRB 20201124A shows that the larger the circular polarization degree the fewer the radio bursts (Jiang et al. 2022). FRB 20220912A with a high event rate of 390 hr −1 show that most bursts appear nearly 100% linear polarization degree and 45% of the bursts have circular polarization with SNR > 3 . Besides, two famous activate FRB repeaters, FRB 121102 and FRB 190520B, also have a few bursts with circular polarization of dozens of percent, but the number fraction of the circularly polarized bursts are much fewer than those of FRB 20201124A and FRB 20220912A (Feng et al. 2022). In this work, we will discuss the physical origin of the observed features of the spectra and polarizations of FRBs from the perspective of radiation mechanisms, aiming to explain the observed narrow spectra and the polarization distribution of FRBs. The paper is organized as follows. In Section 2, we discuss the spectral bandwidth distribution and possible physical origins of narrow spectra of FRBs. In Section 3, we generally analyze the features of the spectra and polarizations of the intrinsic radiation mechanisms, including the radiation mechanisms with the particle's deflection angle larger than the radiation beaming angle in Section 3.1 and the opposite scenario in Section 3.2, and the possible astrophysical scenarios are discussed in the Section 3.3. The results are summarized and discussed in Section 4. The convention = /10 is adopted in cgs units unless otherwise specified. NARROW SPECTRA OF FRBS: OBSERVATION AND PHYSICAL ORIGIN Spectral bandwidth distribution of an FRBs repeater Some FRBs, e.g., FRB 20190711A, FRB 20201124A, and FRB 20220912A, appear extremely narrow spectra within the bandwidth of a telescope (e.g., Kumar et al. 2021;Zhou et al. 2022;Zhang et al. 2023), implying that the spectra of at least some FRBs are intrinsically narrow. In this section, we will discuss the implication of the observed narrow bandwidths of FRBs, and emphasize that the observed relative spectral bandwidth mainly depends on the intrinsic spectral shape, definition of spectral bandwidth (e.g., flux-thresholddependent bandwidth, full width at half maximum, full width at tenth maximum, etc.), and the telescope's bandwidth. We first consider that the intrinsic spectra of radio bursts from an FRB source have a general form described by a broken power-law distribution, = ,0            0 , for 0 , 0 − ℎ , for > 0 ,(1) where ,0 corresponds to the maximum flux at the peak frequency 0 . Notice that > 0 and ℎ > 0 are assumed here, considering that the flux vanishes at → 0 or ∞. We define the telescope's threshold flux as ,th , then the signal-to-noise ratio in the frequency domain is / = ,0 / ,th if the telescope's bandwidth is assumed to be infinite. Thus, the flux-threshold-dependent relative spectral bandwidth of a radio burst is Δ 0 1 ℎ − − 1 . ( We can see that the relative spectral bandwidth Δ / 0 depends on / and the intrinsic spectral shape that is described by the above two power-law indexes. In practical measurement, one usually defines the spectral bandwidth via the full width at half maximum (FWHM) (e.g., Kumar et al. 2021;Zhang et al. 2023) or via the full width at tenth maximum (FWTM) (e.g., Pleunis et al. 2021) instead of that based on / , which replace / with 2 and 10 in Eq.(2), respectively. For example, for FWHM, the relative spectral bandwidth of a radio burst is Δ 0 2 1 ℎ − 2 − 1 . (3) Since such a defined (FWHM or FWTM) relative spectral bandwidth only depend on the intrinsic spectral shape, the Δ / 0 distribution of an FRB repeater should be narrow for a given intrinsic radiation mechanism. Most intrinsic radiation mechanisms involved in various astrophysical scenarios have a low-frequency spectral index of < 3, see Table 1 in detail. For example, the synchrotron selfabsorption has a low-frequency spectral index of = 5/2 (Rybicki & Lightman 1986) and the curvature radiation by a bunch-cavity (or electron-positron) pair has a low-frequency spectral index of = 8/3 Yang & Zhang 2023). Thus, for these radiation mechanisms with < 3, the relative spectral bandwidth for FWHM should satisfy Δ 0 > 0.2,(4) according to Eq.(3). Assuming that the intrinsic spectrum is consistent with that of the curvature radiation, one approximately has 2/3 and ℎ → ∞ (Yang & Zhang 2018b, 2023. For a radio burst with / ∼ 10, the flux-threshold-dependent relative spectral bandwidth is predicted to be Δ / 0 0.97, which corresponds to a wide spectrum. According to Eq.(3), the relative spectral bandwidth for FWHM is Δ / 0 0.65. If the signal-to-noise ratio is relatively small with / ∼ 5, the flux-threshold-dependent relative spectral bandwidth would be Δ / 0 0.91 that appears narrower than that for / ∼ 10, but the relative spectral bandwidth for FWHM keeps unchanged. In particular, for FRB 20190711A with an extremely narrow FWHM spectral bandwidth of Δ / 0 ∼ 0.05 (Kumar et al. 2021), one has ( , ℎ ) 14, implying an extremely narrow intrinsic spectrum that should involve some special mechanisms, see the discussion in Section 2.2. In reality, the bandwidth Δ of a radio telescope is usually narrow Curvature radiation by a single charged particle 2/3 Jackson (1998); Yang & Zhang (2018b) Curvature radiation by a bunch-cavity (or electron-positron) pair 8/3 ; Yang & Zhang (2023) Curvature radiation by fluctuating bunches 0 Yang & Zhang (2023) Synchrotron radiation by particles with a random pitch-angle distribution 1/3 Jackson (1998); Rybicki & Lightman (1986) Synchrotron radiation by particles with a narrow pitch-angle distribution 2/3 Yang & Zhang (2018a) Synchrotron self-absorption 5/2 Rybicki & Lightman (1986) Jitter radiation 1 Medvedev (2000); Dermer & Menon (2009) Blackbody radiation 2 Rybicki & Lightman (1986) Bremsstrahlung radiation 0 Rybicki & Lightman (1986) Inverse Compton scattering Depend on incident photon spectrum Rybicki & Lightman (1986); Zhang (2022b) compared with the telescope's central frequency 0, , Δ 0, . For example, the L band of FAST is from 1 GHz to 1.5 GHz, i.e., 0, = 1.25 GHz and Δ = 0.5 GHz. Due to the limited telescope's bandwidth, many observed spectra of FRBs are usually incomplete. In particular, if the peak frequency of a radio burst is outside the telescope's bandwidth, the FWHM (or FWTM) bandwidth would become invalid to describe the intrinsic spectral shape, because the observed flux is peaked at one end of the telescope's band rather than the intrinsic peak frequency. In this case, the definition of the fluxthreshold-dependent spectral bandwidth might be more meaningful in observation. If a radio burst is observable for a certain telescope, its emission must be in the telescope's bandwidth, leading to the following condition: 0 + Δ 2 > 0, − Δ 2 and 0 − Δ 2 < 0, + Δ 2 .(5) The observed peak frequency 0,obs (not the intrinsic peak frequency) of an observable FRB is usually thought to be in the telescope's bandwidth, 0,obs ∈ 0, − Δ 2 , 0, + Δ 2 ,(6) although the intrinsic peak frequency 0 might be estimated outside the telescope's bandwidth based on the observed spectral shape. The observed spectral bandwidth of an observable FRB is Δ obs = min 0 + Δ 2 , 0, + Δ 2 − max 0 − Δ 2 , 0, − Δ 2 .(7) Since the telescope's bandwidth is usually narrow, the distribution of the intrinsic peak frequency 0 of an FRB repeater could be approximately assumed to be uniform near the telescope's bandwidth, i.e., the distribution function of 0 could be approximately described by ( 0 ) ∼ const.(8) We make a Monte Carlo simulation to generate the distribution of the observed spectral bandwidth Δ obs of the radio bursts from an FRB repeater, as shown in Figure 1. The distribution of Δ 0,obs / obs is approximately consistent with the distribution of Δ obs due to 0,obs ∼ . We take 0, = 1.25 GHz and Δ = 0.5 GHz that are The simulated observed spectral bandwidth distribution of radio bursts from an FRB repeater. The purple, yellow, and blue bars correspond to the different relative spectral bandwidths with Δ / 0 = 0.1, 0.5 and 1, respectively. We take 0, = 1.25 GHz and Δ = 0.5 GHz for the telescope's parameters. The intrinsic peak frequencies 0 of the radio bursts are assumed to be uniformly distributed near the telescope's bandwidths, i.e., ( 0 ) = const. consistent with the L band of FAST. For Δ / 0 = 0.1, the observed spectral bandwidth Δ obs of most bursts are consistent with the intrinsic ones due to Δ < Δ . For Δ / 0 = 1, since many bursts have Δ > Δ , the observed spectral bandwidth Δ obs of most bursts would be constrained by the bandwidth of the telescope, leading to Δ obs ∼ Δ . Very few bursts have Δ obs < Δ , because only a part of one end (low-frequency end or high-frequency end) of the intrinsic spectral bandwidth Δ is within the telescope's bandwidth. Considering that many radio bursts are incomplete in the frequency domain due to the narrow telescope's bandwidth, the observed spectral bandwidth distribution of radio bursts from an FRB repeater could be used to test whether most of the bursts' spectra of an FRB source are intrinsically narrow. Physical origin of FRB narrow spectra Next, we discuss the possible physical origin of the intrinsic narrow spectra of FRBs. We consider that a finite pulse of electromagnetic wave has the form of ì ( ) = ì ( ) + ì ⊥ ( ), where ì ( ) and ì ⊥ ( ) are a pair of orthogonal components of ì ( ). The properties of ì ( ) vary with time and vanishes sufficiently rapidly for → ±∞, and the power spectrum of ì ( ) satisfies | ( )| 2 = | ( )| 2 + | ⊥ ( )| 2 . Since the two orthogonal components are independent, let us treat only one of the two components, ì ( ) with = , ⊥. In particular, if an observed spectrum | ( )| 2 appears narrow, the main component between | ( )| 2 and | ⊥ ( )| 2 must also be narrow. Without loss of generality, the spectrum of the main component could be roughly described by a rectangular profile with a central frequency 0, and a spectral bandwidth Δ , | ( )| 2 ∝ rect − 0, Δ ,(9) where rect( ) is the rectangular function that is defined as rect( ) ≡ 1 for | | 1/2 and rect( ) ≡ 0 for | | > 1/2. Thus, ( ) as the Fourier transformation of ( ) is given by ( ) ∝ rect − 0, Δ ,(10) where is a phase argument. Generally, cannot be directly obtained only based on the information of the power spectrum | ( )| 2 , but for a selected appropriate pair of orthogonal components, ( ) and ⊥ ( ) might be a pair of real and imaginary numbers, i.e., Im[ ( )] = ( ) and Re[ ⊥ ( )] = ⊥ ( ), see the following discussions in Section 3.1 and Section 3.2, leading to ( ) = /2 with ∈ Z. In this case, one may take = const. According to the properties of the Fourier transform, the corresponding pulse profile is ( ) ∝ Δ √ 2 sinc Δ 2 − ( 0, − ) ,(11) where sinc( ) ≡ sin / . In Figure 2, we plot the pulse profile ( ) based on Eq.(11). The top panel shows the scenario with a narrow spectrum of Δ / 0, = 0.1 and the bottom panel shows the scenario with a wide spectrum of Δ / 0, = 1. We can see that a narrow spectrum with Δ / 0, 1 implies that the electromagnetic signal would be quasi-sinusoid with a frequency of ∼ 0, and a typical pulse duration of ∼ 4 /Δ . There are some possibilities to generate the quasi-sinusoid electromagnetic pulse in physics: (1) The quasi-sinusoid signal is produced by the intrinsic radiation mechanism of a single accelerated charged particle, and the particle motion is required to be periodic during its radiation beam pointing to the observer. We will mainly discuss this scenario in Section 3.2. (2) The quasi-sinusoid signal is generated by the coherence of the radiation process of multiple charged particles with some special distributions. One of the possibilities is that the charged particle distribution is periodic in the time domain. For example, we consider a medium containing multiple radiating particles with a number of . Each radiating particle emits a pulse of 0 ( ) with the same shape but with different arrival times. Thus, the total field from the multiple radiating particles is given by ( ) = ∑︁ 0 ( − ).(12) According to the time-shifting property of the Fourier transform, the power spectrum of multiple radiating particles is | ( )| 2 = | 0 ( )| 2 ∑︁ 2 ,(13) where | 0 ( )| 2 corresponds to the power spectrum of the first radiating particle. The coherence properties of the radiation by the multiple particles are determined by the factor of | exp( )| 2 . If the multiple particles are periodically distributed, one has = / , where 1/ = const. corresponds to the time separation between two adjacent particles in the time domain. We define = exp( / ), the modulus square of the sum of the phase factor in Eq.(13) is calculated by ∑︁ =1 2 = − 1 − 1 2 = 2 − − * 2 − − * = 1 − cos( / ) 1 − cos( / ) = sin 2 ( /2 ) sin 2 ( /2 ) .(14) where * is the conjugation of the complex number , and the geometric sequence summation is used in the above calculation. Therefore, the power spectrum by the multiple particles with a periodic distribution is given by | ( )| 2 = | 0 ( )| 2 sin 2 2 sin −2 2 .(15) The radiation is coherently amplified for sin 2 ( /2 ) ∼ 0, leading to the coherent peak frequencies at / = 2 with ∈ Z + . In Figure 3, we plot the power spectrum of the multiple radiating particles with a periodic distribution according to Eq.(15). The spectrum of a single particle is assumed to be | 0 ( )| 2 ∝ 2/3 exp(− / ) (corresponding to the spectrum of the curvature radiation by a single particle) as an example, and = 10 4 and = are taken. We can see that the coherent energy is radiated into multiples of 2 with narrow bandwidths. Note that the wave at a very low frequency is also significantly coherent due to the finite duration of the particle . The spectrum of multiple radiating particles with a periodic distribution in the time domain. A curvature-radiation-like spectrum for a single particle, | 0 ( ) | 2 ∝ 2/3 exp(− / ), is taken as an example in this figure. We take = and = 10 4 . The peak frequency is at 2 with ∈ Z + . train, but it is very likely to be absorbed due to various absorption mechanisms, e.g., plasma absorption, synchrotron-self absorption, etc. (3) The quasi-sinusoid signal is attributed to the plasma resonant process. In unmagnetized plasma, the resonance condition for a particle with velocity ì follows by considering the wave phase at the position of the particle to be time-independent, leading to − ì ( ) · ì = 0.(16) For example, Cherenkov radiation is confined within the Cherenkov cone and moves outward in a direction normal to the cone, and the radiation energy is mainly at the resonant frequency given by Eq.(16). It is noteworthy that the charged particle is not required to be accelerated for Cherenkov radiation, which is different from that discussed in the case (1). Furthermore, in magnetized plasma, the gyroresonance condition is given by − ì ( ) · ì − = 0 with ∈ Z,(17) where = / is the cyclotron frequency, is the magnetic field strength, is the particle's Lorentz factor. The spectral bandwidths Δ in these scenarios depend on the distributions of particles' velocity and magnetic fields, which is beyond the scope of this work. RADIATIONS BY ACCELERATED CHARGED PARTICLES: SPECTRA AND POLARIZATIONS In this section, we generally analyze the features of the spectra and polarizations of the radiation by charged accelerated particles, which are applicable to most radiation mechanisms in various astrophysical scenarios. We consider that a particle with a charge moves on a trajectory ì 0 ( ) with velocity ì ( ) and acceleration ì ( ) , where is the retarded time. The line-of-sight direction is ì and the distance between the particle and the observer is . The radiation field at (ì, ) is given by (e.g., Rybicki & Lightman 1986) ì (ì, ) = ì 3 × {( ì − ì ) × ì } rec , ì (ì, ) = ì × ì rec ,(18) where ≡ 1 − ì · ì , the quantities in the square bracket, [...] rec , are evaluated at the retarded time . The radiation energy per unit frequency interval per unit solid angle is E ≡ Ω = 2 4 2 ∫ ∞ −∞ −3 ì × {( ì − ì ) × ì } rec 2 (19) = 2 2 4 2 ∫ ì × ( ì × ì ) exp ( − ì · ì 0 ( )/ ) 2 (20) = 2 2 4 2 − ì ( ) + ì ⊥ ⊥ ( ) 2 ,(21) where and ⊥ are two orthogonal components perpendicular to the line of sight. The linear polarization degree is = ( * − ⊥ * ⊥ ) 2 + ( * ⊥ + ⊥ * ) 2 1/2 * + ⊥ * ⊥ ,(22) and the circular polarization degree is = 1 * ⊥ − ⊥ * * + ⊥ * ⊥ .(23) It should be noted that the E corresponds to the total radiation energy in an entire pulse. If the radiation pulse repeats on average time , the above radiation energy could be converted to the radiation power (Rybicki & Lightman 1986) P ≡ 1 Ω = E .(24) The radiation of a charged particle with Lorentz factor undergoing arbitrary is a coherent superposition with the contributions from the accelerations parallel to and perpendicular to the particle's velocity. For comparable parallel and perpendicular forces, the radiation from the parallel component is of order 1/ 2 compared to that from the perpendicular component. Thus one usually neglects the parallel acceleration 1 . The radiation spectrum of the perpendicular component depends on the relation between the particle's deflection angle and the radiation beaming angle ∼ 1/ (Landau & Lifshitz 1975), as shown in Figure 4. The particle's deflection angle is determined as follows. The particle's momentum is ∼ , and the change in the perpendicular momentum due to a transverse force ⊥ is ⊥ ∼ ⊥ Δ acc (Δ acc is the time during which the particle acceleration changes significantly). Thus, the particle's deflection angle is ∼ ⊥ ∼ ⊥ Δ acc , leading to ∼ ⊥ Δ acc .(25) If 1, i.e., the particle's deflection angle is much larger than the radiation beaming angle, the observer will see radiation from a short segment of the electron's trajectory that is nearly parallel to the line of sight, as shown in the panel (a) of Figure 4, which corresponds to the scenarios of curvature radiation (e.g., Jackson 1998;Yang & Zhang 2018b, 2023, traditional (large-pitch-angle) synchrotron radiation (e.g., Ginzburg & Syrovatskii 1969;Jackson 1998;Rybicki & Lightman 1986), etc. If 1, i.e., the particle's deflection angle is much smaller than the radiation beaming angle, the particle's entire trajectory would be seen by the observer, as shown in the panel (b) of Figure 4, which corresponds to the small-pitch-angle synchrotron radiation (e.g., Epstein 1973), jitter radiation (e.g., Medvedev 2000), etc. In the following discussion, we will discuss the case of 1 in Section 3.1 and the case of 1 in Section 3.2. Deflection angle larger than radiation beaming angle In the scenario with 1, the radiation is equivalent to the radiation by the particle moving instantaneously at constant speed on an appropriate circular path (Jackson 1998), as shown in the panel (a) of Figure 4. We consider that the acceleration curvature radius is , the angle between the line of sight and the trajectory plane is , and the radiation angular frequency is . The radiation energy per unit frequency interval per unit solid angle is given by the above equation with (Jackson 1998) ( ) = 2 √ 3 1 2 + 2 2/3 ( ),(26)⊥ ( ) = 2 √ 3 1 2 + 2 1/2 1/3 ( ),(27) and ≡ 1 2ˆ 1 + 2 2 3/2 withˆ≡ ,(28) where = 3 3 /2 is the typical radiation frequency. The radiation energy per unit frequency interval per unit solid angle is E = 3 2 4 2 2ˆ2 (1 + 2 2 ) 2 2 2/3 ( ) + 1 1/ 2 2 + 1 2 1/3 ( ) .(29) The spectrum E of a single radiating particle is intrinsically wide (Jackson 1998;Yang & Zhang 2018b), Δ / 0 ∼ 1 (see Section 2.1), as shown in Figure 5. The spectrum satisfies the power-law distribution with E ∝ˆ2 /3 at the low frequency and appears an exponential decay at the high frequency. Meanwhile, the larger the viewing angle , the lower the cutoff frequency. A relatively narrow spectrum could be generated by the coherent curvature radiation from wave due to the parallel acceleration (Beloborodov 2022 Figure 5. The spectrum given by Eq.(29) that is applicable for the scenario with the particle's deflection angle larger than the radiation beaming angle, 1. The black, red, and blue lines correspond to = 0.1, 1 and 3, respectively. The unit of E is arbitrary. a structured bunch Yang & Zhang 2023), but it is still hard to explain the observed extremely narrow spectrum of FRB 20190711 with Δ / ∼ 0.05. In particular, the dynamic fluctuation of the bunches would also make the spectrum show a white noise, especially at the low-frequency band (Yang & Zhang 2023). One possibility to generate a narrow spectrum via the curvature radiation is that the bunches along a field line are periodically distributed due to some plasma effects, like resonant Langmuir wave, in which scenario, the radiation could be amplified at some harmonic frequencies as discussed above, see Eq.(15) and Figure 3. ; Zhang 2022b; Qu & Zhang 2023). -5 -4 -3 -2 -1 0 1 2 -4 -3 -2 -1 0 1 Log@ẁ D Log@E w êa.u.D According to Eq.(29) and using the property of Bessel function, ( ) ∼ √︁ /2 exp(− ) for 1, the radiation energy falls off in angle approximately as E ∼ E ,0 exp −ˆ3 3 ,(30) where E ,0 ≡ E ( = 0) and E ∝ 2 ( ) is used. We consider that only the burst with radiation energy E > −1 E ,0 with 1(31) could be observed due to the constraint of a detector's sensitivity, leading to < th ≡ 1 ln 1/3 .(32) Only the bursts with viewing angle < th are observable. Notice that the distribution of the intrinsic burst energies is neglected, and here we are mainly interested in the suppression effect by the viewing direction. According to Eq.(22), Eq.(23), Eq.(26) and Eq. (27), the linear polarization degree is = 2 2/3 ( ) − 1/(1/ 2 2 + 1) 2 1/3 ( ) 2 2/3 ( ) + 1/(1/ 2 2 + 1) 2 1/3 ( ) ,(33) and the circular polarization degree is = 2/(1/ 2 2 + 1) 1/2 2/3 ( ) 1/3 ( ) 2 2/3 ( ) + 1/(1/ 2 2 + 1) 2 1/3 ( ) . Similar to the spectrum given by E , both and are also the functions as the variables (ˆ, ). In Figure 6, we plot the linear and circular polarization degrees Π with = and as the functions of the dimensionless frequencyˆand the viewing angle , respectively. For a certain viewing angle , the higher the frequencyˆ, the lower (higher) the linear (circular) polarization degree. For a certain frequencyˆ, the larger the viewing angle , the lower (higher) the linear (circular) polarization degree. Thus, the high circular polarization degree should be attributed to the off-beam observation (Wang et al. 2022a;Liu et al. 2023;Qu & Zhang 2023). Furthermore, we consider that multiple radiating particles are uniformly distributed in a fan beam with an opening angle Θ , and the viewing angle is Θ. The spectrum of the radiation by multiple particles is usually wide with the exception of the particles with some special (e.g., periodic) distributions as pointed out above. Since the spectrum by multiple particles has been discussed in detail in Yang & Zhang (2018b, 2023), we will not repeat it here. In the following discussion, we mainly focus on the polarization properties of multiple particles. According to the polarization properties given by Eq.(33) and Eq.(34), the polarization is 100% linear when the viewing direction is on the trajectory plane, and most radiation energy is emitted near the trajectory plane. The larger the viewing angle, the higher the circular polarization degree. Since the circular polarizations on the different sides of the trajectory plane are opposite, in the particles' beam center the coherent sum of the circular polarizations will be canceled, leading to the linear polarization dominated. A detailed analysis was also discussed in Wang et al. (2022b) and Liu et al. (2023). Therefore, the linear polarization degree could be approximately given by Π 1, for Θ Θ , (Θ − Θ ), for Θ > Θ ,(35) and the circular polarization degree could be approximately given by Π 0, for Θ Θ , (Θ − Θ ), for Θ > Θ ,(36) Since the view direction related to the trajectory plane is random, the number of the bursts emitting within (Θ, Θ + Θ) is 2 (Θ) Θ = 1 (Θ + th ) Θ,(37) where th is given by Eq.(32), and Θ + th corresponds to the threshold angle above which the observed flux would be less than the telescope's flux threshold. Therefore, the cumulative distribution of the linear and circular polarization degrees are (> Π ) = Θ + ( )(Π ) Θ + th ,(38)(> Π ) = th − ( )(Π ) ( Θ + th ) ,(39) for ( )(Π ) th with = , , respectively, where ( ) (Π ) is the inverse function of ( ) given by Eq.(33) and Eq.(34). In Figure 7, we plot the cumulative distributions of the linear and circular polarization degrees according to Eq.(38) and Eq.(39). The cumulative distributions of the polarization degrees depend on the telescope's flux threshold , the particles' beaming angle Θ , and observed frequencyˆ. We can see that: (1) The higher the telescope's sensitivity, the lower the number fraction between the linearly and circularly polarized bursts. The reason is that most high circularly polarized bursts have relatively low fluxes due to large values of . (2) The larger the particles' beaming angle, the higher the number fraction between the linearly and circularly polarized bursts. If Θ 1, most bursts would have Π ∼ 1 and Π ∼ 0. A moderate number fraction between linearly polarized bursts and circularly polarized bursts as shown in FRB 20201124A (Xu et al. 2022;Jiang et al. 2022) requires that Θ ∼ 1. (3) The higher the observed frequency, the higher the number fraction between the linearly and circularly polarized bursts. The reason is that the threshold viewing angle th is significantly suppressed at the high frequency (see Eq.(32)), leading to the relative number of the bursts from the particle beaming angle Θ becoming larger. Deflection angle smaller than radiation beaming angle In the scenario with 1, the charged particle moves along the line of sight with an almost constant velocity ì but with a varying acceleration ì , as shown in the panel (b) of Figure 4. According to Eq.(19), the radiation energy per unit frequency interval per unit solid angle could be written as (also see Landau & Lifshitz 1975 ) E = 2 4 2 ˜ 4 ì × ( ì − ì ) × ì˜ 2(40) 2 Notice that the distribution of the viewing direction should be not (Θ) Θ = sin Θ Θ in this scenario, because the viewing direction is related to the trajectory plane of the accelerated particle, see Figure 14.9 and Section 14 in Jackson (1998) for a detail discussion. Figure 7. The cumulative distribution of the linear (top panels) and circular (bottom panels) polarization degrees for the radiation by multiple particles with 1. Notice that the cumulative distribution of the circular polarization degrees is (> Π = 0) = 1 at Π = 0 and decreases significantly once Π > 0. The component of (> Π = 0) = 1 at Π = 0 is not shown in the bottom panels. The distribution function of the viewing direction is assumed to satisfy Eq.(37). In the left panels: the solid, dashed, and dotted lines correspond to = 10, 100 and 1000, respectively forˆ= 1 and Θ = 10. In the middle panels: the solid, dashed, and dotted lines correspond to Θ = 3, 10 and 100, respectively forˆ= 1 and = 100. In the right panels: the solid, dashed, and dotted lines correspond toˆ= 0.1, 1 and 3, respectively for Θ = 10 and = 100. with ì˜≡ ∫ ∞ −∞ ì˜ and˜≡ (1 − ì · ì ) .(41) In the ultrarelativistic case, the longitudinal acceleration is smaller than the transverse acceleration, ì / ì ⊥ ∼ 1/ 2 1. Thus, ì and ì are approximately perpendicular to each other, ì ⊥ ì . Since both ì and ì are approximately constant in the above equation, the properties of the spectrum and polarization are mainly determined by the acceleration ì . An estimate of the typical frequency in which the radiation is mainly concentrated is made as follows (Landau & Lifshitz 1975): the Fourier component ì˜i s significantly different from zero only if 1/˜is of the same order as the time Δ acc during which the particle acceleration changes significantly, leading to ∼ (1 − ) −1 Δ −1 acc ∼ 2 2 Δ −1 acc .(42) For example, if the particle's acceleration is due to the Lorentz force by the magnetic field , one has Δ −1 acc ∼ = / , where is the cyclotron frequency, leading to the typical radiation frequency to be ∼ / , see the following discussion in detail. Next, we discuss the general properties of the polarization in the scenario with 1. Choosing a coordinate system withdirection pointing toward the observer and with the particle velocity on the − plane, thus ì = (0, 0, 1) and ì = (0, sin , cos ). In the coordinate system with -direction pointing toward the particle velocity ì and -axis parallel with -axis, the acceleration could be written as ì = ( cos , sin , 0), where = | ì | and is the azimuth angle of ì in the − plane that is perpendicular to -direction. Thus, the acceleration in the coordinate system is ì = ( cos , sin cos , − sin sin ). According to Eq.(40), the radiation polarization property is determined by ì × ( ì − ì ) × ì = [− cos (1 − cos ), sin (1 − cos ), 0] ,(43) leading to (˜) ∝ − ∫ ( ) cos ( )˜,(44)⊥ (˜) ∝ ∫ ( ) sin ( )˜.(45) Based on Eq.(22) and Eq.(23), we can easily prove that: (1) If the acceleration is always on a straight line perpendicular to ì , i.e., = const., the polarization is fully linear with = 1; (2) If the acceleration rotates with a constant angular velocity Ω on the plane perpendicular to ì , i.e., ( ) = Ω and ( ) = const, the polarization is fully circular with = 1. In order to obtain the accurate spectrum and polarization, a simpler and more intuitive processing method is to calculate the radiation in the particle comoving frame with velocity = cos related to the observer frame , then transfer the radiation to the frame via Lorentz and Doppler transformations. In the frame, the particle moves with the velocity ⊥ (1 − 2 ) = ( 2 − 1) 1/2 sin cos 2 + 2 sin 2 1. Thus, the particle in the frame is non-relativistic for 1. In many astrophysical scenarios, the perpendicular acceleration of a charged particle is usually attributed to the Lorentz force by magnetic fields, meanwhile, the intrinsic variation timescale of the magnetic field is longer than Δ acc . In this case, the radiation in the frame is cyclotron-like. We consider that in the frame the acceleration curvature radius is , and the angle between the line of sight and the trajectory plane is . We define ≡ sin (47) with as the harmonic number, then in the frame, the radiation power per unit solid angle in the -th harmonic is (Landau & Lifshitz 1975;Jackson 1998) Ω = 2 4 0 2 8 3 − ì ( ) + ì ⊥ ⊥ ( ) 2 (48) with ( ) = 2 ( ) ,(49)⊥ ( ) = 2 cot ( ),(50) where the fundamental frequency is 0 = .(51) In particular, if the gyration motion is caused by the Lorentz force of the magnetic field, one has 0 = = / , where is the cyclotron frequency in the frame. According to Eq.(22), Eq.(23), Eq.(49) and Eq.(50), the linear polarization degree is = [ ( )/ ] 2 − (cot 2 / 2 ) 2 ( ) [ ( )/ ] 2 + (cot 2 / 2 ) 2 ( ) ,(52) and the circular polarization degree is = 2[ ( )/ ] (cot / ) ( ) [ ( )/ ] 2 + (cot 2 / 2 ) 2 ( ) .(53) Due to 1, the emission but the fundamental frequency = 1 can be neglected, leading to an extremely narrow spectrum. Using the properties of Bessel function ( ) ∼ [1/Γ( + 1)] ( /2) for 0 < ( + 1) 1/2 , the radiation power reduces to P ≡ Ω = 2 2 0 2 8 (1 + cos 2 ) ( − 0 ).(54) The linear polarization degree is = 1 − cos 2 1 + cos 2 ,(55) and the linear polarization degree is = 2 cos 1 + cos 2 . (56) Using the following transformations 3 cos = cos − 1 − cos 1 − 2 2 1 + 2 2 ,(57)= 1 − cos (1 − 2 ) 1/2 2 (1 + 2 2 ),(58)P = P (1 − 2 ) 3/2 (1 − cos ) 3 P 8 3 (1 + 2 2 ) 3 .(59) 3 Note that the radiation power P is emphasized to be the received specific power in the frame, and a factor of 3 (1 + cos ) 3 should be corrected from the emitted specific power, see Section 4.8 of Rybicki & Lightman (1986) and involve = (1 + cos ) to calculate the specific radiation power. We definē ≡ 0 ,(61) then the received radiation power can be rewritten as P = 2 5 2 0 4¯4 1 −¯+ 1 2¯2 ¯− 2 1 + 2 2 ,(62) and the radiation only occurs at the direction with = 2 − 1 1/2 .(63) Notice that the particle's deflection angle only affects the normalized radiation power but not the typical frequency and the spectral shape. According to Eq.(62), for a certain viewing direction , the emission is only at the frequency¯= 2/(1 + 2 2 ). Thus, the radiation spectrum of a single particle is extremely narrow. Since most radiation energy is emitted with the direction satisfying 1, the typical radiation frequency is¯∼ a few (corresponding to ∼ 0 ), which is consistent with the above result estimated by Eq.(42). Since the polarization degree is Lorentz invariance, according to Eq.(55), Eq.(56) and Eq. (57), the linear and circular polarization degrees are = 2 2 2 1 + 4 4 = 2¯−¯2 2 − 2¯+ 2 ,(64)= 1 − 4 4 1 + 4 4 = 2¯− 2 2 − 2¯+ 2 .(65) In particular, > 1 and < 1 correspond to the opposite (left and right) circular polarization, respectively. In Figure 8, we plot the linear and circular polarization degrees Π with = and as the functions of the dimensionless frequency¯and viewing angle , respectively. The blue and red lines correspond to the linear and circular polarization degrees, respectively. The top and bottom panels show the polarization degree Π as the functions of the dimensionless frequency¯and the viewing angle , respectively. We can see that high linear polarization (low circular polarization) mainly occur at ∼ 1 and ∼ 1, otherwise, low linear polarization (high circular polarization) is dominant. Furthermore, we consider that multiple radiating particles are uniformly distributed in a three-dimensional beaming with an opening angle Θ . The number of the charged particles within (Θ, Θ + Θ) is (Θ) Θ = ,0 sin Θ Θ.(66) When the viewing direction points to the beaming center Θ = 0, the radiation spectrum by multiple particles might be approximately given by P (Θ = 0) ∝ ∫ Θ 0 P ( Θ) sin Θ Θ ∝¯4 − 2¯3 + 2¯2 for¯m in = 2 1 + 2 Θ 2 <¯< 2,(67) where Θ 1 is assumed in the above equation. Outside the particles' beam Θ Θ , the radiation spectrum by multiple particles might be approximately given by P (Θ) ∝ ∫ Θ Θ−Θ P ( Θ) Θ ∝¯5 − 2¯4 + 2¯3 (2¯−¯2) 1/2 for¯m in = 2 1 + 2 Θ 2 <¯<¯m ax = 2 1 + 2 (Θ − Θ ) 2 .(68) Notice that a factor of sin Θ should be involved in the above equation for the viewing direction outside the particles' beam. In Figure 9, we plot the spectra for Θ = 0 and Θ Θ given by Eq.(67) and Eq.(68), respectively. For the case of Θ = 0, the peak frequency is at¯∼ 2 near which the spectrum is narrow due to P ∝¯4. For the case of Θ Θ , the spectrum is much narrower, i.e., Δ¯/¯ 2Θ /Θ 1. Meanwhile, the larger the viewing angle Θ, the narrower the spectrum and the lower the peak radiation power. At last, we should notice that the above discussion assumes that all radiating particles have the same Lorentz factor . The distribution of for multiple particles would make the spectra relatively wider. The observed linear and circular polarization degrees depend on the gyration directions of the radiating particles and whether the viewing direction is within Θ . If all radiating particles have the same gyration directions, clockwise or anticlockwise, the linear and circular polarization degrees of the multiple particles might be written as Figure 9. The spectra that are applicable for multiple radiating particles with 1. The blue and red line corresponds to the on-beam case given by Eq.(67) and the off-beam case given by Eq.(68), respectively. The red solid, dashed and dotted lines correspond to Θ = 3, 5 and 8, respectively. Here Θ = 1 is taken. The unit of P is arbitrary. For easy comparison with different scenarios, the spectra of the off-beam cases are multiplied by an arbitrary factor in this figure. and Π 1, for Θ Θ , Π 0, for Θ Θ , (Θ − Θ ), for Θ > Θ ,(69)(Θ − Θ ), for Θ > Θ ,(70) respectively, similar to the discussion in Section 3.1. Notice that different from the above scenario with 1, the on-beam radiation for 1 is dominated by the ∼ 100% circular polarization. On the other hand, if the gyration directions of the particles are random, their polarizations would cancel out. In the following, we are mainly interested in the former scenario. Since the viewing direction is random in the solid angle, the number of the observable burst within (Θ, Θ + Θ) is given by (Θ) Θ = sin Θ Θ for 0 Θ Θ + th ,(71) otherwise, (Θ) Θ = 0, where Θ + th corresponds to the lower limit of the frequency¯or the lower limit of P . According the polarization properties given by Eq.(64) and Eq.(65), the polarization is ∼ 100% circular except ∼ 1. Therefore, the cumulative distribution of the linear polarization degrees for multiple radiating particles is (> Π ) sin Θ Δ 1 − cos(Θ + th ) ,(72) where Δ = ,2 − ,1 with , solved by Eq.(64), and Δ 1 is used in the above equation. Δ = 1 Π 1/2 1 + √︃ 1 − Π 2 1/2 − 1 − √︃ 1 − Π 2 1/2 .(73) Similarly, the cumulative distribution of the circular polarization degree is (> Π ) 1 − sin Θ Δ 1 − cos(Θ + th )(74) where Δ is solved by Eq.(65), leading to Δ = 2 1 − Π 1 + Π 1/4 .(75) In Figure 10, we plot the cumulative distribution of the linear (top panel) and circular (bottom panel) polarization degrees, respectively. We can see that for multiple radiating particles, the polarization of this radiation mechanism is almost 100% circular polarization. Thus, if the radiation mechanism of FRBs corresponds to this scenario with 1, the observed moderate number fraction between linearly polarized bursts and circularly polarized bursts of FRB 20201124A (Jiang et al. 2022) might be due to the propagation effects, because the intrinsic polarization of this mechanism is predicted to be ∼ 100% circular polarized. Notice that this result is based on the assumption that all radiating particles have the same gyration direction. If the gyration directions of the particles are random, their polarizations would cancel out. A unified picture in different astrophysical scenarios The above two general radiation processes with 1 and 1 could appear in various astrophysical scenarios, e.g, in the magnetosphere of a neutron star or in the synchrotron radiation process in a magnetized shocked medium. The radiation processes in the magnetosphere are shown in Figure 11. In the inner region of the magnetosphere, due to the strong magnetic field, the electrons move almost along the curvature field lines and produce curvature radiation. The accelerations of the electrons are essentially by the Lorentz forces of the drift velocities perpendicular to the field lines. In the picture of the curvature radiation, since the deflection angle (i.e., the deflection angle of the field line) is much larger than the radiation beaming angle, the condition of 1 is satisfied. In the outer region of the magnetosphere with the relatively weak magnetic field, Neutron Star Magnetic Field Lines Radiating Particles Small-pitch-angle synchrotron radiation Curvature radiation Figure 11. Two radiation mechanisms with 1 and 1 in the magnetosphere of a neutron star. In the inner region with a relatively strong magnetic field, the radiation mechanism is curvature radiation. In the outer region with a relatively weak magnetic field, the radiation mechanism is the small-pith-angle synchrotron radiation. the electrons would move in the field lines with a spiral trajectory and the corresponding radiation mechanism is the small-pitch-angle synchrotron radiation. The deflection angle just corresponds to the pitch angle of the synchrotron radiation, leading to 1. One should notice the difference in the deflection angles in the above two scenarios due to the different mechanisms. The critical conditions between the curvature radiation and the small-pitch-angle synchrotron radiation could be obtained as follows: when a charged particle with a Lorentz factor slides along a curved magnetic field line with a curvature radius of , the observer will see the radiation with the emission cone of angular width 1/ around the observer direction and the typical timescale of the radiating process is = / . The gyration period of an electron under a magnetic field is = 2 / = 2 / . If the radiation process is dominated by the small-pitch-angle synchrotron, the number of times for the electron's gyration during the time of must be much larger than once, which requires that the gyration period of the electron is much shorter than , < , leading to the first necessary condition: > cr,1 = 2 2 2 1.1 G 2 2 −1 8 .(76) According to the Larmor formula, the radiation power of the smallpitch-angle synchrotron radiation is = 2 3 4 2 2 2 ⊥ 2 3 2 3 4 2 2 2 2 3 ,(77) where ⊥ = sin for ∼ 1 and 1. The cooling timescale is estimated by cool ∼ 2 , and one obtains cool ∼ 3 3 5 2 4 2 2 5.2 × 10 6 s −1 2 −2 1 −2 −1 .(78) If the small-pitch-angle synchrotron radiation is not significantly cooling during the electron moving along the field line, the cooling timescale cool must be larger than , leading to the second necessary condition: < cr,2 = 3 3 6 2 4 2 1/2 3.9 × 10 6 G −1/2 8 −1 −1 .(79) Based on Eq.(76) and Eq. (79), the small-pitch-angle synchrotron radiation finally requires the conditions: 1.1 G 2 2 −1 8 3.9 × 10 6 G −1/2 8 −1 −1 .(80) Accordingly, the condition of the curvature radiation is 3.9 × 10 6 G −1/2 8 −1 −1 . As we expected, the small-pitch-angle synchrotron radiation is preferred to be in the outer region of the magnetosphere, and the curvature radiation is preferred to be in the inner region of the magnetosphere. Next, we make some comments about the observable properties of the small-pitch-angle synchrotron radiation in the magnetosphere. As discussed in Section 3.2, according to Eq.(42) the typical radiation frequency of the small-pitch-angle synchrotron radiation is ∼ = 2 2.8 GHz 2 1 .(82) Thus, the small-pitch-angle synchrotron radiation in the outer magnetosphere could be emitted at the GHz band. The polarization properties of the small-pitch-angle synchrotron radiation depend on the gyration motions of the charged particles in the magnetosphere. Before the charged particles enter the magnetosphere's outer region, in addition to the parallel velocities along the field line, the charged particles have a small drift velocity perpendicular to the field lines, providing an additional Lorentz force to make the particles move along the curved paths. Since the direction of the drift velocity only depends on the curved field line, the charged particles tend to have the same gyration direction when they enter the outer region of the magnetosphere. Thus, the small-pitch-angle radiation is expected to be highly circularly polarized. Besides, since the spectral shape and the typical frequency of the small-pitch-angle radiation is independent of the particle's pitch angle, for multiple particles, the spectral bandwidth is mainly determined by the distribution of the particles' Lorentz factors. The above two general radiation processes with 1 and 1 might also occur in the magnetized shocked medium, which corresponds to the traditional synchrotron radiation and small-pitchangle synchrotron radiation, as shown in the panel (a) and the panel (b) of Figure 12. The critical condition between the two scenarios depends on the relative directions between the magnetic field, the particles' injection, and the viewing direction. If the direction of the particles' injection is almost parallel to the field lines, the radiation mechanism would be the small-pitch-angle synchrotron radiation, as shown in the panel (b) of Figure 12. However, due to the shock compression, the magnetic field in the shocked medium usually has a significant component parallel to the shock surface. Thus, the smallpitch-angle injection process seems to be fine-turned. Besides, in the magnetized shocked medium, since the directions of the particles' gyration motion are random, the small-pitch-angle synchrotron would be significantly depolarized for multiple particles. DISCUSSIONS AND CONCLUSIONS The observed properties of the spectra and polarizations are important clues to reveal the radiation mechanism of FRBs. In this work, two key observed features of FRBs are focused on: (1) Some radio bursts appear significantly intrinsic narrow spectra in the telescope's bandwidth (Pleunis et al. 2021;Kumar et al. 2021;Zhou et al. 2022), which could be used to identify the radiation mechanism; (2) Some FRB repeaters are found to behave a certain proportion of circularly polarized bursts ( which might originate from the polarization distribution at different viewing directions. We have investigated the physical features of the spectra and polarization of FRBs from the perspective of radiation mechanisms, and the following conclusions are drawn: 1. Without considering the finite bandwidth of a telescope, the fluxthreshold-dependent relative spectral bandwidth of a burst, Δ / 0 , depends on the intrinsic spectral shape and the / in the frequency domain. In practical measurement, the spectral bandwidth is often measured via FWHM. The relative spectral bandwidth Δ / 0 for FWHM only depend on the intrinsic spectral shape, thus, the Δ / 0 distribution of an FRB repeater should be enough narrow for a given intrinsic radiation mechanism. Most radiation mechanisms with lowfrequency spectral index < 3 would lead to Δ / 0 > 0.2 for FWHM. Some bursts (e.g., FRB 20190711A, Kumar et al. (2021)) with Δ / 0 0.2 within the whole telescope's bandwidth suggest that their spectra must be intrinsically narrow and some special radiation mechanisms should be involved. In reality, the finite and narrow telescope's bandwidth usually makes some observed bursts' spectra incomplete, which leads to an observed spectral bandwidth less than the intrinsic one. Thus, the distribution of the observed relative spectral bandwidths of radio bursts depends on both Δ / 0 and the telescope's bandwidth. For example, we consider that a telescope has a central frequency 0, = 1.25 GHz and a bandwidth Δ = 0.5 GHz. For Δ / 0 = 0.1, the observed spectral bandwidth Δ obs of most bursts are consistent with the intrinsic ones. For Δ / 0 = 1, the observed bandwidth Δ obs of most bursts would be constrained by the bandwidth of the telescope, leading to Δ obs ∼ Δ . 2. An intrinsic narrow spectrum with Δ / 0 1 ( 0 is the peak frequency and Δ is the spectral bandwidth) implies that the electromagnetic wave is quasi-sinusoid with a typical frequency of ∼ 0 and a typical pulse duration of ∼ 4 /Δ , which might be produced by: (1) The intrinsic radiation mechanism of a single accelerated charged particle with periodic motion during its radiation beam pointing to the observer; (2) The coherent process by the radiation of multiple charged particles with some special (e.g., periodic) distributions; (3) The plasma resonant process. 3. We generally discuss the spectral shapes and polarization dis-tributions from the perspective of radiation mechanisms. For the radiation mechanisms involving the relativistic particle's perpendicular acceleration, the radiation features (including the spectrum and polarization) depend on the relation between the particle's deflection angle and the radiation beaming angle 1/ . The scenarios with 1 and 1 lead to different features of the spectrum and polarization. 4. If 1, the observer would see radiation from short segments of the particle's trajectory that are nearly parallel to the line of sight. Such a scenario is applicable to curvature radiation and traditional (large-pitch-angle) synchrotron radiation. The intrinsic spectrum of this mechanism is usually wide, and the intrinsic linear/circular polarization degree mainly depends on the angle between the viewing direction and the trajectory plane. The larger the viewing angle, the lower (higher) the linear (circular) polarization degree. Besides, the higher the observed frequency, the lower (higher) the linear (circular) polarization degree. We further discuss the scenario that multiple radiating particles are uniformly distributed in a fan beam, and calculate the cumulative distributions of the linear and circular polarization degrees due to the random viewing direction. Important conclusions for the cumulative polarization distributions include: (1) The higher the telescope's sensitivity, the lower the number fraction between the linearly and circularly polarized bursts. (2) The larger the particles' beaming angle, the higher the number fraction between the linearly and circularly polarized bursts. (3) The higher the observed frequency, the higher the number fraction between the linearly and circularly polarized bursts. 5. If 1, the particle's entire trajectory would be seen by the observer during a long term. Such a scenario is applicable to small-pitch-angle synchrotron radiation and jitter radiation. For a particle with an acceleration timescale Δ acc and a Lorentz factor , the typical radiation frequency is ∼ 2 2 Δ −1 acc , and the polarization depends on the acceleration direction. If the acceleration is always on a straight line perpendicular to the particle's velocity, the polarization is fully linear. If the acceleration rotates with a constant angular velocity on the plane perpendicular to the particle's velocity, the polarization is fully circular. 6. For the scenario with 1, we particularly discuss the radiation from a particle with its perpendicular acceleration attributed to the Lorentz force by magnetic fields. In the comoving frame, the particle's motion is non-relativistic and produces cyclotron radiation, which leads to an extremely narrow spectrum with a peak frequency at the cyclotron frequency 0 = / , where is the magnetic field strength. In the observer frame, the radiation spectrum still keeps extremely narrow and the peak frequency is about at ∼ 0 . In particular, for a certain viewing direction, the radiation is only emitted at a certain frequency within an extremely narrow band. We further discuss the radiation properties of multiple particles uniformly distributed in a three-dimensional beam. Due to the beaming distribution of multiple particles, the radiation spectrum becomes relatively wider compared with that of a single particle, but the relative spectral bandwidth is still narrow, Δ / 0 1. Besides, once the viewing direction is outside the particle beam, the larger the viewing angle, the narrower the spectrum and the lower the peak radiation power. The polarization properties of the multiple particles depend on the gyration directions of the radiating particles. In particular, if all radiating particles have the same gyration directions, the polarization of this radiation mechanism would be almost 100% circular polarization, otherwise, their polarizations would cancel out. 7. We discuss some astrophysical scenarios that might involve radiation processes with 1 and 1. In the magnetosphere of a neutron star, the radiation process with 1 occurs in the inner magnetosphere and the corresponding radiation mechanism is the curvature radiation and the radiation process with 1 occurs in the outer magnetosphere and the corresponding radiation mechanism is the small-pitch-angle synchrotron radiation. One should notice the difference in the deflection angles in the above two scenarios due to the different mechanisms. In particular, the smallpitch-angle radiation is expected to be highly circularly polarized and appears a narrow spectrum. On the other hand, both scenarios with 1 and 1 might occur in the magnetized shocked medium, which corresponds to the traditional synchrotron radiation and the small-pitch-angle radiation, respectively. However, the generation of the small-pitch-angle radiation requires that the direction of the particles' injection is almost parallel to the field lines, which seems fine-turned. Based on the above results, we have the following conclusions for the observed spectra of FRBs: (1) If the observed narrow spectra of FRB are attributed to the intrinsic radiation mechanism that does not involve the coherent process, the particle's deflection angle must be much smaller than the radiation beaming angle, i.e. 1. The number fraction between linearly polarized bursts and circularly polarized bursts of FRBs might be due to the propagation effects (see Qu & Zhang 2023). (2) Coherent process involving some special (like periodic) particle distribution can lead to a narrow spectrum even for the radiation mechanism with 1, but the mechanism for the particles' special distribution need to be further analyzed in detail. (3) If the number fraction between linearly polarized bursts and circularly polarized bursts is attributed to the radiation mechanism with 1, the cumulative distribution of the polarization degree would mainly depend on the particle's beaming angle, and the observed data could be used to constrain the geometric configuration of the FRB emission region. At last, we should emphasize that the above discussion is based on the assumption that the radiation is produced by the perpendicular acceleration of charged particles, which is applicable to most radiation mechanisms in various astrophysical scenarios. However, for the scattering process that involves the acceleration process by the oscillating electric field of an incident electromagnetic wave, both the spectrum and polarization will depend on the properties (including the intensity (strong or weak) and the polarization state) of the incident electromagnetic wave. One special scenario is the scattering process under a strong magnetic field near the surface of a neutron star, in which case, the charged particles can only be accelerated along the field line and the radiation polarization of a single particle becomes linear, see Zhang (2022b) and Qu & Zhang (2023) in detail. Figure 1 . 1Figure 1. The simulated observed spectral bandwidth distribution of radio bursts from an FRB repeater. The purple, yellow, and blue bars correspond to the different relative spectral bandwidths with Δ / 0 = 0.1, 0.5 and 1, respectively. We take 0, = 1.25 GHz and Δ = 0.5 GHz for the telescope's parameters. The intrinsic peak frequencies 0 of the radio bursts are assumed to be uniformly distributed near the telescope's bandwidths, i.e., ( 0 ) = const. Figure 2 . 2The electric field evolution of a pulse of electromagnetic wave with a rectangular power spectrum given by Eq.(9). The top panel corresponds to the scenario with a narrow spectrum of Δ / 0, = 0.1, and the bottom panel corresponds to the scenario with a wide spectrum of Δ / 0, = 1. The phase argument is taken as = 0 here. Figure 3 3Figure 3. The spectrum of multiple radiating particles with a periodic distribution in the time domain. A curvature-radiation-like spectrum for a single particle, | 0 ( ) | 2 ∝ 2/3 exp(− / ), is taken as an example in this figure. We take = and = 10 4 . The peak frequency is at 2 with ∈ Z + . Figure 4 . 4Emission from various points along the trajectory of a relativistic particle. Panel (a) 1: the emission from some parts (bold portions) of the trajectory is observable. Panel (b) 1: emission from the entire trajectory is observable. Figure 6 . 6The relations between the polarization degree Π and the dimensionless frequencyˆand the viewing angle for a single radiating particle with 1. The blue and red lines correspond to the linear and circular polarization degrees, respectively. The top panel shows the polarization degree Π as the function of the dimensionless frequencyˆ. The solid, dashed, and dotted lines correspond to = 0.1, 1 and 3, respectively. The bottom panel shows the polarization degree Π as the function of the viewing angle . The solid, dashed, and dotted lines correspond toˆ= 0.1, 1 and 3, respectively. Figure 8 . 8The relations between the polarization degree Π and the dimensionless frequencyˆand the viewing angle for a single radiating particle with 1. The blue and red lines correspond to the linear and circular polarization degrees, respectively. The top panel shows the polarization degree Π as the function of the dimensionless frequency¯. The bottom panel shows the polarization degree Π as the function of the viewing angle . The dimensionless frequency¯and the viewing angle are related viā = 2/(1 + 2 2 ). Figure 10 . 10The cumulative distribution of the linear (top panel) and circular (bottom panel) polarization degrees. The distribution function of the viewing direction is assumed to satisfy Eq.(66). The black solid, dashed and dotted lines correspond to Θ = 3, 10 and 30, respectively for th = 100. The solid red, blue and black lines correspond to th = 10, 30 and 100, respectively for Θ = 3. = 100 is taken here. Different fromFigure 7, the -axis is log[ (> Π ) ] here. Figure 12 . 12Xu et al. 2022;Jiang et al. 2022;Feng et al. 2022), Two radiation mechanisms with 1 and 1 in the magnetized shocked medium. Panel (a) corresponds to traditional synchrotron radiation. Panel (b) corresponds to the small-pitch-angle synchrotron radiation, in which scenario, the emission region has a magnetic field almost parallel to the line of sight and the direction of the particles' injection is almost parallel to the field lines. Table 1 . 1Summaryof the low-frequency spectral index for various radiation mechanisms Radiation mechanisms Low-frequency spectral index References MNRAS 000, 1-14(2023) We should notice that the parallel acceleration could be dominant under the scattering process. For example, if the incident electromagnetic wave is linear polarized and weak (the Lorentz force by the magnetic field component is much weaker than the electric field force), the charged particle would be linearly accelerated by the oscillating electric field (the scenario for strong wave could be seen in). Besides, under the magnetosphere of a neutron star, even if the incident wave is circularly polarized or strong, the charged particle can only oscillate along the field lines due to the existence of a strong background magnetic field and emit the scattering MNRAS 000, 1-14(2023) ACKNOWLEDGEMENTSWe thank Bing Zhang and Yue Wu for reading the manuscript and for their helpful comments, and acknowledge the discussions with Yi Feng, Jin-Lin Han, Kejia Lee, Ze-Nan Liu, Yuanhong Qu, Wei-Yang Wang, and Yong-Kun Zhang. This work is supported by the National Natural Science Foundation of China grant No.12003028 and the National SKA Program of China (2022SKA0130100).DATA AVAILABILITYThis theoretical study did not generate any new data. The code performed for the calculations is available upon request. . A M Beloborodov, 10.3847/2041-8213/aa78f3ApJ. 84326Beloborodov A. M., 2017, ApJ, 843, L26 . A M Beloborodov, 10.1103/PhysRevLett.128.255003Phys. Rev. Lett. 128255003Beloborodov A. M., 2022, Phys. 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[ "arXiv:physics/0409115v1 [physics.gen-ph] On Cremonian Dimensions Qualitatively Different from Time and Space", "arXiv:physics/0409115v1 [physics.gen-ph] On Cremonian Dimensions Qualitatively Different from Time and Space" ]
[ "Metod Saniga \nAstronomical Institute\nSlovak Academy of Sciences\n05960Tatranská Lomnica\n\nLaboratoire de Physique et Métrologie des Oscillateurs\nSlovak Republic and Institut FEMTO-ST\nCNRS\n32 Avenue de l'ObservatoireF-25044BesançonFrance\n" ]
[ "Astronomical Institute\nSlovak Academy of Sciences\n05960Tatranská Lomnica", "Laboratoire de Physique et Métrologie des Oscillateurs\nSlovak Republic and Institut FEMTO-ST\nCNRS\n32 Avenue de l'ObservatoireF-25044BesançonFrance" ]
[]
We examine a particular kind of six-dimensional Cremonian universe featuring one dimension of space, three dimensions of time and other two dimensions that cannot be ranked as either time or space. One of these two, generated by a one-parametric aggregate of (straight-)lines lying on a quadratic cone, is more similar to the spatial dimension. The other, represented by a singly-parametrical set of singular space quartic curves situated on a proper ruled quadric surface, bears more resemblance to time. Yet, the two dimensions differ profoundly from both time and space because, although being macroscopic, they are not accessible to (detectable by) every Cremonian observer. This toy-model thus demonstrates that there might exist extra-dimensions that need not necessarily be compactified to remain unobservable.
10.1016/j.chaos.2005.01.010
[ "https://export.arxiv.org/pdf/physics/0409115v1.pdf" ]
476,355
physics/0409115
b4abe451909b35f158598479b00843048cf00bb7
arXiv:physics/0409115v1 [physics.gen-ph] On Cremonian Dimensions Qualitatively Different from Time and Space 22 Sep 2004 Metod Saniga Astronomical Institute Slovak Academy of Sciences 05960Tatranská Lomnica Laboratoire de Physique et Métrologie des Oscillateurs Slovak Republic and Institut FEMTO-ST CNRS 32 Avenue de l'ObservatoireF-25044BesançonFrance arXiv:physics/0409115v1 [physics.gen-ph] On Cremonian Dimensions Qualitatively Different from Time and Space 22 Sep 2004 We examine a particular kind of six-dimensional Cremonian universe featuring one dimension of space, three dimensions of time and other two dimensions that cannot be ranked as either time or space. One of these two, generated by a one-parametric aggregate of (straight-)lines lying on a quadratic cone, is more similar to the spatial dimension. The other, represented by a singly-parametrical set of singular space quartic curves situated on a proper ruled quadric surface, bears more resemblance to time. Yet, the two dimensions differ profoundly from both time and space because, although being macroscopic, they are not accessible to (detectable by) every Cremonian observer. This toy-model thus demonstrates that there might exist extra-dimensions that need not necessarily be compactified to remain unobservable. There are a number of features of the macroscopic physical world that still remain substantially beyond grasp of theoretical physics. Among them, the non-trivial structure of time and the observed dimensionality of the universe obviously represent a case in question. As we found [1,2] and have repeatedly stressed [3][4][5], the two properties seem to be intimately intertwined and ask, therefore, for a conceptually new approach to be properly understood. A (very promising) piece of such a formalism is undoubtedly the concept/theory of Cremonian space-times [6][7][8][9][10][11][12][13][14]. The Cremonian picture of space-time is indeed remarkable in several aspects. The first, and perhaps most notable, fact is that without employing any concept of metric (measure), this approach fundamentally distinguishes the time dimension(s) from spatial ones and, in its most trivial form, it straightforwardly leads to their observed number (4) and respective ratio (1+3) as well [6,7,9,10]. Second, it demonstrates that these dimensions are not primordial, but emerge from more fundamental algebraic geometrical structures [13]. Third, it indicates that the universe with the inverse signature might evolutionary be intimately connected with our universe [12]. And last, but not least, when the observer (subject) is concerned, it qualitatively reproduces our ordinary perception of time as well as a whole variety of altered/non-ordinary forms of mental space-times [10,15]; moreover, every observer in this basic Cremonian universe is found to face an intricate 2+1 break-up among the space dimensions themselves [14]. In this paper, we introduce and examine a particular kind of a more complex, six dimensional Cremonian universe whose spatio-temporal sector is still four dimensional, yet featuring three dimensions of time and just a single spatial coordinate. The character of other two dimensions is neither that of space nor time; in addition, these dimensions are only conditionally observable/accessible. This Cremonian universe sits in the 3-dimensional projective space over the fields of the real numbers ℜ and is generated by the configuration of fundamental elements of a homaloidal web of cubic (i.e., third-order) surfaces that share a proper conic, Q, a (straight-)line, L, incident with the conic and not lying in its plane, and three different non-collinear points, B i (i=1,2,3), none of them incident with either L or the plane of Q. The cubics of the web have also two double points, D 1 and D 2 , in common; both the points lie on L, the former being the intersection of L and Q [16,17]. Selecting an allowable system of homogeneous coordinatesz α (α=1,2,3,4) in such a way that L :z 1 = 0 =z 2 ,(1)Q :z 4 = 0 = −2z 1z2 +z 1z3 +z 2z3 ≡ C,(2)B 1 : ̺z α = (0, a, b, c), a, c = 0,(3)B 2 : ̺z α = (f, 0, g, h), f, h = 0,(4)B 3 : ̺z α = (k, k, l, m), k, l, m = 0,(5)D 1 : ̺z α = (0, 0, 1, 0),(6)D 2 : ̺z α = (0, 0, 0, 1),(7) where ̺ is a non-zero proportionality factor, and assuming, without any substantial loss of generality, that a c = f h = 2 k(l − k) lm ≡ −Θ,(8) the web in question is given by W(η α ) : η 1z1z4 k(gz 1 − fz 3 ) f l − gk +z 2 + η 2z2z4 z 1 + k(bz 2 − az 3 ) al − bk + η 3z1 D + η 4z2 D = 0, (9) with D ≡ −2z 1z2 +z 1z3 +z 2z3 + Θz 3z4 = C + Θz 3z4 ,(10) and η α ∈ ℜ. This web generates the following Cremona transformation [16] ̺z ′ 1 =z 1z4 k(gz 1 − fz 3 ) f l − gk +z 2 ,(11)̺z ′ 2 =z 2z4 z 1 + k(bz 2 − az 3 ) al − bk ,(12)̺z ′ 3 =z 1 D,(13)̺z ′ 4 =z 2 D,(14) wherez ′ α are the homogeneous coordinates of an allowable system in the second ("primed") projective space. Our forthcoming task is to find the fundamental elements of W(η α ). To begin with, we recall [6,16] that the fundamental element of a Cremona transformation between two 3-dimensional projective spaces is a curve, or a surface, in one space whose corresponding image in the other space is a single point; 1 the loci of fundamental elements of the same kind being mapped, in general, into curves, i.e., one-dimensional geometrical objects, of the second space. Employing Eqs.(11)- (14), it is quite a straightforward task to spot that in our present case the loci of such elements are the plane of the conic Q, the three planes B i L, the quadric D=0 and the quadratic cone C=0, for their images in the second space are indeed curves, namely lines (for the planes and the quadric) and/or a twisted cubic (for the cone) [17]. To be more explicit, the plane of Q host a pencil (i.e., linear, single parametrical set) of fundamental lines (ϑ 1,2 ∈ ℜ) L(ϑ 1,2 ) : ϑ 1z1 + ϑ 2z2 = 0 =z 4 ,(15) whose point of concurrence is the point D 1 ; a line from this pencil has for its primed counterpart a point of the line s L ′ :z ′ 1 = 0 =z ′ 2 , the latter being single (hence the superscript "s") on the surfaces of the inverse homaloidal web. The three planes B i L contain each a pencil of fundamental conics whose four base (i.e., shared by all the members) points are D 1 , D 2 , B i and K i -the last one being the point, not on L, in which the plane in question cuts the conic Q; in particular, Q i=1 (ϑ 1,2 ) :z 1 = 0 = ϑ 1z4 (bz 2 − az 3 ) + ϑ 2z3 (z 2 + Θz 4 ),(16)Q i=2 (ϑ 1,2 ) :z 2 = 0 = ϑ 1z4 (gz 1 − fz 3 ) + ϑ 2z3 (z 1 + Θz 4 ),(17)Q i=3 (ϑ 1,2 ) :z 1 −z 2 = 0 = ϑ 1z4 (lz 2 − kz 3 ) + ϑ 2 (−2z 2 2 + 2z 2z3 + Θz 3z4 ).(18) It is readily verified that a conic of Q i (ϑ 1,2 ) corresponds to a point of the line d L ′ i , where d L ′ i=1 :z ′ 1 = 0 =z ′ 3 , d L ′ i=2 :z ′ 2 = 0 =z ′ 4 , and d L ′ i=3 :z ′ 3 −z ′ 4 = 0 = (l − gk/f )z ′ 1 − (l − bk/a)z ′ 2, respectively; all these lines are double ("d") on a generic homaloid of the inverse web. The intrinsic structure and mutual coupling between these four pencils are depicted in Figure 1. If one compares this configuration with the one introduced and studied in detail in [6], which is associated with a homaloidal web of quadric surfaces and which reproduces what is macroscopically observed, one finds that the two configurations are prefect inverses of each other; it was, among other things, also this feature that motivated us to examine thoroughly this particular kind of Cremonian universe. Now we turn to quadratic loci of fundamental elements. The quadric D=0, which is proper (i.e., non-composite) and ruled (i.e., containing infinity of lines), accommodates a single parametrical aggregate of fundamental quartics, i.e., curves of order four, F (ϑ 1,2 ) : ϑ 1z1 k(gz 1 − fz 3 ) f l − gk +z 2 + ϑ 2z2 z 1 + k(bz 2 − az 3 ) al − bk = 0 = D;(19) these quartics share the five points B i (i=1,2,3), D 1 and D 2 , and, in the primed space, they correspond to the points of the line q L ′ :z ′ 3 = 0 =z ′ 4 , the latter being of multiplicity four ("q") on the inverse homaloids. All the proper quartics in the set are singular, D 2 being their common double point, and, as it is also obvious from Figure 2, they are genuine space curves. There are just three composite quartics within this aggregate, each comprising a pair of conics, namely (ϑ ≡ ϑ 2 /ϑ 1 ) F (ϑ = 0) ≡ F ⊙ 0 :z 1 = 0 = D ∪ k(gz 1 − fz 3 ) f l − gk +z 2 = 0 = D,(20)F (ϑ = ∞) ≡ F ⊙ ∞ :z 2 = 0 = D ∪z 1 + k(bz 2 − az 3 ) al − bk = 0 = D,(21)F (ϑ = ℘) ≡ F ⊙ ℘ :z 1 −z 2 = 0 = D ∪ agz 1 + bfz 2 − afz 3 = 0 = D,(22) where ℘ ≡ −f (al − bk)/a(f l − gk). F ⊙ 0 (dot-dashed), F ⊙ ∞ (dotted) and F ⊙ ℘ (dashed). In the case of the quadric cone, C=0, the fundamental elements are again lines, forming the following singly-infinite family L C (ϑ) :z 1 − ϑz 2 = 0 = (ϑ + 1)z 3 − 2ϑz 2 ,(23) with the parameter ϑ running through all the real numbers and infinity as well; for substituting the last equation into Eqs. (11)-(14) yields (ς = 0) ςz ′ 1 = ϑ ϑ(ϑ + 1) gk f l − gk + (ϑ + 1) − 2ϑ f k f l − gk ,(24)ςz ′ 2 = ϑ(ϑ + 1) + (ϑ + 1) bk al − bk − 2ϑ ak al − bk ,(25)ςz ′ 3 = 2Θϑ 2 ,(26)ςz ′ 4 = 2Θϑ,(27) which say that a generic line of L C (ϑ) corresponds indeed to a single point of the second projective space. The structure of this aggregate can be discerned from Figure 3. At this point we invoke our fundamental "Cremonian" postulate [6][7][8][9][10] which says that each single parametrical sets of fundamental elements generates/represents a unique dimension of a Cremonian universe, with pencil-borne aggregates having a special status of generating space (lines) and time (conics). Hence, the Cremonian universe under discussion is six-dimensional, with three dimensions of time ( Q i ), one dimension of space ( L) and two additional dimensions ( F and L C ) of a different nature. It is these two extra-dimensions which are of our next concern. It is obvious that the F -dimension bears more resemblance to time, for its constituents are, like conics, of non-linear character, whereas the L C -one, whose elements are lines, is more similar to the spatial dimension. Yet, there exists a profound difference between the four pencil-dimensions and these two "extra-"dimensions. This difference stems from the fact that the former are all planar configurations, whereas the latter are both located on quadratic surfaces, and acquires its most pronounced form when a particular Cremonian observer is concerned. For a Cremonian observer is represented by a line [10,14,15], and whilst any line in a three-dimensional projective space is incident with any plane (see, e.g., Refs. [18,19]), this is no longer the case for a pair comprising a line and a non-composite quadric; given any non-composite quadric (whether proper, or a cone), there exist an infinite number of lines incident with it, but also an infinity of lines which have no intersection with this quadric. If we take this incidence relation as a synonym of the observer's awareness of the particular dimension, then we see that the four "planar" dimensions (i.e., time and space) will be observed by (accessible to) every observer, while the two "quadratic" ones not! In other words, for each of these two non-planar dimensions, there exist two distinct groups of the observers; one comprising observers who perceive this dimension, the other those who do not. This finding thus amounts to saying that these two extra-dimensions are observable only conditionally. It is of crucial importance to realize here that the conditional observability of these extradimensions has nothing to do with their length (compactification), as no concept of the measure/metric has so far been introduced into our model. Instead, it is of a purely algebraic geometrical origin, based solely on the incidence relations between relevant geometrical objects and intimately linked with the fact that the ground field of the background projective space, taken to be that of the real numbers, is not algebraically closed; for the geometrical problem of finding the common points of a line and a quadric in a 3-dimensional projective space defined over a given ground field reduces to the algebraic one of solving/factoring a quadratic equation in the given field, which is not always possible unless the field is algebraically closed (see, e.g., Ref. [20]). This toy-model thus demonstrates that there might exist extra-dimensions that need not necessarily be compactified/curled-up to remain unobservable. And this is a truly serious implication, especially for cosmology and high energy particle physics. Figure 1 : 1A schematic sketch of the structure of the configuration of the four "fundamental" pencils defined by Eqs. (15)-(18). In each pencil, out of an infinite number of its members only several are drawn. This configuration represents the space-time "sector" of the corresponding Cremonian manifold, with three time dimensions ( Q i ) and a single space one ( L). Figure 2 Figure 2 : 22illustrates the shape of this aggregate for a generic case where each composite quartic comprises a pair of proper conics. (This property does not hold in our constrained case (see Eq. (8)), where one of the conics of both F ⊙ 0 and F ⊙ ∞ is composite (a line pair).) A schematic sketch of the structure of the singly-infinite set of quartics defined by Eq. (19) for the generic case where the constraint imposed by Eq. (8) is relaxed. As in the previous figure, only a few quartics are illustrated, including the composites Figure 3 : 3A schematic sketch of the structure of the singly-infinite set of lines "sweeping" the quadric cone C=0. As in the previous two figures, only a finite number of lines are drawn. Equivalently, a fundamental element associated with a given homaloidal web of surfaces is, in general, any curve/surface whose only intersections with a generic member of the web are the base (i.e., common to all the members) elements of the latter[16]. Arrow of time and dimensionality of space. M Saniga, Cosmological constant and the evolution of the universe. Sato K, Suginohara T, Sugiyama NTokyoUniversal Academy Press IncSaniga M. Arrow of time and dimensionality of space. In: Sato K, Suginohara T, Sugiyama N, editors. Cosmological constant and the evolution of the universe. Tokyo: Universal Academy Press Inc.; 1996. p. 283-4. On the transmutation and anihilation of pencil-generated spacetime dimensions. M Saniga, Astrophysics & Space Science. 244Saniga M. On the transmutation and anihilation of pencil-generated spacetime dimensions. Astro- physics & Space Science 1996;244:283-90. Pencils of conics: a means towards a deeper understanding of the arrow of time. M Saniga, Chaos, Solitons & Fractals. 9Saniga M. Pencils of conics: a means towards a deeper understanding of the arrow of time?. Chaos, Solitons & Fractals 1998;9:1071-86. Geometry of psycho(patho)logical space-times: a clue to resolving the enigma of time?. M Saniga, Noetic Journal. 2Saniga M. Geometry of psycho(patho)logical space-times: a clue to resolving the enigma of time?. Noetic Journal 1999;2:265-74. Algebraic geometry: a tool for resolving the enigma of time. M Saniga, Studies on the structure of time: from physics to psycho(patho)logy. Buccheri R, Di Gesù V, Saniga MNew YorkKluwer Academic/Plenum PublishersSaniga M. 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[ "On the evolution of the particle distribution and the cascade in a moving, expanding emission region in blazar jets", "On the evolution of the particle distribution and the cascade in a moving, expanding emission region in blazar jets" ]
[ "Michael Zacharias [email protected] \nLaboratoire Univers et Théories\nObservatoire de Paris\nUniversité PSL\nUniversité Paris Cité\nCNRS\nF-92190MeudonFrance\n\nCentre for Space Research\nNorth-West University\n2520PotchefstroomSouth Africa\n" ]
[ "Laboratoire Univers et Théories\nObservatoire de Paris\nUniversité PSL\nUniversité Paris Cité\nCNRS\nF-92190MeudonFrance", "Centre for Space Research\nNorth-West University\n2520PotchefstroomSouth Africa" ]
[]
Aims. There is a large variety in the models explaining blazar flares. Here, we study the flare profile induced by a moving and expanding blob with special emphasize on the γ-γ pair production. Methods. We first develop a simple semi-analytical model to study the evolution of the particle distribution in the expanding blob and show the influence of the pair production. In a second step, we produce a realistic simulation using the OneHaLe code based upon parameters of PKS 1510-089.Results.The semi-analytical model shows that the pair production significantly influences the flare evolution, while the opening angle and the expansion can prolong flares considerably. The simulation based on PKS 1510-089 indicate that flares of a moving expanding blob result in strongly wavelength dependant light curves including delayed, secondary flares. Conclusions. A moving, expanding blob can cause significant flaring events with a large variety in light curve profiles. High-cadence multiwavelength observations are necessary to derive the details causing the flare. Extended observations beyond the initial burst may provide important information on the opening angle and the particle content due to delayed secondary flares in some energy bands.
10.1051/0004-6361/202244683
[ "https://export.arxiv.org/pdf/2211.12283v1.pdf" ]
253,761,432
2211.12283
23b2f1671de487f33558717d536f53164ea569b5
On the evolution of the particle distribution and the cascade in a moving, expanding emission region in blazar jets November 23, 2022 Michael Zacharias [email protected] Laboratoire Univers et Théories Observatoire de Paris Université PSL Université Paris Cité CNRS F-92190MeudonFrance Centre for Space Research North-West University 2520PotchefstroomSouth Africa On the evolution of the particle distribution and the cascade in a moving, expanding emission region in blazar jets November 23, 2022Received ? / accepted ?Astronomy & Astrophysics manuscript no. OneHaLe_expandingsource ©ESO 2022radiation mechanisms: non-thermal -galaxies: active -galaxies: jets -gamma-rays: galaxies Aims. There is a large variety in the models explaining blazar flares. Here, we study the flare profile induced by a moving and expanding blob with special emphasize on the γ-γ pair production. Methods. We first develop a simple semi-analytical model to study the evolution of the particle distribution in the expanding blob and show the influence of the pair production. In a second step, we produce a realistic simulation using the OneHaLe code based upon parameters of PKS 1510-089.Results.The semi-analytical model shows that the pair production significantly influences the flare evolution, while the opening angle and the expansion can prolong flares considerably. The simulation based on PKS 1510-089 indicate that flares of a moving expanding blob result in strongly wavelength dependant light curves including delayed, secondary flares. Conclusions. A moving, expanding blob can cause significant flaring events with a large variety in light curve profiles. High-cadence multiwavelength observations are necessary to derive the details causing the flare. Extended observations beyond the initial burst may provide important information on the opening angle and the particle content due to delayed secondary flares in some energy bands. Introduction The emission of blazars -active galaxies with the jet pointing at Earth -is typically explained with the so-called onezone model, where a single zone is responsible for most of the source's radiative output. The spectral energy distribution (SED) is characterized by two broad humps. The low-energy one is explained by electron-synchrotron emission, while the high-energy hump can be explained by inverse-Compton emission or hadronically induced processes, such as proton synchrotron or synchrotron emission from the leptonic cascade. An important role is played by photon fields external to the jet, such as the accretion disk (AD), the broad-line region (BLR), and the dusty torus (DT), as these fields may provide ample seed photons for particle-photon and photon-photon interactions (see, e.g., Böttcher 2019; Cerruti 2020, for detailed reviews). The one-zone model is well justified in flares, where the variability time scale restricts the size of the emission region. However, while the particle flow in this region is relativistic, the emission region itself is typically assumed to remain stationary with respect to the black hole (e.g., H.E.S.S. Collaboration et al. 2019), which allows one to ignore various complications, like varying (external) photon fields, adiabatic expansion and cooling, etc. On the other hand, the observation of stationary features in radio VLBI observations (and other wavelengths where jets have been resolved) suggests standing recollimation shocks (e.g., Weaver et al. 2022), where particles may be accelerated and radiate. In such a situation, the emission region would indeed be stationary with respect to the black hole. Now at: Landessternwarte, Universität Heidelberg, Königstuhl 12, D-69117 Heidelberg, Germany The same radio VLBI observations further reveal moving components launched somewhere upstream from the radio core. The interaction of such moving features with the standing features has been connected with multiwavelength flaring events (e.g., Ahnen et al. 2017;H.E.S.S. Collaboration et al. 2021). Indeed, numerical simulations by Fichet de Clairfontaine et al. (2021 show that the interaction of moving and standing shocks can induce rapid flaring events. It is, thus, important to study the flaring characteristics of a moving, expanding emission region (or "blob"). Recently, Boula & Mastichiadis (2022) and Tramacere et al. (2022) discussed this model in detail and derived the expected time delays between the γ-ray and the radio band expected from the progressive optical thinning at radio frequencies of the expanding source (see also Saito et al. 2015). In the present work, we specifically consider the effect of γ-γ pair production on the evolution of the particle distribution and the photon fluxes. Especially, bright external photon fields can have a major influence as they provide a significant amount of absorption for the γ rays of the emission region (e.g., Zacharias 2021). However, as these photon fields are only present in the direct vicinity of the black hole, the optical thickness for γ rays changes with time as the blob moves down the jet. Additionally, the opening angle will strongly influence the escape time of both photons and particles owing to the increase in radius of the emission region. This could potentially prolong the pair production as the (internal) photons have more time to interact before escape. This naturally competes with the thinning of the photon density due to expansion. In any case, one may obtain variability simply through the motion of an expanding blob through the various radiation fields due to the varying injection of pairs. Firstly, we derive a simple semi-analytical model that describes the time-dependent evolution of the particle distribution (Sec. 2). In this section we introduce the basic assumptions about the escape time scale and the implication of the expansion. We then derive the particle distribution without pair injection, followed by a very simple linear cascade model for a few exemplary cases. In Sec. 3 we use the numerical code OneHaLe (Zacharias 2021;Zacharias et al. 2022) to derive the light curves of a realistic simulation based on the parameters of PKS 1510-089, which is known for its bright external photon fields. We conclude in Sec. 4. A simple electron evolution model We derive simple semi-analytical models for the time-dependent evolution of the electron distribution ignoring all energy dependencies. While this is a major simplification, it allows us to study three distinct cases with an at-most linear cascade. Using the time-dependent, one-zone radiation code OneHaLe, we reproduce well two of the three cases. We discuss the failure of the third case, as well as the influence of the choice of parameters on the solutions. Escape time The blob travels with constant speed β Γ c corresponding to the bulk Lorentz factor Γ = (1 − β 2 Γ ) −1/2 along the z-axis. Within a conical jet, the opening angle is constant. VLBI observations suggest an opening angle ∝ α/Γ with α ∼ 0.26 (Pushkarev et al. 2017), but we use α as a free parameter. The radius R of the blob thus evolves as a function of comoving time 1 t and jet coordinate z : R(t) = R 0 + (z (t) − z 0 ) tan (α/Γ) = R 0 + Γβ Γ ct tan (α/Γ) ≈ R 0 + αct,(1) where in the last line we approximate for small angles and β Γ ≈ 1. Boula & Mastichiadis (2022) expressed this equation by an "expansion speed" u exp , which relates to our equation as u exp = αc. In their paper, they used α between 0.01 and 0.2. It should be noted that one of the common estimates for jet expansion, α = 1 (e.g., Ackermann et al. 2016), implies a radial expansion with the speed of light, while α > 1 implies a superluminal expansion and thus a causal disconnection of regions within the blob/jet. On average, particles escape the emission region on an energy-independent time scale t esc (t) = η esc R(t)/c(2) with η esc > 1. This resembles an advective motion of the plasma below the speed of light mimicking the trapping of charged particles in the magnetized blob. Calculating the average escape time scale for photons, one obtains η esc,ph = 3/4 (Böttcher & Chiang 2002). In an expanding blob, the escape time increases. In turn, particles and photons take longer to escape and only efficiently do so once the intrinsic time t since launch surpasses ∼ t esc (t). Namely, t > η esc c (R 0 + αct) ⇔ t > t esc (0) 1 − η esc α ,(3) with t esc (0) = η esc R 0 /c. In case of η esc α → 1, particles are effectively trapped in the blob without a chance of a meaningful escape. In this case the escape of photons is also significantly slowed down. Hence, a significant cascade could still materialize at far distances from the black hole. It also suggests that a cascade that has begun developing closer to the black hole (say, within the external photon fields), can continue to grow even at far distances. Particle density evolution without secondary injection We are interested in the time-dependent evolution of the total particle density, but not its detailed energy-dependent evolution, as this does not change the overall density. Considering only time-dependent primary injection and escape, but no secondary injections, the kinetic equation for the particle distribution n(t) becomes ∂n(t) ∂t + n(t) t esc (t) = Q(t)(4) with the analytical solution n(t) = exp − t dt t esc (t ) t 0 Q(t ) exp t dt t esc (t ) dt .(5) The particle injection rate Q(t) is coupled to the particle injection luminosity L inj (t) in the form Q(t) = L inj (t) V(t)E(γ) = q 0 1 + η esc α t esc (0) t −3−p ,(6) where E(γ) is a function of the injection energy spectrum, which is of no concern to us, while V(t) is the volume of the spherical blob. With the assumption L inj (t) = L 0 [R 0 /R(t)] p = L 0 1 + η esc α t esc (0) t −p and the definition q 0 := L 0 /[V(0)E(γ)], the second equality of Eq. (6) is readily achieved. The power-law index p describes the decrease of the injection luminosity as a function of R(t). As the jet power is proportional to R(t) 2 , p = 2 implies a constant jet particle injection power. While p is a free parameter, we will mostly use p = 2 below. For constant injection and escape -that is a straight jet, α = 0 -the solution of Eq. (5) is n = q 0 t esc (0) as expected. With Eq. (2), the integrals in the exponentials of Eq. (5) are easily solved: n(t) = q 0 1 + η esc α t esc (0) t − 1 ηescα t 0 1 + η esc α t esc (0) t 1 ηescα −(3+p) dt = q 0 t esc (0) 1 − η esc α(2 + p)         1 + η esc α t esc (0) t −(2+p) − 1 + η esc α t esc (0) t − 1 ηesc α         .(7) Article number, page 2 of 15 M. Zacharias: Particle evolution in expanding blobs This equation is positive for all times and p. For p = (η esc α) −1 −2 the density becomes n(t) = q 0 t esc (0) η esc α 1 + η esc α t esc (0) t − 1 ηesc α ln 1 + η esc α t esc (0) t(8) Equation (7) holds for all particle species as long as there is no secondary injection or particle destruction. Below, we refer to Eq. (7) as the "standard solution". Figure 1 exemplifies Eq. (7) for two values of p, and various values of η esc α. The parameters q 0 and R 0 are chosen to resemble typical one-zone parameters, while η esc = 3 implies α ∈ [0.03, 0.3] resembling the typical range of this parameter (Pushkarev et al. 2017;Boula & Mastichiadis 2022). Interestingly, both p and η esc α have a similar effect. A larger p implies a faster decrease in the injection luminosity and thus a faster decrease of the density. More importantly though, larger opening angles imply a slower escape of particles from the emission region, and thus a slower decline of the density [see Eq. (3)]. The peak in each curve is attained at roughly t ∼ t esc (0) (red vertical line), however with a slightly earlier peak time for larger opening angles because of the quicker injection density decrease. For α → 0, the lines would approach a constant as the blob approaches the steady-state. The solid lines in Fig. 1 are derived from simulations using OneHaLe, and confirm the analytical result. The dash-double-dotted line in Fig. 1 indicates n ∼ t −3 , which is the expected evolution of the density if it were solely due to the increase in volume of the blob. Most model lines are harder than this line implying a continuous increase of particle number in the emission region, even for p = 2. Only for small opening angles (black and blue cases) for p = 2, the model lines are softer and the total number of particles decreases at large times. This can be easily understood from Eq. (7), as the evolution at late times is governed by the power-law index: (2 + p) or 1/η esc α, whichever is smaller. If the injection switches off entirely, the density drops ∝ t −1/η esc α , which for large opening angles is the same behavior as with continuous injection. This shows the aforementioned trapping of particles in the (rapidly) expanding blob. Linear cascade evolution In this section we treat the additional injection of electrons through γ-γ pair production of a γ ray colliding with a soft photon. We do so by first deriving three simple (semi-)analytical scenarios. These are compared to simulations in Sec. 2.3.4, where we discuss both the success and failure of the approach. We continue to ignore the energy dependency of the process, as we are only interested in the rough time dependency of the total particle density. In other words, we assume that pair production takes place. This clearly is a very strong simplification, which may not hold in many cases. In this scenario, the injection rate of pairs can be written as Q γγ (t) = ξn soft (t)n γ (t),(9) with the soft photon density, n soft (t), the γ-ray photon density, n γ (t), and the correlation factor ξ ∼ σ T c absorbing all constants and energy dependencies. Both photon distributions can have various underlying production processes. The soft photons are most likely electronsynchrotron photons, or thermal photons from external sources, such as the AD, the BLR and the DT. The γ rays can be produced from electron-inverse-Compton emission, or proton-dependent processes, such as proton-synchrotron or neutral-pion decay. The internal photon densities thus depend on the underlying particle distribution implying n phot (t) ∼ n e/p (t), with the exception of SSC radiation, which depends quadratically on the electron distribution. As we only want to treat a linear cascade evolutionthat is, Q γγ shall depend at most linearly on the electron distribution -we will ignore from now on the following combinations: electron-synchrotron and electron-inverse-Compton, as well as external photon fields and electron-SSC. In fact, the simplest case is hadronically induced γ rays absorbed by external photon fields, as this does not depend on the electron distribution at all. Cases with a linear dependency on the electron distribution are electron-synchrotron absorbing hadronically induced γ rays, as well as external photons absorbing electron-inverse-Compton radiation. This leaves us with three distinct cases: Q γγ (t) ∼ ξn ext (t)n p (t) (10) Q γγ (t) ∼ ξn e (t)n p (t) (11) Q γγ (t) ∼ ξn ext (t)n e (t).(12) The external photon fields comprising AD and isotropic sources, such as the BLR, are by themselves assumed to be timeindependent. However, the motion of the blob changes the distance to these external sources implying n ext (t) = n AD 1 + Γc R AD t 2 + n i H z i Γc − t .(13) The first summand describes the AD field, which is roughly constant close to the accretion disk, and falls off with distancesquared once the blob has travelled a distance corresponding roughly to the AD radius, R AD (Dermer & Schlickeiser 2002). The second summand represents isotropic photon fields (in the black hole frame), such as the BLR and the DT, within a given distance z i from the black hole. Case 1: External and proton-induced photons In this case, Eq. (10) is simply added to Eq. (4) and in Eq. (5). The solution is thus separated into the primary injection following Eq. (7), and the secondary injection leading to n γγ e (t) = ξq 0,p t esc (0) 1 − η esc α(2 + p) 1 + η esc α t esc (0) t − 1 ηesc α × t 0         1 + η esc α t esc (0) t 1 ηesc α −(2+p) − 1         ×           n AD 1 + Γc R AD t 2 + n i H z i Γc − t           dt .(14) While the isotropic contribution can be easily integrated, the accretion disk contribution leads to a hypergeometric integral without a simple analytical solution. For isotropic external photons, that is n AD = 0, the analytical result for Eq. (14) n γγ e (t) = ξq 0,p [t esc (0)] 2 n i 1 + η esc α t esc (0) t − 1 ηescα [1 − η esc α(2 + p)] [1 − η esc α(1 + p)] ×            1 + η esc α t esc (0) t 1 ηescα −(1+p) − 1 − t for t ≤ z i Γc 1 + η esc α t esc (0) z i Γc 1 ηescα −(1+p) − 1 − z i Γc for t > z i Γc .(15) The sum of Eqs. (7) and (15) is shown in Fig. 2. Compared to the standard solution, Eq. (7), there are significantly more parti- Fig. 2, except that the absorber is AD photons with n AD = 1 × 10 11 cm −3 , and R AD = 1 × 10 16 cm. Further parameters are p = 2, q 0,e = q 0,p = 1 cm −3 s −1 , R 0 = 5 × 10 15 cm, η esc = 3, and Γ = 10. cles injected within the boundaries of the external field, z i . The boundary z i is noticeable by the break in the dash-dotted and solid lines. The peak of the particle density is attained later for larger opening angles compared to the standard solution. Beyond z i , the decay of the density again depends strongly on the opening angle. The standard solution is only reached for small opening angles. The semi-analytical solutions of Eq. (14) for the AD, that is n i = 0, are shown in Fig. 3. The behavior is comparable to the isotropic case except for an earlier peak owing to R AD z i . In both figures, 2 and 3, the horizontal dotted lines mark the maximum level of the standard solution. They indicate the time delay for the semi-analytical solution to drop below this density. Naturally, this happens later for the isotropic field than for the AD field, and depends strongly on the opening angle. In fact, the Article number, page 4 of 15 M. Zacharias: Particle evolution in expanding blobs time delay can be orders of magnitude implying a significantly more pronounced and longer high-state. Case 2: Electron-synchrotron and proton-induced photons Adding Eq. (11) to Eq. (4) we obtain with a slight rearrangement ∂n e (t) ∂t + 1 t esc (t) − ξn p (t) n e (t) = Q e (t).(16) The linear cascade acts as a "catastrophic" injection with time scale t γγ (t) = [ξn p (t)] −1 . We directly obtain the generalisation of Eqs. (5) and (7) as n e (t) = exp − t dt t esc (t ) + ξ t n p (t ) dt × t 0 Q e (t ) exp t dt t esc (t ) − ξ t n p (t ) dt dt = q 0,e 1 + η esc α t esc (0) t − 1 ηescα exp ξ t n p (t ) dt × t 0 1 + η esc α t esc (0) t 1 ηesc α −(3+p) exp −ξ t n p (t ) dt dt .(17) As the protons follow Eq. (7), the integrals in the exponentials can be performed: t n p (t ) dt = q 0,p t esc (0) 2 [1 − η esc α(2 + p)] (1 + p) (1 − η esc α) ×        1 − 1 η esc α 1 + η esc α t esc (0) t −(1+p) + (1 + p) 1 + η esc α t esc (0) t 1− 1 ηescα         .(18) With this solution, the remaining integral in Eq. (17) can be solved numerically. The result is shown in Fig. 4. The cascade develops only after t esc (0), while the opening angle dictates the further evolution. Small opening angles result in a high number density and a quick decay, while large opening angles seemingly keep the cascade going for very long times without any significant decay. This is also indicated by the dotted lines. Case 3: External and electron-IC photons This case is similar to the case 2, except that we have to replace n p (t) with n ext (t), and the γ rays stem from IC scattering of the external photon fields. Then the electron density becomes 20) and (21), with n i = 4 × 10 7 cm −3 , and z i = 1 pc. Further parameters are p = 2, q 0,e = 1 cm −3 s −1 , R 0 = 5 × 10 15 cm, η esc = 3, and Γ = 10. n e (t) = q 0,e 1 + η esc α t esc (0) t − 1 ηescα × exp         − ξn AD R AD Γc 1 + Γc R AD t + ξn i tH t, z i Γc         × t 0 1 + η esc α t esc (0) t 1 ηescα −(3+p) × exp         ξn AD R AD Γc 1 + Γc R AD t − ξn i t H z i Γc − t         dt ,(19)] esc =0.1 esc =0.3 esc =0.5 esc =0.7 esc =0.9 Fig. 6. Same as Fig. 2, but for secondary electrons produced from ICinduced γ rays absorbed by AD photons with n AD = 1 × 10 10 cm −3 , and R AD = 1 × 10 16 cm. Further parameters are p = 2, q 0,e = 1 cm −3 s −1 , R 0 = 5 × 10 15 cm, η esc = 3, and Γ = 10. n e t ≤ z i Γc = q 0,e ξn i 1 + η esc α t esc (0) t − 1 ηescα η esc α ξn i t esc (0) 1 ηescα −(3+p) × exp ξn i t esc (0) η esc α 1 + η esc α t esc (0) t × Γ 1 η esc α − (2 + p), ξn i t esc (0) η esc α −Γ 1 η esc α − (2 + p), ξn i t esc (0) η esc α 1 + η esc α t esc (0) t (20) n e t > z i Γc = q 0,e 1 + η esc α t esc (0) t − 1 ηescα exp ξn i z i Γc ×            exp ξn i t esc (0) η esc α η esc α ξn i t esc (0) 1 ηescα −(3+p) ξn i × Γ 1 η esc α − (2 + p), ξn i t esc (0) η esc α −Γ 1 η esc α − (2 + p), ξn i t esc (0) η esc α 1 + η esc α t esc (0) z i Γc + t esc (0) 1 − η esc α(2 + p)         1 + η esc α t esc (0) t 1 ηescα −(2+p) − 1 + η esc α t esc (0) z i Γc 1 ηesc α −(2+p)                ,(21) with the incomplete Gamma-function Γ(q, x). The result is shown in Fig. 5. Compared to Fig. 2, a strong pile-up is visible, which results from ξn i t > 1 close to z i . This maybe due to an extreme choice of parameters, but displays the importance of the immediate feedback of the cascade on the γ rays in this case compared to case 1. We note that the pile-up for η esc α = 0.1 in Fig. 5 might be influenced by some approximation inaccuracies for small values of η esc α, and a smaller pile-up is to be expected in this case. For AD photons, we need to numerically integrate Eq. (19) with the result shown in Fig. 6. The result is fairly similar to case 1, which may again be due to parameter choices, but also because the argument of the exponential is a decreasing function with time. The latter reduces the amount of absorbing photons before the γ rays have fully developed resulting in a reduced cascade compared to the isotropic case. In both cases, the dotted lines are roughly as long as in case 1. In the isotropic case, this is significantly influenced by the pile-up close to z i . With a slightly smaller size of the isotropic field, and a resulting smaller pile-up in case 3, the duration of the high-state in case 3 would be shorter than in case 1. No such strong influence is expected in the AD case. Simulations We use the OneHaLe code (Zacharias 2021;Zacharias et al. 2022, see the appendix for a brief description) to model the timedependent evolution of the blob. In order to comply with some of the assumptions above, we have to slightly tweak the code. The γ-γ pair production injection rate is given by (Aharonian et al. 1983;Cerruti et al. 2021) Q γγ (γ) = 2 3σ T c 32 ∞ γ d n ph ( ) 3 1 4γ( −γ) d˜ n ph (˜ ) 2 × 4 2 γ( − γ) ln 4γ˜ ( − γ) − 8 ˜ + 2 2 (2 ˜ − 1) γ( − γ) − 1 − 1 ˜ 2 γ( − γ) 2        ,(22) with the photon distribution n ph , the photon energies and˜ normalized to the electron rest energy, and the electron Lorentz factor γ. Eq. (22) requires 1/˜ . This and˜ < 1 ensure an absorber with low photon energies for the γ rays. In order to suppress the severe cooling -due to the extreme parameter settings, see Tab. 1 -we switch off electron IC cooling. While this is a massive change, it is necessary to create sufficient synchrotron or IC photons. Furthermore, the magnetic field is kept constant throughout these simulations. Lastly, as we are only interested in the additional injection of pairs from γ-γ processes, the secondary injections from muon decay and Bethe-Heitler pair production have also been disabled. The input parameters for the simulations are given in Tab. 1, while the results are shown as solid lines in Figs. 2 to 6. We emphasize that these parameters are not chosen for realism but merely to reproduce as close as possible the semi-analytical models. In this regard, the external fields labeled "BLR" and "DT" should just be considered as two separate external fields without any intended resemblance to actual BLR and DT photon fields. In case 1, the external fields act as absorbers. In case 3 they are needed for the IC process, while the absorption is done by electron-synchrotron emission. Cases 1 and 3 can be reproduced reasonably well. In case 1, the isotropic photon field is made of two thermal fields with different temperature and luminosity, while in the corresponding case in case 3 only one thermal field is needed. In the latter case, the pile-up close to the edge of the external field is present, but not as strong as in the analytical model 2 . The accretion disk field M. Zacharias: Particle evolution in expanding blobs Table 1. OneHaLe input parameters to simulated cases 1 to 3. The first five parameters are the same in all simulations. The electron distribution parameters of case 1 correspond to the simulation used to reproduce the standard solution in Fig. 1 γ e,min 2 × 10 1 2 × 10 1 2 × 10 1 2 × 10 1 2 × 10 1 Max. e Lorentz factor γ e,max 2 × 10 4 2 × 10 4 2 × 10 5 2 × 10 5 2 × 10 5 e spectral index s e 2.5 2.5 2.5 1.5 1.5 p injection luminosity L p [erg/s] 1.2 × 10 50 6 × 10 49 1 × 10 50 --Min. p Lorentz factor γ p,min 2 2 2 × 10 3 --Max. p Lorentz factor γ p,max 2 × 10 10 2 × 10 10 1 × 10 9 -p spectral index s p 1.5 Fig. 7. Small parameter study using OneHaLe. The black lines correspond in style to those in Figs. 2 to 6, but for the case η esc α = 0.5. The solid colored lines show simulations varying the input parameter as labeled. In the bottom panels, the variation in the electron parameters (magenta lines) preserves the initial injection rate of ∼ 1 cm −3 s −1 . is represented by a Shakura-Sunyaev disk Shakura & Sunyaev (1973) with luminosity L AD = η Edd L Edd . In case 3, the AD field absorbs IC emission scattering both isotropic fields (but not the AD field). stepping in the simulation does not cover enough time steps to fully recreate the pile-up. We have found no parameter set that could reproduce the semi-analytical result in case 2. The simulations with OneHaLe with the parameters given in Tab. 1 show a pronounced maximum shortly after t esc (0) followed by a decrease depending on the opening angle. This seems reasonable, as the dashed lines roughly correspond to the evolution of the γ rays (psynchrotron). In turn, the number of "absorbees" (ie., γ rays) quickly reduces. The long evolution suggested by the semi-analytical model would only be achievable if the absorber were to increase in such a way that progressively a higher fraction of the γ rays were absorbed. Then, at late times, practically all γ rays would be absorbed to keep the cascade going. This is implausible and marks the limit of the simple analytical model that ignores all the energy-dependencies of the cross-section. The influence of the parameters on the simulations is shown in Fig. 7, where 3 variations for each case have been derived. Clearly, reproducing the analytical model required a specific set of parameters. This also shows the limits of the approach. It's nonetheless reassuring that 2 out of 3 cases have been reproduced. Interlude The previous sections have shown the influence of the expansion of the moving blob on the development of the pair cascade. The effects are two-fold. Firstly, a significantly higher density can be achieved through the pair production compared to no secondaries. Secondly, a larger opening angle can drag out the peak of the density evolution. While the densities for the standard solution peak around t esc (0), the curves with pair injection typically peak later than this time scale. The exception to this rule are secondary injections involving AD photons, because of the time scale R AD /Γ/c < t esc (0). For small values of z i , this could become the case as well for the isotropic photon field. For internal photon fields (case 2), however, the peak is always attained after t esc (0), as the internal fields only reach their maximum at this time, and the developing cascade reinforces the interaction. Nonlinear cascades, which we have not treated here, might increase this effect, as both absorber and absorbee are increased at later times. The dotted lines in Figs. 2 to 6 mark the peak density of the standard solution, and indicate at which time the models with pair injection drop below this level. It typically happens at least an order of magnitude after t esc (0). This, of course, depends strongly on the amount of pairs produced, the parameters, and the opening angle. This indicates the importance of pair production and the geometry of the jet for the development and duration of a flux high state. Naturally, the inclusion of the energy dependency on the particle distribution, as well as the pair production process can have a significant influence. In the absence of efficient reacceleration of particles, the particle cooling will lead to a drop in the pair production process, as less energetic γ-ray photons will be produced. The adiabatic expansion of the blob will also cause a reduction in the energy density of the particles and the photon fields, similarly reducing the pair production. Hence, the time evolution might be shorter than envisaged in the present semianalytical model. Modeling based on PKS 1510-089 The flat spectrum radio quasar (FSRQ) PKS 1510-089, located at a redshift of 0.361, is one of the most famous blazars. It was detected at very-high-energy γ rays with H.E.S. S. in 2009S. in (H. E. S. S. Collaboration et al. 2013, and has received significant attention since (e.g., Ahnen et al. 2017;H.E.S.S. Collaboration et al. 2021). Modeling typically requires both the BLR and DT photon fields (Barnacka et al. 2014), or multiple zones (Nalewajko et al. 2012;Prince et al. 2019). While no modeling of the source or a specific light curve is attempted here, we use this source to study the impact of the pair cascade in a moving, expanding blob to enhance flaring events in a realistic setting. The modeling parameters are given in Tab. 2 and are based upon the works by Nalewajko et al. (2012) and Barnacka et al. (2014). We derive multiple modelings: 1) a steady-state model similar to the aforementioned works in order to provide a baseline flux that may describe PKS 1510-089 on average; 2) a moving blob using only leptonic processes similar to Saito et al. (2015); and 3) a moving blob including hadronic interactions. For the moving blobs we again derive models for various opening angles as before. We also derive specifically models without γ-γ pair production to show the difference. For the moving-blob models 2) and 3) the launching position is chosen relatively close to the black hole, where the jet has not expanded much (Zacharias et al. 2022). At this position the blob is still small in order to fit into the jet, and its magnetic field is high compared to the steady-state one-zone model, which is located further down in the jet. One important addition to the simulations of the previous section is the consideration of magnetic flux conservation. That is, B(z) = B 0 R 0 /R(z) assuming a dominating toroidal guide field (Kaiser 2006). This also implies that the larger the opening angle, the faster the magnetic field drops. This may have a significant influence on the synchrotron emission compared to the expectation from the simple model above. Light curves The multi-wavelength light curves are shown in Fig. 8. The wavebands are the very-high-energy γ-ray (VHE) band (E > 30 GeV 3 , corresponding to the energy band of the forthcoming CTA observatory), the high-energy γ-ray (HE) band (100 MeV< E < 100 GeV, corresponding to the energy band of the Fermi- , HE γ rays (2nd row), X-rays (3rd row) and the optical R-band (bottom row) for purely leptonic processes (left column) and including hadronic processes (right column). Displayed models are either with (thick dashed) or without (solid) γ-γ pair production. Colors are for various opening angles as labeled. The vertical lines mark t esc (0) (red), the crossing of the BLR edge (blue) and of the DT edge (magenta); all times in the observer's frame. For clarity, the launch of the blob takes place at t obs = 0.1 d. LAT), the X-ray band (2 − 10 keV, corresponding to the range of Swift-XRT), and the optical R-band. It is immediately obvious that the flare evolution depends strongly on the wavelength and the presence of relativistic protons. The models without pair production (solid lines) do not show major differences between the leptonic and hadronic simulations. Except for the VHE band, which is fully absorbed at early times, all light curves rise quickly to a maximum at the initial escape time scale t esc (0), followed by a subsequent decay. The switch from IC/BLR cooling to IC/DT or synchrotron cooling during the crossing of the BLR edge 4 results in a minor bump in the X-rays and the R-band, which is absent in the HE band. The VHE light curve shows a minor bump at this point, as the environment is now optically thin to VHE photons. Except for the HE band, the decay of the light curve is faster for larger opening angles, which is a consequence of the diminishing magnetic field, and hence a faster decay of the synchrotron emission. One should note here that the X-ray emission from the blob is also influenced by electron-synchrotron emission, as the (initial) high magnetic field shifts the synchrotron peak to higher energies compared to the steady-state model. As the HE band is governed by inverse-Compton emission of external photon fields, the magnetic field has basically no effect, and the decay of the light curve is fully governed by the evolution of the electron density. In the hadronic model and for large opening angles, the VHE light curve shows an additional bump around the time of the DT-edge crossing. This radiation is electron-synchrotron emission from secondary particles injected from muon decay. Inclusion of the pair production has almost no consequence for the VHE band, except for hadronic simulations of large opening angles, where the DT-edge-crossing bump is much more pronounced and shows a plateau phase lasting for a few hours. At that time, a peak also shows up in the X-ray domain for large opening angles. It is induced by pair production, and is also visible -although less pronounced -in the leptonic simulations. The peak flux in the main flare is significantly higher compared to the case without pair production. Similarly, the HE band exhibits higher peak fluxes compared to the solid lines. In both bands, HE and X-rays, the peak fluxes are much more pronounced in the hadronic simulations indicating the significant impact of relativistic protons on the pair production. The R-band shows no change between pair or no pair production implying that pairs are produced predominantly at higher energies. While cooling will reduce the energy of the secondary pairs, their density is insufficient to add to the synchrotron emission of the primaries in the R-band. The most important effects in the light curves shown in Fig. 8 take place within a few days in the observer's frame. Figure 9 displays the first 100 d in the evolution of the radio band at 15 GHz. The rise of the light curve begins after a few days, when the flares in the other bands have already ceased (see also Boula & Mastichiadis 2022;Tramacere et al. 2022). For small opening angles, the radio flare diminishes within 50 d, while for large opening angles the light curve keeps rising. While not shown, for η esc α = 0.9 the peak is reached only after more than 1000 d. There is barely a difference between the cases with and without pair production for the same reason as in the R-band. 4 The crossing of the BLR edge after only ∼ 0.1 d in the observer's frame is a consequence of the relativistic motion of the blob. The foreshortening of the external radiation fields in the blob frame, Eq. (1), and the boosting of the blob emission into the observer's frame result in ∆t obs = ∆z /(δΓβ Γ c). Luminosity evolution An important measure for any blazar model is the luminosity of the emission region. Power is contained in particles, magnetic field, and radiation. The evolution of the luminosities as a function of time is shown in Fig. 10. By design, the magnetic field luminosity is constant with a value of L obs B = 1.8 × 10 44 erg/s, which is why we do not display it in Fig. 10. The total luminosity in the bottom row of that figure is the sum of the electron, proton, magnetic field, and radiation luminosities. The individual luminosities are calculated from the respective energy densities in the comoving frame u i (t) according to L obs i (t) = u i (t)Γ 2 πR(t) 2 c deriving the bulk Lorentz factor Γ from the assumption Γ = δ. For small opening angles (black and blue lines), the respective luminosities quickly rise to the maximum as the source is being filled with particles and radiation, and then decrease. In turn, the total luminosity reaches a bottom value which is given by the constant magnetic field luminosity. It also implies that the initial particle and radiative powers are much larger than the magnetic field power. For η esc α = 0.5, the particle powers reach a constant level implying that the injection and escape of particles is fully countered by the expansion. In the presence of protons (right column in Fig. 10), the electrons slowly continue to rise in this case if pair production is enabled (magenta dashed line). This implies continuous pair creation even at very late times. For large opening angles (orange and yellow lines), the particle luminosities keep rising, as the rapid expansion does not allow for a meaningful escape of particles. The radiative luminosity in this case continues to decrease initially, but reverses this trend for late time, as the continuing accumulation of particles also increases the amount of photons, which also remain longer and longer in the emission region. Interestingly, the total luminosity is initially dominated by radiation, and only after about 1 d in the observer's frame does it change to particle or magnetic field dominance. It implies that transfer of power into radiation is initially very efficient, and that the particles are efficiently cooled. With the decrease in the magnetic field strength, and the decrease of density of internal and external photon fields due to expansion and motion, the radiation production becomes less efficient over time and its luminosity drops more rapidly than that of the particles. In all cases, the comparison between particles and magnetic fields suggests that the emission region is initially particle dominated. For large opening angles, this is maintained throughout, while for small opening angles the magnetic field dominates the particles at late times. This is contrary to most models, where an initial magnetic field dominance is expected which changes to particle dominance later (e.g., Zacharias et al. 2022, and references therein). The reason is that we chose a relatively modest initial magnetic field of only 20 G along with a linear decrease with radius. If the magnetic field were initially mostly poloidal, one would expect a quadratic decrease with radius (Kaiser 2006), and a much faster decrease. In this case, the magnetic power would also drop as a function of time. While our initial value is in line with the measurement in M 87 (Event Horizon Telescope Collaboration et al. 2021), theoretical works suggest much larger values along with a rapid decrease (Zdziarski et al. 2022). Testing such scenarios is left for future work. The black horizontal lines in Fig. 10 mark the Eddington luminosity and the AD luminosity, respectively. As the accretion process fuels the jet, the jet luminosity must be compared with these values. Apparently, all models remain initially close to or below the limit of the Eddington luminosity. In case of protons and pair production (right column, dashed lines), the Eddington 9. Same as Fig. 8, but for the radio band at 15 GHz. luminosity is initially surpassed by a factor 5. This is, however, due to the large radiative power, which includes also the IC scattering of the external fields. While the BLR and the DT are less luminous than the AD, they are strongly beamed in the comoving frame. Thus, the short excess of the Eddington luminosity can be expected, and is within reasonable bounds. The continued increase of jet power for large opening angles has already been discussed, and naturally surpasses the Eddington limit by far. Clearly, the model is not realistic in these cases. Conclusions In this paper we have discussed the particle evolution within a moving, expanding emission region within the jet of a blazar (Boula & Mastichiadis 2022;Tramacere et al. 2022). With a simple semi-analytical model, we have shown that the opening angle has an important influence on the particle distribution and the electron-positron cascade. With increasing opening angle the high-state (including the cascade) may last much longer owing to the increased escape time scale for both particles and photons. With the help of the OneHaLe code (Zacharias 2021;Zacharias et al. 2022), we have shown that the simple model can be reproduced for linear cascades induced by external photon fields, while the linear cascade resulting from internal processes was not reproduced successfully. Naturally, due to the neglect of the energy dependencies of the particle evolution and the cascade, it required quite specific, and extreme parameters in the simulation. Nonetheless, the exercise has demonstrated the importance of the opening angle on the evolution of the particle distribution. The simple model may be improved by considering deltafunction approximations to the energy dependency of the particle distribution and the cascade. This may provide a more reasonable view on the evolution providing a broader range of application. These calculations are left to the interested reader. In order to obtain realistic simulations, we have made use of parameters from PKS 1510-089 (Nalewajko et al. 2012;Barnacka et al. 2014). The leptonic and hadro-leptonic simulations were conducted for various opening angles. From the resulting light curves in Fig. 8 we can derive the following conclusions. A moving blob induces a rapid flare. The evolution depends critically on the blob's speed and on the initial size at launch. Observations in the X-ray and VHE γ-ray are crucial to determine the opening angle and the particle composition, as for large opening angles these bands may (or may not) show a delayed flare when the blob moves out of the DT. These secondary flares are significantly influenced by the pair production process and are more pronounced if a hard and highly energetic proton distribution is present. Interestingly, the optical band is also useful to obtain information on the opening angle, as a small opening angle induces a longer decay phase due to the slower decay of the magnetic field. In the HE γ-ray band, the flare is bright and short with barely a difference between the various opening angles. Importantly, the HE band peaks slightly before the other bands (cf. the red vertical line in Fig. 8), whereas the X-ray and R-band peak at the time corresponding to t esc (0). In all bands, the (primary) flare is asymmetric with a fast rise and a slow decay (Saito et al. 2015;Boula & Mastichiadis 2022;Tramacere et al. 2022). Truly simultaneous and high-cadence observations are paramount to obtain all the potential information. The main flare lasts just for a few hours, and the subtle differences between peak times and decay profiles might be hidden in the statistical errors or a delay in observation. It should also be noted that it is worthwhile to continue observing for a few days in the X-ray and VHE γ-ray band to search for the secondary flare. Radio observations may also provide indications for the opening angle. Large values of η esc α result in years-long variations, while small opening angles result in flares lasting a few tens of days. Single-dish radio observations every few days for up to 100 d after the event at higher energies should provide sufficient information on the opening angle. The downside could be the higher chance of multiple events superposing each other. The jet luminosities are mostly within reasonable bounds compared to the Eddington luminosity. However, for large opening angles, the particle luminosities keep increasing due to the lack of escape, which results in unreasonable jet powers. Additionally, the magnetic field evolution is such that its power remains constant. In turn, the initially partical-dominated emission region becomes magnetic-field-dominated at later times and small opening angles. This is contrary to standard expectations. A scenario, which is more in line with the usual jet evolution, will be discussed elsewhere. The above mentioned evolutionary details of the flare depend on the type of object. The given results are based on parameters describing the FSRQ PKS 1510-089. In BL Lac objects with much weaker or even absent external fields, the effects induced from the pair cascade are probably much less pronounced (Zacharias et al. 2022). In this case, the solid lines in Fig. 8 might already be an adequate description, and the secondary flares in the X-ray and VHE γ-ray band are absent. Similarly, the details of the magnetic field decay may have a significant influence on . Electron luminosity (top row), proton luminosity (2nd row), radiation luminosity (3rd row), and total luminosity (bottom row) as a function of time for pure electron (left column) and electron-proton simulations (right column) for different opening angles (color code as labeled) using the parameters of PKS 1510-089 as in Fig. 8. Simulations without pair production are marked by solid lines, while dashed lines refer to simulations with pair production. The vertical lines have the same meaning as in Fig. 8. The black dot-dashed and black dash-double-dotted lines mark the Eddington and AD luminosities, respectively. Note the logarithmic time axis. Article number, page 12 of 15 M. Zacharias: Particle evolution in expanding blobs the evolution of the flux especially in the R-band, and needs to be discerned from the expansion profile. Lastly, using the standard one-zone description implies that at all times an instantaneous particle spectrum was used. While shock acceleration can quickly accelerate particles (Böttcher & Baring 2019), it would be useful to properly include the particle acceleration in this model as it might have an important influence on the light curve evolution at early, but also at late times. At early times, the delayed injection of high-energetic particles would slow down the flare evolution, and might also reduce the strength of the cascade development. At late times, the weakened magnetic field might not be able to accelerate particles to the highest energies, which might reduce or even inhibit the development of the secondary flare. It would also substantially reduce the particle luminosities. In order to adequately account for these effects a two-zone model is needed (Weidinger & Spanier 2015;Chen et al. 2015;Dmytriiev et al. 2021), which is beyond the scope of this paper. (0). Free parameters are η esc α = 0.5, η esc = 3, R 0 = 5 × 10 15 cm, b = 1, p = 2, Γ = 10 and z ext = 3 × 10 18 cm. Fig. 1 .Fig. 2 . 12Particle density as a function of time for various cases of η esc α and p as labeled. The dashed lines are the analytical result, Eq. (7). The vertical red line is t esc (0). The dash-double-dotted line marks t −3 . Further parameters are q 0 = 1 cm −3 s −1 , R 0 = 5 × 10 15 cm, and η esc = 3. The solid lines are from simulations using OneHaLe. Electron density as a function of time for various cases of η esc α as labeled. Secondary electrons are produced from proton-induced γ rays absorbed by isotropic external photons. The thick dash-dotted line is the analytical solution of Eq. (14) assuming only one isotropic photon field with n i = 1 × 10 10 cm −3 , and z i = 1 pc. The dashed lines are the standard solution, Eq. (7). The vertical red line is t esc (0), while the horizontal dotted lines mark the maximum level of the standard solution and when they are reached by the dash-dotted lines. Further parameters are p = 2, q 0,e = q 0,p = 1 cm −3 s −1 , R 0 = 5 × 10 15 cm, η esc = 3, and Γ = 10. The solid lines mark simulations with OneHaLe. Fig. 3 . 3Same as Fig. 4 . 4Same asFig. 2, but for secondary electrons produced from proton-induced γ rays absorbed by electron-synchrotron photons, Eq. (17). Further parameters are p = 2, q 0,e = 1 cm −3 s −1 , q 0,p = 1 × 10 3 cm −3 s −1 , R 0 = 5 × 10 15 cm, and η esc = 3. Fig. 5 . 5Same as Fig. 2, but for secondary electrons produced from IC-induced γ rays absorbed by isotropic external photons, Eqs. ( Fig. 8 . 8Model light curves based upon parameters describing PKS 1510-089 for VHE γ rays (top row) Fig. 10. Electron luminosity (top row), proton luminosity (2nd row), radiation luminosity (3rd row), and total luminosity (bottom row) as a function of time for pure electron (left column) and electron-proton simulations (right column) for different opening angles (color code as labeled) using the parameters of PKS 1510-089 as in Fig. 8. Simulations without pair production are marked by solid lines, while dashed lines refer to simulations with pair production. The vertical lines have the same meaning as in Fig. 8. The black dot-dashed and black dash-double-dotted lines mark the Eddington and AD luminosities, respectively. Note the logarithmic time axis. Fig. A. 1 . 1Time scales in arbitrary units as labeled versus time in the comoving frame. The lines are indicative of the time-dependency. The vertical red line marks t esc Article number, page 1 of 15 arXiv:2211.12283v1 [astro-ph.HE] 22 Nov 2022 A&A proofs: manuscript no. OneHaLe_expandingsource becomes A&A proofs: manuscript no. OneHaLe_expandingsourceArticle number, page 3 of 15 with the modified Heaviside functionH [x, a] = 1 for x ≤ a, andH [x, a] = a/x for x > a.For isotropic external photon fields, that is n AD = 0, Eq. (19) can be analytically integrated:Article number, page 5 of 15 A&A proofs: manuscript no. OneHaLe_expandingsource .Definition Symbol Case 1 Case 2 Case 3 ISO AD ISO AD Initial distance to black hole z 0 [cm] 1.0 × 10 15 Initial blob radius R 0 [cm] 5 × 10 15 Escape time scaling η esc 3 Doppler factor δ 10 Black hole mass M BH [M ] 3 × 10 8 Initial magnetic field B 0 [G] 10 10 0.3 0.1 0.1 e injection luminosity L e [erg/s] 1.86 × 10 43 1.86 × 10 43 2.3 × 10 43 1 × 10 45 8 × 10 44 Min. e Lorentz factor Table 2 . 2OneHaLe input parameters for the PKS 1510-089 simulations. The steady-state simulation uses a stationary emission region. Parameters below the horizontal line are used in each simulation.Symbol steady-state leptonic hadronic z 0 [cm] 7 × 10 17 1 × 10 15 1 × 10 15 R 0 [cm] 7 × 10 16 5 × 10 14 5 × 10 14 η esc 10 3 3 p - 2 2 δ 22 22 22 B 0 [G] 0.5 20 20 L e [erg/s] 2.2 × 10 43 1.5 × 10 43 1.5 × 10 43 γ e,min 900 900 900 γ e,max 1 × 10 5 1 × 10 5 1 × 10 5 s e 3.4 3.4 3.4 L p [erg/s] - - 1.5 × 10 43 γ p,min - - 2 γ p,max - - 1 × 10 9 s p - - 2.2 M BH [M ] 1.6 × 10 8 η Edd 0.25 L BLR [erg/s] 5 × 10 44 T BLR [K] 1.1 × 10 5 R BLR [cm] 1.2 × 10 17 L DT [erg/s] 1 × 10 45 T DT [K] 1.8 × 10 3 R DT [cm] 1.94 × 10 18 Parameters in the observer's frame are marked by "obs", in the black hole frame by a prime, while parameters in the comoving frame and invariants are not marked. This is probably due to the resolution of the time steps in the code. Given the steepness of the analytical pile-up, the logarithmic time- With this threshold definition, the EBL absorption plays a minor role and we ignore it. Article number, page 14 of 15 M. Zacharias: Particle evolution in expanding blobs AcknowledgementI would like to thank Andreas Zech, Markus Böttcher, and Anton Dmytriiev for fruit-and beerful discussions on the content of the manuscript. I extend my gratitude to the anonymous referee for valuable comments that helped to improve the manuscript. MZ acknowledges postdoctoral financial support from LUTH, Observatoire de Paris. Simulations for this work have been performed on the TAU-cluster of the Centre for Space Research at North-West University, Potchesftroom, South Africa. The OneHaLe code is available upon request to the author.Appendix A: The OneHaLe codeThe OneHaLe code is a time-dependent, one-zone, hadroleptonic radiation code that solves the Fokker-Planck equation of the particle distribution for protons, charged pions, muons and electrons (including positrons). In each time step, the radiation transport equation is solved allowing for the direct feedback of the particle and photon interactions. The Fokker-Planck equation for the particle distribution of species i isThe distributions are given as a function of normalized momentum χ = γβ, with the particle Lorentz factor γ and its corresponding speed β normalized to the speed of light. The first term on the right-hand-side is momentum diffusion representing Fermi-II acceleration using hard-sphere scattering with the ratio a of shock to Alfvèn speed. The second term marks continuous momentum gains and losses. Gains are achieved through Fermi-I acceleration, while losses depend on the particle species and include synchrotron, adiabatic, Bethe-Heitler, pion production, and inverse-Compton processes. The third term marks the injection term, while the forth term marks the catastrophic escape of particles. The last term is the decay term for unstable particles. The acceleration time scale is given as a multiple of the escape time scale: t acc = η acc t esc . It merely marks the reacceleration of particles in the acceleration zone, and does not provide "firstprinciple" acceleration, which typically requires a much smaller zone (e.g.Weidinger & Spanier 2015;Chen et al. 2015;Dmytriiev et al. 2021). The initial acceleration is mimicked through the primary injection term Q i , which for protons and electrons takes the form of a power-law between minimum and maximum Lorentz factors, γ i,min and γ i,max , respectively, with spectral index s i . The injection of pions and muons is directly calculated from the respective interactions (using the template approach ofHümmer et al. 2010) and decays. Secondary electrons are injected from muon decay, Bethe-Heitler pair production, and γ-γ pair production.Further details are given in Zacharias(2021)andZacharias et al. (2022). These references provide the gory details about the particle and photon integrals solved in the code.A.1. Acceleration and cooling time scalesIn this section we briefly discuss the influence of the expanding blob scenario on the acceleration and cooling time scales for illustrative purposes. These are not necessarily considered in the discussion and simulations of the main part.For the (re-)acceleration of particles, OneHaLe includes acceleration terms based on hard-sphere scattering. This is parameterized by an energy-independent time scale t acc (t) = η acc t esc (t), where η acc is a free parameter. Hence, t acc (t) = η acc t esc (0) 1 + η esc α t esc (0) t .(A.2)The cooling of particles also becomes time-dependent, as several important variables are a function of radius. For instance, with the conservation of magnetic flux, the magnetic field B(t) can be written aswith the initial magnetic field B 0 , and the power b ∈ [1, 2] of the radial dependence (b = 1 for a toroidal guide field, b = 2 for poloidal guide field,Kaiser 2006). The synchrotron cooling time scale then becomes:with the particle Lorentz factor γ, the rest mass of the electron m e , the rest mass of the particle m (m p for protons), and the Thomson cross section σ T . Following the prescription ofSchlickeiser (2009), the SSC cooling time scale in the Thomson regime can be written aswith c 1 = 0.684, c 2 = 1.856 × 10 −20 erg 1/2 cm 3/2 , and P 0 = 2 × 10 24 erg −1 s −1 . The functiondepends on the instantaneous electron distribution function n e (γ, t), which may have a complicated time-dependency resulting in a non-linear evolution of the particle distribution(Zacharias & Schlickeiser 2012). If we assume that the energy part of the particle energy distribution is time-independent, the time-dependency derived above for the particle distribution, Eq.(7), can be included here (cf.Fig. A.1). The expansion of the blob results in adiabatic cooling with time scalewhere we followed the prescription ofZdziarski et al. (2014). As we assume Γ = const., the cooling due to external isotropic photon fields such as the BLR is constant in time, as long as the blob is within the distance of the external field:with the energy density of the external photon field in the black hole frame u ext , and the radius of the external photon field z ext . We approximated again β Γ ≈ 1.Figure A.1 shows the evolution with time of the cooling and escape time scale. The adiabatic and acceleration (not shown) time scales behave exactly like the escape time scale, while the external-Compton cooling simply stops once the blob leaves the respective region. 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[]
[ "Orbital-selective Mott-Hubbard transition in the two-band Hubbard model", "Orbital-selective Mott-Hubbard transition in the two-band Hubbard model" ]
[ "R Arita \nMax-Planck-Institut für Festkörperforschung\n70569StuttgartGermany\n", "K Held \nMax-Planck-Institut für Festkörperforschung\n70569StuttgartGermany\n" ]
[ "Max-Planck-Institut für Festkörperforschung\n70569StuttgartGermany", "Max-Planck-Institut für Festkörperforschung\n70569StuttgartGermany" ]
[]
Recent advances in the field of quantum Monte Carlo simulations for impurity problems allow -within dynamical mean field theory-for a more thorough investigation of the two-band Hubbard model with narrow/wide band and SU(2)-symmetric Hund's exchange. The nature of this transition has been controversial, and we establish that an orbital-selective Mott-Hubbard transition exists. Thereby, the wide band still shows metallic behavior after the narrow band became insulating -not a pseudogap as for an Ising Hund's exchange. The coexistence of two solutions with metallic wide band and insulating or metallic narrow band indicates, in general, first-order transitions.
10.1103/physrevb.72.201102
[ "https://export.arxiv.org/pdf/cond-mat/0504040v2.pdf" ]
119,083,091
cond-mat/0504040
2fc6df993794872e59d9f72747d30ee765edd7ca
Orbital-selective Mott-Hubbard transition in the two-band Hubbard model 27 Jan 2006 (Dated: October 18, 2005) R Arita Max-Planck-Institut für Festkörperforschung 70569StuttgartGermany K Held Max-Planck-Institut für Festkörperforschung 70569StuttgartGermany Orbital-selective Mott-Hubbard transition in the two-band Hubbard model 27 Jan 2006 (Dated: October 18, 2005) Recent advances in the field of quantum Monte Carlo simulations for impurity problems allow -within dynamical mean field theory-for a more thorough investigation of the two-band Hubbard model with narrow/wide band and SU(2)-symmetric Hund's exchange. The nature of this transition has been controversial, and we establish that an orbital-selective Mott-Hubbard transition exists. Thereby, the wide band still shows metallic behavior after the narrow band became insulating -not a pseudogap as for an Ising Hund's exchange. The coexistence of two solutions with metallic wide band and insulating or metallic narrow band indicates, in general, first-order transitions. PACS numbers: 71.27.+a, 71.30.+h,71. 10.Fd By virtue of dynamical mean field theory (DMFT) [1,2], our understanding of the Mott-Hubbard transition [3] in the one-band Hubbard model has greatly improved in the last years. The bandwidth-controlled Mott-Hubbard transition is, at least within DMFT [2,4], of first-order at low temperatures (T ) and becomes a smooth crossover for temperatures above a critical point, which terminates the first-order line. A further complication arises exactly at zero temperature where two solutions coexist like for low T s. But at T = 0, the insulating solution is always higher in energy than the metallic one, i.e., the insulating solution is metastable throughout the whole coexistence region. The DMFT Mott-Hubbard transition is of second order at T = 0 despite the coexistence of two solutions. For making contact with experiments, orbital realism has to be taken into account, e.g., within the merger of local density approximation and DMFT (LDA+DMFT approach [5]). In the case of transition metal oxides, typically either the three t 2g or the two e g bands cross the Fermi energy. At the very least, these orbitals should be included. For degenerate orbitals, the situation seems to be clear, at least within DMFT: there is a first-order Mott-Hubbard transition [6]. Most transition metal oxides are, however, non-cubic. Hence, the orbital degeneracy is lifted. Take, for example, the unconventional superconductor Sr 2 RuO 4 [7] which has a wide d xy band and narrow d xz,zy bands [8] and which becomes a Mott-Hubbard insulator upon substituting Sr by Ca [9]. For such a situation with wide and narrow bands the details of the Mott-Hubbard transitions are so far inconclusive, even within DMFT and even for a simple two-band Hubbard model with Coulomb interaction U and Hund's exchange J between the two bands: Koga et al. [10] employed the so-called exact diagonalization (ED) method to solve the DMFT equations and obtain two Mott-Hubbard transitions: first the narrow band becomes insulating, then the wide band. This scenario has been coined orbital-selective Mott-Hubbard transition [11]. In contrast, Liebsch [12] uses quantum Monte Carlo (QMC) simulations and the iterated perturbation theory (IPT) to solve the DMFT equations and finds a single first-order Mott-Hubbard transition with similar changes in both bands. On the insulating side, the wide band has a pseudogap which gradually amplifies to a real gap with increasing U . In principle, the QMC is more suitable for addressing the Mott-Hubbard transition since ED only gives discrete peaks in the spectra, making it difficult to unambiguously identify a gap. However, the QMC simulations are restricted to relatively high temperatures and there is a sign-problem [13] if the Hund's exchange coupling is taken into account in full, i.e., not only the Ising but also the spin-flip component. Since the same limitations as in [12] also apply to all previous LDA+DMFT(QMC) calculations [5], there is an urgent need to clarify whether and how the details of the Mott-Hubbard transition are affected. Another important aspect is whether two solutions coexist. Liebsch finds two coexisting solutions at a single transition, while Koga et al. [10] do not address this question. If there was a first-order transition two consecutive transitions might even be bridged into a single one. In this paper, we study this transition by employing the most recent advances in the field of QMC simulations for DMFT. The recently introduced projective QMC (PQMC) method [14] enables us to address T = 0. Furthermore, the new Hubbard-Stratonovich decoupling of [15] allows for the calculation with the full SU(2)symmetric Hund's exchange at a still-manageable signproblem, in particular in combination with PQMC. Model. Starting point is the two-band Hubbard model H = − 2 m=1 t m i,j σĉ † imσĉjmσ (1) +U i mσn im↑nim↓ + i;σ<σ ′ (U ′ −δ σσ ′ J)n i1σni2σ ′ − J 2 iσ;l =m c † ilσĉilσĉ † imσĉimσ − J 2 iσ;l =m c † ilσĉ † ilσĉimσĉimσ ≡H2 . Here,ĉ † imσ andĉ imσ are creation and annihilation operators for electrons on site i within orbital m and with spin σ. The first line describes the kinetic energy for which we employ the semi-elliptic non-interacting density of states N 0 (ε) = 1 πW 2 m /8 (W m /2) 2 − ε 2 (orbital-dependent hopping amplitudes t m on a Bethe lattice). For the following calculations, we use different widths for the two bands: W 1 = 4 for the wide and W 2 = 2 for the narrow band as in [10,12]. Note that there is no hopping/hybridization between the two bands. The second line describes the intra-(U ) and inter-orbital (U ′ ) Coulomb interaction as well as the Ising-component of the Hund's exchange J (U ′ = U −2J by symmetry; we set J = U/4 as in [10,12]). The third line consists of the spin-flip contribution to the Hund's exchange (yielding together with the second line an SU(2)-symmetric contribution which can also be written as JS i1 S i2 , where S im denotes the spin for orbital m and site i). The last term represents a pair-hopping term of same strength J. Method. QMC calculations which take the spinflip component of Hund's exchange term into account have been a challenge. Although a straight-forward Hubbard-Stratonovich decoupling, exp(J∆τ c † 1 c 2 c † 3 c 4 ) = (1/2) s=±1 exp[s √ J∆τ (c † 1 c 2 −c † 3 c 4 )] , is possible, it has been recognized that it leads to a serious sign problem [13]. Therefore, it was neglected in almost every DMFT(QMC) calculation so far, including [12]. To overcome this problem, several attempts have been made [15,16,17]. Among these, Sakai, RA, and Aoki proposed a new discrete transformation for the spin-flip contribution of the exchange and pair-hopping term [15]: e −∆τ H2 = 1 2 r=±1 e λr(f ↑ −f ↓ ) e a(N ↑ +N ↓ )+bN ↑ N ↓ ,(2)Here, λ ≡ 1 2 log(e 2J∆τ + √ e 4J∆τ −1), a ≡ − log(cosh(λ)), b ≡ log(cosh(J∆τ )), f σ ≡ c † 1σ c 2σ +c † 2σ c 1σ , N σ ≡ n 1σ +n 2σ − 2n 1σ n 2σ . The advantage of this decoupling is that the auxiliary field r is real in contrast to that of [16]. Hence, it is expected to yield better statistics in general [15]. However, even with this decoupling, we note that the usual Hirsch-Fye QMC algorithm [18] does not work very well for DMFT calculation in the strong coupling regime or at low temperatures. For instance, for Hamiltonian (1) and J = U/4, we found it to be infeasible to obtain a self-consistent DMFT solution for U > 2.2 when β(= 1/T ) > 50 because the Green function G(τ ) has a large statistical error at τ ∼ β/2. Therefore, it is difficult to clarify without ambiguity whether an orbital selective Mott transition indeed occurs in multi-orbital systems at low T by means of finite-temperature Hirsch-Fye QMC calculations; also see [19]. Another recent advancement was the development of a new projective QMC (PQMC) algorithm by Feldbacher, KH, and Assaad [14]. In this algorithm, ground state expectation values Ψ 0 |O|Ψ 0 / Ψ 0 |Ψ 0 of an arbitrary operator O are calculated as: O 0 = lim θ→∞ lim β→∞ Tr e −βHT e −θH/2 Oe −θH/2 Tr e −βHT e −θH ,(3) where H T is an auxiliary Hamiltonian (its ground state |Ψ T is the trial wave function which is assumed to be non-orthogonal to the ground state |Ψ 0 of H [14]). For H T , it is convenient to take the one-body part of the Hamiltonian, because the limitβ → ∞ can be taken analytically in this case. Then, the starting point is the zero-temperature non-interacting Green function G 0 (τ, τ ′ ) truncated to 0 ≤ τ, τ ′ ≤ θ and discretized as an L × L matrix (L = θ/∆τ ). From this G 0 (τ, τ ′ ), the zerotemperature interacting Green function G(τ, τ ′ ) is obtained via the same updating equations for the auxiliary Hubbard-Stratonovich fields as for the finite-temperature Hirsch-Fye algorithm. While PQMC gives G(τ ) only for a limited number of (not too large) τ -points, we need G(iω) to solve the DMFT self-consistency cycle. To this end, the maximum entropy method (MEM) is employed to yield the spectral function A(ω) which allows for calculating G(iω n ) = dω A(ω) iωn−ω at any frequency iω n . This makes a crucial difference to finite-temperature calculations. The large statistical errors occurring at τ ∼ β/2 for finite temperatures now occur for rather large τ 's. But even if there is a large statistical error for larger τ 's, the maximum entropy method can extract sufficient information from the first τ points, discarding the larger τ 's with excessive statistical error. One of the main advantages of the PQMC method is that the convergence w.r.t. θ is much faster than that w.r.t. β in the Hirsch-Fye algorithm [14]. (Note that the calculation time increases cubically for θ and β.) Hence, we take in the following PQMC calculations a finite θ = 20 (L = 64), which should be sufficiently close to the T = 0 result: quantitatively small deviations are expected for larger θ's; qualitatively the behavior should not change anymore as in [14]. Similarly as in [14], the central L = 20 are for measurement and P = (L − L)/2 = 22 time slices on the right and left side of the measuring interval for projection. Typically, we performed 7 × 10 6 to 3 × 10 7 QMC sweeps. Results. An indicator for the Mott-Hubbard transition is the quasiparticle weight Z which is plotted in Fig. 1(a). We clearly see that for the narrow band Z = 0 for U ≥ 2.6, while Z is still finite for the wide band. This means that there is a first Mott-Hubbard transition in which only the narrow band becomes insulating at U ≈ 2.5. This is consistent with the result of the DMFT(ED) calculation of [10], in which the critical U c is estimated to be about 2.6. In contrast, there is a single first-order Mott-Hubbard transition at a smaller value is taken into account (at T = 0.03; between U c ≈ 1.8 and 2.1 there are two coexisting solutions/hysteresis) [12]. In our DMFT(PQMC) results, the wide band is still metallic at U = 2.7. But eventually, also the wide band has to become insulating at larger Coulomb interactions, since in the atomic limit both bands are insulating. (The calculation for larger Coulomb interactions unfortunately became computationally too expensive as even in the PQMC the statistical error brought about by the spin flip term of Hund's exchange increases dramatically.) Nonetheless, we can conclude from the data available that there are two different Mott-Hubbard transitions in which first the narrow and then the wide band become insulating. We have an orbital-selective Mott-Hubbard transition. In Fig. 1(b), the double occupancy D = n ↑ n ↓ for the two different bands is plotted as a function of U . We see that for the narrow band D ≈ 0.01 for U ≥ 2.6. A similar value of D ≈ 0.01 was reported [4] for the one-band Hubbard model above the Mott-Hubbard transition, i.e., for U/W ≥ 5.9/4. This suggests a Mott-Hubbard transition very similar to the one-band Hubbard model, albeit only for the narrow band. Final evidence for the orbital-selective Mott-Hubbard transition is obtained from the spectral functions shown in Fig. 2: We can unambiguously say that the wide band is still metallic at U = 2.6, whereas the narrow band is al- ready insulating with a pronounced gap. While the wide band shows a pseudo gap for an Ising-type of Hund's exchange [12], our SU(2) symmetric result reveals a metallic peak in Fig. 2. Let us now study the possibility of first-order Mott-Hubbard transitions. The first question is whether at U = 2.6 (where we find a metallic wide and insulating narrow band) a second solution in which both bands are insulating (co)exists. Starting the DMFT self-consistency cycle with an insulating self-energy for the wide-band, we obtain however the very same (single) solution as in Figs. 1 and 2. This demonstrates that the orbital-selective Mott-Hubbard transition is not merged into a single firstorder transition. There are two distinct Mott-Hubbard transitions. The second question is, Are the orbital-selective Mott-Hubbard transitions (generally) of first-order? In this case, two solutions should coexist for U 2.6: one with a metallic and one with an insulating narrow band. Special care is necessary for insulating solutions in the PQMC with a very narrow charge gap. For such small charge gaps, θ might not be sufficiently large to project -via e −θH/2 -from the metallic trial wave function onto the insulating ground state solution. We then note systematic errors even for intermediate τ 's, and substantial noise appears in the charge gap of the spectrum calculated with the maximum entropy method. This makes the stabilization of a small-gap insulating solution delicate. This problem can be mitigated however by doing the maximum entropy calculation with a reduced number of τ points. Therefore, we used τ points up to τ c ∼ 3.2 and ∼ 1.6 for the following results. For almost all values of U , only a metallic or only an insulating solution is obtained for both τ c ∼ 3.2 and ∼ 1.6. However, for U = 2.4, we find both a solution with a metallic and with an insulating narrow band (the wide band is metallic in both solutions with only minor differences). In Fig. 3, the spectral function of these two solutions are shown; the value of Z and D for the insulating solution is plotted in Fig. 1 as open circles and squares. The DMFT(PQMC) data suggest that two solutions with metallic and insulating narrow band coexists for U ∼ 2.4, so that the Mott-Hubbard transition in which the narrow band becomes insulating (and in which the wide band stays metallic) is in general of first-order. Possibly, the insulating solution is metastable at T = 0. Discussion. For understanding the DMFT results it is instructive to remind ourselves of what is known for the two-orbital Anderson impurity model (AIM). If J > T K (the AIM Kondo temperature) the impurity spins of the two orbitals form a steadfast spin-1 (triplet). For such an AIM and inequivalent orbitals it is known that this spin-1 is screened in two stages: first only by one orbital to a spin-1 2 at T 1 K , and then by the second orbital to a spin-0 at T 2 K [20]. Within DMFT we now have to solve AIMs self-consistently: The AIM's T K 's of one DMFT iteration (crudely T K ≈ ZW ) sets the hybridization strength for the next DMFT iteration. Hence, we can interpret our DMFT results as following: Given the two inequivalent Kondo scales of the AIM, there is a U -interval where only the hybridization strength (and T K ) of the narrow orbital is renormalized to zero by the DMFT iterations. Only the narrow band is insulating. If only the Ising-component of Hund's exchange is taken into account, the behavior of the AIM is completely different. Instead of a triplet, the impurity spins allign to S Z = ±1 (no S Z = 0 component). For J > T K (J ≈ 0.5 and T K ≈ ZW ≈ 0.4 at the Mott-Hubbard transition of [12]), there is no spin Kondo effect any more since it requires a spin-flip of the conduction electrons and, hence, a change of S Z by ±1. As soon as one orbital becomes insulating, there is also no orbital Kondo effect anymore: the whole system is unscreened, i.e., insulating. It is certainly interesting to study whether this kind of physics is relevant for magnetically anisotropic materials. Conclusion. Taking the full SU(2)-symmetry of Hund's exchange into account in the PQMC calculation, we conclude that there are two consecutive Mott-Hubbard transitions, whereby -at least around the first transition-two solutions coexist. By clarifying the the-oretical side, we hope to stimulate further experiments on the orbital-selective Mott-Hubbard transition, e.g., in Sr 2 RuO 4 where results were so-far negative in this respect [21]. We acknowledge very fruitful discussions with M. Feldbacher, S. Sakai, and A. Toschi as well as support by the Alexander von Humboldt foundation (RA) and the Emmy Noether program of the Deutsche Forschungsgemeinschaft (KH). During the completion of our work, we learned about several related studies [19,22]. FIG. 1 : 1U c ≈ 2.1 if only the Ising-component of Hund's exchange (Color online) (a) Quasiparticle weight Z and (b) double occupancy D as a function of U (J = U/4). Red (blue) squares (circles) are the data for the narrow (wide) band. For U = 2.4, two solutions are found: the wide band is metallic for both solutions whereas the narrow band is metallic (closed symbols) or insulating (open symbols). The solid triangle in (a) and (b) is the Uc estimate from DMFT(ED) [10]; the inset enlarges the behavior around the transition. FIG. 2 : 2(Color online) Spectral functions A(ω) for (a) the wide band and (b) the narrow band. For U = 2.6, the narrow band is insulating while the wide band is metallic. FIG. 3 : 3(Color online) Spectral functions A(ω) for (a) the wide and (b) the narrow band at U = 2.4. Two solutions with insulating/metallic narrow band coexist. . W Metzner, D Vollhardt, Phys. Rev. Lett. 62324W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989); . G Kotliar, D Vollhardt, Physics Today. 5753G. Kotliar and D. Vollhardt Physics Today 57, 53 (2004). . A Georges, Rev. Mod. Phys. 6813A. Georges et al., Rev. Mod. Phys. 68, 13 (1996). The Mott Metal-Insulator Transition. F Gebhard, SpringerBerlinF. Gebhard, The Mott Metal-Insulator Transition (Springer, Berlin, 1997); . M Imada, A Fujimori, Y Tokura, Rev. Mod. Phys. 701039M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. 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A Liebsch, Phys. Rev. Lett. 91165103Phys. Rev. BA. Liebsch, Phys. Rev. Lett. 91, 226401 (2003), Phys. Rev. B 70, 165103 (2004). Generally, the statistical error in QMC simulations becomes huge if positive and negative contributions almost compensate. K Held, D Vollhardt, Eur. Phys. J. B. 5473This is the so-called sign-problemK. Held and D. Vollhardt, Eur. Phys. J. B 5, 473 (1998). Generally, the statistical error in QMC simulations be- comes huge if positive and negative contributions almost compensate. This is the so-called sign-problem. . M Feldbacher, K Held, F F Assaad, Phys. Rev. Lett. 93136405M. Feldbacher, K. Held, and F. F. Assaad, Phys. Rev. Lett. 93, 136405 (2004). . S Sakai, R Arita, H Aoki, Phys. Rev. B. 70172504S. Sakai, R. Arita, and H. Aoki, Phys. Rev. B 70, 172504 (2004). . Y Motome, M Imada, J. Phys. Soc. Jpn. 663199Y. Motome and M. Imada, J. Phys. Soc. Jpn. 66, 1872 (1997); ibid 67, 3199 (1998). . J E Han, Phys. Rev. B. 7054513J. E. Han, Phys. Rev. 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[ "A Node-collaboration-informed Graph Convolutional Network for Precise Representation to Undirected Weighted Graphs", "A Node-collaboration-informed Graph Convolutional Network for Precise Representation to Undirected Weighted Graphs" ]
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An undirected weighted graph (UWG) is frequently adopted to describe the interactions among a solo set of nodes from real applications, such as the user contact frequency from a social network services system. A graph convolutional network (GCN) is widely adopted to perform representation learning to a UWG for subsequent pattern analysis tasks such as clustering or missing data estimation. However, existing GCNs mostly neglects the latent collaborative information hidden in its connected node pairs. To address this issue, this study proposes to model the node collaborations via a symmetric latent factor analysis model, and then regards it as a node-collaboration module for supplementing the collaboration loss in a GCN. Based on this idea, a Node-collaboration-informed Graph Convolutional Network (NGCN) is proposed with three-fold ideas: a) Learning latent collaborative information from the interaction of node pairs via a nodecollaboration module; b) Building the residual connection and weighted representation propagation to obtain high representation capacity; and c) Implementing the model optimization in an end-to-end fashion to achieve precise representation to the target UWG. Empirical studies on UWGs emerging from real applications demonstrate that owing to its efficient incorporation of node-collaborations, the proposed NGCN significantly outperforms state-of-the-art GCNs in addressing the task of missing weight estimation. Meanwhile, its good scalability ensures its compatibility with more advanced GCN extensions, which will be further investigated in our future studies.
10.48550/arxiv.2211.16689
[ "https://export.arxiv.org/pdf/2211.16689v1.pdf" ]
254,096,246
2211.16689
13022e15bc016c54248b1938a51d23266f8f5b9c
A Node-collaboration-informed Graph Convolutional Network for Precise Representation to Undirected Weighted Graphs A Node-collaboration-informed Graph Convolutional Network for Precise Representation to Undirected Weighted Graphs Index Terms-Undirected Weighted GraphCollaborative InformationGraph Convolutional NetworkWeight Estimation An undirected weighted graph (UWG) is frequently adopted to describe the interactions among a solo set of nodes from real applications, such as the user contact frequency from a social network services system. A graph convolutional network (GCN) is widely adopted to perform representation learning to a UWG for subsequent pattern analysis tasks such as clustering or missing data estimation. However, existing GCNs mostly neglects the latent collaborative information hidden in its connected node pairs. To address this issue, this study proposes to model the node collaborations via a symmetric latent factor analysis model, and then regards it as a node-collaboration module for supplementing the collaboration loss in a GCN. Based on this idea, a Node-collaboration-informed Graph Convolutional Network (NGCN) is proposed with three-fold ideas: a) Learning latent collaborative information from the interaction of node pairs via a nodecollaboration module; b) Building the residual connection and weighted representation propagation to obtain high representation capacity; and c) Implementing the model optimization in an end-to-end fashion to achieve precise representation to the target UWG. Empirical studies on UWGs emerging from real applications demonstrate that owing to its efficient incorporation of node-collaborations, the proposed NGCN significantly outperforms state-of-the-art GCNs in addressing the task of missing weight estimation. Meanwhile, its good scalability ensures its compatibility with more advanced GCN extensions, which will be further investigated in our future studies. I. INTRODUCTION ECENT years have witnessed the great success of graph convolutional network (GCN) [1] in representation learning for graph-structure data. Owing to its effectiveness in graph representation learning, a GCN has been applied to solve a plethora of real-world problems competently. For instance, James [8] proposes a novel spatial-temporal GCN, which adopts a spatial-temporal attention mechanism for traffic flow forecasting effectively. Yu et al. [9] propose an adversarial GCN, which adopts adversarial training for precise recommendation. Azadifar et al. [10] utilize a deep GCN for gene essentiality prediction. An undirected weighted graph (UWG) is frequently encountered graph-structure data in real applications [6,[11][12][13]15]. In particular, a UWG assigns a quantifiable index to the interaction between two nodes as its weight, which represents the correlation or strength of the interaction between a pair of nodes. For instance, the weight can describe the citation counts between authors [17] in citation network, the confidence of interactome between proteins [18] in a protein network, and the contact rate between users [19] in a social network. From this point of view, we clearly see that node-node interactions in a UWG contain massive useful knowledge to help us better understand inter-relationship among nodes. However, when addressing a UWG, a GCN heavily relies on node features without considering the latent collaborative information hidden in its connected node pairs, which is crucial for missing weight estimation [20]. Therefore, present empirical GCN show poor performance in missing weight estimation of a UWG, and is even outperformed by a linear representation learning model [21]. In line with the aforementioned discoveries, we have encountered the research question: RQ. Is it possible to incorporate collaborative information learned from the interaction of node pairs, thereby improving the GCN's performance on missing weight estimation? To answer this question, this study proposes a Collaborative-informed Graph Convolutional Network (NGCN), which can exploit the latent collaborative information from the interactions of node pairs effectively. Specifically, it consists of the following three-fold ideas: a) Learning latent collaborative information from the interaction of node pairs via a nodecollaboration module, b) Building the residual connection and weighted representation propagation to obtain high representation capacity, and c) Implementing the model optimization in an end-to-end fashion to achieve precise representation to the target UWG. This paper mainly contributes in the following perspectives: a) We propose an NGCN model. Different from popular GCNs that rely solely on smoothed node features, it is able to incorporate learnable collaborative information hidden in the interactions of node pairs to enhance the representation ability on a UWG, and b) The proposed NGCN are comprehensively investigated on extensive benchmark datasets demonstrate it outperforms the state-of-the-art baselines in missing weight estimation. Section 2 states the preliminaries. Section 3 presents the proposed NGCN. Section 4 conducts the empirical studies. Section 5 is the related work. Finally, Section 6 concludes this paper. II. PRELIMINARIES A. Problem Formulation Note that a UWG is defined as [25,26]: Definition 1: Let V={vi | i=1, 2, …, N} represent a set of N nodes, E={eij | vi, vj∈V} represent a set of edges where the nodes vi, vj∈V are connected. In particular, for a UWG G=(V, E), assigns a quantifiable index to the edge between two nodes as its weight. Hence, it is denoted the adjacency matrix A=[aij]∈R N×N , where aij is the non-zero weight if eij∈E and zero otherwise. Each node of G is described by a feature vector xi∈R f , where X∈R N×f is the collection of such vectors in a matrix form. The missing weight estimation for a UWG is defined as [21]: Definition 2: Given a UWG G=(V, E), a representation learning model learns meaningful representation hi∈R f for each node i by an iterative learning process, where H∈R N×f is the collection of such vectors in a matrix form. Subsequently, H is used for predicting the link weight of each link (i, j) as follows: ˆ= , , ij i j a h h (1) where <· ,· > denotes the inner product of two vectors, âij denotes the predicted weight. Note that we estimate the missing weight rather than the probability of link between two nodes. Thus, the non-liner activation function is omitted here [28,29]. B. Graph Convolutional Network As described in prior studies [30,31], a GCN learns low dimensional representations of nodes via aggregation of features from neighbors using non-linear transformations. Its forward propagation for node representation is defined as:     ( 1) 0.5 -0.5 ( ) ( ) + l l l H D A I D H W   (2) where H (0) is the node feature matrix X, H (l) denotes the representation of nodes at l-th layer, H (l+1) denotes the output representation after stacking l GCN layers, W (l) ∈R d(l)×d(l+1) is a trainable weight matrix, I denotes the identity matrix, D denotes the degree matrix of (A+I), σ(•) denotes the activation function. III. METHODS We now introduce the proposed NGCN model. It consists of the following three main components: a) The residual and weighted representation propagation module employs the residual connection and weighted representation propagation into GCN to offer better node representation capacity, b) The collaborative information learning module extracts the useful collaborative information from the interactions of node pairs via a symmetric latent factor analysis model, and c) The estimation module combines collaborative information and node representation to estimate the missing weight. A. Weight Representation Propagation Module Given a UWG in Definition 1, this module takes the node feature matrix X and weighted adjacency matrix A as the input. Following with the forward propagation process in (2), we have that: (0) =, HX (3a)   ( 1) 0.5 0.5 ( ) ( ) ReLU , l l l H D AD H W     (3b) ( 1) ( 1) ( ) = + , lll HHH  (3c) where we adopt ReLU as the activation function in this module,   1 l better representation capacity. Moreover, this operation considers the effect of self-connection. Hence, the self-connection in (3b) is neglected to avoid the numerical instabilities of representation scale. In order to better understanding (3a)-(3c), we reformulated them into a vector form. Thus, for node i, we have that:   0 =, ii hx (4a)       1 () 1 ReLU , ll l i ij j j N i ii jj h a h W dd             (4b)       +1 1 = + , l l l i i i h h h  (4c) where N(i) denotes the directly-connected neighbor set of node i, dii is the weighted diagonal degree of node i,   1 l i h  ,   1 l i h  and   i l h denote the i-th row vector corresponding to   1 l H  ,   1 l H  and   l H , respectively. Note that the task of this study is missing weight estimation. Hence, (4a)-(4c) denotes a weighted representation propagation process, which is different from that of traditional GCN designed for node classification and link estimation in the following aspects: a) For (4a), the node feature x i is randomly initialized vector and needs to be trained; and b) For (4b), d ii denotes sum of weights between node i and its directly-connected neighbors instead of computing its number of neighbor |N(i)|. B. Collaborative Information Learning Module As aforementioned, the proposed NGCN is able to make use of the latent collaborative information from the interactions of node pair, which can be learned outside of GCN. Next, we propose a simple but effective way for NGCN to capture latent collaborative information. Let S=[s ij ]∈R N×N denote the collaborative information matrix, where its element s ij is the collaborative information between nodes i and j in a UWG. It can be achieved with the following generalized generating function:       arg min , , ij ij ij ij S s a S    (5) where Φ(· ) denotes a distance metric, ψ(·) denotes an operator to construct the approximation of A=[a ij ]. In this study, we adopt symmetric latent factor analysis (SLFA) [21][22][23][24] to achieve this purpose. The main reason is that SLFA can extract latent collaborative information from the interactions of node pair accurately since it considers the intrinsic symmetry property of the target UWG. Following the principle of SLFA, let Φ(·) be the Euclidean distance-based metric and ψ(S) ij =(YY T ) ij , we have that:       T 2 T arg min , ij ij ij ij YY s a YY (6) where we use a latent matrix Y∈R N×d to approximate S to reduce the computational burden, d denotes the latent dimension of Y. Note that sij is encoded via (YY T )ij and it indicates that a higher collaboration level sij learned by (6) means that nodes i and j are more similar. It could lead to better node representation in GCN. C. Estimation Module Based on the inferences given in Sections 3.1 and 3.2, NGCN combines collaborative information and node representation to estimate the missing weight. NGCN computes the link weight â ij as follows:   = , 1 , , ij i j i j a h h y y     (7) where yi and yj denote the i-th and j-th row vector of Y, ω is trainable parameter to balance the importance of collaborative information and node representation. In order to inject the collaborative information in an end-to-end fashion, we jointly train NGCN by considering the loss of each module. In this study, we utilize the commonly-adopted Euclidean distance to design the objective function: where Λ denotes the training dataset, λ denotes the regularization coefficient to control the L2 regularization strength to prevent overfitting, Θ={ω, X, Y, {W} L l=1 } denotes all trainable model parameters. We adopt mini-batch Adam [33] to optimize the estimation model and update the model parameters IV. EMPIRICAL STUDIES A. General Settings Evaluation Protocol. This paper concerns the missing weight estimation for a UWG. Hence, we adopt the estimation accuracy as the evaluation protocol [4,[34][35][36]38,39]. Commonly, root mean squared error (RMSE) and mean absolute error (MAE) [5,7,32,37,40,41] are commonly adopted to measure a model's estimation accuracy: 2ˆ , | | ij ij ij ij ij ij abs aa RMSE a a MAE a a where |•| calculates the cardinality of an enclosed set, |•|abs denotes the absolute value of an enclosed number, and Γ denotes testing dataset. Datasets. Four UWG datasets [42] are adopted in our experiments, whose details are summarized in Table 1. Note that we randomly split the known edges set of each UWG dataset into ten disjoint and equally-sized subsets, where seven subsets are chosen as the training set, one as the validation set, and the remaining two as the testing set for 70%-10%-20% train-validationtest setting. The above process is sequentially repeated five times for five-fold cross-validation. Baselines. To demonstrate the effectiveness, we compare our proposed NGCN with the state-of-the-art models for missing weight estimation. Table 2 records all the compared models. Table 2: Details of compared models. No. Model Description M1 MF A matrix factorization-based model [43]. M2 NeuMF A deep neural network-based MF model [44]. M3 GCN A standard graph convolutional network [1]. M4 LR-GCCF A linear residual GCN [45]. M5 DGCN_HN A deep GCN with hybrid normalization [46]. M6 GCMC A graph auto-encoder network [47]. M7 LightGCN A Simplifying GCN [48]. M8 NGCN The proposed NGCN of this study. Training Settings. For achieving the objective results, following general settings are applied to all involved models: a) All the compared models are deployed on a GPU platform with two NVIDIA GeForce RTX 3090 GPU cards; b) We adopt an Adam optimizer and the batch-size is set as 2048; c) The termination conditions are consistent for all the compared models, i.e., the iteration threshold is 1000, or the training will be terminated after the minimum error of 30 iterations; d) The node feature matrix of each model are initialized with the same randomly generated arrays, and the dimension of node feature f=128; e) For the proposed NGCN, the latent matrix's latent dimension d=128, the number of propagation layers L=2; f) For all the compared models, we apply a grid search for the learning rate η={0.00005, 0.0001, 0.0005, 0.001, 0.005, 0.01, 0.05} and L2 regularization coefficient λ={0.00001, 0.00005, 0.0001, 0.0005, 0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1} to achieve the optimal results. B. Comparison Performance We start by comparing the performance of all the methods. Table 3 show the comparison results on RMSE/MAE. From them, we achieve some important verdicts: a) NGCN possesses remarkable advantage in missing weight estimation of a UWG. As depicted in Table 3, M8, i.e., the proposed NGCN, achieves the lowest estimation errors on seven testing cases out of eight in total. For instance, on D1, M8 achieves the optimal RMSE at 0.03520, which is 14.63% lower than 0.04123 achieved by M1, 8.19% lower than 0.03834 achieved by M2, 8.52% lower than 0.03848 achieved by M3,12.57% lower than 0.04026 achieved by M4, 10.64% lower than 0.03939 achieved by M5, 7.81% lower than 0.03818 achieved by M6, and 8.74% lower than 0.03857 achieved by M7. Similar outcomes are also found on MAE. The situation is a little different on RMSE of D2. For this case, M8 is only slightly outperformed by M7. One possible interpretation is that D2 contains valuable linear characteristics. In general, M8's representation ability to the target UWG is impressive. The main reason is that M8 incorporates the information of neighbors by the weighted representation forward propagation process, thereby improving the representation learning to the target UWG. Moreover, compared with the GCN-based models, i.e., M3-M7, M8 not only utilizes the information of neighbors, but also captures collaborative information from the interactions of node pairs, which is crucial for missing weight estimation. Hence, M8 still outperforms them when estimating the missing weight. b) NGCN's performance gain has statistical significance across our experiments. To illustrate this point, the statistical analyses, i.e., the Friedman test and Wilcoxon signed-rank test, are conducted. Firstly, the Friedman test is commonly adopted to validate multiple models' performance on multiple datasets. The Friedman statistical results are recorded in Table 4, where it accepts the hypothesis that involved models have significant differences with a significance level of 0.05. From it, we clearly see that M8, i.e., the proposed NGCN, achieves the lowest Rank value, which means it outperforms other compared models in terms of estimation accuracy. In addition, the Wilcoxon signed-ranks test is effective in checking whether NGCN has a significantly better performance than each compared model. Note that three indicators are adopted in the Wilcoxon signed-ranks test, where large R+ value denotes performance gain, large R-denotes performance loss, and small p-value denotes high significance level. Table 5 records the corresponding results, which demonstrates that NGCN's estimation accuracy for missing weight of a UWG is significantly higher than that of its peers. V. RELATED WORK To date, diverse GCNs [30,31,49] have been proposed. For instance, Abu-El-Haija et al. [30] propose a mixhop GCN, which repeatedly mixes feature representations of neighbors at various distances. Pei et al. [31] propose a geometric GCN, which maps a graph to a continuous latent space via node representations, and then use the geometric relationships defined in the latent space to build structural neighborhoods for aggregation. Bo et al. [49] propose a deep structural GCN, which combines multiple structures from low-order to high-order. Fatemi et al. [16] propose a self-supervision GCN, which can provide more supervision for inferring a graph structure through self-supervision. Due to the iterative aggregation step, these noteworthy GCNs can achieve node representations for various downstream tasks. Obviously, estimating the missing weight of a UWG [2,[20][21][22][23]] is a typical downstream task, which is focused in this paper. However, when performing missing weight estimation, the existing GCNs neglect the latent collaborative information hidden in the interactions of node pairs, which is crucial for missing weight estimation. According to prior researches [3,14,[21][22][23][24]27], SLFA can learn the latent collaborative information from the concerned interaction of node pair, thereby implementing a precise representation of symmetry topology and numerical characteristics of a UWG. For instance, Li et al. [21] propose a second-order SLFA model, which designs an efficient second-order learning algorithm to achieve collaborative information with affordable computational burden. Luo et al. [22] propose an Alternating-Direction-Method of Multipliers (ADMM)-based SLFA model, which incorporates the ADMM principle into its learning scheme for fast and accurate extraction of collaborative information. Song et al. [23] propose an improved SLFA model, which adopts a triple factorization technique to achieve collaborative information with high precision. VI. CONCLUSIONS In this paper, we propose a Collaborative-informed Graph Convolutional Network (NGCN) that learns latent collaborative information from the interactions of node pairs outside of the GNN. It offers flexible incorporation of both node features and latent collaborative information of node pairs, thereby achieving significant performance improvements on missing weight estimation of a UWG. In future, we will further improve the effectiveness of NGCN by incorporating attention mechanism to measure the importance of latent collaborative information for different node-pairs. Moreover, we plan to theoretically analyze why latent collaborative information of node pairs can help GCN break the limitation of expressive power, whose upper-bounded is proven by the 1-Weisfeiler-Lehman (1-WL) graph isomorphism test. Ying Wang, Ye Yuan, Xin Luo R  Y. Wang is with the School of Computer Science and Technology, Chongqing University of Posts and Telecommunications, Chongqing 400065, China, and also with the Chongqing Key Laboratory of Big Data and Intelligent Computing, Chongqing Engineering Research Center of Big Data Application for Smart Cities, and Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing 400714, China (e-mail: [email protected]).  Y. Yuan and X. Luo are with the College of Computer and Information Science, Southwest University, Chongqing 400715, China (e-mail: [email protected], [email protected]). Table 1 : 1Experimental dataset details.No. Name Edges Nodes Density D1 plantsmargin_12NN 25482 1600 1.00% D2 yaleB_10NN 34808 2414 0.60% D3 MISKnowledgeMap 57022 2427 0.96% D4 yeast_30NN 63250 1484 2.87% Table 3 : 3The comparison results on RMSE/MAE, where ✪ indicates that NGCN is outperformed by the compared models.Case M1 M2 M3 M4 M5 M6 M7 M8 D1 RMSE 0.04123 ±5.E-5 0.03834 ±2.E-4 0.03848 ±2.E-4 0.04026 ±2.E-3 0.03939 ±3.E-4 0.03818 ±1.E-4 0.03857 ±5E-5 0.03520±8.E-5 MAE 0.02312 ±1.E-4 0.02317 ±5.E-4 0.02320 ±8.E-5 0.02298 ±9.E-5 0.02171 ±3.E-5 0.02280 ±4.E-4 0.02168 ±1.E-5 0.02068±2.E-4 D2 RMSE 0.07388 ±1.E-4 0.07029 ±5.E-4 0.06921 ±8.E-4 0.07205 ±8.E-4 0.07046 ±1.E-5 ✪0.06796±4.E-4 0.06912 ±8.E-5 0.06873 ±2.E-4 MAE 0.03445 ±4.E-5 0.03508 ±6.E-4 0.03504 ±9.E-4 0.03774 ±6.E-4 0.03336 ±3.E-5 0.03595 ±5.E-4 0.03341 ±5.E-5 0.03191±8.E-5 D3 RMSE 0.07355 ±7.E-5 0.06121 ±2.E-4 0.07339 ±6.E-4 0.09000 ±2.E-3 0.07453 ±2.E-4 0.06238 ±2.E-4 0.07614 ±3.E-4 0.06050±4.E-4 MAE 0.05030 ±1.E-4 0.04527 ±5.E-4 0.05564 ±6.E-4 0.06150 ±1.E-3 0.05062 ±2.E-5 0.04651 ±3.E-4 0.05030 ±4.E-5 0.04368±5.E-4 D4 RMSE 0.08809 ±1.E-4 0.08619 ±2.E-4 0.08576 ±1.E-3 0.08716 ±3.E-4 0.08203 ±4.E-5 0.08231 ±7.E-4 0.08141 ±5.E-5 0.07323±2.E-4 MAE 0.04523 ±4E-5 0.05366 ±8.E-4 0.05181 ±9.E-4 0.05046 ±4.E-4 0.04227 ±5.E-5 0.04796 ±4.E-4 0.04224 ±3.E-5 0.04028±2.E-4 ✪Loss/Win 0/8 0/8 0/8 0/8 0/8 1/7 0/8 - Compared with the MF-based models, i.e., M1 and M2, M8 show significant accuracy improvements. 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[ "Electronic Structure Calculations using Quantum Computing", "Electronic Structure Calculations using Quantum Computing" ]
[ "Nouhaila Innan [email protected]†[email protected] \nFaculty of Sciences Ben M'sick\nQuantum Physics and Magnetism Team, LPMC\nHassan II University of Casablanca\nMorocco\n\nQuantum Formalism Fellow\nZaiku Group Ltd\nLiverpoolUnited Kingdom\n", "Muhammad Al-Zafar ", "Khan \nQuantum Formalism Fellow\nZaiku Group Ltd\nLiverpoolUnited Kingdom\n\nRobotics\nSchool of Computer Science and Applied Mathematics\nAutonomous Intelligence, and Learning Laboratory (RAIL)\nUniversity of the Witwatersrand\n1 Jan Smuts Ave2000Braamfontein, GautengJohannesburgSouth Africa\n", "Mohamed Bennai \nFaculty of Sciences Ben M'sick\nQuantum Physics and Magnetism Team, LPMC\nHassan II University of Casablanca\nMorocco\n" ]
[ "Faculty of Sciences Ben M'sick\nQuantum Physics and Magnetism Team, LPMC\nHassan II University of Casablanca\nMorocco", "Quantum Formalism Fellow\nZaiku Group Ltd\nLiverpoolUnited Kingdom", "Quantum Formalism Fellow\nZaiku Group Ltd\nLiverpoolUnited Kingdom", "Robotics\nSchool of Computer Science and Applied Mathematics\nAutonomous Intelligence, and Learning Laboratory (RAIL)\nUniversity of the Witwatersrand\n1 Jan Smuts Ave2000Braamfontein, GautengJohannesburgSouth Africa", "Faculty of Sciences Ben M'sick\nQuantum Physics and Magnetism Team, LPMC\nHassan II University of Casablanca\nMorocco" ]
[]
The computation of electronic structure properties at the quantum level is a crucial aspect of modern physics research. However, conventional methods can be computationally demanding for larger, more complex systems. To address this issue, we present a hybrid Classical-Quantum computational procedure that uses the Variational Quantum Eigensolver (VQE) algorithm. By mapping the quantum system to a set of qubits and utilising a quantum circuit to prepare the ground state wavefunction, our algorithm offers a streamlined process requiring fewer computational resources than classical methods. Our algorithm demonstrated similar accuracy in rigorous comparisons with conventional electronic structure methods, such as Density Functional Theory and Hartree-Fock Theory, on a range of molecules while utilising significantly fewer resources. These results indicate the potential of the algorithm to expedite the development of new materials and technologies. This work paves the way for overcoming the computational challenges of electronic structure calculations. It demonstrates the transformative impact of quantum computing on advancing our understanding of complex quantum systems.(2)where F is the Fock operator consisting of the external potential energy, V (r), of nuclei; the Coulomb operator, J (r), which describes the interaction between electrons over the volume distribution Ω with charge distribution ρ; K(r) is the exchange operator which accounts for the antisymmetry of the electron wavefunction; i is the energy of the i th electron; and Ψ i (r)is the molecular orbital of the i th electron. Typically, the Fock operator is expressed as the density matrix in Eq.(3):
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[ "https://export.arxiv.org/pdf/2305.07902v1.pdf" ]
258,685,534
2305.07902
ecaafc85b85c744c5d8757d39a7aecc9054965e5
Electronic Structure Calculations using Quantum Computing 13 May 2023 Nouhaila Innan [email protected]†[email protected] Faculty of Sciences Ben M'sick Quantum Physics and Magnetism Team, LPMC Hassan II University of Casablanca Morocco Quantum Formalism Fellow Zaiku Group Ltd LiverpoolUnited Kingdom Muhammad Al-Zafar Khan Quantum Formalism Fellow Zaiku Group Ltd LiverpoolUnited Kingdom Robotics School of Computer Science and Applied Mathematics Autonomous Intelligence, and Learning Laboratory (RAIL) University of the Witwatersrand 1 Jan Smuts Ave2000Braamfontein, GautengJohannesburgSouth Africa Mohamed Bennai Faculty of Sciences Ben M'sick Quantum Physics and Magnetism Team, LPMC Hassan II University of Casablanca Morocco Electronic Structure Calculations using Quantum Computing 13 May 20231Electronic Structure CalculationsQuantum ComputingQuantum AlgorithmVaria- tional Quantum Eigensolver * The computation of electronic structure properties at the quantum level is a crucial aspect of modern physics research. However, conventional methods can be computationally demanding for larger, more complex systems. To address this issue, we present a hybrid Classical-Quantum computational procedure that uses the Variational Quantum Eigensolver (VQE) algorithm. By mapping the quantum system to a set of qubits and utilising a quantum circuit to prepare the ground state wavefunction, our algorithm offers a streamlined process requiring fewer computational resources than classical methods. Our algorithm demonstrated similar accuracy in rigorous comparisons with conventional electronic structure methods, such as Density Functional Theory and Hartree-Fock Theory, on a range of molecules while utilising significantly fewer resources. These results indicate the potential of the algorithm to expedite the development of new materials and technologies. This work paves the way for overcoming the computational challenges of electronic structure calculations. It demonstrates the transformative impact of quantum computing on advancing our understanding of complex quantum systems.(2)where F is the Fock operator consisting of the external potential energy, V (r), of nuclei; the Coulomb operator, J (r), which describes the interaction between electrons over the volume distribution Ω with charge distribution ρ; K(r) is the exchange operator which accounts for the antisymmetry of the electron wavefunction; i is the energy of the i th electron; and Ψ i (r)is the molecular orbital of the i th electron. Typically, the Fock operator is expressed as the density matrix in Eq.(3): I. INTRODUCTION Atoms are the constituent building blocks of all organic and inorganic organisms. Many atoms group together to form molecules, and arrangements of these molecules in certain configurations give rise to the complex diversity of nature and, by extension, to inanimate objects. Electronic structure calculations are used to determine the properties and behaviours of these atoms and molecules. Since these calculations are performed at the atomic scales, an intrinsic paradigm and framework for determining these properties is quantum mechanics, specifically via the generalised, time-dependent, Schrödinger Eq. (1): ı ∂Ψ(r, t) ∂t = − 2 2m ∇ · ∇Ψ(r, t) + V (r, t)Ψ(r, t) ⇐⇒Ê |ψ(r, t) =Ĥ |Ψ(r, t) , where Ψ(r, t) is the wavefunction that is dependent on three-dimensional space r and time t, V (r, t) is the potential energy function, ≈ 1.054×10 −34 J.s is the reduced Planck constant, m is the particle mass, and ı = √ −1. The equivalency relation renders the generalised Schrödinger equation in terms of the energy operator:Ê ∆ = ı ∂ ∂t , and the Hamiltonian operator:Ĥ ∆ = − 2 2m ∇ 2 + V (r, t) . The Schrödinger equation is a nonlinear partial differential equation (PDE) that does not have an explicit solution for general potential energies. Thus, the need to approximate this solution has arisen. Within the context of electronic structure calculations, the most famous "self-consistent" field method introduced by Hartree, 1928 and by Fock, 1930 known famously as "Hartree-Fock theory" (HF), is used to approximate the Schrödinger equation for many-electron systems via Eq. (2): (F − i ) Ψ i (r) = 0, F = − 1 2 ∇ 2 + V (r) + J (r) − K(r) , J (r) = Ω ρ(r ) |r − r | d 3 r . Density Functional Theory (DFT), introduced by Hohenberg & Kohn, 1964, makes use of the Kohn-Sham equations to determine the electronic structure of molecules and works exceptionally well in solids. By theoretically demonstrating via the Hohenberg-Kohn theorem that the ground state electron density of a system is a unique functional of the external potential, and vice versa, the total energy of the system, and all other properties, can be inferred. Mathematically, these ideas are encapsulated in Eq. (4): − 1 2 ∇ 2 + V (r) ρ(r) = i ρ(r),(4) where ρ(r) serves as the electron density, V (r) is the effective potential, and i is the energy of the i th electron. Modern DFT builds upon the work of Hohenberg & Kohn, 1964 and has become a fundamental tool in the arsenal of the contemporary Computational Chemist and Materials Scientist when determining the electronic structure of molecules and solids. Many-body Perturbation Theory (MBPT), introduced by Luttinger & Ward, 1960, and expanded upon by Baym, 1962 andHedin, 1965, built upon the work of Schwinger, 1951 and Matsubara, 1955, combines perturbation theory with Green's functions to calculate the electronic structure of many-electron systems. Mathematically, the ideas of MBPT are encapsulated by the Dyson equation (Dyson, 1949) as expressed in Eq. (5): G(r , r) = G 0 (r , r) + G 0 (r , r)ΣG(r , r), where G 0 is the non-interacting Green's function, G is the full Green's function, and Σ is the self-energy -which serves as a summation over all particle-hole excitations expressed as Eq. (6): Σ = −V d 3 r d 3 r v(|r − r |)X (r, r )G 0 (r , r), X (r, r ) = − 1 V ∂ 2 E ∂F 2 ,(6) where V is the system volume, E is the total energy of the system, v(|r − r |) is the Coulomb interaction potential between particles, X (r, r ) is the rank-2 polarisability tensor that relates the induced dipole moment to the strength and direction of the external electric field, and F is the strength of the external electric field. the technique is expressed in terms of the exponential approximation as Eq. (7): |Ψ CC = e T |Φ 0 ,(7) where |Ψ CC is the CC wavefunction, |Φ 0 is the Hartree-Fock reference state which defines single-particle orbitals and occupation numbers, and T is the cluster operator expressed as Eq. (8): T = i T i ,(8) where T i is the i th -fold excitation and the summation runs over all excitations. Configuration Interactions (CI) is conceptually the simplest method for solving the timeindependent Schrödinger equation under the Born-Oppenheimer approximation (Sherrill & Schaefer III, 1999). The method constructs a trial wavefunction as a linear combination of Slater determinants that correspond to different electron configurations of the system under examination. Using combinatorics, the Slater determinants are constructed by apportioning the electrons to the available orbitals. The CI method is broken up into two sub-methods: The full CI (FCI) method, which considers all possible configurations, and the truncated CI (TCI) method, which considers subsets of the configurations. The wavefunction for the CI method is constructed using Eq. (9): Ψ = n i=0 c i φ i ,(9) where c i are coefficients of the Slater determinants φ i , for 0 ≤ i ≤ n. By solving a system of linear equations, the coefficients can be obtained. Once the coefficients are determined, potential, is described by the Kohn-Sham equations' time-dependent, coupled PDE system. This approach is widely adopted in the studies of theoretical spectroscopy (Besley & Asmuruf, 2010), photochemistry (Matsuzawa et al., 2001), and energy transference. The goal is to approximate the solutions to the time-dependent Schrödinger equation (1) above. Analogously, the time-dependent DFT equation is given by Eq. (11): Eρ(r, t) = Ĥ , ρ(r, t) ,(11) where ρ(r, t) is the time-dependent electron density. The corresponding time-dependent Kohn-Sham equation is expressed as Eq. (12): EΨ i (r, t) = Ĥ KS (t) +Σ(r, t) Ψ i (r, t).(12) whereĤ KS (t) is the time-dependent Kohn-Sham Hamiltonian given by Eq. (13): H KS (t) = − 1 2 n i=1 ∇ 2 i + ρ(r, t) r − r dr + v ext (r, t)ρ(r, t) dr + v Hxc (r, t)ρ(r, t) dr + v xc δρ(r, t) dr,(13) where v ext (r, t) is the time-dependent external potential, v Hxc (r, t) is the time-dependent Hartree potential, v xc (r, t) is the time-dependent exchange-correlation potential, and δρ(r, t) is the deviation of the energy density from the ground state. The exchange-correlation selfenergy operator,Σ(r, t) is given by Eq. (14): Σ(r, t) = d 3 r dt δE xc [ρ] δρ(r , t ) δ(r − r )δ(t − t ),(14) where E xc [ρ] is the exchange-correlation energy functional; δExc[ρ] δρ(r ,t ) is the functional derivative of the exchange-correlation energy with respect to the energy density; δ(r − r ), and δ(t − t ) are the spatial and temporal Dirac delta functions. As one may gauge, TDDFT is highly accurate in describing the ground state energies of many-electron systems; it is mathematically laborious to implement and computationally Method Uses Computational Complexity Hartree-Fock Theory • Electron correlation effects. • Closed-shell systems. • Systems with a large number of electrons. • Used to calculate electronic structure, stability, and reactivity. Many-Body Perturbation Theory • Used extensively in condensed matter physics. These properties can be exploited for a plethora of real-world applications. We discuss some of these implementations below. While these classical methods have been highly successful for decades, and have numerous applications, as discussed above, they possess many drawbacks. We discuss them below. as an alternative method that can be applied as a stand-alone method or in parallel with a classical method. O(N k × n m e − × n p v ), where N is The field of QC is rapidly advancing and delving into the potential of quantum mechanics to process information in ways that classical computers are limited in their ability to achieve. Originally introduced by Feynman, 1982, who catechised the idea of whether a computer could simulate quantum systems based on the fundamental principles of quantum mechanics. This paved the way for the idea of qubits -the amalgamation of "quantum" and "bits", that can exist in a superposition of states and can be entangled with one another. Unlike classical bits that can only exist in one of two states, qubits can exist in a continuum of states, providing QC with the ability to perform certain computations faster than classical computers. One of the most promising QC technologies is based on superconducting circuits that create and manipulate qubits. These circuits operate using microwave signals and are cooled to temperatures close to absolute zero. QC has emerged as a promising paradigm for various applications, from Cryptography = i µ i P i , P i ∈ {X, Y, Z} ,(15) where µ i are coefficients, and P i are the Pauli matrices {X, Y, Z}. Using a Parameterised Quantum Circuit (PQC), U (θ), a trial state, |Ψ(θ) is prepared as Eq. (16): |Ψ(θ) = U (θ) |0 ⊗n .(16) The energy of the system, E, given by Eq. (17): E(θ) = Ψ(θ)|Ĥ |Ψ(θ) θ ,(17) is used to determine the minimum energy of the system, E min , given by Eq. (18): E min = min θ E(θ).(18) The parameters are then optimised using a classical optimiser such as Gradient Descent : As we have discussed the limitations of classical methods used for electronic structure calculations, this research aims to investigate the potential of quantum computing, specifically focusing on the VQE algorithm, to address these challenges and limitations. We believe that the usage of QC for electronic structure calculations offers several advantages over the classical methods; we delineate these below: E 0 = min i µ i .(19) 1. Accurate Modelling of the Behaviour of the Electrons in the System Under Consideration: Since QC is inherently concerned with quantum mechanics, the modelling of quantum mechanical systems in order to predict properties is a more natural and well-furnished method, and can therefore mitigate the inaccuracies of the classical approximation methods. Has the Potential to Offer Exponential Speed-up Over Classical Methods: Since quantum computers have the potential to offer significant speed-ups over classical computers -prospectively exponential; see for example Grover's algorithm for searching unsorted databases (Grover, 1996), and Shor's algorithm used in Cryptography for factoring large numbers and finding discrete logarithms (Shor, 1997) -quantum computers can be used for modelling and simulating larger electron systems. Expandability and Scalability of the Algorithms Used to Model and Sim- ulate the Systems Being Studied: Classical methods become computationally infeasible for large-electron systems, whereas quantum computers have the potential to scale as the molecular systems grow. Therefore, the applications in this paper are used to show that QC-inspired methods are easier to implement, and produce accurate results within a tolerable margin. This paper is structured as follows: In §II, we conduct a comprehensive review of prior research to provide a foundation for our study. In §III, we introduce the new algorithm based on the VQE architecture, and its steps for energy calculation. In §IV, we present our experimental procedures and findings, wherein we compare the performance of VQE against traditional methods such as DFT and HF and demonstrate its proficiency in computing the energies of diverse molecules. Finally, in §V, we summarise our findings and elucidate the potential of the VQE to transform the fields of Condensed Matter Physics, and Computational Chemistry. II. LITERATURE REVIEW Several researchers and groups have attempted to calculate atomic properties using QC. Below, we tabulate a non-exhaustive summary of quintessential research papers and provide descriptions of the fruition of their findings (III, IV, and V). Literature Summary Whitfield et al., 2011 This research highlights the drawbacks of current methods for simulating molecular systems, namely the computational complexity associated with modelling n-particle quantum systems. It is shown how pre-computed molecular integrals can be used to obtain the energy of the n-particle system using quantum phase estimation, the caveat being that such a system must have a fixed nuclear configuration. For the Hydrogen gas (H 2 ) molecule, the simulation of the chemical Hamiltonian is exhibited on a quantum computer. Carleo & Troyer, 2017 This landmark paper used perceptrons with varying hidden neurons to introduce a variational representation of quantum states for one-and two-dimensional spin models with interaction. The ground state was determined using a Reinforcement Learning-inspired scheme, and the unitary temporal evolution dynamics were described. properties. Nevertheless, the study also highlights certain challenges and constraints in scaling the algorithm for larger molecules. III. THEORY In order to perform electronic structure calculations of molecules on a quantum computer, we require the following steps: 1. Write down the Born-Oppenheimer Hamiltonian,Ĥ, for the molecule in terms of creation, a † i , and annihilation, a i , operators. Convert this Hamiltonian into matrix form by applying a suitable fermionic trans- formation that take the creation and annihilation operators into single-bit quantum gates. There are several methods of doing this, and Tab. VI outlines these techniques. We denote the Pauli spin matrices as X, Y, Z for the x, y, z unitary evolutions, respectively, and I for the 2 × 2 identity matrix. 3. Solve for the ground state, and excited states of the molecular system, The current state-of-the-art is the VQE method, which we will adopt in all subsequent calculations. Notwithstanding, the Phase Estimation Algorithm (PEA) -see Kitaev, 1995;Kitaev, 1997 -is also widely used. a † i −→ 1 2 {(X i − ıY i ) ⊗ i−1 j=1 Z j , a i −→ 1 2 (X i + ıY i ) ⊗ i−1 j=1 Z j . For a 2n-electron system, the computational complexity of this method is O(n). Binary code transformation (Steudtner & Wehner, 2018) a † i −→ 1 2 X U (i) ⊗ 1 + Z F (i) ⊗ Z P(i) , a i −→ 1 2 X U (i) ⊗ 1 − Z F (i) ⊗ Z P(i) , where U (i) is the qubit update set when the creation and annihilation operators are applied to orbital i, F (i) is a checking function that ascertains whether the creation and annihilation operators yield 0, and P(i) is a parity-check function that checks the phase change when the creation and annihilation operators are applied to spin orbital i. The computational complexity will be dependent on the value of U (i), F (i), and P(i). Parity transformation (Seeley et al., 2012) a † i −→ 1 2 ⊗ n j=1+1 X j ⊗ X i ⊗ Z i−1 − ı ⊗ n j=1+1 X j ⊗ Y i , a i −→ 1 2 ⊗ n j=1+1 X j ⊗ X i ⊗ Z i−1 + ı ⊗ n j=1+1 X j ⊗ Y i , for a 2n-electron system. The computational complexity of this method is O(n). Bravyi-Kitaev transformation 2015) a † i −→ 1 2 X U (i) ⊗ X i ⊗ Z P(i) − ıX U (i) ⊗ Y i ⊗ Z P(i) , a i −→ 1 2 X U (i) ⊗ X i ⊗ Z P(i) + ıX U (i) ⊗ Y i ⊗ Z P(i) , where U (i) and P(i) have their regular meanings. This method combines the Jordan-Wigner and Parity transformations, and for a 2n-electron system, it has a quasilinear computational complexity O(n log n). Below we computationalise the calculational steps for electronic structure calculations of molecular systems using quantum computers in the form of the algorithm 2 below. Algorithm 2 QElectra(Ψ 0 ) // This algorithm takes in a ground state, Ψ 0 , and returns the ground and excited state energies input basis functions χ i , χ j , χ k , χ l ; creation operators a † i , a † j , annihilation operators a j , a k , a l ; tolerance E ; perturbation factor λ initialise E 0 // initialising the ground state energy E 0 for each atom in the molecule do calculate the nuclear repulsion / Coulomb energy h 0 = 1 4πε 0 Natom i=1 Natom j=i+1 z i z j |r i − r j | calculate the one-electron operator F = − 1 2 i 1 ω i ∇ 2 i − A,i Z A r iA + i<j 1 r ij calculate the one-electron orbital integral h ij = χ i |F |χ j = +∞ −∞ χ * i (r)F (r)χ j (r) dr calculate the two-electron orbital integral h ijkl = +∞ −∞ +∞ −∞ χ i (r 1 )χ j (r 1 )χ k (r 2 )χ l (r 2 ) r 12 dr 1 dr 2 calculate the second quantisation Hamiltonian operator H = h 0 + i,j h ij a † i a j + i,j,k,l h ijkl a † i a † j a k a l end for map the Hamiltonian operatorĤ to single-bit quantum gates using a suitable transformation as described in Tab. VI Algorithm 2 QElectra (Ψ 0 ) -Part 2 while ||E i+1 0 − E i 0 || > E do calculate the ground state energy, and excited states using the VQE ansatz In algorithm 2 above, the first step entails calculating the nuclear binding energy h 0 . The variables in its associated formula are: ε 0 , the permittivity of free space; N atom , the number of atoms in the molecular system being studied; z i , z j , the atomic numbers of the i th and j th atoms respectively; and r i and r j , the position vectors of the i th and j th atoms respectively, in reference to a defined coordinate system. E 0 = Ψ 0 (θ)|Ĥ |Ψ 0 (θ) Ψ 0 (θ) |Ψ 0 (θ) | , E k = E 0 + kλ update θ i+1 ←− θ i Secondly, we calculate the one-electron operator,F . The variables associated with the formula are: ω i = κm e − , the effective mass of the i th electron (the rest mass of the electron, m e − , scaled by some factor κ); Z, the charge of the A th nucleus; r iA , the distance between the i th and A th nucleus; and r ij , the distance between the i th and j th nucleus. Thirdly, we calculate the one-electron orbital integral, h ij . The variables associated with the formula are: χ i and χ j , the i th and j th basis functions respectively. In the fourth step, we calculate the two-electron integral, h ijkl , and the variables associated with this formula have the same meaning as the variables above. In the sixth step, we use the previously calculated variables in order to compute the Hamiltonian operator for the system,Ĥ. Lastly, whilst the successive ground state energies are greater than some tolerance threshold E 0 in each iteration, the VQE approximation method is used to calculate the ground state energy E 0 , and the energy of the k th excited state, E k , for some perturbation factor λ ∈ [0, 1], with λ 0.1 for small perturbations, and λ 1 for large perturbations. Finally, once the stopping criterion is met, i.e. when the difference between successive ground state energies is smaller than the tolerance, the algorithm returns the ground state and k th excited state energies. ‡ We unequivocally point out the caveat that while we do not claim to introduce a revolutionary, paradigm-shifting algorithm that is groundbreaking in the form of 2, the original contribution, and saliency of the method championed, in this paper is the consolidation of, and systemization of ideas and methodologies adopted by researchers working in the field, who use hybrid Classical-Quantum (CQ) approaches to perform electronic structure calculations -as seen in the literature cited -into one synthesised form, which is comprehensive. We would also like to mention the modularity property of Algorithm 2: We have used the VQE approach to find the minimum energy, however, we have seen in the literature that some researchers have used the PEA. IV. EXPERIMENTS, RESULTS, AND DISCUSSION This study aimed to determine the most precise and efficient method for computing elec- For the DFT calculations, we generated the electronic density of the system using a selected exchange-correlation functional and initial atomic coordinates. We then solved the Kohn-Sham equations (4), yielding the electronic wavefunctions and energies. This was accomplished through iterative Self-Consistent Field (SCF) calculations, repeated until convergence. Similarly, in the HF calculations, the processes was initiated with an initial guess for the wavefunction. The Hartree-Fock equations (2) were subsequently solved in order to obtain the self-consistent wavefunction and energies, and again utilised iterative SCF calculations until convergence was achieved. The total energy was calculated for both cases, and con- θ k ←− θ k −a k∇θ k J(θ k ), where θ k and θ k+1 are the parameter vectors at iteration k and k +1, a k is the step size,∇ θ k J(θ k ) is the estimated gradient of the cost function at θ k , and J(θ k ) is the cost function itself. The VQE calculations were performed on the AER simulator backend provided by IBM. In summary, this research paper has delved into the promising potential of QC, with specific emphasis on the usage of the VQE algorithm, for electronic structure calculations. This study has identified the limitations and complexities by thoroughly comparing traditional electronic structure calculation methods, including HF theory, DFT, and CC. Our findings indicate that the VQE algorithm provides a more efficient solution to these limitations due to quantum parallelism. The theory section of this paper has elaborated on the VQE algorithm in detail, including the creation and annihilation of operator mappings to single-bit quantum gates. This study has also showcased the power of the VQE by calculating the energies of five different molecules, and comparing values obtained from traditional methods. Our research indicates that the VQE can achieve similar energy values using fewer computational resources. The implications of our research affirms that QC has the potential to revolutionise the field of Computational Chemistry, providing a new paradigm in electronic structure calculations with wide-ranging applications in Materials Science, and Physics. Furthermore, this study highlights the need for continued development of QC algorithms and hardware to fully realise this technology's potential. In conclusion, this research paper provides a comprehensive introduction to electronic structure calculations and a particular QC approach to these calculations, highlighting the potential of the VQE as a robust algorithm in this domain. The results of this study have expensive. Thus, the impetus for an easier technique. Quantum Monte Carlo (QMC), see Foulkes et al., 2001; Frank et al., 2019, is a statistical precude used to sample a many-body wavefunction of a system. At the heart of the method is the generation of a large number of random samples generated from the wavefunction. There are several QMC methods used for the simulation of molecules. These include: The Metropolis algorithm (Metropolis et al., 1953), Variational MC (VMC) -see Kalos, 1962, Diffusion MC (DMC) -see Barnett & Whaley, 1993, Green's Function MC (GFMC) -see Trivedi & Ceperley, 1990, Auxiliary Field Quantum MC (AFQMC) -see Lee et al., 2022, Projector Quantum MC (PQMC) -see Hetzel et al., 1997, Path Integral MC (PIMC) -see Barker, 1979 and Cazorla & Boronat, 2017, Stochastic Reconfiguration -see Sorella, 1998, and Population Annealing MC (PAMC) -see Weigel et al., 2021, amongst others. Below, we present the simplest of these methods, the Metropolis algorithm 1. Algorithm 1 1Metropolis(s,s, E(s), E(s), T ) // the algorithm takes in a state s, candidate states, corresponding energies E(s) and E(s) respectively, and temperature T input the number of samples n repeat calculate the change in energy ∆E = E(s) − E(s) if ∆E < 0 then accept candidate states set P ←− 1 // set the probability to 1 set s ←− s else if ∆E > 0 then accept candidate states set P ←− exp − ∆E T // set the probability to a decaying exponential of the ratio of the energy difference and temperature set s ←− s else pass end if until n samples are obtained return a set of configurations of the system being studied Molecular Dynamics (MD), introduced by the landmark paper by Alder & Wainwright, 1959, and expanded upon by the Nobel Laureate, Martin Karplus (1930-), and his research group. The aim of MD is to use the laws of Classical and Quantum Mechanics to study the motion of molecules over time. Since the mathematical equations are well known, it will be an exercise in futility to restate them here; see the excellent source by Smit & Frenkel, 1996 for a detailed discussion. Some of the important features that these methods can be used to calculate are tabulated below (I & II). • Ground 3 ),where N is the number of atoms in the system. •• Used Used to describe small systems. O(N 4 )−O(N 6 ), where N is the number of atoms in the system / basis functions used to describe the system. For second-order perturbations: O(N 5 ), for third-order perturbations: O(N 6 ), etc. • the number of spin-orbitals, n e − is the number of electrons, n v is the number of virtual orbitals, m is the number of singly-excited determinants in the calculation, and p is the number of doubly-Used to study electronic excitations, chemical reactions, and charge transfer processes. O(N 3 ), where N is the number of electrons in the system.• Used to study phonons, excitons, and plasmons. The usage and importance of the classical and neoclassical algorithms described above can be broken up into the study of the following properties of molecules. 1. Molecular Geometry: This describes the physical arrangement of atoms in space that constitute a molecule via bond angles and electron pair arrangements. This property determines the intermolecular forces, the polarity, and the molecule's reactivity. 2. Magnetic Properties: This property determines the behaviour of a molecule in the presence of a magnetic field. The spin and orbital motion of the constituent electrons in the atoms create magnetic moments which influence the electronic structure, the symmetry, and the molecular geometry of the molecule. 3. Electronic Spectra: This property facilitates the molecule's absorption and emission of electromagnetic radiation (ER). When the constituent atoms absorb energy, the electrons enter an excited state with higher energy. Conversely, the electrons go into lower energy states when the atoms radiate ER. 4. Chemical Reactivity: This refers to the ability of the molecule to undergo chemical reactions and is therefore influenced by the presence of chemical functional groups and steric hindrances such as obstructions and overlapping electron cloud repulsions between atoms. ( GD), Stochastic Gradient Descent (SGD), or any of the other classical optimisers. This iterative process is repeated until a convergence criterion is met, and the final parameters θ * are utilised to calculate the ground state energy of the Hamiltonian according to Eq.(19) FIG. 1 : 1Architecture of the VQE. ( Seeley et al., 2012; Tranter et al., according to the chosen optimisation method (GD, SGD, Adam, etc.) end while return E 0 , E k tronic energies through a comparative analysis of three approaches: HF, DFT, and VQE. The investigation focused on assessing the efficacy of QC in this process, aiming to demonstrate the superiority of quantum methods over classical approaches. The efficacy of several techniques were evaluated through the computation of energy values for diverse molecules, including Water (H 2 O), Lithium Hydride (LiH), Methane (CH 4 ), Ammonia (NH 3 ), and Carbon Dioxide (CO 2 ). This can be attributed to the inherent approximations used in the DFT method, which may not accurately capture the exact behaviour of the electrons in the system. These findings have important implications for Computational Chemistry, as we have demonstrated that QC can provide a more efficient and accurate method for computing electronic energies than classical approaches. FIG. 4 : 4Electronic structure calculations for Methane molecule (CH 4 ): Method comparison. FIG. 5 :FIG. 6 : 56Electronic structure calculations for Ammonia molecule (NH3): Method comparison. Electronic structure calculations for Carbon Dioxide molecule (CO 2 ): Method comparison. .02679364497443 J −76.33340861478466 J −76.02657123746106 J Lithium Hydride (LiH) −7.981767664359352 J −8.068192292902214 J −7.979985984912321 J Methane (CH 4 ) −40.19870325538812 J −40.44299420579781 J −40.19911992417514 J Ammonia (NH 3 ) −56.18109675851954 J −56.46351100537343 J −56.172108720433144 J Carbon Dioxide (CO 2 ) −187.65110770987644 J −188.4094301538952 J −187.6573437805891 J V. CONCLUSION Coupled Cluster (CC) is an advancement of the Hartree-Fock method, described above, introduced by Coester & Kümmel, 1960, serves as a highly accurate method for computing molecular properties. Initially developed for calculating nuclear binding energies and several other properties, the technique is now used to model the electronic wavefunction of solidstate systems, precisely, transition metal complexes, bond dissociation energies, excitation energies, dipole moments, potential energy surfaces, and reaction barriers. Mathematically, TABLE I : IUses of the various electronic structure calculation methods, and computational complexities -Part I. TABLE II : IIUses of the various electronic structure calculation methods, and computational complexities -Part II.Method Uses Computational Complexity • Used to describe specific re- gions of large systems. Coupled Cluster • Considered the gold stan- dard for predicting molecular properties. O(N 6 ) for the Coupled Cluster Singles and Doubles (CCSD) method, where N is the number of molecular orbitals. • Used to predict bond break- ing, reaction pathways, and excited state energies. The exponential computational complex- ity can be reduced by employing reduced Coupled Cluster methods or by parallelis- ing the calculations. Configuration Interaction • Well-suited for systems with strong electron correlation. 1 . 1Materials Science and Engineering: Used for designing novel materials with pertinent properties for a particular application. For predicting the properties of new materials and gaining a deeper understanding of existing materials.2. Nanotechnology: Used for designing and understanding materials in the nanometre scale ranging from 1 − 100 nm. These include nanowires, nanotubes, nanopores, nanocapsules, nanorods, nanofibers, nanopillars, nanostructured membranes, nanocomposites, and dendrimers. 3. Energy Research: For designing novel energy conversion and storage materials. These include state-of-the-art batteries and solar cells.4. Physical Chemistry and Chemical Physics: Perhaps the most ubiquitous application. It is used to study the properties of molecules, the outcomes of chemical reactions, and understand reaction mechanisms. 5. Condensed Matter Physics: Used to determine materials' magnetic, electronic, and optical properties. 6. Biochemistry and Drug Design: Used for modelling the structure of DNA, RNA, proteins, biomolecules, and the design of pharmaceutical-grade drugs. 7. Environmental Research: Used to study pollutants, and their impact on ecosystems. Additionally, for the design of green-materials for environmental remediation and pollution-mitigation. 1 . 1Producing Inaccurate Results: Since these methods are numerical approximations of the Schrödinger equation's description of the electron, errors can easily be carried over and compounded, producing unreliable and imprecise values. 2. The Inability to Capture Quantum Mechanical Effects: These classical methods do not account for Quantum Entanglement or Quantum Tunnelling. Since these phenomena have a bearing on macroscopic physical and chemical properties, this results in coarse-grained results which deviate from experiments in many cases. 3. Inadequacy of Models to be Modular and Transferable: Computer models / simulations are specific to molecules, and require domain expertise in order to edit code and adjust it for the study of other molecules. In addition, adding new parameters to the model is not a trivial exercise and requires a significant overhaul of the code. 4. Limitations and Constraints in the Scope of the Models: The models are limited to small-electron systems. For larger electron systems, the models become computationally intractable. 5. The Insufficiency in Capturing Theoretical Subtleties and Chemical Accuracy in Reactions: Classical methods are limited in their Inability to account for abstruse differences in structural dissimilarities, and energy variations. These facets are important in predicting reaction mechanisms.Thus, we advocate for, and try to galvanise, the idea of using Quantum Computing (QC) and Optimisation problems, to Electronic Structure calculations. Various approaches to QC have been explored, including trapped ions, quantum dots, and topological qubits with Quntinuum's recent success in creating non-Abelian anyons -nonabelions(Iqbal et al, 2023) in the pursuit of fault-tolerant quantum computers being a noteworthy example. However,the realisation of large-scale and dependable quantum computers remains a significant tech- nical challenge that must be overcome to fully exploit QC's potential. One promising approach to quantum computation is the Variational Quantum Eigensolver (VQE) algorithm, which was introduced by Peruzzo et al., 2013. It is a quantum algorithm used to find the lowest eigenvalue of a given Hamiltonian, which corresponds to the ground state energy of a quantum system. VQE is a hybrid algorithm that combines classical and quantum computing resources to determine the lowest eigenvalue of a given Hamiltonian, corresponding to a quantum system's ground state energy. Notably, VQE is designed to run on noisy intermediate scale quantum (NISQ) computers -see Preskill, 2018 -which have a limited number of qubits, and high error rates. The algorithm represents a promising solution for determining the ground state energies of molecules and materials. The VQE works by computing the expectation value of the Hamiltonian,Ĥ, which is given by Eq. (15):Ĥ TABLE III : IIIResearch papers that have applied QC to electronic structure calculations -Part I. TABLE IV : IVResearch papers that have applied QC to electronic structure calculations -Part II.Literature Summary Xia & Kais, 2018 In this seminal work, the need for the hybridisation of QC with ML is delineated in order to achieve more accurate results in electronic struc- ture calculations with reduced computational times. A composite model, consisting of a Restricted Boltzmann Machine (RBM), was employed to ascertain the electronic ground state energy for small molecular systems (the demonstrable use case of small molecular systems was chosen be- cause of the current NISQ-era QC technology available). The examples of the Hydrogen (H 2 ), Lithium Hydride (LiH), and Water (H 2 O) molecules were chosen for potential energy surface calculations. Sureshbabu et al., 2021 An IBM quantum computer was used to calculate electronic structure on two-dimensional crystal structures -specifically monolayer hexagonal Boron Nitride (h − BN) and monolayer Graphene (h − C) -using a hy- brid method comprising a restricted Boltzmann machine and a quantum machine learning (QML) algorithm. The results were consistent with traditional results from classical methods. Rossmannek et al., 2021 An effective Hamiltonian was constructed by incorporating a mean-field potential on a restricted Action Space (AS) via an embedding scheme. Using the VQE algorithm, the ground state of the AS Hamiltonian,Ĥ 0 , was calculated. Song et al., 2023 Using VQE circuits on Quantinuum's ion-trap quantum computer H1- 1, small molecules were simulated with plane-wave basis sets. Using a small number of iterations, the results from Correlation Optimised Virtual Orbitals (COVO) were replicated within the tolerable range of TABLE V : VResearch papers that have applied QC to electronic structure calculations -Part III. This study implements the VQE using Qiskit / IBM Quantum to determine the ground state energy of a Hydrogen (H) molecule, the results reveal that VQE is highly effective in accurately computing molecularLiterature Summary Naeij et al., 2023 This study used a VQE to calculate the ground state energy of Proto- nated Molecular Hydrogen (H + 3 ), Hydroxide (OH − ), Hydrogen Fluoride (HF), and Borane (BH 3 ). The Unitary Coupled Cluster for Single and Double excitations (UCCSD) is used to construct an ansatz with the fermion-to-qubit and parity transformation. Lastly, this hybrid VQE method was benchmarked against the Unrestricted Hartree-Fock (UHF) and FCI classical methods, and high fidelity between the classical and quantum approaches is shown. Qing & Xie, 2023 TABLE VI : VIVarious mappings of creation, a † i , and annihilation, a i , operators to single-bit quantum gates.Method Mapping Jordan-Wigner transformation (Jordan & Wigner, 1928; Whitfield et al., 2011; Fradkin, 1989) vergence of the electronic wavefunctions and energies was accomplished. The results from the QC approach were then compared with the results obtained through the DFT and HF methods.For the VQE calculations, firstly the electronic structure of each molecule was obtained, and the electronic Hamiltonian was successively mapped to a qubit Hamiltonian using the parity mapper. Using the qubit converter, the qubit Hamiltonian was converted to the Pauli basis, expressed in terms of fermionic operators, to a qubit Hamiltonian exhibited in terms ofPauli operators. This conversion is necessary because quantum computers typically operate with qubits. Using the Jordan-Wigner transformation method, the qubit converter maps the fermionic problem to the qubit problem. The result is a qubit Hamiltonian expressed as a linear combination of Pauli operators. the Unitary Coupled Cluster (UCC) ansatz was deployed with single and double excitations to construct a trial wavefunction for the VQE computation.This ansatz involves representing the wavefunction as a linear combination of exponentially parameterised unitary operators acting on a reference state, which can be expressed as: |Ψ UCC = e T |Φ , where T is a cluster operator that generates the single and double excitations from the reference state |Φ . In order to carry out the VQE calculation, the Simultaneous Perturbation Stochastic Approximation (SPSA) optimiser was utilised with a fixed number of iterations; this optimiser estimates the gradient of the cost function using two randomly perturbed function evaluations, and updates the parameters of the ansatz in the direction that minimises the estimated gradient, as given by the following update rule: The results obtained demonstrate that the VQE method consistently outperformed the HF and DFT methods in terms of both accuracy and efficiency. For example, in the case of Water, the energy values computed using the HF, DFT, and VQE methods were−76.02679364497443 J, −76.33340861478466 J, and −76.02657123746106 J, respectively, as shown in Fig. 2. The VQE energy value agrees with the HF value, and is much more accurate than the DFT value. This trend is consistent across all the molecules studied, as shown in Figures 3, 4, 5, and 6, indicates that the VQE is a robust and reliable method for computing electronic energies. Based on the graphs and Tab. VII, which summarised the energy values obtained using the HF, DFT, and VQE methods for each molecule, these results demonstrate that the VQE method can potentially provide a more efficient and ac-curate approach to determining the energy of electronic structures, especially for complex molecules where DFT may not provide accurate results. Energy of Water Molecule (H2O) FIG. 2: Electronic structure calculations for Water molecule (H 2 O): Method comparison.Notably, the DFT energy values are consistently lower than the HF and VQE values.0 10 20 30 40 50 Iteration 76.2 76.0 75.8 75.6 75.4 E(J) VQE Hartree Fock DFT TABLE VII : VIIEnergy values obtained using the HF, DFT, and VQE methods for each molecule. mE h (milli-Hartree) ≈ 0.2993 eV. Nature Communications, 9 (1), pp. 1-6. Computational Chemistry, and demonstrate the transformative potential of QC in this area. Lastly, this study sets the stage for future research and development in this exciting field. implications for the field of Computational Chemistry, and demonstrate the transformative potential of QC in this area. Lastly, this study sets the stage for future research and development in this exciting field. . Vi, References, VI. REFERENCES . B J Alder, T E Wainwright, Studies in Molecular Dynamics. I. General Method. The Journal of Chemical Physics. 312Alder, B. J., & Wainwright, T. E. (1959). Studies in Molecular Dynamics. I. General Method. The Journal of Chemical Physics, 31 (2), pp. 459-466. A Quantum-Statistical Monte Carlo Method; Path Integrals with Boundary Conditions. J A Barker, The Journal of Chemical Physics. 706Barker, J. A. (1979). A Quantum-Statistical Monte Carlo Method; Path Integrals with Boundary Conditions. The Journal of Chemical Physics, 70 (6), pp. 2914-2918. 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[ "Nearly Optimal Policy Optimization with Stable at Any Time Guarantee", "Nearly Optimal Policy Optimization with Stable at Any Time Guarantee" ]
[ "Tianhao Wu ", "Yunchang Yang ", "Han Zhong ", "Liwei Wang ", "Simon S Du ", "Jiantao Jiao " ]
[]
[]
Policy optimization methods are one of the most widely used classes of Reinforcement Learning (RL) algorithms. However, theoretical understanding of these methods remains insufficient. Even in the episodic (time-inhomogeneous) tabular setting, the state-of-the-art theoretical result of policy-based method in Shani et al.(2020)where S is the number of states, A is the number of actions, H is the horizon, and K is the number of episodes, and there is a √ SH gap compared with the information theoretic lower bound Ω( √ SAH 3 K) (Jin et al., 2018). To bridge such a gap, we propose a novel algorithm Reference-based Policy Optimization with Stable at Any Time guarantee (RPO-SAT), which features the property "Stable at Any Time". We prove that our algorithm achieves O( √ SAH 3 K + √ AH 4 K) regret. When S > H, our algorithm is minimax optimal when ignoring logarithmic factors. To our best knowledge, RPO-SAT is the first computationally efficient, nearly minimax optimal policybased algorithm for tabular RL.
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[ "https://export.arxiv.org/pdf/2112.10935v3.pdf" ]
245,353,767
2112.10935
e75ee46c70e9fc56a59b5c783c3070f28791c218
Nearly Optimal Policy Optimization with Stable at Any Time Guarantee 3 Dec 2022 Tianhao Wu Yunchang Yang Han Zhong Liwei Wang Simon S Du Jiantao Jiao Nearly Optimal Policy Optimization with Stable at Any Time Guarantee 3 Dec 2022 Policy optimization methods are one of the most widely used classes of Reinforcement Learning (RL) algorithms. However, theoretical understanding of these methods remains insufficient. Even in the episodic (time-inhomogeneous) tabular setting, the state-of-the-art theoretical result of policy-based method in Shani et al.(2020)where S is the number of states, A is the number of actions, H is the horizon, and K is the number of episodes, and there is a √ SH gap compared with the information theoretic lower bound Ω( √ SAH 3 K) (Jin et al., 2018). To bridge such a gap, we propose a novel algorithm Reference-based Policy Optimization with Stable at Any Time guarantee (RPO-SAT), which features the property "Stable at Any Time". We prove that our algorithm achieves O( √ SAH 3 K + √ AH 4 K) regret. When S > H, our algorithm is minimax optimal when ignoring logarithmic factors. To our best knowledge, RPO-SAT is the first computationally efficient, nearly minimax optimal policybased algorithm for tabular RL. Introduction Reinforcement Learning (RL) has achieved phenomenal successes in solving complex sequential decision-making problems (Silver et al., 2016;2017;Levine et al., 2016;Gu et al., 2017). Most of these empirical successes are driven by policy-based (policy optimization) methods, such as policy gradient (Sutton et al., 1999), natural policy gradient (NPG) (Kakade, 2001), trust region policy optimiza-tion (TRPO) (Schulman et al., 2015), and proximal policy optimization (PPO) (Schulman et al., 2017). For example, Haarnoja et al. (2018) proposed a policy-based state-ofthe-art reinforcement learning algorithm, soft actor-critic (SAC), which outperformed value-based methods in a variety of real world robotics tasks including manipulation and locomotion. In fact, Kalashnikov et al. (2018) observed that compared with value-based methods such as Q-learning, policy-based methods work better with dense reward. On the other hand, for sparse reward cases in robotics, value-based methods perform better. Motivated by this, a line of recent work (Fazel et al., 2018;Bhandari & Russo, 2019;Liu et al., 2019;Wang et al., 2019;Agarwal et al., 2021) provides global convergence guarantees for these popular policy-based methods. However, to achieve this goal, they made several assumptions. Agarwal et al. (2021) assumes they have the access to either the exact population policy gradient or an estimate of it up to certain precision for all states uniformly compared with the state distribution induced by π * , bypassing the hardness of exploration. They showed that even with this stringent assumption, the convergence rate would depend on the distribution mismatch coefficient D ∞ = max s d π * s 0 (s) µ(s) , where µ is the starting state distribution of the algorithm and d π * s0 (s) is the stationary state distribution of the optimal policy π * starting from s 0 . This dependency is problematic since D ∞ is small only when the initial distribution has a good coverage of the optimal stationary distribution, which may not happen in practice. However, in online value-based RL, algorithm such as Azar et al. (2017) can achieve fast convergence rate (or regret) independent of the distribution mismatch coefficient, or equivalently, without the coverage assumption. Though value-based methods have achieved the information theoretical optimal regret in tabular (Azar et al., 2017) and linear MDPs settings (Zanette et al., 2020), it remains unclear whether policy-based methods can achieve information theoretically optimal regret in the same settings. To address this issue, Cai et al. (2020) proposed the idea of optimism in policy optimization, which seems similar to the valuebased optimism but different in nature, since it encourages optimism for Q π instead of Q * (Section 4). With this new idea, Cai et al. (2020); Shani et al. (2020) managed to establish the regret guarantees without additional assumptions, though the regret is suboptimal. In this work, we focus on the same setting as in Shani et al. (2020): episodic tabular MDPs with unknown transitions, stochastic rewards/losses, and bandit feedback. In this setting, the state-of-the-art result of policy-based method is O( √ S 2 AH 4 K) (Shani et al., 2020). Here S and A are the cardinality of states and actions, respectively, H is the episode horizon, and K is the number of episodes. Compared with the information theoretic limit (Jaksch et al., 2010;Azar et al., 2017;Jin et al., 2018;Domingues et al., 2021), there is still a gap √ SH. Main Contributions In this paper we present a novel provably efficient policy optimization algorithm, Reference-based Policy Optimization with Stable at Any Time guarantee (RPO-SAT). We establish a high probablity regret upper bound O( √ SAH 3 K + √ AH 4 K) for our algorithm. Importantly, if S > H, the main term in this bound matches the information theoretic limit Ω( √ SAH 3 K) (Jin et al., 2018), up to some lower order term O(poly(S, A, H)K 1/4 ). We introduce our algorithmic innovations and analytical innovations as follows: Algorithmic innovations: • We introduce a novel reference V estimator in our algorithm. It is conceptually simple and easy to implement as it just updates the reference value to be the mean of empirical V values when some conditions are triggered (cf. Algorithm 1 Lines 18-20). • We carefully incorporate the reference V estimator into our bonus term, in the way of adding a weighted absolute difference between the estimated V values and the reference V values to control the instability of the estimation process. Readers may refer to Section 4 for more details. • Another highlight is that we modify the policy improvement phase of our algorithm to meet a novel property, which we called Stable at Any Time (SAT). (For a detailed definition see Equation (10) in Section 4.) More specifically, instead of using the KLdivergence regularization term proposed in Shani et al. (2020), we use the ℓ 2 regularization term. This is crucial to ensure SAT. Analytical innovations: • We prove that our algorithm satisfies the SAT property. The analysis is done by two steps: First, we establish a 1-st step regret bound O( √ S 2 AH 4 K). Second, we use a new technique "Forward Induction" to prove the same for all the h-th step regret. Here the h-step regret is defined in (4). Readers may refer to Section 4 for more details of the "Forward Induction" technique. • We show that the combination of the SAT property and the simple reference V estimator yields a precise approximation of V * . We use this property to derive a O( √ SAH 3 K) upper bound for the sum of bonus terms, which leads to a √ S reduction in terms of regret. Related Work Our work contributes to the theoretical investigations of policy-based methods in RL (Cai et al., 2020;Shani et al., 2020;Lancewicki et al., 2020;Fei et al., 2020;He et al., 2021;Zhong et al., 2021;Luo et al., 2021;Zanette et al., 2021). The most related policy-based method is proposed by Shani et al. (2020), who also studies the episodic tabular MDPs with unknown transitions, stochastic losses, and bandit feedback. It is important to understand whether it is possible to eliminate this gap and thus achieve minimax optimality, or alternatively to show that this gap is inevitable for policy-based methods. We also provide interesting practical insights. For example, the usage of reference estimator and ℓ 2 regularization with decreasing learning rate to stabilize the estimate of V and Q. The use of reference estimator can be traced to (Zhang et al., 2020b). They use the reference estimator to maximize the data utilization, hence reduce the estimation variance. However, our usage of reference estimator is different from theirs. The reason why they can reduce a √ H factor is because the bottleneck term is only estimated using 1/H fraction of data, while the usage of reference can make use of all the data, hence fully utilizing available data. However, for policy-based RL, there is no such problem of data utilization. In fact, the bottleneck of policy-based methods is the instability of Q estimation, therefore we use the reference estimator to stabilize the estimation process and reduce a √ S factor in the regret. Readers may turn to Section 4 for a detailed explanation for "instability". Our work is also closely related to another line of work on value-based methods. In particular, Azar et al. Menard et al. (2021) have shown that the value-based methods can achieve O( √ SAH 3 K) regret upper bound, which matches the information theoretic limit. Different from these works, we are the first to prove the (nearly) optimal regret bound for policy-based methods. Preliminaries A finite horizon stochastic Markov Decision Process (MDP) with time-variant transitions M is defined by a tu- ple (S, A, H, P = {P h } H h=1 , c = {c h } H h=1 ) , where S and A are finite state and action spaces with cardinality S and A, respectively, and H ∈ N is the horizon of the MDP. At time step h, and state s, the agent performs an action a, transitions to the next state s ′ with probability P h (s ′ | s, a), and suffers a random cost C h (s, a) ∈ [0, 1] drawn i.i.d from a distribution with expectation c h (s, a). A stochastic policy π : S × [H] → ∆ A is a mapping from states and time-step indices to a distribution over actions, i.e., ∆ A = π ∈ R A : a π(a) = 1, π(a) ≥ 0 . The performance of a policy π when starting from state s at time h is measured by its value function, which is defined as V π h (s) = E H h ′ =h c h ′ (s h ′ , a h ′ ) | s h = s, π, P .(1) The expectation is taken with respect to the randomness of the transition, the cost function and the policy. The Qfunction of a policy given the state action pair (s, a) at timestep h is defined by Q π h (s, a) = E H h ′ =h c h ′ (s h ′ , a h ′ ) | s h = s, a h = a, π, P . (2) By the above definitions, for any fixed policy π, we can obtain the Bellman equation Q π h (s, a) = c h (s, a) + P h (· | s, a)V π h+1 (·), V π h (s) = Q π h (s, ·), π h (· | s) .(3) An optimal policy π * minimizes the value for all states s and time-steps h simultaneously, and its corresponding optimal value is denoted by V * h (s) = min π V π h (s), for all h ∈ [H]. We consider an agent that repeatedly interacts with an MDP in a sequence of K episodes such that the starting state at the k-th episode, s k 1 , is initialized by a fixed state s 1 * . In this paper we define the notion of h-th step regret, Regret h , as follows: Regret h (K) = K k=1 V π k h (s k h ) − V * h (s k h ) .(4) When h = 1, this matches the traditional definition of regret, which measures the performance of the agent starting from s 1 . In this case we also use Regret(K) for simplicity. * Our subsequent analysis can be extended to the setting where the initial state is sampled from a fixed distribution. Notations and Definitions We denote the number of times that the agent has visited state s, state-action pair (s, a) and state-action-transition pair (s, a, s ′ at the h-th step by n k h (s), n k h (s, a) and n k h (s, a, s ′ ) respectively. We denote by X k the empirical average of a random variable X. All quantities are based on experience gathered until the end of the k-th episode. We denote by ∆ A the probability simplex over the action space, i.e. ∆ A = {(p 1 , ..., p |A| ) | p i ≥ 0, i p i = 1}. We use O(X) to refer to a quantity that depends on X up to a poly-log expression of a quantity at most polynomial in S, A, K, H and δ −1 . Similarly, represents ≤ up to numerical constants or poly-log factors. We define X ∨ Y := max{X, Y }. RPO-SAT: Reference-based Policy Optimization with Stable at Any Time guarantee Algorithm 1 Reference-based Policy Optimization with Stable at Any Time guarantee (RPO-SAT) for ∀s, a ∈ S × A do 7: 1: initialize Q h (x, a) ← 0, V h (x, a) ← 0 and V ref h (x, a) ← 0, C 0 = √ S 3 AH 3 2: for episode k = 1, ..., Calculate u k h as in (5) 8: b k h (s, a) = min{u k h (s, a), 2 ln 2SAHT δ ′ n k h (s,a) + H 4S ln 3SAHT δ ′ n k h (s,a) } 9: Q k h (s, a) = max{c k h (s, a) + P k h (· | s, a)V k h+1 (·) − b k h (s, a),0} 10: for ∀s ∈ S do 11: V k h (s) = Q k h (s, ·), π k h(V ref h (s) = 1 n k h (s) k i=1 V i h (s i h )1[s i h = s] 20: end for 21: end for In this section, we present our algorithm RPO-SAT (Reference-based Policy Optimization with Stable at Any Time guarantee). The pseudocode is given in Algorithm 1. We start by reviewing the optimistic policy optimization algorithms OPPO and POMD proposed by Cai et al. (2020);Shani et al. (2020). In each update, OPPO and POMD involve a policy improvement phase and a policy evaluation phase. In policy evaluation phase, OPPO and POMD explicitly incorporate a UCB bonus function into the estimated Q-function to promote exploration. Then, in the policy improvement phase, OPPO and POMD improve the policy by Online Mirror Descent (OMD) with KLregularization, where the estimated Q-function serves as the gradient. Compared with existing optimistic policy optimization algorithms in Shani et al. (2020);Cai et al. (2020), our algorithm has three novelties. First, in policy evaluation phase, we introduce the reference V estimator (Lines 18-20). Specifically, for any s ∈ S, if the number of visiting s satisfies the condition in Line 18, we update the reference value to be the empirical mean estimator of V (Line 19). Zhang et al. (2020b) also adopts a reference value estimator. Nevertheless their update conditions and methods are different from us, and they reduce a √ H factor while we reduce a √ S factor. Second, we make some modifications in policy improvement phase. Specifically, the policy optimization step in the h-th step of the k-th episode is π k+1 h (·|s) = argmin π h η k Q k h (·, s), π h − π k h + D(π h , π k h ), where η k is the stepsize in the k-th episode, D is some distance measure and Q k h is the estimated Qfunction in the h-th step of the k-th episode. Different from Shani et al. (2020);Cai et al. (2020) choosing D(π h , π k h ) = KL(π h , π k h ), we choose D(π h , π k h ) = π h − π k h 2 2 . In this case the solution of the OMD is: π k+1 h (·|s) = Π ∆A π k h (·|s) − η k Q k h (·, s) where Π ∆A is the Euclidean projection onto ∆ A , i.e. Π ∆A (x) = argmin y∈∆A x − y 2 . Unlike previous works, we also adopt a decreasing learning rate schedule instead of a fixed learning rate. These modifications are necessary for our analysis since they ensure the SAT property, making it possible to learn a good reference V value. Finally, we design a novel bonus term which carefully incorporate the reference V estimator mentioned before by adding a weighted absolute difference Term (iii), which is also referred to as the instability term. Specifically, we define u k h (s, a) = Term(i) + Term(ii) + Term(iii) + Term(iv) where Term(i) = 2 ln 2SAHT δ ′ n k h (s, a) , Term(ii) = 6V Y ∼P k h (·|s,a) V ref h+1 (Y ) ln 2SAHK δ ′ n k h (s, a) + 4H ln 2SAHK δ ′ S · n k h (s, a) + 8 √ SH 2 ln 2SAHK δ ′ 3n k h (s, a) , Term(iii) = y∈S    2P k h (y | s,a)(1−P k h (y | s,a))ln 2SAHK δ ′ n k h (s, a) − 1 + 7 ln 2SAHK δ ′ 3n k h (s, a) |V k h+1 (y) − V ref h+1 (y)|, Term(iv) = 4H ln 3SAHT δ ′ n k h (s, a) + 7SH ln 2SAHK δ ′ 3n k h (s, a) + 2S 3/2 A 1/4 H 7/4 K 1/4 ln 2SAHK δ ′ n k h (s, a) . where (i) is the estimation error for reward functions and (ii)-(iv) are estimation errors for transition kernels. We will provide more explanations for this seemingly complicated term in Section 4. Furthermore, we set the bonus as b k h (s, a) = min u k h (s, a), 2 ln 2SAHT δ ′ n k h (s, a) + H 4S ln 3SAHT δ ′ n k h (s, a)    .(6) With this carefully chosen bonus function, we can also achieve optimism like previous work in optimistic policy optimization (Shani et al., 2020;Cai et al., 2020). Notably, our bonus function is smaller than that in Shani et al. (2020), thus leads to our tighter regret bound. Now we state our main theoretical results for RPO-SAT. Theorem 3.1 (Regret bound). Suppose in Algorithm 1, we choose η t = O( 1/(H 2 At)), and C 0 = √ S 3 AH 3 , then for sufficiently large K, we have Regret(K) ≤ O( √ SAH 3 K + √ AH 4 K+ S 5/2 A 5/4 H 3/2 K 1/4 ). We provide a proof sketch in Section 5. The full proof is in appendix B. Note that previous literature shows that the regret lower bound is Ω( √ SAH 3 K) (Jin et al., 2018). Hence our result matches the information theoretic limit up to logarithmic factors when S > H. Technique Overview In this section, we illustrate the main steps of achieving near optimal regret bound and introduce our key techniques. Achieving Optimism via Reference. Similar to previous works (Cai et al., 2020;Shani et al., 2020), to achieve optimism, a crucial step is to design proper bonus term to upper bound (P k h − P h )(· | s, a)V k h+1 (·) . For example, Jaksch et al. (2010); Shani et al. (2020) bound this term in a separate way: (P k h − P h )(· | s, a)V k h+1 (·) (7) ≤ (P k h − P h )(· | s, a) 1 · V k h+1 (·) ∞ ≤ O SH 2 n k h (s, a) , which will leads to an additional √ S factor due to the absence of making use of the optimism of Q k . This issue is later addressed by value-based algorithm UCBVI (Azar et al., 2017). They divide the term into two separate terms: (P h − P h )(· | s, a)V k h+1 (·) = (P k h − P h )(· | s, a)V * h+1 (·) + (P k h − P h )(· | s, a)(V k h+1 − V * h+1 )(·).(8) They bound the first term using straightforward application of Chernoff-Hoeffding inequality, which removes the √ S factor since V * is deterministic. Thanks to the fact that V k h ≤ V * h for any (k, h) ∈ [K] × [H] , they can bound the second term successfully (see Appendix C for more details). However, this approach is not applicable for policybased methods which improve the policy in a conservative way instead of choosing the greedy policy (when the stepsize η → ∞, the OMD update becomes the greedy policy, and for any η < ∞ this update is "conservatively" greedy). This key property of policy-based methods makes it only possible to ensure V k h ≤ V π k h , the optimism V k h ≤ V * h doesn't hold in general. To tackle this challenge, we notice that as long as V k h converges to V * h sufficiently fast (at least on average), (P k h − P h )(· | s, a)(V k h+1 − V * h+1 )(·) can be bounded by a term related to the rate of convergence. This leads to the important notion called Stable at Any Time (SAT). Precisely, we introduce the reference V esti- mator V ref = {V ref h } h∈[H] and decompose (P k h − P h )(· | s, a)V k h+1 (·) as (P k h − P h )(· | s, a)V k h+1 (·) = (P k h − P h )(· | s, a)V * h+1 (·) (a) + (P k h − P h )(· | s, a)(V k h+1 − V ref h+1 )(·) (b) + (P k h − P h )(· | s, a)(V ref h+1 − V * h+1 )(·) (c) .(9) By standard concentration inequalities, we can bound the first term by the quantity depending on the variance of V * h+1 . Once we have SAT, an easy implication is that |V ref h (s) − V * h (s)| ≤ O( H/S) for any n k h (s) large enough (Lemma 5.3). We can replace the variance of V * h+1 (unknown) by the variance of V ref h+1 (known) and bound the second and third term in an analogous way of (7) and remove the factor √ S as desired. Specifically, we can upper bound Terms (a), (b), (c) in (9) by Terms (ii), (iii), (iv) in (5), respectively. Stable at Any Time. The high-level idea is that in or- Zhang et al. (2020b) shows that in the senario of greedy policy, this term converges to zero as the number of visit goes to infinity. However, due to the nature of conservative policy update scheme, we cannot guarantee the convergence, unless we make the coverage assumption as in Agarwal et al. (2021). Fortunately, it's possible to obtain an average convergence guarantee, which we called SAT. Specifically, we say that an algorithm satisfy the property of SAT if for all K ′ and h, der to control (P k h − P h )(· | s, a)(V k h+1 − V * h+1 )(·), term V k h+1 − V * h+1 must satisfy some properties. For example,K ′ k=1 |V k h (s k h ) − V * h (s k h )| ≤ O( √ K ′ ).(10) The meaning of Stable at Any Time can be interpreted as follows: The above inequality implies that the estimation V k h varies around the fixed value V * h , hence "stable". And since we impose that the inequality holds for all h and K ′ , hence "at any time". For this reason, we also name y∈S P k h (y|s,a) n k h (s,a) |V k h+1 (y) − V ref h+1 (y)| in the bonus term (5) as the instability term, since it's an upper bound of (P k h − P h )(· | s, a)(V k h+1 − V ref h+1 )(·) , which measures the instability of V k with respect to the fixed reference V ref . Another nice implication is that we can obtain a precise estimate of V * if SAT is satisfied. Specifically, combining the update rules of reference V estimator and the fact that n k h (s) ≥ C 0 √ k, we have: |V ref h (s) − V * h (s)| ≤ 1 n k h (s) k i=1 |V i h (s i h ) − V * h (s i h )|1[s i h = s] ≤ O(1/C 0 ). Coarse Regret Bound. We are able to show that (10) can be implied by Regret 1 (K ′ ) ≤ O( √ K ′ ).(11) for any K ′ ∈ [K]. See Lemma 5.3 for details. We point out that establishing Equation (10) To this end, we use the following two novel techniques: 1. Replace the unbounded KL-divergence regularization term by the bounded ℓ 2 regularization term, which allows us to choose varying stepsize which depends on the current time step instead of K. 2. Forward Induction, that is, deriving the regret upper bound for the 1-st step and obtaining the regret upper bounds for h-th (2 ≤ h ≤ H) step by induction. See Section 5 for more details. Putting these together, we can remove the additional factor √ S as desired. If we also adopt the Bernstein variance reduction technique (Azar et al., 2017), we can further improve a √ H factor in the main term. Proof Sketch of Theorem 3.1 In this section, our goal is to provide a sketch of the proof of Theorem 3.1. One key ingredient in regret analysis in RL is the optimism condition, namely −2b k h (s, a) ≤ Q k h (s, a)−c h (s, a)−P h (· | s, a)V k h+1 (·) ≤ 0.(12) This condition guarantees that Q k h is an optimistic estimation of Q π k h (or Q * h ), and that the regret can be bounded by the sum of bonus terms (Jaksch et al., 2010;Azar et al., 2017;Cai et al., 2020;Shani et al., 2020). In our case, it does not appear straightforward to show such condition hold or not since the bonus term b k h depends on the value of V ref , which we do not know in advance. Dealing with this problem needs additional effort. But let's put it aside for a while, and assumes that (12) holds temporarily. In Section 5.1, we first demonstrate the intuition of the proof under this assumption. Then in Section 5.2, we show how to remove this assumption, with additional techniques. A warm-up: the Optimism Assumption In this section, we first demonstrate the intuition of the proof, under the assumption that optimism condition (12) holds. We first recall the useful notion of h-th step regret in episode K ′ as follows: Regret h (K ′ ) = K ′ k=1 V π k h (s k h ) − V * h (s k h ) . The notion aligns with the normal regret if h = 1, in other words, we have Regret 1 (K ′ ) = Regret(K ′ ). We first state the following lemma, which bounds the estimation error using bonus function: Lemma 5.1 (Bounding estimation error). Suppose that op- timism holds, namely for ∀k ≤ K, h ≤ H, s ∈ S, a ∈ A, −2b k h (s, a) ≤ Q k h (s, a) − c h (s, a) − P h (·|s, a)V k h+1 ≤ 0. Then for ∀K ′ ≤ K, h ′ ≤ H, it holds with high probability that K ′ k=1 V π k h (s k h ) − V k h (s k h ) ≤ O K ′ k=1 H h=h ′ b k h (s k h , a k h ) + O( √ H 3 K ′ ). Proof. See §B.1 for a detailed proof. The above lemma is useful since it tells us that the sum of the estimation error V π k h (s k h ) − V k h (s k h ) can be roughly viewed as the sum of bonus function. By the choice of our bonus term, we have b k h (s, a) ≤ SH 2 n k (s,a) . Following standard techniques from (Shani et al., 2020), we get a easy corollary K ′ k=1 V π k h (s k h ) − V k h (s k h ) ≤ O( √ S 2 AH 4 K ′ ). Next, we show that our algorithm satisfies a coarse regret bound: Lemma 5.2 (Coarse regret bound). With the same assumptions and notations in Lemma 5.1, we have for ∀K ′ ≤ K: Regret(K ′ ) = K ′ k=1 V π k 1 (s k 1 )−V * 1 (s k 1 ) ≤ O( √ S 2 AH 4 K ′ ).(13) Proof. We first decompose the regret term in the following way: K ′ k=1 V π k 1 (s 1 ) − V * 1 (s 1 ) = K ′ k=1 V π k 1 (s 1 ) − V k 1 (s 1 ) (i) + K ′ k=1 V k 1 (s 1 ) − V * 1 (s 1 ) (ii)(14) Term (i) can be bounded by O( √ S 2 AH 4 K ′ ) using the corollary of Lemma 5.1. We take a closer look at Term (ii), by standard regret decomposition lemma (Lemma A.1): Term (ii) = k,h E Q k h (s h , ·) , π k h (· | s h ) − π * h (· | s h ) | s 1 , π * , P (iii) + k,h E Q k h (s h ,a h )−c h (s h ,a h )−P h (· | s h ,a h )V k h+1 | s 1 ,π * ,P(iv) . Term (iii) is also called the OMD term. Using Lemma B.6, we have Term (iii) ≤ O( √ AH 4 K ′ ). We note that the reason why we change the Bregman term and the learning rate schedule is to ensure a Term (iii) ≤ O( √ K ′ ) type bound. In (Shani et al., 2020), they use the KL-divergence as the Bregman term and a fixed learning rate that is a function of K, hence their bound is Term (iii) ≤ O( log(A)H 4 K). Although our choice of the ℓ 2 penalty term and decreasing learning rate leads to a larger dependence on A, but it has a better dependence on K ′ , which is crucial to ensure that we can learn a good reference function. Combining with the fact that optimism holds, we have Term (iv) ≤ 0, therefore the lemma is proved. Now we have a Regret(K ′ ) = O( √ K ′ ) bound for all K ′ ≤ K. We show that this immediately implies the following key theorem, which we called the Average Convergence Lemma. The following lemma guarantees that the reference value is a good approximation of V * : Lemma 5.3 (Average convergence of V k ). With the same assumptions and notations in Lemma 5.1, we have for all K ′ ≤ K, h ≤ H, Regret h (K ′ ) ≤ O( √ S 2 AH 4 K ′ ).(15) As a consequence, SAT holds, for ∀K ′ ≤ K, h ≤ H, K ′ k=1 |V k h (s k h ) − V * h (s k h )| ≤ O( √ S 2 AH 4 K ′ ). Proof. For the proof of Equation (15), we use a novel technique called Forward Induction, starting from the case when h = 1, Regret 1 (K ′ ) = Regret(K ′ ) ≤ O( √ S 2 AH 4 K ′ ) , which is true according to Lemma 13. Using induction we can proof Equation (15) for all h, which we will leave the details to Appendix B.3. For the second statement, we note that |V k h (s k h ) − V * h (s k h )| ≤|V k h (s k h ) − V π k (s k h )| + |V π k h (s k h ) − V * h (s k h )| = V π k h (s k h ) − V k h (s k h ) + V π k h (s k h ) − V * h (s k h ) . These two terms can be handled by Lemma 5.1 and Equation (15) separately, hence finished the proof of Lemma 5.3. As mentioned before, Theorem 5.3 is significant in the sense that it implies the reference value V ref h (s) = 1 n k h (s) k i=1 V i h (s i h )1[s i h = s] is close to the optimal value if n k h (s) is large enough, in other words, if n k h (s) ≥ C 0 √ k = √ S 3 AH 3 k: |V ref h (s) − V * h (s)| ≤ 1 n k h (s) k i=1 |V i h (s i h ) − V * h (s i h )|1[s i h = s] ≤ 1 n k h (s) k i=1 |V i h (s i h ) − V * h (s i h )| ≤ O( H S ) This explains the reason why we choose C 0 = √ S 3 AH 3 in the algorithm, the 1/ √ S bound serves a key role in reducing the √ S factor in the regret. Finally, we are ready to prove Theorem 3.1. Combining Lemma 5.1 and Equation (14), K k=1 V π k 1 (s k 1 ) − V * 1 (s k 1 ) ≤ O K k=1 H h=1 b k h (s k h , a k h ) + O( √ AH 4 K).(16) The only thing left is to prove that K k=1 H h=1 b k h (s k h , a k h ) ≤ O( √ SAH 3 K + S 5 2 A 5 4 H 11 4 K 1 4 ). By Lemma B.7, this is true. Therefore we have finished the proof. How to prove optimism In this section, we show how to remove the optimism assumption in Lemma 5.1, Lemma 13 and Lemma 5.3. In fact, we will first prove the following lemma , which shows that both optimism and coarse regret bound holds for all episodes. Lemma 5.4 (Coarse analysis of regret). For any K ′ ∈ [K], we have the following regret bound: Regret(K ′ ) = K ′ k=1 V π k 1 (s k 1 )−V * 1 (s k 1 ) ≤ O( √ S 2 AH 4 K ′ ). (17) and the optimism holds, namely −2b k h (s, a) ≤ Q k h (s, a)−c h (s, a)−P h (· | s, a)V k h+1 (·) ≤ 0.(18) The full proof is in Appendix B.2. Here we mainly explain the core idea. To prove Lemma 5.4 we use induction. For the first few episodes of the algorithm, the bonus term is dominated by 2 ln 2SAHT δ ′ n k h (s,a) + H 4S ln 3SAHT δ ′ n k h (s,a) , which is the same as in Shani et al. (2020). Therefore the optimism and coarse regret bound automatically holds. In the induction step, assume that optimism and coarse regret holds for all k ≤ K ′ and h. Then by optimism we have that 1-step regret is bounded by O( √ S 2 AH 4 K ′ ). Then by Lemma 5.3, we have that Regret h (K ′ ) ≤ O( √ S 2 AH 4 K ′ ) . This means that if in the K ′ + 1th episode, a new reference function is calculated, then this new reference function must be H/S accurate, i.e. |V ref (s) − V * (s)| ≤ O( H/S). Hence, the carefully designed bonus guarantees that optimism still holds for k = K ′ + 1, which finishes the proof. The induction process is shown in Figure 1. Finally, thanks to optimism, we can bound the regret by K k=1 H h=1 b k h (s k h , a k h ) . The rest follows from the same arguments in the previous section. Optimism for ∀(k, h, s, a) ∈ [K ′ ] × [H] × S × A Q k h (s, a) − c h (s, a) − P h (·|s, a)V k h+1 (·) ≤ 0 1-st step regret: K ′ k=1 V π k 1 (s k 1 ) − V * 1 (s k 1 ) ≤ O( √ K ′ ) h-th step regret: K ′ k=1 V π k h (s k h ) − V * h (s k h ) ≤ O( √ K ′ ) For any V ref (s) updated before K ′ + 1 episode |V ref (s) − V * (s)| ≤ O( H/S) Optimism for ∀(k, h, s, a) ∈ [K ′ + 1] × [H] × S × A Forward Induction h ← h + 1 Figure 1. Proving optimism and h-th step regret by induction Conclusion and Future Work In this paper, we proposed the first optimistic policy optimization algorithm for tabular, episodic RL that can achieve regret guarantee O( √ SAH 3 K + √ AH 4 K + poly(S, A, H)K 1/4 ). This algorithm improves upon previous results (Shani et al., 2020) and matches the information theoretic limit Ω( √ SAH 3 K) when S > H. Our results also raise a number of promising directions for future work. Theoretically, can we design better policy-based methods that can eliminate the constraint S > H? Practically, can we leverage the insight of RPO-SAT to improve practical policy-based RL algorithms? Specifically, can we design different regularization terms to stabilize the V /Q estimation process to make the algorithm more sample efficient? We look forward to answering these questions in the future. Zhang, Z., Ji, X., and Du, S. S. Is reinforcement learning more difficult than bandits? a near-optimal algorithm escaping the curse of horizon. arXiv preprint arXiv:2009.13503, 2020a. Conference on Machine Zhang, Z., Zhou, Y., and Ji, X. Almost optimal modelfree reinforcement learningvia reference-advantage decomposition. Advances in Neural Information Processing Systems, 33, 2020b. Zhong, H., Yang, Z., Wang, Z., and Szepesvári, C. Optimistic policy optimization is provably efficient in nonstationary mdps. arXiv preprint arXiv:2110.08984, 2021. A. Regret Decomposition and Failure Events A.1. Regret Decomposition For any (k, h) ∈ [K] × [H], we define ζ 1 k,h = [V k h (s k h ) − V π k h (s k h )] − [Q k h (s k h , a k h ) − Q π k h (s k h , a k h )], ζ 2 k,h = [P h (· | s k h , a k h )V k h+1 (·) − P h (· | s k h , a k h )V π k h+1 (·)] − [V k h+1 (s k h+1 ) − V π k h+1 (s k h+1 )].(19) By the definition, we have that ζ 1 k,h and ζ 2 k,h represent the randomness of executing a stochastic policy π k h (· | s k h ) and the randomness of observing the next state from stochastic transition kernel P h (· | s k h , a k h ), respectively. With these notations, we have the following standard regret decomposition lemma (Cai et al., 2020;Shani et al., 2020). Lemma A.1 (Regret Decomposition). For any (K ′ , h ′ ) ∈ [K] × [H], it holds that Regret h ′ (K ′ ) = K ′ k=1 V π k h ′ (s k h ′ ) − V k h ′ (s k h ′ ) (i) + K ′ k=1 V k h ′ (s k h ′ ) − V * h ′ (s k h ′ ) (ii) = K ′ k=1 H h=h ′ c h (s k h , a k h ) + P h (· | s k h , a k h )V k h+1 (·) − Q k h (s k h , a k h ) (i.1) + K ′ k=1 H h=h ′ (ζ 1 k,h + ζ 2 k,h ) (i.2) + K ′ k=1 H h=h ′ E Q k h (s h , ·), π k h (· | s h ) − π h (· | s h ) | s h ′ = s k h ′ , π, P (ii.1) + K ′ k=1 H h=h ′ E Q k h (s h , a h ) − c h (s h , a h ) − P h (· | s h , a h ) V k h+1 (·) | s h ′ = s k h ′ , π, P (ii.2) Proof. See Cai et al. (2020) for a detailed proof. A.2. Failure Events Definition A.2. We define the following failure events: Note that the martingales ζ 1 k,h and ζ 2 k,h defined in (19) satisfy |ζ 1 k,h + ζ 2 k,h | ≤ 4H. By Azuma-Hoeffding inequality, we have Pr(F 0 ) ≤ δ ′ . F 0 k = ∃h : k k ′ =1 H h ′ =h (ζ 1 k ′ ,h ′ + ζ 2 k ′ ,h ′ ) ≥ 16H 3 K ln 2H δ ′ , F 1 k =    ∃s, a, h : c h (s, a) − c k h (s, a) ≥ 2 ln 2SAHT δ ′ n k h (s, a)    , F 2 k =    ∃s, a, h : P h (· | s, a) − P k h (· | s, a) 1 ≥ 4S ln 3SAHT δ ′ n k h (s, a)    , F 3 k =    ∃s, a, h : P h (· | s, a) − P k h (· | s, a) V * h (·) ≥ H 4 ln 3SAHT δ ′ n k h (s, a) + 2H ln 2SAHT δ ′ 3n k h (s, a)    , F 4 k =      ∃s ′ , s, a, h : P k h (s ′ | s, a) − P h (s ′ | s, a) ≥ 2P k h (s ′ | s, a)(1 − P k h (s ′ | s, a)) ln 2SAHK δ ′ n k h (s, a) − 1 + 7 ln 2SAHK δ ′ 3n k h (s, a)      , F = K k=1 F 0 k ∪ F 1 k ∪ F 2 k ∪ F 3 k ∪ F 4 k , where δ ′ = δ 5 . Lemma A.3. It holds that Pr(F ) ≤ δ. Let F 1 = K k=1 F 1 k . By Hoeffding's inequality, we have Pr c h (s, a) − c k h (s, a) ≥ 2 ln 1 δ ′ n k h (s, a) ≤ δ ′(20) Using a union bound over all s, a and all possible values of n k (s, a) and k., we have Pr {F c } ≤ δ ′ . Let F 2 = K k=1 F 2 k . Then Pr F 2 ≤ δ ′ , which is implied by (Weissman et al., 2003) while applying union bound on all s, a and all possible values of n k (s, a) and k. Let F 3 = K k=1 F 3 k . According to (Azar et al., 2017), we have that with probability at least 1 − δ ′ P h − P k h V * h (s, a) ≤ 2H 2 ln 2HK δ ′ n k h (s, a) + 2H ln 2HK δ ′ 3n k h (s, a) .(21) Take a union bound over s, a, k, we have Pr F 3 ≤ δ ′ . Let F 4 = K k=1 F 4 k . The Empirical Bernstein inequality (Theorem 4 in (Maurer & Pontil, 2009)) combined with a union bound argument on s, a, s ′ ,n k h (s, a) also implies the following bound holds with probability at least 1 − δ ′ : P k h (s ′ | s, a) − P h (s ′ | s, a) ≤ 2P (s ′ | s, a)(1 − P (s ′ | s, a)) ln 2SAHK δ ′ n k h (s, a) − 1 + 7 ln 2SAHK δ ′ 3n k h (s, a)(22) Therefore, Pr{F 4 } ≤ δ ′ Finally, take a union bound with δ ′ = δ 5 , we have Pr {F } ≤ δ. Below we will assume that the failure event F does not happen, which is with high probability. B. Missing Proofs for Section 5 B.1. Proof of Lemma 5.1 Proof. By Lemma A.1, we have K ′ k=1 V π k h (s k h ) − V k h (s k h ) + K ′ k=1 H h=h ′ E Q k h (s h , ·), π k h (· | s h ) − π h (· | s h ) | s h ′ = s k h ′ , π, P + K ′ k=1 H h=h ′ (ζ 1 k,h + ζ 2 k,h ). Here ζ 1 k,h and ζ 2 k,h are martingales defined in (19) satisfying |ζ 1 k,h + ζ 2 k,h | ≤ 4H. Under Azuma-Hoeffding inequalities and the optimistic assumption that −2b k h (s, a) ≤ Q k h (s, a) − c h (s, a) − P h (·|s, a)V k h+1 ≤ 0, we finish the proof of Lemma 5.1. B.2. Full Proof of Theorem 3.1 As discussed in Section 5, we only need to prove the coarse regret in Lemma 5.4, but this time without the optimism assumption. We restate the lemma for ease of reading. Lemma B.1 (Coarse analysis of dynamic regret, restatement of Lemma 5.4). Conditioned on F c , for any K ′ ∈ [K], we have the following regret bound: Regret(K ′ ) = K ′ k=1 V π k 1 (s k 1 ) − V * 1 (s k 1 ) ≤ O( √ S 2 AH 4 K ′ ).(23) and the optimism holds, namely −2b k h (s, a) ≤ Q k h (s, a) − c h (s, a) − P h (· | s, a)V k h+1 (·) ≤ 0.(24) Remark B.2. In our algorithm, we change the Bregman penalty term from KL-divergence d KL (π h π k h ) to π h − π k h 2 2 , we also let the learning rate to be of the form η t = O( 1 √ t ) instead of a constant depending on K. We note that although this will cause more regret in the OMD term, it is crucial to obtain a Regret( K ′ ) = O( √ K ′ ) bound instead of Regret(K ′ ) = O(K ′ ). Proof. First, by Lemma A.1, we decompose the regret in the following way. K ′ k=1 V π k 1 (s 1 ) − V * 1 (s 1 ) = K ′ k=1 H h=1 c h (s k h , a k h ) + P h (· | s k h , a k h )V k h+1 (·) − Q k h (s k h , a k h ) (i) + K k=1 H h=1 (ζ 1 k,h + ζ 2 k,h ) (ii) + K ′ k=1 H h=1 E Q k h (s h , ·), π k h (· | s h ) − π h (· | s h ) | s 1 = s k 1 , π, P (iii) + K ′ k=1 H h=1 E Q k h (s h , a h ) − c h (s h , a h ) − P h (· | s h , a h ) V k h+1 (·) | s 1 = s k 1 , π, P(iv) . We first prove (23) and (24) for early episodes, namely when n k h < C 0 √ k for all s (which is true at the beginning of the algorithm), then the regret bound holds. Note that in this case, conditioned on F c , we have SH √ C0K ′ 1/4 n k h (s,a) ≥ H 2 S n k h (s,a) , which implies b k h (s, a) = 2 ln 2SAHT δ ′ n k h (s,a) + H 4S ln 3SAHT δ ′ n k h (s,a) . In this case, by Lemma B.3, we have −2b k h (s, a) ≤ Q k h (s, a) − c h (s, a) − P h (· | s, a)V k h+1 (·) ≤ 0, which further implies that Term (i) ≤O K ′ k=1 H h=1 b k h (s k h , a k h ) ≤O K ′ k=1 H h=1 2 ln 2SAHT δ ′ n k h (s, a) + H 4S ln 3SAHT δ ′ n k h (s, a) ≤ O( √ S 2 AH 4 K ′ ), and Term (iv) ≤ 0. Also, by Lemma B.3, we know that (24) holds in this period. Applying Lemma B.6 to Term (iii), we have Term (iii) ≤ O( √ AH 4 K ′ ) Therefore, under event F c , we have K ′ k=1 V π k 1 (s 1 ) − V * 1 (s 1 ) ≤ O( √ S 2 AH 4 K ′ ) Next, we prove (23) and (24) for the remaining episodes. We prove this claim by induction. In fact, we will prove the following claim: for each episode, (23) and (24) hold. We have shown that this claim holds for the first episodes. Assume that for k = K ′ − 1, we have 1)), we want to prove that for k = K ′ , we have K ′ −1 k=1 (V π k 1 (s 1 ) − V * 1 (s 1 )) ≤ O( S 2 AH 4 (K ′ −K ′ k=1 (V π k 1 (s 1 ) − V * 1 (s 1 )) ≤ O( √ S 2 AH 4 K ′ ). If there exists (h, s, a) such that n K ′ h (s) ≥ C 0 √ K ′ (i.e. we construct V ref h (s) in this episode), then since (23) and (24) holds for previous episodes, using Lemma 5.3 we have V ref h (s) − V * h (s) = 1 n K ′ h (s) K ′ i=1 V i h (s i h ) − V * h (s) 1[s i h = s] ≤ 1 n K ′ h (s) K ′ i=1 V i h (s i h ) − V * h (s) 1[s i h = s] ≤ 1 C 0 √ K ′ K ′ i=1 V i h (s) − V * h (s) ≤ 1 C 0 √ K ′ √ S 2 AH 4 K ′ = √ S 2 AH 4 C 0 Therefore, by Lemma B.4 we know that Q k h is an optimistic estimation of Q * h . Together with Lemmas A.1 and B.6, we have K ′ k=1 V π k 1 (s 1 ) − V * 1 (s 1 ) ≤O K ′ k=1 H h=1 b k h (s k h , a k h ) + O( √ AH 4 K ′ ) ≤O K ′ k=1 H h=1 2 ln 2SAHT δ ′ n k h (s, a) + H 4S ln 3SAHT δ ′ n k h (s, a) + O( √ AH 4 K ′ ) ≤ O( √ S 2 AH 4 K ′ ), which concludes our proof. B.3. Proof of Lemma 5.3 Proof. We have K ′ k=1 |V k h (s k h ) − V * h (s k h )| ≤ K ′ k=1 |V k h (s k h ) − V π k h (s k h )| + K ′ k=1 |V π k h (s k h ) − V * h (s k h )| = K ′ k=1 (V π k h (s k h ) − V k h (s k h )) + K ′ k=1 (V π k h (s k h ) − V * h (s k h )) where in the last step we use optimism and definition of V * . By Lemma 5.1, the first term K ′ k=1 (V π k h (s k h ) − V k h (s k h )) can be bounded as K ′ k=1 V π k h (s h ) − V k h (s h ) ≤O K ′ k=1 H h ′ =h b k h ′ (s k h ′ , a k h ′ ) + O( √ H 3 K ′ ) ≤O K ′ k=1 H h ′ =h H 2 S n k h ′ (s k h ′ , a k h ′ ) + O( √ H 3 K ′ ) ≤ O( √ S 2 AH 4 K ′ ).(25) For the second term K ′ k=1 (V π k h (s k h ) − V * h (s k h )), we use a forward induction trick: First we have K ′ k=1 (V π k h (s k h ) − V * h (s k h )) ≤ O( √ S 2 AH 4 K ′ ) holds for h = 1. By induction, if the claim holds up to h, then we have K ′ k=1 V π k h (s k h ) − V * h (s k h ) = K ′ k=1 Q π k h (s k h , ·), π k h (·|s k h ) − Q * h (s k h , ·), π * h (·|s k h ) ≥ K ′ k=1 Q π k h (s k h , ·), π k h (·|s k h ) − Q * h (s k h , ·), π k h (·|s k h ) = K ′ k=1 P h (s k h , ·)(V π k h+1 − V * h+1 ), π k h (·|s k h ) = K ′ k=1 V π k h+1 (s k h+1 ) − V * h+1 (s k h+1 ) + P h (·|s k h , π k h (s k h ))(V π k h+1 − V * h+1 )(·) − (V π k h+1 − V * h+1 )(s k h+1 ) (a) , where Term (a) is a martingale. Using Azuma-Hoeffding inequality, we have Term (a) ≤ O( √ H 2 K ′ ). Therefore by induction, for all h ∈ [H] we have K ′ k=1 V π k h+1 (s k h+1 ) − V * h+1 (s k h+1 ) ≤ K ′ k=1 V π k 1 (s k 1 ) − V * 1 (s k 1 ) + O(h √ H 2 K ′ ) ≤ O( √ S 2 AH 4 K ′ ), which concludes the proof of Lemma 5.3. , we have B.4. Useful that −2b k h (s, a) ≤ Q k h (s, a) − c h (s, a) − P h (· | s, a)V k h+1 (·) ≤ 0.(26) Proof. Recall that Q k h takes form Q k h (s, a) = max{c k h (s, a) + P k h (· | s, a)V k h+1 (·) − b k h (s, a), 0}.(27) We have c h (s, a) + P h (· | s, a)V k h+1 (·) − Q k h (s, a) ≤c h (s, a) − c k h (s, a) + P h (· | s, a)V k h+1 (·) − P k h (· | s, a)V k h+1 (·) + b k h (s, a).(28) Under the event F c , we have c h (s, a) − c k h (s, a) ≤ 2 ln 2SAHT δ ′ n k h (s, a) ,(29) and P h (· | s, a)V k h+1 (·) − P k h (· | s, a)V k h+1 (·) ≤ P k h (· | s h , a h ) − P h (· | s h , a h ) 1 V k h+1 (·) ∞ ≤H · P k h (· | s h , a h ) − P h (· | s h , a h ) 1 ≤H 4S ln 3SAHT δ ′ n k h (s, a) .(30) Here the first inequality follows from Cauchy-Schwartz inequality, the second inequality uses the fact that V k h (·) ∞ ≤ H for any (k, h) ∈ [K] × [H], and the last inequality holds conditioned on the event F c . Plugging (29) and (30) into (28), together with the definition of b k h , we obtain c h (s, a) + P h (· | s, a)V k h+1 (·) − Q k h (s, a) ≤ 2b k h .(31) Similarly, by the definition of Q k h in (27) , we have Q k h (s, a) − c h (s, a) − P h (· | s, a)V k h+1 (·) ≤ max c k h (s, a) − c h (s, a) + P k h (· | s, a)V k h+1 (·) − P h (· | s, a)V k h+1 (·) − b k h (s, a), 0 ≤ max{b k h − b k h , 0} = 0,(32) where the last inequality follows from (29) and (30). Combining (31) and (32), we finish the proof. Lemma B.4. Conditioned on the event F c , when the reference function V ref satisfies |V ref (s) − V * (s)| ≤ √ S 2 AH 4 C0 where C 0 = √ S 3 AH 3 , we have −2b k h (s, a) ≤ Q k h (s, a) − c h (s, a) − P h (· | s, a)V k h+1 (·) ≤ 0. Proof. When F does not happen, we have c h (s, a) − c k h (s, a) ≤ 2 ln 2SAHT δ ′ n k h (s, a) and P h (· | s, a)V k h+1 (·) − P k h (· | s, a)V k h+1 (·) ≤ H 4S ln 3SAHT δ ′ n k h (s, a) . Meanwhile, we have P h (· | s, a)V k h+1 (·) − P k h (· | s, a)V k h+1 (·) ≤ P k h (· | s, a) − P h (· | s, a) V * h+1 (·) + P k h (· | s, a) − P h (· | s, a) (V k h+1 − V ref h+1 )(·) + P k h (· | s, a) − P h (· | s, a) (V ref h+1 − V * h+1 )(·)(33) For the first term, we have P k h (· | s, a) − P h (· | s, a) V * h+1 ≤ 2V Y ∼P h (·|s,a) V * h+1 (Y ) ln 2SAHK δ ′ n k h (s, a) + 7 ln 2SAHK δ ′ 3n k h (s, a) ≤ 4V Y ∼P h (·|s,a) V ref h+1 (Y ) ln 2SAHK δ ′ n k h (s, a) + 4H ln 2SAHK δ ′ S · n k h (s, a) + 7 ln 2SAHK δ ′ 3n k h (s, a) ,(34) where the last inequality follows from the fact that 2V(X) ≤ 2 V(Y ) + V(Y − X) ≤ 2 V(Y ) + 2 V(Y − X) and |V * h+1 − V ref h+1 | ≤ H/S. Under the event F c , we have P k h (y | s, a) − P h (y | s, a) ≤ 2P k h (y | s, a)(1 − P k h (y | s, a)) ln 2SAHK δ ′ n k h (s, a) − 1 + 7 ln 2SAHK δ ′ 3n k h (s, a) , By AM-GM inequality, we have 1 2 P k h (y | s, a) + ln 2SAHK δ ′ n k h (s, a) ≥ 2P k h (y | s, a) ln 2SAHK δ ′ n k h (s, a) ≥ 2P k h (y | s, a) 1 − P k h (y | s, a) ln 2SAHK δ ′ n k h (s, a) − 1 which further implies that P h (y | s, a) − 3 2 · P k h (y | s, a) ≤ 10 ln 2SAHK δ ′ 3n k h (s, a) . Then, we have V Y ∼P h (·|s,a) V ref h+1 (Y ) = y∈S P h (y | s, a) V ref h+1 (y) − P k h (· | s, a)V ref h+1 (·) 2 ≤ y∈S P h (y | s, a) V ref h+1 (y) − P k h (· | s, a)V ref h+1 (·) 2 ≤ y∈S 3 2 P k h (y | s, a) + 10 ln 2SAHK δ ′ 3n k h (s, a) · V ref h+1 (y) − P k h (· | s, a)V ref h+1 (·) 2 ≤ 3 2 V Y ∼P k h (·|s,a) V ref h+1 (Y ) + 10SH 2 ln 2SAHK δ ′ 3n k h (s, a) .(35) Plugging (35) into (34), we obtain P k h (· | s, a) − P h (· | s, a) V * h+1 ≤ 6V Y ∼P k h (·|s,a) V ref h+1 (Y ) ln 2SAHK δ ′ n k h (s, a) + 4H ln 2SAHK δ ′ S · n k h (s, a) + 8 √ SH 2 ln 2SAHK δ ′ 3n k h (s, a) .(36) Nearly Optimal Policy Optimization with Stable at Any Time Guarantee Meanwhile, it holds that P k h (· | s, a) − P h (· | s, a) (V k h+1 − V ref h+1 ) ≤ y∈S P k h (y | s, a) − P h (y | s, a) · |V k h+1 (y) − V ref h+1 (y)| ≤ y∈S   2P h (y | s, a)(1 − P h (y | s, a)) ln 2SAHK δ ′ n k h (s, a) − 1 + 7 ln 2SAHK δ ′ 3n k h (s, a)   |V k h+1 (y) − V ref h+1 (y)|,(37) where the last inequality follows from the definition of event F c . Plugging (36) and (37) into (33), we have P h (· | s, a)V k h+1 (·) − P k h (· | s, a)V k h+1 (·) ≤ 6V Y ∼P k h (·|s,a) V ref h+1 (Y ) ln 2SAHK δ ′ n k h (s, a) + 4H ln 2SAHK δ ′ S · n k h (s, a) + 8 √ SH 2 ln 2SAHK δ ′ 3n k h (s, a) + y∈S   2P h (y | s, a)(1 − P h (y | s, a)) ln 2SAHK δ ′ n k h (s, a) − 1 + 7 ln 2SAHK δ ′ 3n k h (s, a)   |V k h+1 (y) − V ref h+1 (y)| + y∈S P k h (· | s, a) − P h (· | s, a) |V ref h+1 (y) − V * h+1 (y)| (b) For Term (b), we divide S into two sets: S 0 = {y ∈ S : n k h (y) ≥ C 0 √ k} and S c 0 . Since |V ref h+1 (s) − V * h+1 (s)| ≤ √ S 2 AH 4 C0 , we have y∈S0 P k h (· | s, a) − P h (· | s h , a h ) |V ref h+1 (y) − V * h+1 (y)| ≤ 4S ln 3SAHT δ ′ n k h (s, a) √ S 2 AH 4 C 0 Nearly Optimal Policy Optimization with Stable at Any Time Guarantee For y ∈ S c 0 , we have y∈S c 0 P k h (· | s, a) − P h (· | s h , a h ) |V ref h+1 (y) − V * h+1 (y)| ≤ y∈S c 0    2P k h (s ′ | s, a)(1 − P k h (s ′ | s, a)) ln 2SAHK δ ′ n k h (s, a) − 1 + 7 ln 2SAHK δ ′ 3n k h (s, a)    |V ref h+1 (y) − V * h+1 (y)| ≤ y∈S c 0    4P k h (s ′ | s, a) ln 2SAHK δ ′ n k h (s, a)    |V ref h+1 (y) − V * h+1 (y)| + 7 ln 2SAHK δ ′ 3n k h (s, a) |V ref h+1 (y) − V * h+1 (y)| = y∈S c 0 4n k h (s, a, y) ln 2SAHK δ ′ n k h (s, a) 2 |V ref h+1 (y) − V * h+1 (y)| + 7SH ln 2SAHK δ ′ 3n k h (s, a) ≤ y∈S c 0 4n k h (y) ln 2SAHK δ ′ n k h (s, a) 2 H + 7SH ln 2SAHK δ ′ 3n k h (s, a) ≤ y∈S c 0 4C 0 √ K ln 2SAHK δ ′ n k h (s, a) 2 H + 7SH ln 2SAHK δ ′ 3n k h (s, a) ≤ 2S √ C 0 K 1/4 H ln 2SAHK δ ′ n k h (s, a) + 7SH ln 2SAHK δ ′ 3n k h (s, a) Therefore, we have a), 0}. Then we can prove that (31) and (32) still hold, which finishes the proof. P k h (· | s h , a h ) − P h (· | s h , a h ) V k h+1 ≤ 6V Y ∼P k h (·|s,a) V ref h+1 (Y ) ln 2SAHK δ ′ n k h (s, a) + 4H ln 2SAHK δ ′ S · n k h (s, a) + 8 √ SH 2 ln 2SAHK δ ′ 3n k h (s, a) + y∈S    2P k h (s ′ | s, a)(1 − P k h (s ′ | s, a)) ln 2SAHK δ ′ n k h (s, a) − 1 + 7 ln 2SAHK δ ′ 3n k h (s, a)    |V k h+1 (y) − V ref h+1 (y)| + 4S ln 3SAHT δ ′ n k h (s, a) √ S 2 AH 4 C 0 + 2S √ C 0 K 1/4 H ln 2SAHK δ ′ n k h (s, a) + 7SH ln 2SAHK δ ′ 3n k h (s, a) By definition, Q k h (s, a) = max{c k h (s, a) + P k h (· | s, a)V k h+1 (·) − b k h (s, B.4.2. MIRROR DESCENT The mirror descent (MD) algorithm (Beck & Teboulle, 2003) is a proximal convex optimization method that minimizes a linear approximation of the objective together with a proximity term, defined in terms of a Bregman divergence between the old and new solution estimates. In our analysis we choose the Bregman divergence to be the l 2 norm. If {f k } K k=1 is a sequence of convex functions f k : R d → R and C is a constraints set, the k-th iterate of MD is the following: x k+1 ∈ arg min x∈C η k g k (x k ) , x − x k + x − x k 2 2 ,(38) where η k is the stepsize. The MD algorithm ensures Regret (K ′ ) = K ′ k=1 f (x k ) − min x f (x) ≤ O( √ K ′ ) for all K ′ ∈ [K]. Then, by the definition of b k h , K k=1 H h=1 b k h (s k h , a k h ) ≤ K k=1 H h=1 O    V Y ∼P k h (·|s,a) V * h+1 (Y ) n k h (s, a) + y∈S P k h (y) n k h |V k h+1 (y) − V ref1 P k h (y)n k h |V k h+1 (y) − V ref h+1 (y)| − 1 P k h (s k h+1 )n k h |V k h+1 (s k h+1 ) − V ref h+1 (s k h+1 )| + K k=1 H h=1 1 P k h (s k h+1 )n k h |V k h+1 (s k h+1 ) − V ref h+1 (s k h+1 )| ≤ O( √ H 3 K) + K k=1 H h=1 1 P k h (s k h+1 )n k h |V k h+1 (s k h+1 ) − V ref h+1 (s k h+1 )| = K k=1 H h=1 1 P k h (s k h+1 )n k h |V k h+1 (s k h+1 ) − V * h+1 (s k h+1 )| the sum ≤ O( √ H 4 S 2 AK) + 1 P k h (s k h+1 )n k h |V * h+1 (s k h+1 ) − V ref h+1 (s k h+1 )| almost ≤ O(1/ √ S) + O( √ H 3 K) ≤ O(S 2 AH √ H 2 S 2 AK) + O( √ S 2 AH 2 K/ √ S) + O( √ H 3 K) = O( √ SAH 3 K) + O(S 5/2 A 5/4 H 3/2 K 1/4 ) The second line is the sum of martingale differences bounded by H, directly applying Azuma-Hoeffding inequality yields the second inequality. The third inequality makes use of Lemma 11 in (Zhang et al., 2020b). (44) So we only have to deal with the first term, in which P k h (y)n k h is large. Thus the martingale can be bounded. The third inequality in The last inequality in (44) As above, we only need to consider the case where P k h (s k h+1 )n k h ≥ 2H 2 SL (n k h is the shorthand for n k h (s k h , a k h )). Then we know the first term is bounded by O( √ SAH 3 K). For the second term, we need to use the Multiplicative Chernoff bound (See the Wiki for Chernoff bound): Pr(X ≤ (1 − δ)µ) ≤ e − δ 2 µ 2 , where 0 ≤ δ ≤ 1 and X = n i=1 X i , X i are independent Bernoulli r.v., E[X] = µ. Set X = n k h (s k h , a k h , s k h+1 ), µ = P k h (s k h+1 )n k h . Taking union bound over all h, k, S with P k h (s k h+1 )n k h ≥ 2H 2 SL, we have that with high probability, P k h (s k h+1 )n k h ≥ 1 2 n k h (s k h , a k h , s k h+1 ). With this we can now apply Lemma 11 in (Zhang et al. 2020). C. Explanation of How UCBVI Uses Optimism In Azar et al. (2017), they need to bound the term (P k h − P h )(· | s, a)(V k h+1 − V * h+1 )(·) using optimism, as mentioned in Section 4. Define ∆ k h def = V * h − V π k h , ∆ k h def = V k h − V π k h , and δ k h def = ∆ k h s k h . We denote by a numerical constant which can vary from line to line. We also use L to represent the logarithmic term L = ln( HSAT /δ). Using Bernstein's inequality, this term is bounded by y P π k y | s k h L P π k y | s k h n k h ∆ k h+1 (y) + SHL n k h . where n k h def = n k s k h , π k s k h . Now considering only the y such that P π k y | s k h n k h ≥ H 2 L, and since 0 ≤ ∆ k h+1 ≤ ∆ k h+1 by optimism, then P π k k − P π k ∆ k,h+1 s k h is bounded by ǫ k h + L P π k s k h+1 | s k h n k h δ k,h+1 + SHL n k h ≤ ǫ k h + 1 H δ k h+1 + SHL n k h . where ǫ k h def = L n k h y P π k y | s k h ∆ k h+1 (y) P π k (y|s k h ) − δ k h+1 P π k (s k h+1 |s k h ) . The sum over the neglected y such that P π k y | s k h n k h < H 2 L contributes to an additional term y P π k y | s k h n k h L n k h 2 ∆ k h+1 (y) ≤ SH 2 L n k h . Then they prove that the sum of ǫ k h is of order O( √ T ) and the sum of 1 n k h is a constant order term. (2017); Zanette & Brunskill (2019); Zhang et al. (2020a;b); Proof. These standard concentration inequalities also appear in Azar et al. (2017); Cai et al. (2020); Shani et al. (2020). For completeness, we present the proof sketch here. K do3: Rollout a trajectory by acting π k 4: Update counters and empirical model n k = {n k h } h∈[H] , c k = {c k h } h∈[H] , P k = {P k h } h∈[H] 5: for Step h = H, ..., 1 do 6: from coarse regret bound is highly non-trivial. There exists two challenges: 1. Previous policy-based methods (Cai et al., 2020; Shani et al., 2020) choose a constant mirror descent stepsize depending on K, which impedes us from obtaining regret bound sublinear on K ′ for all K ′ ∈ [K]. 2. For the step 2 ≤ h ≤ H, the state s k h is not fixed, which makes the OMD term difficult to bound. Learning, pp. 263-272. PMLR, 2017. Beck, A. and Teboulle, M. Mirror descent and nonlinear projected subgradient methods for convex optimization.Fei, Y., Yang, Z., Wang, Z., and Xie, Q. Dynamic regret of policy optimization in non-stationary environments.Menard, P., Domingues, O. D., Shang, X., and Valko, M. Ucb momentum q-learning: Correcting the bias without forgetting. arXiv preprint arXiv:2103.01312, 2021.Orabona, F. A modern introduction to online learning. arXiv preprint arXiv:1912.13213, 2019.Operations Research Letters, 31(3):167-175, 2003. Bhandari, J. and Russo, D. Global optimality guar- antees for policy gradient methods. arXiv preprint arXiv:1906.01786, 2019. Cai, Q., Yang, Z., Jin, C., and Wang, Z. Provably efficient exploration in policy optimization. In International Con- ference on Machine Learning, pp. 1283-1294. PMLR, 2020. Domingues, O. D., Ménard, P., Kaufmann, E., and Valko, M. Episodic reinforcement learning in finite mdps: Min- imax lower bounds revisited. In Algorithmic Learning Theory, pp. 578-598. PMLR, 2021. Fazel, M., Ge, R., Kakade, S., and Mesbahi, M. Global convergence of policy gradient methods for the linear quadratic regulator. In International Conference on Ma- chine Learning, pp. 1467-1476. PMLR, 2018. arXiv preprint arXiv:2007.00148, 2020. Gu, S., Holly, E., Lillicrap, T., and Levine, S. Deep rein- forcement learning for robotic manipulation with asyn- chronous off-policy updates. In 2017 IEEE interna- tional conference on robotics and automation (ICRA), pp. 3389-3396. IEEE, 2017. Haarnoja, T., Zhou, A., Hartikainen, K., Tucker, G., Ha, S., Tan, J., Kumar, V., Zhu, H., Gupta, A., Abbeel, P., et al. Soft actor-critic algorithms and applications. arXiv preprint arXiv:1812.05905, 2018. He, J., Zhou, D., and Gu, Q. Nearly optimal regret for learn- ing adversarial mdps with linear function approximation. arXiv preprint arXiv:2102.08940, 2021. Jaksch, T., Ortner, R., and Auer, P. Near-optimal regret bounds for reinforcement learning. Journal of Machine Learning Research, 11(4), 2010. Jin, C., Allen-Zhu, Z., Bubeck, S., and Jordan, M. I. Is q-learning provably efficient? arXiv preprint arXiv:1807.03765, 2018. Kakade, S. M. A natural policy gradient. Advances in neural information processing systems, 14, 2001. Kalashnikov, D., Irpan, A., Pastor, P., Ibarz, J., Herzog, A., Jang, E., Quillen, D., Holly, E., Kalakrishnan, M., Van- houcke, V., et al. Qt-opt: Scalable deep reinforcement learning for vision-based robotic manipulation. arXiv preprint arXiv:1806.10293, 2018. Lancewicki, T., Rosenberg, A., and Mansour, Y. Learn- ing adversarial markov decision processes with delayed feedback. arXiv preprint arXiv:2012.14843, 2020. Levine, S., Finn, C., Darrell, T., and Abbeel, P. 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Op- timistic policy optimization with bandit feedback. In In- ternational Conference on Machine Learning, pp. 8604- 8613. PMLR, 2020. Silver, D., Huang, A., Maddison, C. J., Guez, A., Sifre, L., Van Den Driessche, G., Schrittwieser, J., Antonoglou, I., Panneershelvam, V., Lanctot, M., et al. Mastering the game of go with deep neural networks and tree search. nature, 529(7587):484-489, 2016. Silver, D., Schrittwieser, J., Simonyan, K., Antonoglou, I., Huang, A., Guez, A., Hubert, T., Baker, L., Lai, M., Bolton, A., et al. Mastering the game of go without hu- man knowledge. nature, 550(7676):354-359, 2017. Sutton, R. S., McAllester, D. A., Singh, S. P., Mansour, Y., et al. Policy gradient methods for reinforcement learning with function approximation. In NIPs, volume 99, pp. 1057-1063. Citeseer, 1999. Wang, L., Cai, Q., Yang, Z., and Wang, Z. Neural policy gradient methods: Global optimality and rates of conver- gence. arXiv preprint arXiv:1909.01150, 2019. Weissman, T., Ordentlich, E., Seroussi, G., Verdu, S., and Weinberger, M. J. Inequalities for the l1 deviation of the empirical distribution. Hewlett-Packard Labs, Tech. Rep, 2003. Zanette, A. and Brunskill, E. Tighter problem-dependent regret bounds in reinforcement learning without domain knowledge using value function bounds. In Interna- tional Conference on Machine Learning, pp. 7304-7312. PMLR, 2019. Zanette, A., Lazaric, A., Kochenderfer, M., and Brunskill, E. Learning near optimal policies with low inherent bellman error. In International Conference on Machine Learning, pp. 10978-10989. PMLR, 2020. Zanette, A., Cheng, C.-A., and Agarwal, A. Cau- tiously optimistic policy optimization and exploration with linear function approximation. arXiv preprint arXiv:2103.12923, 2021. The second inequality follows fromLemma 19 in Zhang et al. (2020b) and standard techniques (e.g.(Shani et al., 2020) or(Zhang et al., 2020b)). Using Lemma B.8, we can further bound Term (v) as:Term (v) ≤ O( √ SAH 3 K) + O(S 5/2 A 5/4 H 3/2 K 1/4 )Therefore, combining Equation(16)and(43), we complete the proof of Theorem 3.1.Lemma B.8.Proof. We haveh+1 (y)| + S 3/2 A 1/4 H 7/4 K 1/4 n k h    ≤ O( √ SAH 3 K) + K k=1 H h=1 y∈S O    P k h (y) n k h |V k h+1 (y) − V ref h+1 (y)|    + O(S 5/2 A 5/4 H 11/4 K 1/4 ) ≤ O( √ SAH 3 K) + K k=1 H h=1 y∈S O    P k h (y) + O( 1/n k h ) n k h |V k h+1 (y) − V ref h+1 (y)|    + O(S 5/2 A 5/4 H 11/4 K 1/4 ) ≤ O( √ SAH 3 K) + K k=1 H h=1 y∈S O P k h (y) n k h |V k h+1 (y) − V ref h+1 (y)| (v) + O(S 5/2 A 5/4 H 11/4 K 1/4 ) (43) K k=1 H h=1 y∈S P k h (y) n k h |V k h+1 (y) − V ref h+1 (y)| ≤ O( √ SAH 3 K) + O(S 5/2 A 5/4 H 3/2 K 1/4 ) K k=1 H h=1 y∈S P k h (y) n k h |V k h+1 (y) − V ref h+1 (y)| ≤ K k=1 H h=1 y∈S P k h (y) This is a technique used in(Azar et al. 2017). We explain the main idea here.First we define the typical state-actions pairs as which means these state-action pairs are visited frequently enough.Define ∆ k h+1 (y) = V k h+1 (y) − V ref h+1 (y) . We haveThe second term can be bounded by[y] k,h def = y : y ∈ S, n k h (s k h , a k h )P (y | s k h , a k h ) ≥ 2H 2 SL y∈S P k h (y) n k h ∆ k h+1 (y) = y∈[y] k,h P k h (y) n k h ∆ k h+1 (y) + y / ∈[y] k,h P k h (y) n k h ∆ k h+1 (y). y / ∈[y] k,h P k h (y)n k h n k h 2 ∆ k h+1 (y) ≤ SH √ 4LH 2 S n k h AcknowledgementLiwei Wang was supported by Exploratory Research Project of Zhejiang Lab (No. 2022RC0AN02), Project 2020BD006 supported by PKUBaidu Fund, the major key project of PCL (PCL2021A12).The following lemma(Theorem 6.8 in (Orabona, 2019)) is a fundamental inequality for analysis of OMD regret, which will be used in our analysis.Lemma B.5 (OMD Regret, Theorem 6.8 in(Orabona, 2019)). Assume η k+1 ≤ η k , k = 1, . . . , K. Then, using OMD with the l 2 norm, learning rate {η k } and uniform initialization x 1 = [1/d, . . . , 1/d], the following regret bounds holdIn our analysis, by adapting the above lemma to our notation, we get the following lemma.Lemma B.6 (OMD in Policy Optimization). Assume η k+1 ≤ η k , k = 1, . . . , K. Then, using OMD with the l 2 norm, learning rate {η k } and uniform initialization π 1 h (· | s) = [1/A, . . . , 1/A], the following regret bounds holdfor any h ∈ [H]. Taking summation over h ∈ [H] concludes our proof.B.5. Sum of BonusLemma B.7. It holds thatProof. For simplicity we let n k
[]
[ "Self-supervised Video Transformer", "Self-supervised Video Transformer" ]
[ "Kanchana Ranasinghe [email protected] \nStony Brook University\n\n\nZayed University of AI\n\n", "Muzammal Naseer \nNational University\n\n\nZayed University of AI\n\n", "Salman Khan \nNational University\n\n\nZayed University of AI\n\n", "Fahad Shahbaz Khan \nLinköping University\n\n", "Michael Ryoo \nStony Brook University\n\n\nZayed University of AI\n\n" ]
[ "Stony Brook University\n", "Zayed University of AI\n", "National University\n", "Zayed University of AI\n", "National University\n", "Zayed University of AI\n", "Linköping University\n", "Stony Brook University\n", "Zayed University of AI\n" ]
[]
In this paper, we propose self-supervised training for video transformers using unlabeled video data. From a given video, we create local and global spatiotemporal views with varying spatial sizes and frame rates. Our selfsupervised objective seeks to match the features of these different views representing the same video, to be invariant to spatiotemporal variations in actions. To the best of our knowledge, the proposed approach is the first to alleviate the dependency on negative samples or dedicated memory banks in Self-supervised Video Transformer (SVT). Further, owing to the flexibility of Transformer models, SVT supports slow-fast video processing within a single architecture using dynamically adjusted positional encoding and supports long-term relationship modeling along spatiotemporal dimensions. Our approach performs well on four action recognition benchmarks (Kinetics-400, UCF-101, HMDB-51, and SSv2) and converges faster with small batch sizes. Code is available at: https://git.io/J1juJ
10.1109/cvpr52688.2022.00289
[ "https://arxiv.org/pdf/2112.01514v2.pdf" ]
244,800,737
2112.01514
0a541dfdd516d00d83c7a974f828477416d65455
Self-supervised Video Transformer Kanchana Ranasinghe [email protected] Stony Brook University Zayed University of AI Muzammal Naseer National University Zayed University of AI Salman Khan National University Zayed University of AI Fahad Shahbaz Khan Linköping University Michael Ryoo Stony Brook University Zayed University of AI Self-supervised Video Transformer Code is available at: https://git.io/J1juJ In this paper, we propose self-supervised training for video transformers using unlabeled video data. From a given video, we create local and global spatiotemporal views with varying spatial sizes and frame rates. Our selfsupervised objective seeks to match the features of these different views representing the same video, to be invariant to spatiotemporal variations in actions. To the best of our knowledge, the proposed approach is the first to alleviate the dependency on negative samples or dedicated memory banks in Self-supervised Video Transformer (SVT). Further, owing to the flexibility of Transformer models, SVT supports slow-fast video processing within a single architecture using dynamically adjusted positional encoding and supports long-term relationship modeling along spatiotemporal dimensions. Our approach performs well on four action recognition benchmarks (Kinetics-400, UCF-101, HMDB-51, and SSv2) and converges faster with small batch sizes. Code is available at: https://git.io/J1juJ Introduction Self-supervised learning enables extraction of meaningful representations from unlabeled data, alleviating the need for expensive annotations. Recent self-supervised methods perform on-par with supervised learning for certain vision tasks [11,12,16,37]. The necessity of self-supervised learning is even greater in domains such as video analysis where annotations are more expensive [40,44,63,65]. At the same time, the emergence of vision transformers (ViTs) [26] and their successful adoption to different computer vision tasks including video understanding [4,9,29,53,67,68] within the supervised setting shows their promise in the video domain. In fact, recent works using simple ViT backbones [9] surpass convolutional neural networks (CNN) for supervised video analysis with reduced compute. Motivated by the ability of self-attention to model long-range dependencies, we propose a simple yet effective method to train video transformers [9] in a self-supervised manner. This process uses spatial and temporal context as a supervisory signal (from unlabelled videos) to learn motion, scale, and viewpoint invariant features. Many existing self-supervised representation learning methods on videos [64,65,88] use contrastive learning objectives which can require larger batch sizes, longer training regimes, careful negative mining and dedicated memory banks. Further, the contrastive objectives require careful temporal sampling [64] and multiple networks looking at similar/different clips to develop attract/repel loss formulations [65]. In contrast, we propose to learn self-supervised features from unlabelled videos via self-distillation [72] by a twin network strategy (student-teacher models) [13,34]. Our proposed approach, Self-supervised Video Transformer (SVT), trains student and teacher models with a similarity objective [13] that matches the representations along spatial and temporal dimensions by space and time attention [9]. We achieve this by creating spatiotemporal positive views that differ in spatial sizes and are sampled at different time frames from a single video (Fig. 1). During training, teacher video transformer parameters are updated as an exponential moving average of the student video transformer. Both of these networks process different spatiotemporal views of the same video and our objective function is designed to predict one view from the other in the feature space. This allows SVT to learn robust features that are invariant to spatiotemporal changes in videos while generating discriminative features across videos [34]. SVT does not depend on negative mining or large batch sizes and remains computationally efficient as it converges within only a few epochs (≈ 20 on Kinetics-400 [14]). In addition to the above advantages, our design allows the flexibility to model varying time-resolutions and spatial scales within a unified architecture. This is a much desired feature for video processing since real-world actions can occur with varying temporal and spatial details. Remarkably, current self-supervision based video frameworks [64,87] operate on fixed spatial and temporal scales which can pose difficulties in modeling the expressivity and dynamic nature of actions. We note that convolutional backbones used in these approaches lack the adaptability to varying temporal resolutions (due to fixed number of channels) and thus require dedicated networks for each resolution [30,45]. To address this challenge, the proposed SVT uses dynamically adjusted positional encodings to handle varying temporal resolutions within the same architecture. Further, the selfattention mechanism in SVT can capture both local and global long-range dependencies across both space and time, offering much larger receptive fields as compared to traditional convolutional kernels [57]. The main contributions in this work are as follows: • We introduce a novel mechanism for self-supervised training of video transformers by exploiting spatiotemporal correspondences between varying fields of view (global and local) across space and time (Sec. 3.2). • Self-supervision in SVT is performed via a joint motion and crossview correspondence learning objective. Specifically, global and local spatiotemporal views with varying frame rates and spatial characteristics (Sec. 3 Our extensive experiments and results on various video datasets including Kinetics-400 [14], UCF-101 [69], HMDB-51 [49], and SSv2 [33] show state-of-the-art transfer of our self-supervised features using only RGB data. Further, our method shows a rapid convergence rate. Related Work Transformers in Vision. Since the initial success of transformers in natural language processing (NLP) tasks [21,77], they have emerged as a competitive architecture for various other domains [46]. Among vision tasks, the initial works focused on a combination of convolutional and selfattention based architectures [10,83,85,89]. A convolution free variant, vision transformer (ViT) [26], achieved competitive performance on image classification tasks. While earlier works proposing ViT [26] depended on large-scale datasets, more recent efforts achieve similar results with medium-scale datasets using various augmentation strategies [71,76]. Later architectures also explore improving computational efficiency of ViTs focusing on transformer blocks [52,67]. ViTs have also been adopted for video classification tasks [4,9,29,67,68]. Our work builds on the TimeSformer backbone [9], a direct adaptation of standard ViTs using separate attention across dimensions. Self-supervised Learning in Images. Early imagebased self-supervised learning work focused on pretext tasks that require useful representations to solve [24,25,48,58,61,78,90]. However, recently contrastive methods have dominated self-supervised learning [6,13,16,17,19,27,37,38,50,55,74,75] . These approaches generally consider two views of a single sample (transformed through augmentations) and pull them (positives) together while pushing away from all other (negative) samples in representation space [6,59]. Key drawbacks of these methods are the necessity for careful mining of positive / negative samples [75] and reliance on large numbers of negative samples (leading to large batch sizes [16] or memory banks [37]). While clustering methods improve on this using cluster targets [3,5,7,11,12,41,73,88], recent regression based methods that predict alternate representations [13,34] eliminate the need for sample mining and negative samples. In particular, Caron et al. [13] explore predicting spatially localglobal correspondences with ViT backbones within the image domain, which we extend in our work to the video domain with suitable improvements. Self-supervised Learning in Videos. While selfsupervised learning in videos were initially dominated by approaches based on pretext tasks unique to the video domain [2,32,43,54,56,62,70,[79][80][81]84], recent work focuses more on contrastive losses similar to the image domain [22,31,35,36,64,65]. A combination of previous pretext tasks over multiple modalities with cross-modality distillation is presented in [63]; SVT differs in how our Figure 2. Our spatiotemporal sampling generates global and local views from a given input video. Global views contain different frame rates and spatial characteristics controlled by sampling strategy and combinations of augmentations. Local views have varying frame rates as well as limited fields of view due to random cropping. One global view is randomly selected and passed through the teacher model to generate a target while other global and local views are passed through the student model. Network weights are then updated by matching the online student local (cross-view correspondences) and global (motion correspondences) views to the target teacher global view. We use a standard ViT backbone with separate space-time attention [9] followed by an MLP for predicting target features from online features. self-distillation operates within a single modality and network. The idea of varying resolution along temporal dimension is explored in [15,40]. They use contrastive losses between different videos at same resolution for speed consistency or the same video at different resolutions for appearance consistency. Unlike these works, we jointly vary spatial and temporal resolutions and use a predictive objective as self-supervision. The idea of views with limited locality is also explored in [8,20,65]. While [8] uses views of varying locality for disentangling the representation space into temporally local and global features using contrastive objectives, our approach uses view locality to learn correspondences along and across dimensions with our predictive objective. A similar predictive objective with temporal locality constrained views is used in [65] and contrastive losses with spatial local-global crops is used in [20]; however our approach focuses on spatio-temporal constraints extending correspondences across dimensions, uses a single shared network for processing alternate views, and additionally combines varying resolutions to generate our alternate views exploiting unique ViT architectural features. Self-supervised Video Transformer In this section, we discuss our Self-supervised Video Transformer (SVT) approach. Unlike contrastive methods, we process two clips from the same video with varying spatial-temporal characteristics (Sec. 3.2) avoiding the need for negative mining or memory banks. Our loss formulation matches the representations from both dissimilar clips to enforce invariance to motion and spatial changes for the same action sequence. A naive objective enforcing invariance would collapse all representations to be the same, however we use a teacher-student network pair where the former acts as a more stable target for the later, enabling convergence of the online student network to learn discriminative representations [34]. This approach simultaneously incorporates rich spatiotemporal context in the representations while keeping them discriminative. In the following, we first introduce the overall architecture of SVT in Sec. 3.1 followed by the self-supervised learning process in Sec. 3.2, our objective functions in Sec. 3.3 and inference in Sec. 3.4. SVT: Architecture We apply separate attention along temporal and spatial dimensions of input video clips using a video transformer [9]. Consider a video X = {x t } N t=1 , where N represents the number of frames. We define a clip (also termed view interchangeably) as a subset of these N frames selected through a sampling strategy. We define H, W , T to be the height, width, and number of frames respectively for the sampled clip. Our sampling methodology (Fig. 2) generates two types of clips, global (g) and local (l) spatiotemporal views. Both g and l are subsets of the video frame set X, with views g = {x t } Kg t=1 , l = {x t } K l t=1 , and |K l | ≤ |K g |. Global views are generated by uniformly sampling a variable number of frames from a randomly selected 90% portion of a video's time axis. We generate two such global spatiotemporal views (g 1 , g 2 ) at low (T = 8) and high (T = 16) frame rates and spatial resolution H = W = 224. Local views are generated by uniformly sampling frames from a randomly selected video region covering 1/8 th of the time axis and ≈ 40% area along spatial axes. We generate eight such local spatiotemporal views with T ∈ {2, 4, 8, 16} and spatial resolution fixed at H = W = 96. Specifically, we randomly sample two global (g 1 , g 2 ) and eight local (l 1 , . . . , l 8 ) spatiotemporal views. Note that both spatial and temporal dimensions within our sampled views differ from those of the original video. We introduce the channel dimension, C, which is fixed at 3 for RGB inputs considered in our case. Our SVT, comprising of 12 encoder blocks, processes each sampled clip of shape During training, we divide each frame within a view into patches [26]. Thus, for a given view of maximum size H = W = 224 and T = 16, each SVT encoder block processes a maximum of 196 spatial and 16 temporal tokens, and the embedding dimension of each token is R 768 [26]. Since the maximum number of spatial and temporal tokens vary due to variable dimensions in our views, we deploy dynamic positional encoding (Sec. 3.2.3) to account for any missing tokens for views of size W < 224, H < 224 and T < 16. Note the minimum spatial and temporal sizes in our proposed views are H = W = 96 and T = 2, respectively. In addition to these input spatial and temporal tokens, we use a single classification token as the feature vector within the architecture [21,26]. This classification token represents the common features learned by the SVT along spatial and temporal dimensions of a given video. Finally, we use a multi-layer perceptron (MLP) as a projection head over the classification token from the final encoder block [13,34]. We define the output of our projection head as f . (C × T × W × H), As illustrated in Fig. 2, our overall approach uses a teacher-student setup inspired from [13,34] for selfdistillation [72]. Our teacher model is an exact architectural replica of the student model. SVT: Self-supervised Training We train SVT in a self-supervised manner by predicting the different views (video clips) with varying spatiotemporal characteristics from each other in the feature space of student and teacher models. To this end, we adopt a simple routing strategy that randomly selects and passes different views through the teacher and student models. The teacher SVT processes a given global spatiotemporal view to produce a feature vector, f gt , which is used as the target label, while the student SVT processes local and global spatiotemporal views to produce feature vectors, f gs , and f (1) ls , ..., f (8) ls , which are matched to the target feature f gt through our proposed loss (Eq. 1). During each training step, we update the student model weights via backpropagation while teacher weights are updated as an exponential moving average (EMA) of the student weights [13]. Our motivation for predicting such varying views of a video is that it leads to modeling the contextual information defining the underlying distribution of videos by learning motion correspondences (global to global spatiotemporal view matching) and cross-view correspondences (local to global spatiotemporal view matching) (Fig. I). This makes the model invariant to motion, scale and viewpoint variations. Thus, our self-supervised video representation learning approach depends on closing the gap between feature representations of different spatiotemporal views from the same video using a self-distillation mechanism. Next, we explain how motion correspondences and cross-view correspondences are learned, followed by our loss formulation. Motion Correspondences A defining characteristic of a video is the frame rate. Varying the frame rate can change motion context of a video (e.g., walking slow vs walking fast) while controlling nuanced actions (e.g., subtle body-movements of walking action). In general, clips are sampled from videos at a fixed frame rate [64,87]. However, given two clips of varying frame rate (different number of total frames for each clip), predicting one from the other in feature space explicitly involves modeling the motion correspondences (MC) of objects across frames. Further, predicting subtle movements captured at high frame rates will force a model to learn motion related contextual information from a low frame rate input. We model this desired property into our training by matching global to global spatiotemporal views. Refer to Appendix A for further details. Cross-View Correspondences In addition to learning motion correspondences, our training strategy aims to model relationships across spatiotemporal variations as well by learning cross-view correspondences (CVC). The cross-view correspondences are learned by matching the local spatiotemporal views processed by our student SVT (f (i) ls : i ∈ [1, 8]) with a global spatiotemporal view representation processed by our teacher SVT model (f gt ). Our local views cover a limited portion of videos along both spatial and temporal dimensions. Our intuition is that predicting a global spatiotemporal view of a video from a local spatiotemporal view in the latent space forces the model to learn high-level contex-tual information by modeling, a) spatial context in the form of possible neighbourhoods of a given spatial crop, and b) temporal context in the form of possible previous or future frames from a given temporal crop. Note that in the crossview correspondences, we predict a global view frame using all frames of a local view by our similarity objective (Eq. 3). Dynamic Positional Embedding Vision transformers [26] require inputs to be converted to sequences of tokens, which allows efficient parallel processing. Positional encoding is used to model ordering of these sequences [57]. Interestingly, positional encoding also allows ViT to process variable input resolution by interpolating the positional embedding for the missing tokens. As mentioned earlier, our motion and cross-view correspondences involve varying spatial and temporal resolutions which results in variable spatial and temporal input tokens during training (Sec. 3.1). We use this property of positional encoding to our advantage by accommodating varying spatial and temporal tokens in our proposed training mechanism. In implementing this, during training we use a separate positional encoding vector for spatial and temporal dimensions and fix these vectors to the highest resolution across each dimension. Similar to [26], our positional encoding is a learned vector. We vary the positional encoding vectors through interpolation during training to account for the missing spatial or temporal tokens at lower frame rate or spatial size. This allows our single SVT model to process inputs of varying resolution while also giving the positional embedding a dynamic nature which is more suited for different sized inputs in the downstream tasks. During slow-fast inference (Section 3.4) on downstream tasks, the positional encoding is interpolated to the maximum frame count and spatial resolution used across all views. We note that our learned positional encoding is implicitly tied to frame number to cue the relative ordering of the sampled frames. Given the varying frame rates of views, it does not encode the exact time stamp (frame rate information). We hypothesize that despite not differentiating frame rates, cuing frame order is sufficient for SVT training. Augmentations In addition to our sampling strategy (temporal dimension augmentations), we also apply standard image augmentations to the spatial dimension, i.e., augmentations are applied to the individual frames sampled for each view. We follow temporally consistent spatial augmentations [64] where the same randomly selected augmentation is applied equally to all frames belonging to a single view. The standard augmentations used include random color jitter, gray scaling, Gaussian blur, and solarization. We also apply random horizontal flips to datasets not containing flip equivariant classes (e.g., walking left to right). SVT Loss We enforce motion and cross-view correspondences by matching our proposed spatiotemporal views within the feature space. Specifically, we match global to global views to learn motion and local to global views to learn cross-view correspondences by minimizing the following objective: L = L lg + L gg .(1) The global and local spatiotemporal views are passed though the student and teacher models to get the corresponding feature outputs f g and f l . These feature vectors are normalized to obtainf as follows: f [i] = exp(f [i])/τ n i=1 exp(f [i])/τ , where τ is a temperature parameter used to control sharpness of the exponential function [13] andf [i] is each element off ∈ R n . Motion Correspondence Loss: We forward pass a global view through the teacher SVT serving as the target feature which is compared with an alternate global view processed by the student SVT to obtain a loss term (Eq. 2). This loss measures the difference in motion correspondences between these two global views. L gg = −f gt · log(f gs ),(2) where,f gs andf gt are the feature outputs of different global spatiotemporal views from the student and teacher network respectively and [·] is dot product operator. Cross-view Correspondence Loss: All local spatiotemporal views are passed through the student SVT model and mapped to a global spatiotemporal view from the teacher SVT model to reduce the difference in feature representation, learning cross-view correspondences (Eq. 3). L lg = k i=1 −f gt · log(f (i) ls ),(3) where the sum is performed over k different local spatiotemporal views (k = 8 used consistently across all experiments) andf (i) ls are the feature outputs for i th local view. Convergence: Given our two separate student (θ) and teacher (ξ) networks, let us view our overall loss, L as a function of their learnable parameters, L θ,ξ . There exists a concern of collapse to a trivial solution (teacher and student outputs always equal a constant) during training. However, we note that SVT parameters do not converge to such a minimum over L θ,ξ because: a) The SVT teacher parameter updates are not in the direction of ∇ ξ L θ,ξ since ξ t+1 ← τ ξ t + (1 − τ )θ t for τ ∈ [0, 1] (EMA update). b) SVT's gradient descent on L θ,ξ does not act jointly over (64, 96, 96), pass through a shared network, and generate two different feature vectors (class tokens). These vectors are combined in a deterministic manner (with no learnable parameters), e.g. summation, to obtain a joint vector that is fed to the downstream task classifier. (θ, ξ). This is similar to BYOL [34] where such a loss acts on the outputs of student and teacher networks. Additionally, as suggested in [13], we also use centering and sharpening of teacher outputs to further facilitate convergence. SVT: Slow-Fast Inference Slow-Fast inference refers to using two different video clips with high spatial but low temporal and low spatial but high temporal resolutions. This allows capturing finerinformation across each dimension with minimal increase in computation. Recent methods [30,45] deploy such inference but use multiple network architectures for processing videos at different resolutions. However, our dynamic positional encodings allow Slow-Fast inference within our single SVT model (Sec. 3.2.3) as illustrated in Figure 3. We use this on downstream tasks for improved performance. Experiments Experimental Setup and Protocols Datasets: We use the Kinetics-400 data [14] (train set) for the self-supervised training phase of SVT. We use its validation set for evaluation. Additionally, we evaluate on three downstream datasets, UCF-101 [69], HMBD-51 [49], and Something-Something v2 (SSv2) [33]. Self-supervised Training: We train our models for 20 epochs on the train set of Kinetics-400 dataset [14] without any labels using a batch size of 32 across 4 NVIDIA-A100 GPUs. This batch size refers to the number of unique videos present within a given batch. We randomly initialize weights relevant to temporal attention while spatial attention weights are initialized using a ViT model trained in a self-supervised manner over the ImageNet-1k dataset [66]. We follow this initialization setup to obtain faster spacetime ViT convergence similar to the supervised setting [9]. We use an Adam optimizer [47] with a learning rate of 5e−4 scaled using a cosine schedule with linear warmup for 5 epochs [18,71]. We also use weight decay scaled from 0.04 to 0.1 during training. Our code builds over the training frameworks from [9,13,28,86]. Downstream Tasks: We perform two types of evaluations on our downstream task of action recognition for each dataset, a) Linear: We train a linear classifier over our pretrained SVT backbone (which is frozen during training) for 20 epochs with a batch size of 32 on a single NVIDIA-V100 GPU. We use SGD with an initial learning rate of 8e − 3, a cosine decay schedule and momentum of 0.9 similar to recent work [13,64]; b) Fine-tuning: We replace the projection head over the SVT with a randomly initialized linear layer, initialize the SVT backbone with our pre-trained weights, and train the network end-to-end for 15 epochs with a batch size of 32 across 4 NVIDIA-V100 GPUs. We use SGD with a learning rate of 5e − 3 decayed by a factor of 10 at epochs 11 and 14, momentum of 0.9, and weight decay of 1e − 4 following [9]. During both training of linear classifier and fine-tuning of SVT, we sample two clips of varying spatiotemporal resolution from each video. During evaluation, we follow our proposed slow-fast inference strategy (Sec. 3.4). We use two clips per video sampled at different spatiotemporal resolutions (T, W, H) ∈ { (8,224,224), (64,96, 96)} with 3 spatial crops each for testing (6 clips in total). This is computationally more efficient in comparison to recent works [64,65] that uniformly sample 10 clips at similar or high resolutions from full-length videos with 3 crops each for testing (total of 30 clips per video). Results We compare SVT with state-of-the-art approaches (trained on RGB input modality for fair comparison) for the downstream task of action recognition. UCF-101 & HMDB-51: Our method out-performs stateof-the-art for UCF-101 and is on-par for HMDB-51 (Tab. 1). While CORP [39] exhibits higher performance on HMDB-51, we highlight how SVT: a) is pretrained for a much shorter duration (20 epochs) with smaller batch-sizes (32); b) uses a single architectural design across all tasks. CORP [39] models are pre-trained for 800 epochs with a batch-size of 1024 using 64 NVIDIA-V100 GPUs and uses different variants (CORP f and CORP m ) to obtain optimal performance on different datasets. Kinetics-400: We present our results on Kinetics-400 [14] in Tab. 2 where our approach obtains state-of-the-art for both linear evaluation and fine-tuning settings. Performance on Kinetics-400 is heavily dependent on appearance attributes, i.e. a large proportion of its videos can be recognized by a single frame [91]. Strong performance of SVT on this dataset exhibits how well our proposed approach learns appearance related contextual information. SSv2: Similarly, we obtain state-of-the-art results on SSv2 Table 1. UCF-101 [69] & HMBD-51 [49]: Top-1 (%) accuracy for both linear evaluation and fine-tuning. All models are pre-trained on Kinetics-400 [14] except ELo [63] which uses YouTube8M dataset [1]. Gray shaded methods use additional optical flow inputs. S-Res and T-Res represent spatial and temporal input resolution, respectively. Our approach shows state-of-the-art or on par performance. UCF-101 [69] HMDB-51 [ [39]. Performance on this dataset indicates how SVT feature representations capture strong motion related contextual cues as well. Ablative Analysis We systematically dissect the contribution of each component of our method. We study the effect of five individual elements: a) different combinations of local and global view correspondences; b) varying field of view along temporal vs spatial dimensions; c) temporal sampling strategy; d) spatial augmentations; e) slow-fast inference. In all our ablative experiments, SVT self-supervised training uses a subset of the Kinetics-400 train set containing 60K videos. Evaluation is carried out over alternate train-set splits of UCF-101 and HMDB-51. We train SVT for 20 epochs and evaluate using the same setup as described in Sec. 4.1. View Correspondences. Learning correspondences between local and global views is the key motivation behind our proposed cross-view correspondences. Since multiple local-global view combinations can be considered for matching and prediction between views, we explore the effect of predicting each type of view from the other in Tab. 4. We observe that jointly predicting local to global and global to global view correspondences results in the optimal performance, while predicting global to local or local to local views leads to reduced performance. We believe this trend exists due to the emphasis on learning rich context in the case of joint prediction, which is absent for individual cases. Further, the performance drop for local to local correspondences (non-overlapping views) conforms with previous findings on the effectiveness of temporally closer positive views for contrastive self-supervised losses [31,64]. Spatial vs Temporal Field of View. The optimal combination of spatiotemporal views in Tab. 4 involves varying the field of view (crops) along both spatial and temporal dimensions (Sec. 3.2.2). We study the effects of these variations (spatial or temporal) in Tab. 5. No variation along the spatial dimension denotes that all frames are of fixed spatial resolution 224 × 224 with no spatial cropping, and no temporal variation denotes that all frames in our views are sampled from a fixed time-axis region of a video. We observe that temporal variations have a significant effect on UCF-101, while variations in the field of view along both spatial and temporal dimension perform the best (Tab. 5). Temporal Sampling Strategy. We study how our proposed temporal sampling strategy for motion correspondences (MC) could be replaced with alternate sampling approaches. To verify the effectiveness of MC, we replace it within SVT with an alternate approach. The temporal interval sampling (TIS) strategy in [64] obtains state-of-the-art performance under their contrastive video self-supervised setting with CNN backbones. Our experiments incorporating TIS in SVT (Tab. 6) highlight the advantage of our proposed MC sampling strategy over TIS. Augmentations: We next explore standard spatial augmentations used on videos. Temporally consistent augmentations (TCA) proposed in [64] lead to improvements in their CNN based video self-supervision approach. We evaluate its effect on our approach in Tab. 7 which shows slight improvements. Given these performance gains, we adopt TCA in our SVT training process as well. Slow-Fast Inference: Finally, we study the effect of our proposed Slow-Fast inference (Sec. 3.4) in Tab. 8. We observe higher gains on HMDB-51 [49], where the classes are easier to separate with motion information [36]. Conclusion We present a video transformer based model trained using self-supervised objectives named SVT. Given an input video sequence, our approach first creates a set of spatiotemporally varying views sampled at different scales and frame rates from the same video. We then define two sets of correspondence learning tasks which seek to model the motion properties and cross-view relationships between the sampled clips. Specifically, our self-supervised objective reconstructs one view from the other in the latent space of student and teacher networks. Our approach is fast to train (converges within only a few iterations), does not require negative samples and large batch sizes that are required by previous contrastive video representation learning methods. Additionally, our SVT allows modeling long-range spatiotemporal dependencies and can perform dynamic slowfast inference within a single architecture. SVT is evaluated on four benchmark action recognition datasets where it performs well in comparison to existing state of the art. Limitations: In this work, we explore SVT within the context of RGB input modality. Given large-scale multimodal video datasets, the additional supervision available in the form of alternate modalities is not used by our current approach. In future work, we will explore how SVT can be modified to utilize multi-modal data sources. A. View Routing and Matching In the main paper (Sec. 3.2), we illustrate the concept of SVT using a single global view passed through the teacher, which generates target for all the other views passed through the student model. However, in practice, multiple global views are all passed through the teacher model, and we separately map each student view (global and local) to the multiple teacher targets. In the case of two global views, g1 (T = 8) and g2 (T = 16), we obtain two targets,f (1) gt andf (2) gt . Both these global views are also passed through the student model to obtainf (1) gs andf (2) gs . We mapf (1) gs tof (2) gt andf (2) gs tof (1) gt . The local views passed through the student that generatesf (1) ls ...f (8) gs which are separately mapped to both teacher targets,f (1) gt andf (2) gt . Our proposed loss is applied over each mapped student-teacher feature pair. B. Comparison to Supervised Training In SVT, we use a standard ViT backbone with split attention across space and time dimensions similar to [9]. We compare SVT with supervised pre-training based initialization for Kinetics-400 training reported in [9]. For fairness, our comparison with [9] includes the highest reported input resolution used in their work since the SVT uses slow-fast inference. These results are presented in Table I. C. Dataset Description We use the Kinetics-400 [14] training set for the SVT self-supervised training and its validation set for evaluation of learned self-supervised representations. Kinetics-400 is a large-scale dataset containing 240k training videos and 20k validation videos belonging to 400 different action classes. On average, these videos are of duration around 10 seconds, with 25 frames per second (i.e., around 250 frames per video). Interestingly, most classes of this dataset are considered to be separable with appearance information alone [91]. In addition to Kinetics-400, we evaluate our approach on three downstream datasets, UCF-101 [69], HMBD-51 [49], and Something-Something v2 (SSv2) [33]. UCF-101 and HMBD-51 are small-scale datasets each containing 13k videos (9.5k/3.7k train/test) belonging to 101 classes and 5k (3.5k/1.5k train/test) videos belonging to 51 classes respectively, while SSv2 is a large-scale dataset heavily focused on motion with 168k training and 24k validation videos belonging to 174 action classes. Unlike UCF101 and HMDB51 which contain action classes similar to Kinetics-400, SSv2 contains very different actions involving complex human object interactions, such as 'Moving something up' or 'Pushing something from left to right'. D. Future Directions As discussed in the main paper, the key limitation of SVT is being constrained to operating within a single modality input (RGB video). We hope to explore how SVT to can improved to utilize alternate modalities (Optical Flow, Audio) for better self supervision in future work. In this work, we focus on evaluating the effectiveness of our proposed cross-view and motion correspondences (that compose the core of SVT) in relation to ViT backbones. The question of applicability of our proposed approach under convolutional neural network (CNN) settings remains unexplored. However, we highlight that the main components (temporal attention, dynamic input sizes, and slowfast inference) of our proposed SVT are designed to leverage some unique characteristics of ViTs, which could not be directly implemented with a CNN backbone. On the other hand, we believe that self-distillation and view matching, also core to SVT, can be applied to CNNs and is an interesting future direction. Another key issue is the significant drop in performance (top-1 accuracy) for linear evaluation in large-scale datasets (Kinetics-400 and SSv2). Particularly in SSv2, our features perform poorly in the linear evaluation setting (in comparison to fine-tune setting). A key reason for this could be the significant domain different between Kinetics-400 and SSv2 (as opposed to UCF-101 and HMDB-51 which contain videos and classes more similar to Kinetics-400). The self-supervised training phase of SVT uses Kinetics-400 only, and the SSv2 experiments use that representation for linear evaluation. An interesting future direction we hope to explore is self-supervised training using the SSv2 dataset itself, which could potentially reveal more interesting insights on representations learned by SVT. E. Attention Visualization Following the approach in [13], we visualize the attention of our classification token (feature vector) towards each spatiotemporal patch token within the final encoder block of SVT for two randomly selected videos. As illustrated in Figure I. Attention Visualization: We uniformly sample four frames from two videos (cols 1-4 and 5-8 respectively) and visualize the attention from the classification token of self-supervised vision transformer DINO [13] (second row) and our SVT (last row). Observe how DINO attention is scattered around multiple objects, while SVT is focused on 'crawling baby' and 'person walking on hands' across frames which are the salient objects for these action. This highlights how SVT learns to pay attention to the motion within a video. Fig. I, SVT attends to the regions of motion in these videos, even in the case of highly detailed backgrounds (right). Attention to the salient moving object in each case qualitatively demonstrates how our proposed cross-view and motion correspondences learn spatiotemporally invariant representations. Figure 1 . 1Our Self-supervised Video Transformer (SVT) learns cross-view and motion correspondences by jointly matching video clips sampled with varying field of view and temporal resolutions. Specifically, Global views (top and bottom right) with different temporal resolutions as well as Local views (bottom left) from different spatiotemporal windows are sampled. The representations of these multiple views are matched in a student-teacher framework to learn cross-view and motion correspondences (middle block). The proposed self-supervised framework can learn highquality spatiotemporal features while converging faster. Figure 3 . 3Slow-Fast Inference: we uniformly sample two clips of the same video at resolutions(8, 224, 244) and Table 4 . 4View Correspondences. Predicting local to global and global to global views remains optimal over any other combination.l → g g → g l → l g → l UCF-101 HMDB-51 84.11 50.72 81.95 49.04 84.64 52.17 83.11 51.23 84.71 51.88 83.69 51.71 Table 5. Spatial vs Temporal variations. Cross-view cor- respondences with varying field of view along both spatial and temporal dimensions lead to optimal results. Tempo- ral variations between views has more effect than applying only spatial variation. Spatial Temporal UCF-101 HMDB-51 73.81 42.91 82.90 42.59 84.64 52.17 Table 6. Temporal Sampling Strategy . We compare our proposed temporal sam- pling strategy, motion correspondences (MC) (Sec. 3.2.1), against the alternate approach of temporal interval sampler (TIS) [64] used with CNNs under contrastive settings. UCF-101 HMDB-51 Ours + TIS [64] 82.24 50.10 Ours + MC 84.64 52.17 Table 7. Augmentations: Using tem- porally consistent augmentations (TCA) [64] applied randomly over the spa- tial dimensions for different views result in consistent improvements on UCF-101 and HMDB-51 datasets. UCF-101 HMDB-51 w/o TCA [64] 84.20 52.10 w TCA [64] 84.64 52.17 Table 8. Slow-Fast Inference: Feed- ing multiple views of varying spatiotempo- ral resolutions to a single shared network (multi-view) results in clear performance gains over feeding single-views across both UCF-101 and HMDB-51 datasets. Slow-Fast UCF-101 HMDB-51 84.64 52.17 84.80 53.22 Table I . IComparison of SVT with supervised pretraining methods containing similar backbone (ViT-B) on Kinetics-400. For each different pre-training strategy, we finetune on Kinetics-400 and report accuracy (top-1) on Kinetics-400 validation set.Pretrain Dataset Supervision Accuracy Random-init - 64.8 [9] ImageNet-1K 75.8 [9] ImageNet-21K 79.7 [9] ImageNet-1K 69.9 Kinetics-400 78.1 Youtube-8m: A large-scale video classification benchmark. 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[]
[ "Improving Chinese Spelling Check by Character Pronunciation Prediction: The Effects of Adaptivity and Granularity", "Improving Chinese Spelling Check by Character Pronunciation Prediction: The Effects of Adaptivity and Granularity" ]
[ "Jiahao Li [email protected] \nUniversity of Science and Technology of China\nHefeiChina\n", "Quan Wang [email protected] \nMOE Key Laboratory of Trustworthy Distributed Computing and Service\nBeijing University of Posts and Telecommunications\nBeijingChina\n", "Zhendong Mao [email protected]@people.cn \nUniversity of Science and Technology of China\nHefeiChina\n", "Junbo Guo \nPeople's Daily Online Co\nBeijingChina\n", "Yanyan Yang \nPeople's Public\nSecurity University of China\nBeijingChina\n", "Yongdong Zhang \nUniversity of Science and Technology of China\nHefeiChina\n" ]
[ "University of Science and Technology of China\nHefeiChina", "MOE Key Laboratory of Trustworthy Distributed Computing and Service\nBeijing University of Posts and Telecommunications\nBeijingChina", "University of Science and Technology of China\nHefeiChina", "People's Daily Online Co\nBeijingChina", "People's Public\nSecurity University of China\nBeijingChina", "University of Science and Technology of China\nHefeiChina" ]
[ "Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing" ]
Chinese spelling check (CSC) is a fundamental NLP task that detects and corrects spelling errors in Chinese texts. As most of these spelling errors are caused by phonetic similarity, effectively modeling the pronunciation of Chinese characters is a key factor for CSC. In this paper, we consider introducing an auxiliary task of Chinese pronunciation prediction (CPP) to improve CSC, and, for the first time, systematically discuss the adaptivity and granularity of this auxiliary task. We propose SCOPE which builds on top of a shared encoder two parallel decoders, one for the primary CSC task and the other for a fine-grained auxiliary CPP task, with a novel adaptive weighting scheme to balance the two tasks. In addition, we design a delicate iterative correction strategy for further improvements during inference. Empirical evaluation shows that SCOPE achieves new state-of-theart on three CSC benchmarks, demonstrating the effectiveness and superiority of the auxiliary CPP task. Comprehensive ablation studies further verify the positive effects of adaptivity and granularity of the task. Code and data used in this paper are publicly available at https: //github.com/jiahaozhenbang/SCOPE.
10.48550/arxiv.2210.10996
[ "https://www.aclanthology.org/2022.emnlp-main.287.pdf" ]
253,018,348
2210.10996
b02e54ac7c13ea76083c9d7f9cd4b3ca1f0cd71f
Improving Chinese Spelling Check by Character Pronunciation Prediction: The Effects of Adaptivity and Granularity December 7-11, 2022 Jiahao Li [email protected] University of Science and Technology of China HefeiChina Quan Wang [email protected] MOE Key Laboratory of Trustworthy Distributed Computing and Service Beijing University of Posts and Telecommunications BeijingChina Zhendong Mao [email protected]@people.cn University of Science and Technology of China HefeiChina Junbo Guo People's Daily Online Co BeijingChina Yanyan Yang People's Public Security University of China BeijingChina Yongdong Zhang University of Science and Technology of China HefeiChina Improving Chinese Spelling Check by Character Pronunciation Prediction: The Effects of Adaptivity and Granularity Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing the 2022 Conference on Empirical Methods in Natural Language ProcessingDecember 7-11, 2022 Chinese spelling check (CSC) is a fundamental NLP task that detects and corrects spelling errors in Chinese texts. As most of these spelling errors are caused by phonetic similarity, effectively modeling the pronunciation of Chinese characters is a key factor for CSC. In this paper, we consider introducing an auxiliary task of Chinese pronunciation prediction (CPP) to improve CSC, and, for the first time, systematically discuss the adaptivity and granularity of this auxiliary task. We propose SCOPE which builds on top of a shared encoder two parallel decoders, one for the primary CSC task and the other for a fine-grained auxiliary CPP task, with a novel adaptive weighting scheme to balance the two tasks. In addition, we design a delicate iterative correction strategy for further improvements during inference. Empirical evaluation shows that SCOPE achieves new state-of-theart on three CSC benchmarks, demonstrating the effectiveness and superiority of the auxiliary CPP task. Comprehensive ablation studies further verify the positive effects of adaptivity and granularity of the task. Code and data used in this paper are publicly available at https: //github.com/jiahaozhenbang/SCOPE. Abstract Chinese spelling check (CSC) is a fundamental NLP task that detects and corrects spelling errors in Chinese texts. As most of these spelling errors are caused by phonetic similarity, effectively modeling the pronunciation of Chinese characters is a key factor for CSC. In this paper, we consider introducing an auxiliary task of Chinese pronunciation prediction (CPP) to improve CSC, and, for the first time, systematically discuss the adaptivity and granularity of this auxiliary task. We propose SCOPE which builds on top of a shared encoder two parallel decoders, one for the primary CSC task and the other for a fine-grained auxiliary CPP task, with a novel adaptive weighting scheme to balance the two tasks. In addition, we design a delicate iterative correction strategy for further improvements during inference. Empirical evaluation shows that SCOPE achieves new state-of-theart on three CSC benchmarks, demonstrating the effectiveness and superiority of the auxiliary CPP task. Comprehensive ablation studies further verify the positive effects of adaptivity and granularity of the task. Code and data used in this paper are publicly available at https: //github.com/jiahaozhenbang/SCOPE. Introduction Chinese Spelling Check (CSC), which aims to detect and correct spelling errors in Chinese texts, is a fundamental task in Chinese natural language processing. Spelling errors mainly originate from human writing errors and machine recognition errors, e.g., errors caused by automatic speech recognition (ASR) and optical character recognition (OCR) systems . With the latest development of deep neural networks, neural CSC methods, 1 1 I think you will finish well. R: 我觉得你们会好好的玩(wan2/w,an,2)。 I think you will play well. W:我以前想要高(gao1/g,ao,1)诉你。 0 2 /3 I tried to high you before. R: 我以前想要告(gao4/g,ao,4)诉你。 I tried to tell you before. W:他收(shou1/sh,ou,1)到山上的时候。 0 1 /3 When he received the mountain. R: 他走(zou3/z,ou,3)到山上的时候。 When he walked up the mountain. W:行为都被蓝(lan2/l,an,2)控设备录影。 0 0 Actions are recorded by blue control devices. R: 行为都被监(jian1/j,ian,1)控设备录影。 Actions are recorded by surveillance devices. (Tseng et al., 2015). For each instance, coarse-/fine-grained pinyin of the misspelled (red) and correct (blue) characters are provided, along with their phonological similarity degree (the fraction of identical components) in terms of these two types of pinyin. in particular those based on encoder-decoder architectures, have become the mainstream of research in recent years (Xu et al., 2021;Liu et al., 2021). Encoder-decoder models regard CSC as a special sequence-to-sequence (Seq2Seq) problem, where a sentence with spelling errors is given as the input and a corrected sentence of the same length will be generated as the output. Previous research has shown that about 76% of Chinese spelling errors are induced by phonological similarity (Liu et al., 2011). Hence, it is a crucial factor to effectively model the pronunciation of Chinese characters for the CSC task. In fact, almost all current advanced CSC approaches have actually exploited, either explicitly or implicitly, character pronunciation. The implicit use takes into account phonological similarities between pairs of characters, e.g., by increasing the decoding probability of characters with similar pronunciation (Cheng et al., 2020) or integrating such similarities into the encoding process via graph convolutional networks (GCNs) (Cheng et al., 2020). The explicit use considers directly the pronunciation, or more specifically, pinyin 1 , of individual characters, encoding the pinyin of input characters to produce extra phonetic features (Xu et al., 2021; or decoding the pinyin of target correct characters to serve as an auxiliary prediction task (Liu et al., 2021;Ji et al., 2021). This paper also considers improving CSC with auxiliary character pronunciation prediction (CPP), but focuses specifically on the adaptivity and granularity of the auxiliary task, which have never been systematically studied before. First, all the prior attempts in similar spirit simply assigned a universal trade-off between the primary and auxiliary tasks for all instances during training, while ignoring the fact that the auxiliary task might provide different levels of benefits given different instances. Take for example the instances shown in Table 1. Compared to the misspelled character "蓝" and its correction "监" in the 4th instance, the two characters "完" and "玩" in the 1st instance are much more similar in pronunciation, suggesting that the spelling error there is more likely to be caused by phonological similarity, to which the pronunciation-related auxiliary task might provide greater benefits and hence should be assigned a larger weight. Second, prior efforts mainly explored predicting the whole pinyin of a character, e.g., "gao1" for "高". Nevertheless, a syllable in Chinese is inherently composed of an initial, a final, and a tone, e.g., "g", "ao", and "1" for "高". This fine-grained phonetic representation can better reflect not only the intrinsic regularities of Chinese pronunciation, but also the phonological similarities between Chinese characters. Consider for example the "高" and "告" case from the 2nd instance in Table 1. These two characters show no similarity in terms of their whole pinyin, but actually they share the same initial and final, differing solely in their tones. Based on the above intuitions we devise SCOPE (i.e., Spelling Check by prOnunciation PrEdiction), which introduces a fine-grained CPP task with an adaptive task weighting scheme to improve CSC. Figure 1 provides an overview of SCOPE. Given a sentence with spelling errors as input, we encode it using ChineseBERT to produce semantic and phonetic features. Then we build on top of the encoder two parallel decoders, one to generate target correct characters, i.e., the primary CSC task, and the other to predict the initial, final and tone of the pinyin of each target character, i.e., the auxiliary fine-grained CPP task. The trade-off between the two tasks can be further adjusted adaptively for each instance, according to the phonological similarity between input and target characters therein. In addition, we design an iterative correction strategy during inference to address the over-correction issue and tackle difficult instances with consecutive errors. We empirically evaluate SCOPE on three shared benchmarks, and achieve substantial and consistent improvements over previous state-of-the-art on all three benchmarks, demonstrating the effectiveness and superiority of our auxiliary CPP task. Comprehensive ablation studies further verify the positive effects of adaptivity and granularity of the task. The main contributions of this paper are summarized as follows: (1) We investigate the possibility of introducing an auxiliary CPP task to improve CSC and, for the first time, systematically discuss the adaptivity and granularity of this auxiliary task. (2) We propose SCOPE, which builds two parallel decoders upon a shared encoder for CSC and CPP, with a novel adaptive weighting scheme to balance the two tasks. (3) We establish new state-of-the-art on three benchmarking CSC datasets. Related Work CSC is a fundamental NLP task that has received wide attention over the past decades. Early work on this topic was mainly based on manually designed rules (Mangu and Brill, 1997;Jiang et al., 2012). After that, statistical language models became the mainstream for CSC (Chen et al., 2013;Yu and Li, 2014;Tseng et al., 2015). Methods of this kind in general followed a pipeline of error detection, candidate generation, and candidate selection. Given a sentence, the error positions are first detected by the perplexity of a language model. The candidates for corrections can then be generated according to similarity between characters, typically by using Figure 1: Overview of SCOPE. Top: The one-encoder-two-decoder structure for CSC and CPP. The input sentence X is fed into the encoder and then, after character-/pronunciation-specific feature projection, two parallel decoders, one to predict the characters, the other to predict the initial, final, and tone of each character in the target sentence. Bottom: Adaptive task weighting between CSC and CPP (detached in the backward pass). The target sentence Y is fed into the encoder and the pronunciation-specific feature projection layer. Then the similarities between input and target sentences on character level are calculated and the adaptive weights are accordingly defined. Note: Only the CSC decoder branch (along with the encoder) will be used at inference time. a confusion set. And the final corrections can be determined by scoring the sentence replaced by the candidates with the language model Xie et al., 2015). In the era of deep learning, especially after Transformer (Vaswani et al., 2017) and pre-trained language models like BERT (Devlin et al., 2019) were proposed, a large number of neural CSC methods have emerged. used Transformer as an encoder to produce candidates and designed a confidence-similarity decoder to filter these candidates. Zhang et al. (2020) designed a detection network based on Bi-GRU to predict the error probability of each character and passed the probabilities to a BERT-based correction network via a soft masking mechanism. Cheng et al. (2020) employed GCNs combined with BERT to further model interdependences between characters. Recent work of (Xu et al., 2021;Liu et al., 2021; proposed to encode phonetic and glyph information in addition to semantic information, and then combine phonetic, glyph and semantic features to make final predictions. As we could see, modeling pronunciation information is prevailing in CSC research , typically via an encoding process to extract phonetic features. Liu et al. (2021) proposed the first work that considered predicting the pronunciation of target characters as an auxiliary task. Their work, however, employed pronunciation prediction in a coarse-grained, non-adaptive manner, which is quite different to ours. Our Approach This section presents our approach SCOPE for the CSC task. Below, we first define the problem formulation and then describe our approach in detail. Problem Formulation The Chinese spelling check (CSC) task is to detect and correct spelling errors in Chinese texts. Given a misspelled sentence X = {x 1 , x 2 , · · · , x n } with n characters, a CSC model takes X as input, detects potential spelling errors on character level, and outputs a corresponding correct sentence Y = {y 1 , y 2 , · · · , y n } of equal length. This task can be viewed as a conditional sequence generation problem that models the probability of p(Y |X). We are further given the fine-grained pinyin of each character y i in the correct sentence Y , represented as a triplet in the form of (α i , β i , γ i ), where α i , β i , and γ i indicate the initial, final, and tone, respectively. Note that such kind of pinyin of the output sentence is required and provided solely during training. 2 SCOPE Architecture The key idea of SCOPE is to employ a fine-grained character pronunciation prediction (CPP) task with an adaptive task weighting scheme to improve CSC. In achieving this SCOPE builds upon a shared encoder two parallel decoders, one for the primary CSC task and the other for the auxiliary CPP task. The trade-off between the two tasks is further determined adaptively based on the phonological similarity between input and target characters. Figure 1 summarizes the overall architecture of SCOPE. Encoder Similar to recent CSC approaches that leverage multimodal information (Liu et al., 2021;Xu et al., 2021), we use ChineseBERT as the encoder to extract semantic, phonetic, and morphologic features as well for the CSC task. ChineseBERT is a pre-trained language model that incorporates both the pinyin and glyph information of Chinese characters. Specifically, for each character x i in the input sentence X, the encoder first produces its char embedding, pinyin embedding, and glyph embedding, all with embedding size D. These three embeddings are then concatenated and mapped to a D-dimensional fused embedding via a fully connected layer. After that, just like in most other pre-trained language models, the fused embedding is added with a position embedding, and fed into a stack of successive Transformer layers to generate a contextualized representation h i ∈ R D for the input character x i . We denote the character representations after this encoding process as H = {h 1 , h 2 , · · · , h n }. As the encoder is not the main concern of this paper, we just provide a brief sketch of the encoder and refer readers to (Vaswani et al., 2017; for details. Decoder for CSC This decoder is to predict the characters in the correct sentence Y based on the encoding output H. Specifically, given each input character x i , we first project its encoding output h i into a character-specific feature space: h (c) i = GeLU W (c) h i + b (c) ,(1) and then predict the corresponding correct characterŷ i based on the projection output: p(ŷ i |X) = softmax W (y) h (c) i + b (y) .(2) Here W (c) ∈ R D×D , b (c) ∈ R D are learnable parameters of the character-specific feature projection layer; W (y) ∈ R V ×D , b (y) ∈ R V are learnable parameters of the character prediction layer; V is the vocabulary size. Decoder for CPP This decoder is to predict the fine-grained pinyin, i.e., the initial, final, and tone, of each character in the correct sentence Y based on the encoding output H. Again, given each input character x i and its encoding output h i , we project h i into a pronunciation-specific feature space: h (p) i = GeLU W (p) h i + b (p) ,(3) and predict the initialα i , finalβ i , and toneγ i of the corresponding correct character based on the projection output: p(α i |X) = softmax W (α) h (p) i + b (α) , (4) p(β i |X) = softmax W (β) h (p) i + b (β) , (5) p(γ i |X) = softmax W (γ) h (p) i + b (γ) . (6) Here W (p) ∈ R D×D and b (p) ∈ R D are learnable parameters of the pronunciation-specific feature projection layer; W (δ) ∈ R U ×D , b (δ) ∈ R U with δ ∈ {α, β, γ} are learnable parameters of the pronunciation prediction layers; U is the total number of pronunciation units (initials, finals, and tones). Adaptive Task Weighting We devise an adaptive task weighting scheme to balance the primary CSC and auxiliary CPP tasks during training. Given an input sentence X, the CSC task aims to match the predicted characters {ŷ i } n i=1 with the ground truth {y i } n i=1 , while the CPP task aims to match the predicted fine-grained pinyin {(α i ,β i ,γ i )} n i=1 with the ground truth {(α i , β i , γ i )} n i=1 . Their loss functions are respectively defined as: L (c) i = − log p(ŷ i = y i |X),(7)L (p) i = − 1 3 δ∈{α,β,γ} log p(δ i = δ i |X), (8) where L (c) i , L(p) i are the character and pronunciation prediction losses associated with the i-th character in the sentence, and the pronunciation prediction loss L (p) i is averaged over the initial, final, and tone prediction. Then as we have discussed earlier in the introduction, the auxiliary CPP task might provide different levels of benefits given different input characters. The more similar the input and target characters are in their pronunciation, the more likely there would be a spelling error caused by phonetic similarity. And to this case the CPP task might provide greater benefits and should be assigned a larger weight. To calculate such adaptive weights, we feed the target correct sentence Y to the encoder and the followup pronunciation-specific projection layer. Then we calculate for each input character x i and its target character y i a cosine similarity cos(h (p) x i , h (p) y i ) based on their pronunciation-specific feature repre- sentations h (p) x i , h (p) y i (see Eq. (3)), and accordingly define the adaptive weight at the i-th position as: w i = e −(cos(h (p) x i ,h (p) y i )−1) 2 .(9) The higher the cosine similarity cos(h (p) x i , h(p) y i ) is, the larger the weight w i will be. Finally, the overall loss is defined as the CSC loss with an adaptively weighted CPP loss: L = 1 n n i=1 L (c) i + w i L (p) i ,(10) where L (c) i and L i are the character-specific CSC and CPP losses defined in Eq. (7) and Eq. (8), respectively. There are two points worth noting here: (1) The branch of encoding and mapping the target sentence Y is employed solely in the forward pass to calculate adaptive weights, and will be detached in the backward pass. (2) The auxiliary CPP task, as well as the adaptive weighting scheme, is introduced solely during training. At inference time, we use the CSC decoder alone for prediction. Constrained Iterative Correction As pointed out by Liu et al. (2022), advanced CSC models based on pre-trained language models (e.g., BERT (Devlin et al., 2019) and ChineseBERT ) typically have poor performance on multi-typo texts, and tend to overcorrect valid expressions to more frequent expressions. To address these deficiencies, we devise a simple yet effective constrained iterative correction strategy during inference. Specifically, at inference time, for each input sentence we detect and correct spelling errors in an iterative fashion. During each iteration, only the corrections that appear in a specified window around each correction position in the previous iteration are allowed. After the iterations, if a position is modified every iteration, we restore this position to its original character without any correction. We empirically set the iteration number to 2 and the window size to 3 (i.e., one position on the left and one on the right of the current position). As we will see later in our case study in Section 4.5, this iterative correction strategy can effectively address the overcorrection issue and tackle difficult instances with multiple, in particular, consecutive errors. Further Pre-training with Confusion Set To obtain better initialization for SCOPE, we perform further pre-training by using a confusion set, as commonly practiced in most recently proposed CSC models (Xu et al., 2021;Liu et al., 2021). We consider wiki2019zh 3 that consists of one million Chinese Wikipedia articles, split these articles into paragraphs, and regard each paragraph as a target sequence with no spelling errors. We further collect easily confused character pairs from a mixture of three publicly available confusion sets (Wu et al., 2013;Wang et al., 2018), and retain only the pairs where both characters appear frequently (top 40%) in the wiki2019zh corpus. Then, for each target sequence, we create a potentially misspelled input sequence by randomly selecting and replacing 15% of the characters. Each selected character is replaced with an easily confused character (if any) 80% of the time, a random character from the vocabulary 10% of the time, and remained unchanged 10% of the time. After that, we pre-train SCOPE on these misspelled and correct sequence pairs before adapting it to target datasets. Experiments and Results In this section, we introduce our experiments and results on SIGHAN benchmarks (Wu et al., 2013;Tseng et al., 2015). We then verify the effectiveness of our model design, in particular the adaptivity and granularity of the auxiliary CPP task, via extensive ablation studies and analyses. Experimental Setups Datasets and Evaluation Metrics As in previous work (Cheng et al., 2020;Liu et al., 2021;Xu et al., 2021), our training data is a combination of (1) manually annotated training examples from SIGHAN13 (Wu et al., 2013), SIGHAN14 (Cheng et al., 2020;Xu et al., 2021) to convert them to simplified Chinese using OpenCC 4 . We further use pypinyin 5 to obtain the pinyin of each character, and segment it into the initial, final, and tone using a pre-defined vocabulary of initials and finals provided by Xu et al. (2021). 6 We use the widely adopted sentence-level precision, recall and F1 as our main evaluation metrics. Sentence-level metrics are stricter than characterlevel metrics since a sentence is considered to be correct if and only if all errors in the sentence are successfully detected and corrected. Metrics are reported on the detection and correction sub-tasks. Besides sentence-level evaluation, we also consider character-level evaluation and the official SIGHAN evaluation. We leave their results to Appendix A Baseline Methods We compare SCOPE against the following baseline methods. All these methods have employed character phonetic information in some manner, and represent current state-of-the-art on the SIGHAN benchmarks. • FASpell (Hong et al., 2019) employs BERT to generate candidates for corrections and filters visually/phonologically irrelevant candidates by a confidence-similarity decoder. • SpellGCN (Cheng et al., 2020) learns pronunciation/shape similarities between characters via GCNs, and combines the graph representations with BERT output for final prediction. • MLM-phonetics jointly fine-tunes a detection module and a correction module on the basis of a pre-trained language model with phonetic features. • REALISE (Xu et al., 2021) models semantic, phonetic and visual information of input characters, and selectively mixes information in these modalities to predict final corrections. • PLOME (Liu et al., 2021) extracts phonetic and visual features of characters using GRU. It also predicts the pronunciation of target characters, but in a coarse-grained, non-adaptive manner. Implementation Details In SCOPE, the encoder is initialized from ChineseBERT-base 7 , while the decoders are randomly initialized. We then conduct further pre-training on wiki2019zh for 1 epoch with a batch size of 512 and a learning rate of 10 −4 . The other hyperparameters are set to their default values as in ChineseBERT . During this pre-training stage, we do not use the adaptive task weighting scheme, and simply set the auxiliary CPP task weight to 1 for all characters for computational efficiency. After that, we fine-tune on the combined training set. We set the maximum sequence length to 192 and the learning rate to 5×10 −5 . The optimal models on SIGHAN13/SIGHAN14/SIGHAN15 are obtained by training with batch sizes of 96/96/ 64 for 20/30/30 epochs, respectively. Other hyperparameters are again set to their default values as in ChineseBERT. All experiments are conducted on 2 GeForce RTX 3090 with 24G memory. Table 3: Sentence-level performance on the test sets of SIGHAN13, SIGHAN14, SIGHAN15, where precision (P), recall (R), F1 (F) for detection (D) and correction (C) are reported (%). Baseline results are directly taken from their respective literatures. Results marked by " †" are obtained by applying a post-processing step on SIGHAN13 which removes all detected and corrected "的", "地", "得" from the model output before evaluation, due to the relatively poor annotation quality about "的", "地", "得" on SIGHAN13 as observed and suggested by Xu et al. (2021). Main Results outperforms the second best performing baseline MLM-phonetics by large margins (+5.3/+6.1 detection/correction F1). We attribute this phenomenon to the fact that the annotation quality is relatively poor on SIGHAN13, with a lot of mixed usage of "的", "地", "得" not annotated (Cheng et al., 2020). We hence follow REALISE (Xu et al., 2021) and remove all detected and corrected "的", "地", "得" from the model output before evaluation. This postprocessing trick is extremely useful on SIGHAN13, and it might even conceal improvements from other strategies on this dataset. Besides sentence-level metrics, we also consider character-level evaluation and the official SIGHAN evaluation, and make further comparison to some other methods that have their results reported in these settings (Ji et al., 2021;Liu et al., 2022). We leave the results to Appendix A, which reveal that SCOPE still performs the best in these new settings. Eliminating Encoder Differences As SCOPE uses a different and potentially more powerful encoder (i.e., ChineseBERT) compared to the baselines, we further conduct experiments to eliminate the effects of different encoders and focus solely on the auxiliary CPP task, which is the main contribution of this work. To do so, we initialize the encoder from a well-trained REALISE model (one of the best performing baselines with its code and model released to the public). Then, we perform further pre-training on wiki2019zh and fine-tune on the combination of SIGHAN benchmarks with our adaptively-weighted, fine-grained CPP task. The pre-training and fine-tuning configurations are the same as those introduced above in Section 4.1. The constrained iterative correction (CIC) strategy is also applied during inference. We call this setting SCOPE (REALISE). Table 4 presents the sentence-level performance of this new setting on the test sets of SIGHAN13, SIGHAN14, and SIGHAN15. We can observe that SCOPE (REALISE) consistently outperforms its direct opponent REALISE on all the datasets. The improvements, in most cases, are rather substantial, except for those on the relatively poorly annotated SIGHAN13. These results verify the effectiveness of our approach irrespective of the encoder. Effects of Adaptivity and Granularity This section then investigates the effects of adaptivity and granularity of the auxiliary CPP task on the overall CSC performance. Adaptivity As for adaptivity, we make comparison among the following three diverse task weighting schemes that balance the CSC and CPP tasks. • Fully-adaptive (Full-adapt) is the scheme we used in SCOPE. It determines the CPP task weights according to phonological similarities between input and target characters, and the similarities are further adjusted dynamically during model training (see Eq. (9)). • Partially-adaptive (Part-adapt) also decides the CPP task weights according to phonological similarities, but the similarities are static, defined as w i = 1 − norm(edit_distance i ), where edit_distance i is the Levenshtein edit distance (Levenshtein et al., 1966) between the pinyin sequences of the i input and target characters and norm(·) is a normalization function. The smaller the edit distance is, the larger the weight will be. • Non-adaptive (Non-adapt) considers no adaptivity and simply sets the CPP task weight to 1 for all characters (w i = 1 for all i). We compare the three settings in the SIGHAN finetuning stage, starting from the same checkpoint after pre-training on wiki2019zh with a non-adaptive task weighting scheme. Here Full-adapt is equivalent to SCOPE. Granularity As for granularity, we consider and make comparison between two types of CPP tasks. • Fine-grained (Fine) is the task we employed in SCOPE that predicts the initial, final, and tone of the pinyin of each target character. • Coarse-grained (Coarse) is a task that predicts the whole pinyin of each target character. For fair comparison, we also introduce further pretraining on wiki2019zh with a coarse-grained CPP task, and use this checkpoint to initialize the Coarse setting during SIGHAN fine-tuning. In both two settings the CPP task is adaptively weighted as in Eq. (9), and Fine is equivalent to SCOPE. Results Table 5 presents the sentence-level performance of these SCOPE variants on the test set of SIGHAN15. The scores of our best performing baseline REALISE as well as SCOPE without the CPP task (denoted as w/o CPP) are also provided for reference. We can see that introducing an auxiliary CPP task always brings benefits to CSC, no matter what level of adaptivity and granularity the task is. As for the adaptivity of task weighting, the Full-adapt scheme that considers dynamic adaptivity performs better than Part-adapt that considers static adaptivity, which in turn performs better than Non-adapt that considers no adaptivity. As for the granularity, a fine-grained CPP task performs better than a coarse-grained one. These results verify the rationality of introducing a fine-grained CPP task with adaptive task weighting to improve CSC. Case Study Table 7 further shows several cases from the SIGHAN15 test set to illustrate how the constrained iterative correction strategy (see Section 3.3) can effectively tackle consecutive spelling errors and address the over-correction issue. For consecutive errors, e.g., "户秃" in the first case, this strategy is able to correct them iteratively, one character at a time, e.g., by modifying "秃" to "涂" in the first round and then "户" to "糊" in the second round. For over-correction where the model makes unnecessary modifications, e.g., "他" to "她" in the third case and "隔" to "葛" in the fourth case, the iterative correction strategy can always change them back most of the time. Ablation and Case Study Conclusions This paper proposes SCOPE, which employs a finegrained Chinese pronunciation prediction (CPP) task with adaptive task weighting to improve the performance of Chinese spelling check (CSC). Our method builds upon a shared encoder two parallel decoders, one to predict target characters i.e., CSC, and the other to predict initials, finals, and tones of target characters, i.e., fine-grained CPP. The two decoders are then balanced adaptively according to the phonetic similarity between input and target characters. An iterative correction strategy is further designed during inference. SCOPE establishes new state-of-the-art on three SIGHAN benchmarks, verifying the effectiveness and superiority of introducing an auxiliary CPP task to improve CSC. Extensive ablation studies further verify the positive effects of dynamic adaptivity and fine granularity of this auxiliary task. Limitations SCOPE introduces an auxiliary CPP task alongside the primary CSC task in the training phase. This auxiliary CPP task causes 28% extra overhead of Tackle Consecutive Errors Input: 我以前想要高诉你，可我忘了。我真户秃。 I tried to high you before, but I forgot. I'm really house bald. Iteration 1: 我以前想要告诉你，可我忘了。我真户涂。 I tried to tell you before, but I forgot. I'm really house painted. Iteration 2: 我以前想要告诉你，可我忘了。我真糊涂。 I tried to tell you before, but I forgot. I'm really muddled. Input: 可是福物生对我们很客气。 But the fortune object man was polite to us. Iteration 1: 可是福务生对我们很客气。 But the fortune business man was polite to us. Iteration 2: 可是服务生对我们很客气。 But the waiter was very polite to us. Address Over-correction Issue Input: 他再也不会撤扬。 He will never withdraw raise again. Iteration 1: 她再也不会撤样。 She will never withdraw appearance again. Iteration 2: 他再也不会这样。 He will never do this again. Input: 幸运地，隔天她带着辞典来学校。 Fortunately, she came to school the next day with a thesaurus. Iteration 1: 幸运地，葛天她带着辞典来学校。 Fortunately, Ge Tian she came to school with a thesaurus. Iteration 2: 幸运地，隔天她带着辞典来学校。 Fortunately, she came to school the next day with a thesaurus. The results are shown in Table 8 and Table 9. We can see that regardless of the evaluation scenarios, SCOPE consistently outperforms all the baselines in almost all metrics, verifying its effectiveness and superiority for CSC. B Hyperparameter Search We conduct a hyperparameter search for learning rate, batch size and epoch. Learning rate is tuned from {2×10 −5 , 5×10 −5 }, batch size from {48, 64, 96} and epoch from {20, 30}. There are 12 hyperparameter search trials in total on each dataset. The optimal configurations are given in Section 4.1. Table 1 : 1Instances from SIGHAN15 ,SIGHAN15 (Tseng et al., 2015), and (2) 271K training examples fromWang et al. (2018) automatically generated by ASR-and OCR-based methods. We employ the test sets of SIGHAN13,Training Set #Sent Avg. Length #Errors SIGHAN15 2,338 31.1 3,037 SIGHAN14 3,437 49.6 5,122 SIGHAN13 700 41.8 343 Wang271K 271,329 42.6 381,962 Test Set #Sent Avg. Length #Errors SIGHAN15 1,100 30.6 703 SIGHAN14 1,062 50.0 771 SIGHAN13 1,000 74.3 1,224 Table 2 : 2Statistics of the datasets, including the number of sentences, the average length of sentences in tokens, and the number of errors in characters. We train on a combination of the training sets, and evaluate separately on each test set. SIGHAN14, SIGHAN15 for evaluation. The statis- tics of the used datasets are shown in Table 2. The original SIGHAN datasets are in traditional Chi- nese. We follow previous work Table 3 3presents the sentence-level performance of SCOPE and its baseline methods on the test sets of SIGHAN13, SIGHAN14, and SIGHAN15. We can see that SCOPE consistently outperforms all the baselines on all the datasets in almost all metrics, verifying its effectiveness and superiority for CSC. The improvements, in most cases, are rather sub- stantial, e.g., +2.5/+2.9 detection/correction F1 on SIGHAN15 and +1.8/+1.4 detection/correction F1 on SIGHAN14. Note that on SIGHAN13, although the improvements over the best performing base- line REALISE are somehow limited, SCOPE still 7 https://github.com/ShannonAI/ChineseBert 4280 Table 4 : 4Performance of SCOPE with the same encoder as REALISE on test sets of SIGHAN13, SIGHAN14, and SIGHAN15. Table 5 : 5Performance of SCOPE with different levels of adaptivity and granularity of the auxiliary CPP task on the test set of SIGHAN15. Ablation StudyWe conduct ablation studies on SIGHAN15 with the following settings: (1) removing the auxiliary CPP task (w/o CPP); (2) removing further pre-training on wiki2019zh (w/o FPT); and (3) removing the constrained iterative correction strategy at inference time (w/o CIC). The results are presented inTable 6. We can see that no matter which component we remove, the performance of SCOPE drops. This fully demonstrates the effectiveness of each component in our method.SIGHAN15 Detection-level Correction-level D-P D-R D-F C-P C-R C-F SCOPE 81.1 84.3 82.7 79.2 82.3 80.7 w/o CPP 79.1 82.4 80.7 76.8 80.0 78.4 w/o FPT 80.2 83.2 81.7 77.5 80.4 78.9 w/o CIC 78.3 82.6 80.4 76.5 80.8 78.6 Table 6 : 6Ablation results on the test set of SIGHAN15. The following changes are applied to SCOPE: removing the CPP task (w/o CPP), removing further pre-training (w/o FPT), and removing constrained iterative correction (w/o CIC). Table 7 : 7Cases from the SIGHAN15 test set to show how the iterative correction strategy can tackle consecutive errors and address the over-correction issue. Erroneous characters are in red, and their SCOPE corrections are in blue and underlined. computation, with the runtime per epoch increasing from 19.32 minutes to 24.68 minutes. But the extra overhead of GPU memory is almost negligible, as the CPP decoder contains only 1M out of the total 148M parameters of the whole model (to which the encoder contributes 146M parameters). Note that the additional overhead caused by CPP is required only in the training phase, but not at inference time.Dataset Model Detection-level Correction-level D-P D-R D-F C-P C-R C-F SIGHAN15 SpellGCN (Cheng et al., 2020) 77.7 85.6 81.4 96.9 82.9 89.4 PLOME (Liu et al., 2021) 85.2 86.8 86.0 97.2 85.0 90.7 CRASpell (Liu et al., 2022) 83.5 89.2 86.3 97.1 86.6 91.5 SCOPE (ours) 86.8 88.9 87.8 97.4 86.6 91.7 Table 8 : 8Character-level performance on the whole test set of SIGHAN15, with baseline results directly taken from their respective literatures.Dataset Model Detection-level Correction-level D-P D-R D-F C-P C-R C-F SIGHAN15 SpellGCN (Cheng et al., 2020) 85.9 80.6 83.1 85.4 77.6 81.3 PLOME (Liu et al., 2021) 87.9 80.9 84.3 87.6 78.3 82.7 GAD (Guo et al., 2021) 86.0 80.4 83.1 85.6 77.8 81.5 SpellBERT (Ji et al., 2021) 87.5 73.6 80.0 87.1 71.5 78.5 SCOPE (ours) 89.4 84.3 86.3 89.2 82.4 85.7 Table 9 : 9Official evaluation results on the whole test set of SIGHAN15, with baseline results directly taken from their respective literatures. Pinyin is the official phonetic system of Mandarin Chinese, which literally means "spelled sounds". In fact, we also use the pinyin of each character xi in the input sentence X during the ChineseBERT encoding process (detailed later), and this kind of pinyin of the input sentence is required and provided during both training and inference. https://github.com/brightmart/nlp_chinese_ corpus https://github.com/BYVoid/OpenCC 5 https://pypi.org/project/pypinyin 6 https://github.com/DaDaMrX/ReaLiSe https://github.com/DaDaMrX/ReaLiSe 9 https://github.com/liushulinle/CRASpell 10 http://ir.itc.ntnu.edu.tw/lre/sighan7csc.html 11 http://ir.itc.ntnu.edu.tw/lre/clp14csc.html 12 http://ir.itc.ntnu.edu.tw/lre/sighan8csc.html AcknowledgementsWe would like to thank all the reviewers for their insightful and valuable suggestions, which significantly improve the quality of this paper. This work is supported by National Natural Science Fundation of China under Grants 61876223, 62222212 and U19A2057, and Science Fund for Creative Research Groups under Grant 62121002.A Character-level and Official EvaluationThis section further compares SCOPE to some recently proposed methods that have not been evaluated with sentence-level metrics, but instead with character-level and/or official evaluation metrics. These baseline methods include:• SpellBERT (Ji et al., 2021) uses a lightweight pre-trained model for CSC, encoding phonetic and visual features with GNNs.• GAD(Guo et al., 2021)models the global dependency between all candidate characters by a global attention decoder.• CRASpell(Liu et al., 2022)constructs a noise modeling module that makes their model robust to consecutive spelling errors, with a copy mechanism to handle over-correction.For reference, we also include two previously compared baselines SpellGCN(Cheng et al., 2020)and PLOME(Liu et al., 2021)that have their results reported on these new metrics. We use the code released byREALISE (Xu et al., 2021)8 for sentencelevel evaluation and the code released by CRASpell(Liu et al., 2022) 9for character-level evaluation. The official evaluation scripts are provided along with the datasets. 10,11,12 A study of language modeling for chinese spelling check. 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[ "https://github.com/ShannonAI/ChineseBert", "https://github.com/brightmart/nlp_chinese_", "https://github.com/BYVoid/OpenCC", "https://github.com/DaDaMrX/ReaLiSe", "https://github.com/DaDaMrX/ReaLiSe", "https://github.com/liushulinle/CRASpell" ]
[ "Uncertainty principle guarantees genuine source of intrinsic randomness", "Uncertainty principle guarantees genuine source of intrinsic randomness" ]
[ "Trina Chakraborty ", "Manik Banik ", "Pinaki Patra " ]
[]
[]
The Born's rule introduces intrinsic randomness to the outcomes of a measurement performed on a quantum mechanical system. But, if the system is prepared in the eigenstate of an observable then the measurement outcome of that observable is completely predictable and hence there is no intrinsic randomness. On the other hand, if two incompatible observables are measured (either sequentially on a particle or simultaneously on two identical copies of the particle) then uncertainty principle guarantees intrinsic randomness in the subsequent outcomes independent of the preparation state of the system. In this article we show that this is true not only in quantum mechanics but for any no-signaling probabilistic theory. Also the minimum amount of intrinsic randomness that can be guaranteed for arbitrarily prepared state of the system is quantified by the amount of (un)certainty.
10.1007/s11128-013-0695-5
[ "https://arxiv.org/pdf/1211.5679v2.pdf" ]
12,865,686
1211.5679
c5e565f3a451c8b913375892a94f211760328b04
Uncertainty principle guarantees genuine source of intrinsic randomness 15 Nov 2013 Trina Chakraborty Manik Banik Pinaki Patra Uncertainty principle guarantees genuine source of intrinsic randomness 15 Nov 2013Received: date / Accepted: dateNoname manuscript No. (will be inserted by the editor) The Born's rule introduces intrinsic randomness to the outcomes of a measurement performed on a quantum mechanical system. But, if the system is prepared in the eigenstate of an observable then the measurement outcome of that observable is completely predictable and hence there is no intrinsic randomness. On the other hand, if two incompatible observables are measured (either sequentially on a particle or simultaneously on two identical copies of the particle) then uncertainty principle guarantees intrinsic randomness in the subsequent outcomes independent of the preparation state of the system. In this article we show that this is true not only in quantum mechanics but for any no-signaling probabilistic theory. Also the minimum amount of intrinsic randomness that can be guaranteed for arbitrarily prepared state of the system is quantified by the amount of (un)certainty. Introduction Heisenberg's Uncertainty Principle [1] is one of the primitive constitutional concepts of Quantum Physics. It makes a fundamental difference between quantum theory of physical world and it's classical counterpart and drastically modifies our classical conceptual framework. The uncertainty principle states that there are incompatible measurements, such as position and momentum, for which there is a trade-off relationship in the degrees of sharpness of the preparation or measurement of their values, such that a simultaneous or sequential determination of the values requires a nonzero amount of unsharpness [2,3]. This principle actually states a fundamental property of quantum systems, and is not a statement about the observational limitation of current technology. Along with uncertainty principle, Born's rule is another important key aspect in quantum mechanics, first stated by Max Born in the context of scattering theory [4]. This rule provides a link between the mathematical formalism of quantum theory and experiment and almost single-handedly responsible for practically all predictions of quantum physics. In the history of science, Born's rule is often seen as a turning point where intrinsic randomness entered into fundamental physics. Note that if the system is prepared in one of the eigenstates of a given observable then the outcome of the given observable is fully deterministic. Thus given an observable acting on a quantum system one can not associate intrinsic randomness to the outcomes independent of preparation state of the system. In other word, Born's rule cannot guarantee intrinsic randomness to the outcomes of a physical process for arbitrarily prepared state of a quantum system. In contrast to this, if two incompatible observables are measured either sequentially on a particle or simultaneously on two identical copies of the particle then according to the uncertainty principle intrinsic randomness is associated with the subsequent outcomes independent of the prepared state of the quantum system. Here we show that existence of such preparation state independent random process is guaranteed by uncertainty principle not only quantum theory but in any general no signaling theory (GNST). Considering a particular form of uncertainty principle, namely fine-grained uncertainty relation recently introduced in [5], we show that the minimum amount of intrinsic randomness that can be guaranteed for arbitrarily prepared state of the given quantum system is determined by the amount of (un)certainty present in quantum mechanics. This quantitative link also holds in all other GNSTs. Fine-grained uncertainty relation Measurements allow us to gain information about the state of a physical system. In quantum mechanics uncertainty principle imposes some limitation on what we can hope to learn about the state of the system. The general form of the Heisenberg's uncertainty relation for two observables A and B, introduced by Robertson [6], looks ∆A∆B ≥ 1 2 | ψ|[A, B]|ψ |(1) where ∆X = ψ|X 2 |ψ − ψ|X|ψ 2 represents the standard deviation which is a measure of uncertainty of the corresponding observable X, for X ∈ {A, B}. For many situations the standard deviation is not a natural way of quantifying uncertainty [7,8]. A modern approach to overcome these issues is to consider entropic measure for quantifying uncertainty [9]. We use the notation p τ (o (m) |m) to denote the probability of obtaining outcome o (m) when a measurement m, chosen from a set of measurements M, is performed on a system in state τ . In quantum theory, the state of a system is described by a density operator acting on a Hilbert space, while for a general theory, one can consider τ as an abstract representation of a state. The Shannon entropy of the distribution over measurement outcomes of measurement m on a system in state τ is given by H τ (m) = − o (m) p τ (o (m) |m) log 2 p τ (o (m) |m)(2) A general entropic uncertainty relation is of the form m p(m)H τ (m) ≥ c M,D(3) where p(m) is any probability distribution over the set of measurements M, and c M,D is some positive constant determined by M and the distribution D = {p(m)} m . Please note that, the lower bound c M,D is independent of the state τ . Entropic uncertainty relations for two observables was first introduced by Deutsch [10], then a improved version was conjectured [11] and then proved [12] (see also [13] for a recent survey about entropic uncertainty relations and references therein). In [5], the authors pointed out that entropic functions are a coarse way of measuring the uncertainty of a set of measurements as they do not distinguish the uncertainty inherent in obtaining any combination of outcomes o(m) for different measurements m. Thus they introduce fine-grained uncertainty relations consisting of a series of inequalities, one for each combination of possible outcomes, which can be written as a string o = (o (1) , ..., o (n) ) ∈ B ×n with n = |M|. That is, for each o, a set of measurements M , and distribution D = {p(m)} m P cert (τ ; o) := m∈M p(m)p τ (o (m) |m) ≤ ζ o (M, D)(4) For a fixed set of measurements, the set of inequalities U = { m∈M p τ (o (m) |m) ≤ ζ o | ∀o ∈ B ×n }(5) thus forms a fine-grained uncertainty relation. These relations dictates that one cannot obtain a measurement outcome with certainty for all measurements simultaneously whenever ζ o < 1. In this fine-grained version of the uncertainty relation, the amount of (un)certainty in a particular theory is characterized by the values of ζ o = max τ m∈M p(m)p τ (o (m) |m)(6) where the maximization is taken over all states allowed for a particular system in the concerned theory. Genuine randomness source in GNST Studying physical theory in general probabilistic framework has motivated recently by the work of L. Hardy [14], and many interesting research works have been done in this field [15,16,17,18,19,20,21,22,23]. Let τ describe the state of a system in general theory (GNST) which belongs in a convex state space Γ . Convexity of the state space Γ implies that any probabilistic mixture of two states is again a possible state of the system. Given any state τ ∈ Γ , a GNST assigns a probability measure p τ (a|A) for obtaining outcomes a ∈ {a 1 , ..., a n } when measurement A is performed on the system. If, for a given τ , p τ (a|A) is different from 1 for all a, then we can say that the measurement process A induce an intrinsic randomness to its outcomes in the concerned GNST, whenever the state of the system is described by τ . We can quantify the randomness of the outcome a resulting from the measurement of observable A on a state τ through the guessing probability [24,25] G(τ, A) = max a p τ (a|A)(7) The guessing probability can be expressed in bits and is then known as the min-entropy [26] H ∞ (τ, A) = − log 2 G(τ, A)(8) Though measurement A induces intrinsic randomness to it's outcomes when the system's state is τ , it does not give security of intrinsic randomness for arbitrarily prepared state of the system. Thus the measurement process A may not be a genuine source of intrinsic randomness (defined following) in the concerned GNST. Definition : In any GNST, a physical process will be called a genuine source of intrinsic randomness if it guarantees nonzero amount of intrinsic randomness for every possible system's states allowed in the concerned GNST. It may happen that the outcomes of the measurement A are random when the system is in the state τ 1 , whereas the outcomes are deterministic when the state of the system is τ 2 . As for example in quantum spin- 1 2 system the outcomes of spin measurement along z-direction, according to Born's rule, is fully random if the system is in one of the the eigenstates of σ x or σ y ; but the same Born's rule assigns deterministic outcomes for σ z measurement if the system is in one of it's eigenstates. Actually for any measurement process in quantum mechanics outcomes are deterministic when the system is in one of it's eigenstates. Thus according to our definition, quantum mechanical measurement processes are not genuine source of intrinsic randomness. Let there arises a situation that one, say Alice, has to produce some (private) randomness by performing a (publicly) known measurement on many copies of identically prepared quantum mechanical system, but an un-trusted party, say Bob, makes supply (with no quantum memory [27,28]) of the quantum system to her. In this situation the postulated Born's rule, alone, cannot guarantee the desired randomness as Bob might prepare the system in one of the eigenstates of the measurement observable. Interestingly in the following we show that the uncertainty principle, independent of any further assumption, can guarantee such randomness not only in quantum mechanics but in all GNST. Uncertainty guarantees genuine source of randomness Consider two two-outcomes incompatible measurements, say m 1 and m 2 , in a GNST with system's state space Γ . Taking D uniform, let the fine-grained uncertainty relation [5] confirms ζ o,m1,m2 amount of (un)certainty in the concerned GNST (o ≡ (o (m1) , o (m2) ), where ζ o,m1,m2 is given by ζ o,m1,m2 = max τ ∈Γ [ 1 2 p τ (o (m1) |m 1 ) + 1 2 p τ (o (m2) |m 2 ) ](9) For notational simplicity from now on we denote outcomes of both the measurements m 1 and m 2 by 0 and 1, i.e. o (m1) , o (m2) ∈ {0, 1}. With this assumed amount of (un)certainty in our hand in the concerned GNST, we state our main result in the following theorem. Theorem : If in any GNST the amount of (un)certainty amounts to ζ, then in the concerned GNST there exists a genuine random process which guarantees at least −2 log 2 ζ bits of intrinsic randomness. Proof : Given many copies of identically prepared system, in any allowed state τ ∈ Γ , we perform measurement m 1 on the 1 st copy of the system producing outcomes 0 and 1 with probabilities p τ (0|m 1 ) and p τ (1|m 1 ) respectively. Measurement m 2 on the 2 nd copy produces outcomes 0 and 1 with probabilities p τ (0|m 2 ) and p τ (1|m 2 ) respectively. As these two measurement processes are independent, it can be considered as a four outcomes process with outcomes denoted by (j, k) with occurrence probabilities p τ (j|m 1 )p τ (k|m 2 ) respectively, where j, k ∈ {0, 1}. This process is repetitive. Among these four probabilities min-entropy of the highest one quantifies (eqn.(8)) the random bits associated with the process. Without loss of generality consider that maximum (un)certainty is achieved for the pair (0,0), i.e. 1 2 p τ (0|m 1 ) + 1 2 p τ (0|m 2 ) ≤ ζ ∀ τ ∈ Γ(10) p τ (0|m 1 ) and p τ (0|m 2 ) are real number lying in the interval [0, 1], we thus have ( p τ (0|m 1 ) − p τ (0|m 2 )) 2 ≥ 0 ⇒ 2 p τ (0|m 1 )p τ (0|m 2 ) ≤ p τ (0|m 1 ) + p τ (0|m 2 ) ⇒ p τ (0|m 1 )p τ (0|m 2 ) ≤ 1 4 {p τ (0|m 1 ) + p τ (0|m 2 )} 2 ⇒ p τ (0|m 1 )p τ (0|m 2 ) ≤ ζ 2 ∀ τ ∈ Γ Therefore the process described above associates −2 log 2 ζ (= − log 2 ζ 2 ) bits of intrinsic randomness to the outcome string. Moreover this amount of intrinsic randomness is associated for any preparation state of the system. Thus uncertainty principle guarantees existence of a genuine random process-hence the theorem follows. Genuine source of randomness in different GNSTs Given a GNST, with system's state space Γ , we can construct hidden variable theory (HVT) [29] (see also [30] for very interesting discussion about HVT). In this HVT, state of the system is described by τ combined with another parameter λ ∈ Λ. Let's denote the state space of the system in this HVT as Γ × Λ. Given the knowledge of the system specified by the pair (τ, λ) ∈ Γ × Λ, the HVT assigns a probability rule p (τ,λ) (o (m) |m) for obtaining outcomes o (m) when measurements m ∈ M is performed on the system. Let the amount of uncertainty in our concerning GNST and the corresponding HVT is quantified by ζ Γ and ζ Γ ×Λ respectively. The minimum amount of intrinsic randomness that can be guaranteed for arbitrarily prepared state in the concerned GNST and the corresponding HVT is quantified by our theorem accordingly. In some cases, in principle, it is possible to construct realistic HVT where there is no uncertainty and therefore intrinsic randomness vanishes in this realistic HVT. In the following we discuss few important theories. Classical Physics : In classical physics there is no uncertainty relation between any pair of observables i.e. for any pair of observables in classical mechanics we always have ζ cl = 1. Therefore in classical mechanics we cannot have genuine source of intrinsic randomness. Quantum mechanics : As discussed in [5] the amount of (un)certainty in quantum mechanics amounts to ζ Q = 1 2 + 1 2 √ 2 . According to our derived formula there exists a genuine random process which certifies at least −2 log 2 ( 1 2 + 1 2 √ 2 ) ∼ = 0.457 bits of intrinsic randomness. Performing σ z and σ x measurements on the 1 st and 2 nd copy of the system respectively this minimum bound of intrinsic randomness can be achieved when the state of the system is one of the eigenstats of (σ z ± σ x )/ √ 2. In quantum mechanics the same process may allow at most 2 bits of intrinsic randomness, which is achieved when the system's is prepared in one of the eigenstates of σ y . For other choices of quantum states intrinsic randomness lies between these two extreme values. Bell-Mermin model for 2-states quantum system : An ontological model for a two dimensional Hilbert space has originally introduced by Bell [31] and then Mermin presented it in a more intuitive form [32] (see [33] for a quick view of this model). The model employs an ontic state space Λ that is a Cartesian product of a pair of state spaces, Λ = Λ ′ × Λ ′′ . Each of Λ ′ and Λ ′′ is isomorphic to the unit sphere. A system prepared according to quantum state ψ is assumed to be described by a product distribution p(λ ′ , λ ′′ |ψ) = p(λ ′ |ψ)p(λ ′′ |ψ) on Λ ′ × Λ ′′ , where p(λ ′ |ψ) is a Dirac-delta function centered at ψ and p(λ ′′ |ψ) is a uniform distribution independent of ψ. When projective measurement associated with the basis {φ, φ ⊥ } is performed Bell-Mermin model associated this measurement with the indicator function p(φ|λ ′ , λ ′′ ) = Θ(φ.(λ ′ + λ ′′ )) (11) where Θ is the Heaviside step function defined by Θ(x) = 1 if x > 0 = 0 if x ≤ 0 This Heaviside step function clearly indicates that Bell-Mermin model is a realistic HVT of 2-states quantum system, and in this theory we have no uncertainty as well as no genuine source of intrinsic randomness. Box world (PR-box) : PR correlation, introduced in [34], has got large attention in recent years to understand quantum non-locality. This correlation is a bipartite non-signaling correlation which achieves the maximum algebraic value of Bell-CHSH expression. If A and B are the binary input of two distance parties with binary outputs a and b respectively, then PR correlation is describe as P (ab|AB) = 1 2 , if a ⊕ b = AB = 0, otherwise(12) here A, B, a, b take values from {0, 1}. In [5] it has pointed out that ζ P R cond = 1 for conditional (collapsed) distributions of a PR box correlation. We can therefore conclude that conditional distributions of PR correlation will not allow any genuine random process. Spekkens's toy theory : Introducing a foundational principle, namely knowledge balance principle, Spekkens constructs a toy theory [35] in defense of the epistemic view of quantum states. A wide variety of phenomena are found to be reproduced within this toy theory analogous to quantum mechanics. Our derived relation certifies a genuine random process in this toy theory where the minimum amount of intrinsic randomness guaranteed by this process differs from that of quantum mechanics. It can be shown that in the framework of fine-grained formalism the amount of uncertainty present in toy theory amounts to ζ toy = 3 4 , therefore there exists a genuine random process which certifies at least 0.83 bits of intrinsic randomness in toy theory. Discussion Quantum mechanics, till date, is the most successful theory to describe physical world. There exists various different aspects like intrinsic randomness [36], uncertainty, nonlocality, steering, entanglement etc. that make fundamental distinction between quantum mechanics and classical mechanics. But all these aspects and possible relations among them are not yet well understood from very foundational perspective. Various interesting results have been proved recently concerning these issues, particularly about randomness, nonlocality and uncertainty [5,24,37,38,39,25]. Pironio et. al. have showed that Bell's theorem can certify random numbers [24]. In [38,39], it has been proved that intrinsic randomness can be amplified. Acin et. al. discussed about possible connection between randomness and nonlocality [25]. The question of intrinsic randomness attracts so much interest as it has practical importance in various areas like cryptography, gambling, numerical and biological simulations using monte-carlo method etc. But mathematical difficulties of characterizing random numbers [40] force us to look for physical process where generation of random number can be relied on unpredictability of that physical event [41,42,43]. So the question of existence of genuine random process in a particular theory demands practical importance along with foundational interest. In this present article, we have pointed out that though Born's rule is considered as one of the milestone postulate which has introduced intrinsic randomness in fundamental physics, it cannot certify quantum measurement process as genuine source of intrinsic randomness in quantum mechanics. On the other hand if uncertainty principle is taken as granted, then genuine source of intrinsic randomness can be certified not only in quantum mechanics but in all probabilistic theory. We also derive a quantitative connection between amount of uncertainty and minimum amount of intrinsic randomness generated from a genuine random source in any GNST. In [5] it has been proved that in any probabilistic theory the amount of nonlocality is determined by the strength of uncertainty accompanied with the strength of steering. In view of this result we can say that the minimum amount of genuine randomness certified in a single party system of a GNST, alone, cannot quantify the amount of nonlocality in a bipartite system of the concerned GNST. Our finding establishes a fundamental quantitative link between two different aspects, namely intrinsic randomness and uncertainty, of any general theory and opens few interesting questions. First of all it is worth interesting to find whether preparation state independent intrinsic randomness can be guaranteed and quantified by complementarity principle, another important feature of quantum mechanics. Our intuition go affirmative in this case. It is also interesting to study whether it is possible to quantify preparation state independent intrinsic randomness, considering other forms of uncertainty relation. Acknowledgments :We like to thank G.Kar for many simulating discussion and giving suggestions. Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik / The Actual Content of Quantum Theoretical Kinematics and Mechanics. W Heisenberg, Z. Phys. J. A. Wheeler, W. H. Zurek4362Princeton University PressN.J.Heisenberg, W.:Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik / The Actual Content of Quantum Theoretical Kinematics and Mechanics. Z. Phys. 43, 172 (1927). 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[ "Contrastive Learning of Coarse-Grained Force Fields", "Contrastive Learning of Coarse-Grained Force Fields", "Contrastive Learning of Coarse-Grained Force Fields", "Contrastive Learning of Coarse-Grained Force Fields" ]
[ "Xinqiang Ding \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUnited States\n", "Bin Zhang [email protected] \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUnited States\n", "Xinqiang Ding \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUnited States\n", "Bin Zhang [email protected] \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUnited States\n" ]
[ "Department of Chemistry\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUnited States" ]
[]
Coarse-grained models have proven helpful for simulating complex systems over long timescales to provide molecular insights into various processes. Methodologies for systematic parameterization of the underlying energy function, or force field that describes the interactions among different components of the system are of great interest for ensuring simulation accuracy. We present a new method, potential contrasting, to enable efficient learning of force fields that can accurately reproduce the conformational distribution produced with all-atom simulations. Potential contrasting generalizes the noise contrastive estimation method with umbrella sampling to better learn the complex energy landscape of molecular systems. When applied to the Trp-cage protein, we found that the technique produces force fields that thoroughly capture the thermodynamics of the folding process despite the use of only α-Carbons in the coarse-grained model.We further showed that potential contrasting could be applied over large datasets that combine the conformational ensembles of many proteins to ensure the transferability of coarse-grained force fields. We anticipate potential contrasting to be a powerful tool for building general-purpose coarse-grained force fields.
10.1021/acs.jctc.2c00616
[ "https://export.arxiv.org/pdf/2205.10861v1.pdf" ]
248,986,136
2205.10861
20056cde42edb137f1fec9b837c21c3ac348b055
Contrastive Learning of Coarse-Grained Force Fields 22 May 2022 Xinqiang Ding Department of Chemistry Massachusetts Institute of Technology 02139CambridgeMassachusettsUnited States Bin Zhang [email protected] Department of Chemistry Massachusetts Institute of Technology 02139CambridgeMassachusettsUnited States Contrastive Learning of Coarse-Grained Force Fields 22 May 2022 Coarse-grained models have proven helpful for simulating complex systems over long timescales to provide molecular insights into various processes. Methodologies for systematic parameterization of the underlying energy function, or force field that describes the interactions among different components of the system are of great interest for ensuring simulation accuracy. We present a new method, potential contrasting, to enable efficient learning of force fields that can accurately reproduce the conformational distribution produced with all-atom simulations. Potential contrasting generalizes the noise contrastive estimation method with umbrella sampling to better learn the complex energy landscape of molecular systems. When applied to the Trp-cage protein, we found that the technique produces force fields that thoroughly capture the thermodynamics of the folding process despite the use of only α-Carbons in the coarse-grained model.We further showed that potential contrasting could be applied over large datasets that combine the conformational ensembles of many proteins to ensure the transferability of coarse-grained force fields. We anticipate potential contrasting to be a powerful tool for building general-purpose coarse-grained force fields. INTRODUCTION Coarse-grained (CG) molecular dynamics simulations are computationally efficient and can simulate long time scale processes that are not accessible to all-atom simulations. [1][2][3][4] They are widely used for understanding dynamical processes in physics, chemistry and biology. [5][6][7][8][9][10][11][12][13][14][15][16][17] The accuracy of these simulations depends on how well the force fields can describe the interactions among various components of the system under investigation. Therefore, algorithms and methodologies that can produce high-quality coarse-grained force fields (CGFF), or CG potential energy, are of key interest. Numerous approaches have been introduced for systematically parameterizing CGFFs. 18,19 Top-down approaches often rely on a set of experimental structural or thermodynamic properties to fine-tune CGFFs and ensure the physical relevance of CG simulations. 3,5,18,[20][21][22][23][24] On the other hand, bottom-up approaches learn CGFFs from an ensemble of atomistic configurations collected using simulations performed at finer resolution, typically with all-atom force fields. [25][26][27][28] From the configurational ensemble, various physical quantities and correlation functions can be computed to serve as targets for recreation with CGFF. 22,23,[29][30][31][32][33][34][35][36][37][38][39][40] In addition, CGFFs can also be optimized to enforce the statistical consistency between their corresponding Boltzmann distributions and the reference configurational distribution with variational methods. [41][42][43][44] The consistency is achieved when the CG potential energy matches the potential of mean force dictated by the all-atom force field and the mapping that connects atomistic and CG configurations. Existing variational methods optimize force field parameters by formulating and solving regression problems or maximizing the likelihood of observing the reference configurations. The force matching method 25,41 and its generalization 45,45,46 belong to the former category and aim to minimize the difference between forces for CG coordinates calculated from the CGFF and target values estimated from all-atom simulations. A perfect match in forces ensures that the CG energy function reproduces the potential of mean force. On the other hand, the relative entropy method, 42 or equivalently maximum likelihood, 34 directly opti-mizes the CG energy function by minimizing the relative entropy and maximizing the overlap between the CG Boltzmann distribution and the configurational distribution from all-atom simulations. The relative entropy is minimized when the CG energy function reproduces the potential of mean force, and the CG Boltzmann distribution assigns high probabilities to configurations from all-atom simulations. While existing force field parameterization methods have found great success in many applications, they are not without limitations. For example, the relative entropy method needs to run simulations to sample from trial CG potentials in every optimization step and can be computationally expensive. While the force matching method can learn the CG potential directly without iterative sampling, it often requires extra atomic force information, and the quality of the resulting potential can be sensitive to the accumulation of errors through the integration of the estimated force. Here we developed a new variational method called potential contrasting for learning CGFFs, and applied it for multi-scale coarse-graining of protein folding. Potential contrasting generalizes the noise contrastive estimation method 47 to formulate force field parameterization into a classification problem. Input for the method is a target ensemble of protein conformations from all-atom simulations, and no atomic force information is required. When applied to the peptide Trp-cage, we found that potential contrasting can produce force fields that accurately reproduce the all-atom conformational ensemble and capture the complex folding landscape. The method also revealed the importance of including many-body potentials in CG models to describe protein biophysics with a reduced degree of freedom and implicit solvation. In addition, we showed that potential contrasting is computationally efficient and trivially parallelizable, enabling the parameterization of transferable force fields using large datasets collected from multiple proteins. METHODS Potential constrasting combines a machine learning method called noise contrastive estimation 47 (NCE) with molecular simulation techniques. In this section, we first introduce NCE using the Müller potential 48 as an example. Then we present how the NCE method is generalized and used in potential constrasting to learn CGFFs for protein folding. Noise contrastive estimation The NCE method 47 learns a probabilistic model on observed data. It is especially useful for learning unnormalized statistical models where the probability density function is only specified up to a normalization constant. It is evident that NCE is connected to bottom-up force field optimization, which aims to parameterize an energy function or an unnormalized Boltzmann distribution from data produced by all-atom simulations. Here we use the Müller potential as an example to show how NCE helps to learn energy functions. Given a set of data ( Figure 1a) drawn from the Müller potential with Markov chain Monte Carlo sampling, NCE aims to approximate their probability distribution with p(x; θ) defined as log p(x; θ) = −β[u p (x; θ) − F p ], where u p (x; θ) is the potential energy parameterized with θ and F p is the free energy. To optimize the parameters θ, NCE performs a logistic regression to discriminate the N p data samples Figure 1a) that are drawn from a noise distribution q(x). Specifically, we assign binary labels of y = 1 and y = 0 to data and noise samples, respectively. NCE parameterizes the energy function by maximizing the following averaged log-likelihood of labels: {x i p } Np i=1 from N q noise samples {x i q } Nq i=1 (ℓ(θ, F p ) = 1 N p Np i=1 log P (y = 1|x i p ) + Nq i=1 log P (y = 0|x i q ) ,(1) with P (y = 1|x) = p(x; θ) p(x; θ) + νq(x) and P (y = 0|x) = νq(x) p(x; θ) + νq(x) ,(2) where ν = P (y = 0)/P (y = 1) = N q /N p . By definition, maximizing the above objective function forces the probability function p(x; θ) to assign high values to data samples (the first term) and low values to noise samples (the second term). In that regard, NCE is similar to the standard maximum likelihood estimation, 49 which assigns high probability on training data. Previous works 47 have proven that the solution θ * for optimizing ℓ(θ, F p ) behaves like the maximum likelihood estimator for large noise sample sizes and p(x; θ * ) converges to the true data distribution. The advantage of NCE over maximum likelihood estimation is that the free energy F p is treated as a free parameter, and the optimization avoids the computationally expensive procedure for evaluating F p rigorously. In addition, a nice property of ℓ(θ, F p ) is that it is a concave function and has a unique maximum point if the potential energy function u p (x; θ) is linear to θ. Treating F p as an independent variable, while being advantageous, also introduces a dependence of NCE's performance on the noise distribution because the noise sample size is always limited in practice. If p(x; θ) is a normalized density with conserved probability mass, as in the maximum likelihood optimization, increasing its value on data samples would implicitly decrease its value on regions outside the data. Such a balance of probability density is not guaranteed in NCE since p(x; θ) is not strictly normalized due to the approximate treatment of F p . The use of a noise distribution remedies this issue by allowing an explicit probability minimization for the region covered by noise samples. While a comprehensive theory is still missing on designing optimal noise distributions, 50 we find that a useful guiding principle is to design the noise distribution such that it covers the phase space occupied by and surrounding the data samples. Without significant overlap between data and noise samples, the objective function, ℓ(θ, F p ), can be trivially optimized by assigning high probability on data samples and low probability on noise samples without forcing p(x; θ) to capture the distributional structure within the data samples. In such cases, both terms in the objective function approach the constant zero, and the gradient on θ vanishes, hindering the optimization. We parameterized the potential energy function u p (x; θ) using a two dimensional cubic spline 51 with 169 spline coefficients. The noise distribution q(x) was chosen as the uniform distribution. 500, 000 samples were generated for both data and noise. We learned the parameters θ by maximizing the NCE objective function ℓ(θ, F ) (Eq. 1) using the L-BFGS algorithm 52 (In practice, we minimize the negative of the NCE objective function). As shown in Figure 1c, u p (x; θ * ) closely matches the underlying Müller potential (Figure 1b), supporting the effectiveness of NCE for learning potential energy functions. Potential contrasting for learning force fields We find that the current formulation of NCE, although theoretically sound, is not effective for learning molecular force fields in practice. Therefore, we developed a new method named Figure 2: Workflow of the potential contrasting method for learning coarse-grained force fields for the Trp-cage protein. The functional form of the potential energy function is chosen as u p (x; θ), where θ is the set of parameters that need to be learned. The ensemble of conformations from all-atom simulations are converted into a coarse-grained ensemble using a predefined CG mapping as data samples. Here we map each amino acid into one coarse-grained particle at the C α position. Based on the data samples, a noise potential u q (x) is designed and used to generate an ensemble of noise conformations and optimize the parameters θ with potential contrasting. potential contrasting by generalizing NCE and introducing a customized way of defining the noise distribution. We present details of the method with applications to protein molecules in mind, for which the development of CGFFs is of great significance but has been challenging. 2,20 However, potential contrasting is general and can be applied to other types of molecules. Generalizing NCE to unnormalized noise distributions. Current formulation of NCE 47 requires specifying noise distributions for which the normalized probability density can be determined at ease. This requirement restricts the choice of noise distributions because the normalization constant is difficult to compute for many probabilistic functions, including Boltzmann distributions defined by complex potentials. Here we propose that this requirement is not necessary, and generalize NCE to use noise distributions specified with a potential energy function u q (x). Specifically, we set q(x) = e −β[uq(x)−Fq] , where F q is the free energy. Similarly to F p , we treat F q as an extra parameter in the optimization instead of computing its value explicitly. As a result, the logistic regression objective function in Eq. 1 becomes ℓ(θ, F p , F q ) = 1 N p Np i=1 log 1 1 + νe −β[uq(x i p )−up(x i p ;θ)+Fp−Fq] + Nq i=1 log 1 1 + ν −1 e −β[up(x i q ;θ)−uq(x i q )+Fq−Fp] .(3) Because the value of ℓ(θ, F p , F q ) in Eq. 3 only depends on θ and the difference between F p and F q , we merge the two free energy into one free parameter ∆F = F p − F q , i.e., ℓ(θ, ∆F ) = 1 N p Np i=1 log 1 1 + νe −β[uq(x i p )−up(x i p ;θ)+∆F ] + Nq i=1 log 1 1 + ν −1 e −β[up(x i q ;θ)−uq(x i q )−∆F ] .(4) Potential contrasting uses ℓ(θ, ∆F ) as the objective function and optimizes the parameters θ by maximizing ℓ(θ, ∆F ). θ * is used to represent optimized parameters. Defining the noise distribution for learning CGFFs of protein folding. As mentioned before, the performance of NCE depends critically on the noise distribution, which should produce samples with sufficient overlap with the training data. For low dimensional systems, a feasible choice for the noise is the uniform distribution used in the Müller potential example. For complex systems such as protein molecules, uniform distributions suffer the dimensionality curse to cover the relevant phase space. Our generalization to unnormalized Boltzmann distributions significantly broadens the choices of noise distributions to facilitate producing complex molecular structures that resemble data samples. We further propose an umbrella sampling procedure to design noise potential energy functions and enhancer overlap between noise and data samples. We design the noise potential energy function such that the noise samples contain both folded and unfolded structures to match the configurational ensemble from all-atom simulations. For a given protein, we start with an energy function that includes terms for bonds, angles and dihedral angles defined as u bonded (x) = L−1 i=1 1 2 k i (b i − b • i ) 2 + L−2 i=1 S angle (a i ; c a i ) + L−3 i=1 S dihedral (d i ; c d i ),(5) where L is the number of residues in the protein, and b i , a i and d We combined configurations sampled from all M umbrella simulations together to construct the noise ensemble. The probability distribution of the generalized ensemble can be [55][56][57][58][59] where u i (x) is the energy function used in the ith umbrella simulation that includes both u bonded (x) and the bias function on the RMSD. v i are adjustable energies that need to be fitted and added to the potential energy u i (x) so that the relative free energies of the M states match the relative populations of structures sampled from these states. Correspondingly, the noise potential function can be described as p gm (x) ∝ M i=1 exp(−β[u i (x)+v i ]),computed as u q (x) = −β −1 log M i=1 exp(−β[u i (x) + v i ]) . More details on the procedure are included in the Supporting Information. Extending potential contrasting to multiple proteins. With the developments outlined above, potential contrasting can be used to parameterize CG energy functions for a specific protein by optimizing ℓ(θ, ∆F ) defined in Eq. 4. It can be further generalized to learn CG potential functions with transferable parameters. Suppose that we can produce data and noise samples for a collection of proteins, the objective function to ensure that the CGFF reproduces the target configurational distribution for each protein can be defined as ℓ tot (θ, {∆F k } K k=1 ) = K k=1 1 N k p N k p i=1 log 1 1 + ν k e −β[u k q (x ki p )−up(x ki p ;θ)+∆F k ] + N k q i=1 log 1 1 + ν −1 k e −β[up(x ki q ;θ)−u k q (x ki q )−∆F k ] .(6) The above expression is a sum of potential contrasting objective functions (Eq. 4) introduced for each individual protein. {x ki p : i = 1, · · ·N k p } and {x ki q : i = 1, · · ·, N k q } represent the data and noise samples for the kth protein, with N k p and N k q corresponding to the respective sample sizes, and ν k = N k q /N k p . While the same energy function u p (x ki p ; θ) with transferable parameters θ is used, different noise potential energy functions, u k q (x ki q ), can be introduced for individual proteins. The aggregated objective function maintains the property of being concave if the CG energy function is linear to θ. We note that the objective function can be generalized straightforwardly if the CGFF introduces non-transferable parameters across proteins, as detailed in the Supporting Information. RESULTS Potential contrasting is a general-purpose method for force field parameterization. We focus on its application to protein folding and show that it can be used to optimize CGFFs for a specific protein and a collection of proteins. Given a sufficiently flexible functional form, the force field produced by potential contrasting can accurately reproduce the configurational distribution of all-atom simulations. We further demonstrate its efficiency by simultaneously optimizing over 12 proteins to derive CG potential functions with transferable parameters. Coarse-grained force field for the Trp-cage protein We applied potential contrasting to learn CGFFs for a 20 amino acids long peptide, Trpcage. As detailed in the Methods Section, potential contrasting parameterizes the force field by maximizing its effectiveness in differentiating data samples from noise samples. We use as data samples a total of N p = 1, 044, 000 conformations from a 208-µs long molecular dynamics simulation with explicit solvents performed in Ref. 60. This fully atomistic simulation captures multiple folding and unfolding events for the peptide. We generated N q = 1, 044, 000 noise samples ( Figure S3) that include both folded and disordered configurations and computed the noise potential u q (x) using the umbrella sampling procedure described in the Methods Section. In the following, we use potential contrasting to learn three CGFFs with different flexibility and complexity. For simplicity, we only use C α atoms to represent protein conformations and define energies, but potential contrasting can be easily generalized to more refined structural models. CGFF with bonded terms and pairwise non-bonded interactions. We first learned a CGFF, u pair p (x; θ), that includes bonded terms and pairwise non-bonded terms defined as u pair p (x; θ) =u bond (x) + u angle (x) + u dihedral (x) + u elec (x) + u contact (x) = L−1 i=1 1 2 k i (b i − b • i ) 2 + L−2 i=1 S angle (a i ; c a i ) + L−3 i=1 S dihedral (d i ; c d i )+ L−4 i=1 L j=i+4 q i q j 4πǫr ij exp(−r ij /λ D ) + L−4 i=1 L j=i+4 S contact (r ij ; c ij ).(7) The bond, angle, and dihedral terms are similarly defined as in Eq. 5. Non-bonded terms include electrostatics u elec (x) and a contact energy term u contact (x), both of which act between pairs of CG particles that are separated by four or more bonds. The electrostatic interaction is modeled using the Debye-Hückel theory, where q i is the net charge of the ith residue, λ D is the Debye screening length, and r ij is the distance between residues i and j. Figure 3: Parameterizing CGFFs for the Trp-cage protein using potential contrasting and allatom simulations. (a-c) Distributions of RMSD with respect to the folded structure for conformations sampled from the all-atom simulation (orange) and CG simulations with learned CG potentials that differ in the representation of the non-bonded interactions (Eq. 7-9). (d) Free energy profiles along the RMSD with respect to the folded structure for conformations sampled from the all-atom simulation and CG simulations with the three different learned potentials. (e-h) Free energy surfaces over the first two tICA coordinates for the all-atom simulation (h) and CG simulations with the three different learned potentials. The three meta-stable states in h are labels as 1, 2, and 3, with the corresponding representative structures shown in part j, k, and l. (i) The many-body potential u mb ss (x; φ * ) as a function of the RMSD with respect to the folded α-helix structure. The non-bonded contact energy is defined with cubic spline functions S contact (r ij ; c ij ) and c ij are spline basis coefficients ( Figure S1). Because bond energies are much stronger than others, the parameters b • i and k i were directly fitted based on the mean and the variance of the ith bond's distribution in the data samples. Therefore, the parameter θ only includes spline basis coefficients, i.e., θ = {c a i , c d i , c ij }. To prevent overfitting, regularization terms on the potential energy u p (x; θ) are added in the optimization to control their smoothness. Details on regularization terms are included in the Supporting Information. Since the en-ergy function depends on the parameters θ linearly, potential contrasting is guaranteed to produce a unique solution θ * . We carried out molecular dynamics simulations (see the Supporting Information for details) with the learned CGFF u pair p (x; θ * ) to evaluate the resulting structural ensemble. Similar to that from the all-atom simulation, the distribution of RMSD with respect to the folded structure for CGFF is bimodal (Figure 3a). Therefore, the learned CG potential function u pair p (x; θ * ) captures both folded and unfolded structures. However, a significant discrepancy exists between the two distributions. The CG simulation produced fewer folded structures, and the two maximums of the corresponding RMSD distribution do not exactly match that of the all-atom result. The discrepancy is more clear if we convert the RMSD distribution histogram into free energy surfaces (Figure 3d). Deviations can also be seen when comparing the free energy surface over the first two components of the time-independent component analysis 61-63 (tICA), which describe the slowest processes observed in the simulation. The all-atom surface has three meta-stable states: one folded state and two different unfolded states that cannot be differentiated using RMSD alone (Figure 3h). Although the CG simulation samples all three meta-stable states (Figure 3e), it produces a smaller population of the folded state and does not capture the cooperative transitions between folded and unfolded structures ( Figure S4). Adding many-body interactions parameterized using neural networks. The discrepancy between the CG and the all-atom simulations could be caused by the pair-wise potential being too restrictive and cannot capture many-body interactions that might arise due to coarse-graining. Next, we learned a more flexible energy function that includes an extra term parameterized using a feed-forward neural network with parameters φ, i.e., u nn p (x; θ, φ) = u pair p (x; θ) + u nn mb (x; φ).(8) The additional energy term, u nn mb (x; φ), is invariant to translations and rotations and takes angles, dihedral angles, and pairwise distances as inputs ( Figure S2). It can represent complex interactions involving multiple residues because the neural network is fully connected to couple different degrees of freedom. 64 A CG simulation performed with the learned potential function u nn p (x; θ * , φ * ) now indeed matches the all-atom results well. The maximums of the RMSD distribution are much better placed (Figure 3c), suggesting that the CG simulation accurately predicts the folded structure. Importantly, the CG simulation reproduces the relative population of the folded structure and the unfolded ensemble and the free energy barrier between them (Figure 3d and S4). Similarly, the free energy surface of the first two tICA coordinates (Figure 3g and 3h) agrees well with the all-atom one. Therefore, despite only using only α-carbons, the CGFF captures the complex folding landscape of the peptide determined from atomistic explicit solvent simulations. Adding secondary structure inspired many-body potentials. Although parameterizing the many-body energy term using a neural network improves the accuracy of the resulting force field, it has a few disadvantages. For instance, the potential function u nn p (x; θ, φ) is not linear to φ, and the optimized parameters depends on initial conditions. Moreover, it is difficult to interpret the many-body energy in simple physical terms. To avoid these issues, we learned a CG potential function with a secondary structure based many-body energy term. Secondary structure biases are frequently incorporated into coarse-grained models as fragment memories for improved quality of structural predictions. 15,22,65 They help account for cooperative effects arising from water molecules involving many residues that are challenging to describe with pair-wise potentials. Specifically, the secondary structure based many-body energy term is defined as u ss p (x; θ, φ) = u pair p (x; θ) + u ss mb (x; φ).(9) It is parameterized using cubic spline functions as u ss mb (x; φ) = S ss (rmsd ss(x, x • ); c ss ). Here rmsd ss(x, x • ) is the RMSD calculated on the α-helix (residue 3 to residue 15) between a given structure x and the folded structure x • . The parameter φ includes all the spline basis coefficients c ss . This design of the energy function in Eq. 9 further ensures linear dependence on parameters and a unique solution for force field optimization. The CG simulation results using the learned potential u ss p (x; θ * , φ * ) are shown in Figure 3b, 3d , 3f, and S4. Although the many-body energy term is restricted within the α-helix, the CG simulation correctly reproduces the relative populations of folded and unfolded states and the free energy barrier. Its performance is almost as good as the potential with a neural network based many-body term defined over the whole protein. The learned many-body potential function u ss mb (x; φ * ) along the α-helix RMSD is shown in Figure 3i. It has a deep well near 0 nm and quickly approaches zero when the RMSD is larger than 0.3 nm. Therefore, the potential only plays a significant role in stabilizing the folded structure when the α−helix is already close to the native state. Its impact is minimal when the α-helix adopts unfolded configurations. Efficient optimization of transferable force fields with data from multiple proteins The above results suggest that potential contrasting is a powerful tool to parameterize flexible CGFFs for specific proteins and capture their complex folding landscapes. Next, we show that the method also allows efficient optimization of transferable force fields using all-atom simulations of 12 fast-folding proteins performed in Ref. 60. The transferable force field for the kth protein is defined using Eq. 9 as u k p (x k ) = u bond (x k ) + u angle (x k ) + u dihedral (x k ) + u contact (x k ) + u elec (x k ) + u ss mb (x k ; φ k ).(10) As a proof of principle, we only shared parameters for pair-wise non-bonded interactions and allowed protein-specific non-transferable parameters for both the bonded term and the many-body term. The pair-wise contact potential is now defined as u contact (x) = L−4 i=1 L j=i+4 S contact (r ij ; c contact IJ ).(11) While S contact (r; c contact IJ ) shares the same functional form as that in Eq. 7, its parameters now only depend on residue types I and J. Because c contact IJ are made to depend on residue types alone, they are transferable among proteins. Our choice of limiting the force field's transferability is due to the well-known challenges of predicting secondary structures in CG models. 2 While potential contrasting allows efficient optimization of all parameters across proteins, the accuracy of the resulting CGFF may be poor. Allowing protein-specific potentials alleviates the challenges in describing secondary structures using CG models with only one particle per residue. Both transferable parameters and non-transferable parameters were learned by optimizing the aggregated objective function defined in Eq. 6. For each of the 12 proteins, we used evenly spaced 250,000 conformations from the corresponding all-atom simulation as data samples. Using the umbrella sampling procedure described in the Methods Section, we generated the same number of noise samples and computed the noise potentials u k q (x). Because the energy function u k p (x) is linear to all parameters, optimizing the aggregated objective function (Eq. 6) converges to a unique solution. In addition, because the aggregated objective function is a weighted sum of objective functions for individual proteins, its computing and optimization can be easily parallelized among proteins. Using 12 Nvidia Volta V100 GPUs, each assigned to calculate the potential contrasting objective function of one protein, we can optimize the aggregated objective function ( Figure S5) and learn all parameters in 30 minutes. CG simulations using the learned potential functions are compared to all-atom simulations (Figure 4 and S6) in terms of the radius of gyration (Rg) and the RMSD from the folded structures. Structures close to the native state are sampled in the CG simulations for all proteins (Figure 4). The lowest RMSD for configurations sampled in CG simulations range from 0.2Å to 5.3Å and are less than 4Å for 10 out of 12 proteins. Because the CG potential functions (Eq. 10) are restricted to share transferable non-bonded interactions, their performances at reproducing all-atom simulations are compromised compared to the potential function u ss p that is specific to the Trp-cage protein and has no transferable parameters. Nonetheless, the CG simulations capture folding and unfolding transitions for all but the NTL9 proteins ( Figure 4). The learned transferable contact potential energy functions between pairs of amino acids are shown in Figure 5 and S7. Although we parameterize these non-bonded contact potentials using cubic splines and do not restrict them to specific mathematical expressions, they all converge to functions that resemble the Lennard-Jones potential widely used in all-atom and CG force fields. CONCLUSION and DISCUSSION By generalizing noise contrastive estimation with unnormalized noise distributions, we developed a new method, potential contrasting, for learning force fields from reference molecular configurations. Potential contrasting combines the advantages of existing variational methods such as force matching and relative entropy minimization. As with the force matching method, it is computationally efficient and does not need sampling during force field optimization. Like the relative entropy method, potential contrasting does not require force information. We showed that the method is effective and succeeds in producing CG energy functions that accurately reproduce configurational distributions obtained from all-atom simulations. In addition, potential contrasting can be trivially parallelized for efficient learning of transferable CGFFs using simulation data of multiple systems. With its efficacy and efficiency, potential contrasting is well-positioned to systematically learn transferable CGFFs based on all-atom force fields, addressing one of the significant challenges in coarse-grained modeling. Although we focused in this study on using potential contrasting to learn CGFFs, the method is general. It can be applied to learning various types of force fields. For instance, potential contrasting can be readily applied to parameterize implicit solvent models using all-atom simulations with explicit water molecules. With further development, it could also be used to improve existing all-atom force fields by incorporating information from quantum mechanical calculations or experimental data. Such applications and development will be investigated in future studies. Our use of unnormalized noise distributions produced with umbrella sampling is essential for parameterizing accurate CGFFs. Unnormalized noise distributions defined with molecular energy functions allow the generation of noise samples that resemble the configurations produced from all-atom simulations. Therefore, significant overlap in the phase space between noise and data samples can be achieved. Such overlap can be difficult to ensure with arbitrary noise distributions since all-atom simulations only sample limited regions of phase space with low energy. We note that, upon training molecular simulation data, probabilistic models parameterized with normalizing flows [66][67][68] have been shown to produce realistic and stable molecular conformations. 46,[69][70][71][72][73] These models have indeed been proposed to serve as noise distributions for contrastive learning to guarantee overlap with data samples. 47,69 However, we found that using flow-based models as noise distributions produced CGFFs with sub-par quality. Similar findings have been reached in other recent studies as well. 50 By optimizing the overlap with data samples, flow-based models may hinder the minimization of probability for regions outside the data. Further research is needed to design optimal noise distributions in NCE. Acknowledgement This work was supported by the National Institutes of Health (R35GM133580). Supporting Information Available Detailed procedure for generating noise samples for learning CGFF of protein folding, learning CG potential functions with both transferable and non-transferable parameters, cubic splines for flexible potential energy parameterization, parameter optimization and regularization, molecular dynamics simulations with the CGFFs, many-body interactions parameterized using neural network, ( Figure S1) cubic spline basis, ( Figure S2) the neural network used for parameterizing the many-body energy term, ( Figure S3) distributions of RMSD with respect to the folded structure of the Trp-cage protein for the noise samples generated using umbrella sampling, ( Figure S4) trajectories of RMSD with respect to the folded structure of the Trp-cage protein for conformations from the all-atom simulation and the CG simulations, ( Figure S5) convergence of the aggregated loss function with weight decay during the optimization of the CG potential functions that have both transferable and non-transferable parameters, ( Figure S6) trajectories of raidus of gyration and RMSD with respect to the folded structure from the all-atom simulation and the CG simulation of all 12 proteins, (Figure S7) learned transferable contact potential energy functions between pairs of amino acids. (Table S1) setup used in umbrella sampling used for generating noise samples. Figure 1 : 1Noise contrastive learning accurately reproduces the Müller potential from sampled data only. (a) Illustration of the noise contrastive estimation method. Representative data samples from Monte Carlo sampling of the Müller potential and noise samples from a noise distribution q(x) are shown in blue and orange, respectively. The target data distribution is parameterized using a potential energy function u p (x; θ), i.e., p(x; θ) ∝ exp(−βu p (x; θ)), where θ is a set of parameters. θ is optimized in a logistic regression to classify the data and noise samples. (b) Contour plot of the Müller potential. Energy is shown in the units of k B T . (c) Contour plot of the potential energy function u p (x; θ * ) learned using noise contrastive estimation. Energy is shown in the units of k B T . i represent the ith bond, angle and dihedral angle. A quadratic function is used for energies on bonds. k i and b • i are the force constant and the equilibrium value for the ith bond. Cubic spline functions, S angle and S dihedral , are used for energies on angles and dihedral angles. c a i and c d i are spline coefficients for the ith angle and dihedral angle, respectively. Using data samples, we fit each bonded energy term in u bonded (x) independently such that it will reproduce the marginal distribution of the corresponding degree of freedom from the data samples. To generate both folded and unfolded structures for noise samples, we further carried out umbrella sampling simulations 53,54 with u bonded (x) by biasing the root-mean-squared-deviation (RMSD) from the folded structure towards different values. Figure 4 :Figure 5 : 45Comparison between all-atom simulations and CG simulations performed with the learned transferable force field. (a) For each of the 12 proteins, we show the folded structure (red) from the all-atom simulation, the structure (blue) from the CG simulation that has the lowest RMSD with respect to the folded structure, and the C α -RMSD (over all residues) between the two structures. The two plots on the right of structures are distributions of RMSD to the folded structure and distributions of Rg (radius of gyration) for conformations sampled from all-atom simulations (orange) and CG simulations (blue). (b) Trajectories of Rg and RMSD with respect to the folded structure for the all-atom simulation (orange) and the CG simulation (blue) of the Protein B. Although the data from all-atom simulations and CG simulations are plotted in the same figure, their time scales are different. Similar plots for other proteins are included in the Supporting Information. Learned transferable contact potential energy functions between representative pairs of amino acids. 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[ "NMR analogues of the quantum Zeno effect", "NMR analogues of the quantum Zeno effect" ]
[ "Li Xiao \nCentre for Quantum Computation, Clarendon Laboratory\nUniversity of Oxford\nParks RoadOX1 3PUOxfordUnited Kingdom\n", "Jonathan A Jones \nCentre for Quantum Computation, Clarendon Laboratory\nUniversity of Oxford\nParks RoadOX1 3PUOxfordUnited Kingdom\n" ]
[ "Centre for Quantum Computation, Clarendon Laboratory\nUniversity of Oxford\nParks RoadOX1 3PUOxfordUnited Kingdom", "Centre for Quantum Computation, Clarendon Laboratory\nUniversity of Oxford\nParks RoadOX1 3PUOxfordUnited Kingdom" ]
[]
We describe Nuclear Magnetic Resonance (NMR) demonstrations of the quantum Zeno effect, and discuss briefly how these are related to similar phenomena in more conventional NMR experiments.
10.1016/j.physleta.2006.06.086
[ "https://arxiv.org/pdf/quant-ph/0506235v2.pdf" ]
15,479,167
quant-ph/0506235
3f77d9daa591b4d13f7eb545e4d66870e946517a
NMR analogues of the quantum Zeno effect arXiv:quant-ph/0506235v2 30 Jun 2006 Li Xiao Centre for Quantum Computation, Clarendon Laboratory University of Oxford Parks RoadOX1 3PUOxfordUnited Kingdom Jonathan A Jones Centre for Quantum Computation, Clarendon Laboratory University of Oxford Parks RoadOX1 3PUOxfordUnited Kingdom NMR analogues of the quantum Zeno effect arXiv:quant-ph/0506235v2 30 Jun 2006NMRquantum zeno effectquantum information PACS: 0367-a0365Xp8256-b We describe Nuclear Magnetic Resonance (NMR) demonstrations of the quantum Zeno effect, and discuss briefly how these are related to similar phenomena in more conventional NMR experiments. Introduction It is often claimed that a watched pot never boils; in quantum mechanics this observation is in fact true, and is known as the quantum Zeno effect [1,2,3]. More specifically it is possible to suppress the coherent evolution of a quantum system by making frequent measurements which project the quantum system onto its eigenstates. This result is most simply described in the context of quantum information theory [4]. Consider a two-level quantum system, or qubit, such as a spin-1/2 particle in a magnetic field, with two states denoted |0 and |1 . Transitions between these states can be induced, for example by the application of a resonant electromagnetic field, causing the system to undergo coherent oscillations of the form |ψ(t) = cos(ωt/2)|0 + i sin(ωt/2)|1 (1) known as Rabi flopping, with a not gate, which interconverts |0 and |1 , occurring when ωt = π. In the language of Nuclear Magnetic Resonance (NMR) [5,6] this corresponds to applying a 180 • x pulse to a spin starting in the thermal equilibrium state I z . Suppose, however, that at time intervals τ a measurement is made in the {|0 , |1 } basis. The first measurement will project the system onto either |0 , with probability cos 2 (ωτ /2), or |1 , with probability sin 2 (ωτ /2), and if the time is short, such that ωτ ≪ 1, the system will almost always be found in the initial state |0 . Subsequent evolution and measurements will have the same effect, with the system being repeatedly reset to the initial state. If n measurements are made at equally spaced times during a not gate, such that τ = π/nω, then the probability that the system will always be found in the initial state is P n = [cos 2 (π/2n)] n ≈ exp(−π 2 /4n) ≈ 1 − π 2 4n(2) showing that frequent measurements can effectively suppress the Rabi flopping. This effect has been discussed in a range of quantum systems (e.g., [7]), but not as yet in NMR. We begin by showing how this can be done, and relate the Zeno effect to well-known phenomena in conventional NMR. An one-spin NMR implementation NMR has provided an excellent toy system for demonstrating many quantum information phenomena [8,9], but it is not immediately obvious how it can be used to demonstrate a quantum Zeno effect. This is because the standard NMR measurement process is not the sort of projective measurement usually considered in quantum theory, but rather a weak ensemble measurement which effectively monitors the spin system without changing it. It is, however, perfectly possible to simulate the effects of projective measurements in NMR by using pulsed magnetic field gradients [10]. The effect of a projective measurement on a single qubit is to project a coherent superposition of the form |ψ = α|0 + β|1 onto |0 with probability |α| 2 and onto |1 with probability |β| 2 . If the outcome of the measurement is lost, then the final state must be described by an incoherent mixture of the form |α| 2 |0 0| + |β| 2 |1 1| showing that the key effect of measurement is just to decohere the state, removing the off-diagonal coherence terms. The same effect can be achieved without performing an explicit measurement by increasing the decoherence rate, that is reducing the spin-spin relaxation time T 2 . In NMR this is conveniently simulated by reducing the apparent coherence time T * 2 by applying a magnetic field gradient [10] which makes the Larmor frequency of the spins vary over the ensemble. As the quantum Zeno effect only depends on the projection process, and not on the result of the measurement, a Zeno effect should also be seen in this system. It could be argued that gradients do not truly decohere the state, as their effects can in principle be reversed, but the effects of diffusion within the sample mean that gradients cannot be fully reversed, and so can be indistinguishable from true decoherence [11]. These ideas are easily explored in an NMR system using a single isolated spin-1/2 nucleus in exact resonance with the frequency of the applied RF. The main part of the NMR pulse sequence comprises a series of small flip-angle RF pulses separated by delays, equivalent to a dante selective excitation sequence [12] except that gradient pulses may be applied during the delays between the RF pulses. This is followed by a gradient crush to remove any xymagnetization, and a 90 • observation pulse. The final NMR signal should be a single line with an intensity proportional to the remaining I z magnetization. In the absence of the optional gradient pulses this signal strength should show cosine oscillations, tracking the underlying Rabi flopping. In the presence of the gradient pulses these oscillations should be suppressed. This sequence was applied to a standard 1 H NMR lineshape sample, comprising 1% CHCl 3 in solution in acetone-d 6 . All experiments were performed on a Varian INOVA 600 MHz spectrometer, at a temperature of 20 • C. This system has a very slow spin-lattice relaxation time (T 1 ≈ 90 s), and so T 1 effects can be ignored during the sequence. The 1 H RF frequency was placed accurately on resonance, and the power reduced so that the RF nutation rate was around 1 • per µs, and 1 µs pulses were applied at 1 ms intervals. The number of repetitions, n, was varied from 0 to 400 in steps of 10. The results are plotted in Figure 1, and show the form expected. In the absence of gradients the observed I z magnetization undergoes cosine modulation, with a period of n ≈ 360, while in the presence of gradients this modulation is almost completely suppressed. The deviations from perfect cosine modulation can be ascribed to a combination of B 1 inhomogeneity and T * 2 decoherence during the dante sequence. The slight exponential decay visible in the data acquired with gradients is consistent with the Zeno effect, and occurs because the interval between the measurements is not quite zero. Discussion The experiment above corresponds to the simplest version of the Zeno effect, in which measurements can be treated as occurring instantaneously at certain points in the evolution, but it is also possible to relax this limit slightly. As long as the measurement occurs very rapidly in comparison with the evolution rate, the overall effect will be similar (although some authors do not consider this to be a true Zeno effect [3]). When measurement is replaced by gradient-induced decoherence the timescale of the "measurement" is inversely proportional to the strength of the gradient field, and so a Zeno-like effect can be seen by attempting to excite a sample with an RF field during the application of a field gradient. Clearly this will be ineffective if the gradient is strong compared with the RF field, that is the measurement is fast compared with the nutation frequency. Indeed, as there is no real difference between frequency variation arising from gradients and that arising from intrinsic interactions, the frequency-selective behaviour of a dante sequence can be seen as an example of the Zeno effect. In the same way, the effect of a spin-lock field in suppressing Zeeman evolution can be seen as Zeno-like. A more thorough discussion of the relationship between quantum Zeno effects and decoupling sequences can be found elsewhere [13]. Despite the comments above, this one-spin demonstration is perhaps too simple to be of any real interest. In particular it is well known that the dynamics of a single isolated spin is indistinguishable from that of a classical magnetic moment, and so the results above can be understood using the classical vector model [6]. To address this we consider a two-qubit system in which something closer to traditional quantum measurements can be performed. The controllednot gate can be considered as a measurement gate, as it causes the target qubit to become correlated with the control qubit, and decoherence of the target qubit (whether spontaneous or artificially induced) completes the measurement process. A suitable quantum circuit is shown in figure 2 where the section enclosed in parentheses is repeated n times. Note that it is not necessary to know the results of the measurements on the target qubit, as these are just discarded, and so it is not necessary to initialize this qubit before the start of the circuit; indeed it is simplest to assume that this qubit starts in the maximally mixed state. The final result is obtained by measuring the control qubit in the computational basis. A two-spin NMR implementation This circuit can be easily implemented using NMR techniques in a heteronuclear two-spin system. The controlled-not gate is normally constructed from pseudo-Hadamard gates [8], single-qubit z-rotations (usually implemented using frame rotations [14]), and periods of evolution under the J-coupling between the two spins. For our experiments this approach can be simplified in two ways. Firstly the fact that the initial state of the target qubit is irrelevant means that the initial pseudo-Hadamard gate can be dropped. Secondly the RF reference frequencies for the two spins can be chosen to be in resonance with the high-frequency components of the two multiplets, such that free evolution implements both the desired J-coupling and the z-rotations simultaneously. The selective decoherence of the target spin can be implemented by applying two magnetic field gradients with equal but opposite strengths separated by a 180 • pulse applied to the target spin. This pulse will also refocus the natural evolution of the two-spin system except for the chemical shift evolution of the control spin; this can be refocused stroboscopically, that is by choosing the total length of the decoherence period as an integer multiple of the inverse of the evolution frequency. This approach can be easily extended to study weak measurements, by replacing the controlled-not gate with its r th root, that is a gate which applies a (180 • /r) x rotation to the target qubit if and only if the control qubit is in the state |1 , which can be implemented in a similar way to a controlled-not gate [8]. The final NMR pulse sequence is shown in figure 3. We chose to use the spin system provided by the 1 H and 13 C nuclei in a sample of 10 mg of 13 C labeled sodium formate (Na + HCO − 2 ) dissolved in 0.75 ml of D 2 O at a temperature of 20 • C [15]. The effects of relaxation are much more important in this system, both because of the shorter relaxation times and because of the length of the measurement (imposed by the requirement to work stroboscopically). For this reason the flip angle θ of the weak pulses was increased from around 1 • to 5 • . This larger flip angle means that the Zeno effect will not be as effective as in the one qubit system. With this more complex pulse sequence it is necessary to think more carefully about the effects of the gradients than was the case with a single qubit. In particular it appears that the negative gradient will act to partly cancel the positive gradient applied during the following iteration [6]. A full analysis shows that the situation is more complex than this, but this is a genuine concern. The problem is further complicated by the presence of diffusion, which acts to make successive gradients partially independent of one another. This helps resolve the problem described above, but can introduce a new problem, as in the presence of strong diffusion the direct effects of the gradients on the control spin will not be refocused. We addressed this compound problem in two ways. Firstly, our gradient pulses do not last for the whole period T g (see figure 3), but are somewhat shorter and are placed tightly around the 180 • pulse. This acts to maximise the effects of diffusion between measurements, while minimising them within measurements. Secondly, we chose to use the 13 C as our control qubit, with the 1 H spin as the target qubit. As the size of diffusive effects depends on the square of the gyromagnetic ratio [11] these effects will be 16 times larger for the 1 H target spin (where we want them to be large) than for the 13 C control spin (where we want them to be small). The results of implementing this sequence are shown in figure 4. Data is shown for strong measurements (r = 1), weak measurements (r = 64) and intermediate measurements (r = 16), and a calculated line, obtained by numerical simulation, is plotted in each case. As expected, strong measurements are effective in suppressing the Rabi oscillations, while weaker measurements have less effect. (The Zeno effect is not as clear in this plot as in Figure 1 because of the larger flip angle used between measurements.) There is broad agreement between experimental and calculated results, but deviations can be seen. These can be largely explained by considering imperfections in the gradients used to implement measurements on the target spin, in particular the effects of diffusion. Firstly diffusion means that the dephasing of the control spin is not perfectly refocused, and so a weak "background" measurement occurs even in the case of very large values of r. Secondly, interactions between gradients on successive measurements cannot be completely ignored. These imperfections mean that strong measurements will not be perfectly strong, and weak measurements will not be perfectly weak, leading to the deviations observed. Conclusions The Zeno effect in NMR experiments has not previously been explored, probably as a result of the difficulty of performing true quantum measurements. It is, however, possible to implement effective measurements using field gradients, allowing the effect to be easily demonstrated. Drawing an analogy between gradients and naturally occurring variations in the Larmor frequency of spins leads to a link between the Zeno effect with weak measurements and the behaviour of frequency selective pulses. This can be explored in more detail using a two spin system where the measurement strength can be easily controlled, and the expected results are seen. Fig. 1 . 1Experimental results demonstrating the quantum Zeno effect in NMR. The x-axis shows the number n of small flip-angle (approximately 1 • ) pulses applied, while the y-axis shows the NMR signal intensity as a fraction of the maximum intensity observed. Stars and circles show data points with and without the application of gradient pulses between the RF pulses. The smooth lines show a cosine oscillation cos(nθ) and an exponential decay exp(−kn), with fitted values θ = 0.978 • and k = 7.5 × 10 −5 . Fig. 2 . 2A quantum circuit for exploring the Zeno effect in a two qubit system. The section enclosed in parentheses is repeated n times. Fig. 3 . 3The NMR pulse sequence used to implement the quantum circuit shown inFigure 2in a spin system comprising a 1 H and a 13 C nucleus. The section enclosed in parentheses is repeated n times. RF pulses are shown in black, with thin, medium and thick boxes indicating flip angles of 5 • , 90 • and 180 • respectively. Gradient pulses are shown as grey boxes on the line labeled G, with height indicating the strength of each pulse. The strength of the measurement is determined by setting the delay T r = 1/2rJ, where J is the scalar coupling constant, to obtain the r th root of controlled-not gate. The period T g , during which gradients are applied, is chosen stroboscopically as 1/J. 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[ "The ALPINE-ALMA [CII] Survey: Multi-Wavelength Ancillary Data and Basic Physical Measurements", "The ALPINE-ALMA [CII] Survey: Multi-Wavelength Ancillary Data and Basic Physical Measurements" ]
[ "A L Faisst [email protected] \nIPAC\nM/C 314-6\n\nCalifornia Institute of Technology\n1200 East California Boulevard91125PasadenaCAUSA\n", "D Schaerer \nObservatoire de Genève\nUniversité de Genève\n51 Ch. des Maillettes1290VersoixSwitzerland\n\nInstitut de Recherche en Astrophysique et Planétologie − IRAP\nCNRS\nUniversité de Toulouse\nUPS-OMP\n14, avenue E. BelinF31400ToulouseFrance\n", "B C Lemaux \nDepartment of Physics\nUniversity of California\nOne Shields Ave95616Davis, DavisCAUSA\n", "P A Oesch \nObservatoire de Genève\nUniversité de Genève\n51 Ch. des Maillettes1290VersoixSwitzerland\n", "Y Fudamoto \nObservatoire de Genève\nUniversité de Genève\n51 Ch. des Maillettes1290VersoixSwitzerland\n", "P Cassata \nDipartimento di Fisica e Astronomia\nUniversità di Padova\nvicolo dell'Osservatorio3 I-35122PadovaItaly\n\nINAF\nOsservatorio Astronomico di Padova\nvicolo dell'Osservatorio 5I-35122PadovaItaly\n", "M Béthermin \nLAM (Laboratoire d'Astrophysique de Marseille)\nUMR 7326\nAix Marseille Université\nCNRS\n13388MarseilleFrance\n", "P L Capak \nIPAC\nM/C 314-6\n\nCalifornia Institute of Technology\n1200 East California Boulevard91125PasadenaCAUSA\n\nThe Cosmic Dawn Center\nUniversity of Copenhagen\nLyngbyvej 2DK-2100Vibenshuset, CopenhagenDenmark\n\nNiels Bohr Institute\nUniversity of Copenhagen\nLyngbyvej 2DK-2100CopenhagenDenmark\n", "O Le Fèvre \nLAM (Laboratoire d'Astrophysique de Marseille)\nUMR 7326\nAix Marseille Université\nCNRS\n13388MarseilleFrance\n", "J D Silverman \nKavli Institute for the Physics and Mathematics of the Universe\nThe University of Tokyo\n277-8583KashiwaJapan\n\n(Kavli IPMU\n\n\nDepartment of Astronomy\nSchool of Science\nThe University of Tokyo\n7-3-1 Hongo113-0033BunkyoTokyoJapan\n", "L Yan \nThe Caltech Optical Observatories\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n", "M Ginolfi \nObservatoire de Genève\nUniversité de Genève\n51 Ch. des Maillettes1290VersoixSwitzerland\n", "D Vergani \nINAF -Osservatorio di Astrofisica e Scienza dello Spazio di Bologna\nvia Gobetti 93/3I-40129BolognaItaly\n", "G Zamorani \nINAF -Osservatorio di Astrofisica e Scienza dello Spazio di Bologna\nvia Gobetti 93/3I-40129BolognaItaly\n", "E Zucca \nINAF -Osservatorio di Astrofisica e Scienza dello Spazio di Bologna\nvia Gobetti 93/3I-40129BolognaItaly\n", "Andreas L Faisst ", "\nSpace Telescope Science Institute\n3700 San Martin Drive21218BaltimoreMDUSA\n", "\nInstituto de Investigación Multidisciplinar en Ciencia y Tecnología\nUniversidad de La Serena\nRaúl Bitrán 1305, La SerenaChile\n", "\nDepartamento de Astronomía\nUniversidad de La Serena\nAv. Juan Cisternas, La Serena1200NorteChile\n", "\nCentro de Astronomía (CITEVA)\nUniversidad de Antofagasta\nAvenida Angamos 601AntofagastaChile\n", "\nDipartimento di Fisica e Astronomia\nUniversità di Bologna\nVia Gobetti 93/2 -I-40129BolognaItaly\n", "\nINAF -Osservatorio Astrofisico di Arcetri\nLargo E. Fermi 5I-50125FirenzeItaly\n", "\nJet Propulsion Laboratory\nCalifornia Institute of Technology\n91109PasadenaCAUSA\n", "\nInstituto de Física y Astronomía\nUniversidad de Valparaíso\nAvda. Gran Bretaña 1111ValparaísoChile\n", "\nCavendish Laboratory\nUniversity of Cambridge\n19 J. J. Thomson AveCB3 0HECambridgeUK\n", "\nKavli Institute for Cosmology\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK\n", "\nMax-Planck-Institut für Astronomie\nKönigstuhl 17D-69117HeidelbergGermany\n", "\nDepartment of Astronomy\nCornell University\nSpace Sciences Building14853IthacaNYUSA\n", "\nCalifornia Institute of Technology\n1200 East California Boulevard249-17, 91125PasadenaMC, CAUSA\n", "\nLeiden Observatory\nLeiden University\nPO Box 95002300 RALeidenThe Netherlands\n" ]
[ "IPAC\nM/C 314-6", "California Institute of Technology\n1200 East California Boulevard91125PasadenaCAUSA", "Observatoire de Genève\nUniversité de Genève\n51 Ch. des Maillettes1290VersoixSwitzerland", "Institut de Recherche en Astrophysique et Planétologie − IRAP\nCNRS\nUniversité de Toulouse\nUPS-OMP\n14, avenue E. BelinF31400ToulouseFrance", "Department of Physics\nUniversity of California\nOne Shields Ave95616Davis, DavisCAUSA", "Observatoire de Genève\nUniversité de Genève\n51 Ch. des Maillettes1290VersoixSwitzerland", "Observatoire de Genève\nUniversité de Genève\n51 Ch. des Maillettes1290VersoixSwitzerland", "Dipartimento di Fisica e Astronomia\nUniversità di Padova\nvicolo dell'Osservatorio3 I-35122PadovaItaly", "INAF\nOsservatorio Astronomico di Padova\nvicolo dell'Osservatorio 5I-35122PadovaItaly", "LAM (Laboratoire d'Astrophysique de Marseille)\nUMR 7326\nAix Marseille Université\nCNRS\n13388MarseilleFrance", "IPAC\nM/C 314-6", "California Institute of Technology\n1200 East California Boulevard91125PasadenaCAUSA", "The Cosmic Dawn Center\nUniversity of Copenhagen\nLyngbyvej 2DK-2100Vibenshuset, CopenhagenDenmark", "Niels Bohr Institute\nUniversity of Copenhagen\nLyngbyvej 2DK-2100CopenhagenDenmark", "LAM (Laboratoire d'Astrophysique de Marseille)\nUMR 7326\nAix Marseille Université\nCNRS\n13388MarseilleFrance", "Kavli Institute for the Physics and Mathematics of the Universe\nThe University of Tokyo\n277-8583KashiwaJapan", "(Kavli IPMU\n", "Department of Astronomy\nSchool of Science\nThe University of Tokyo\n7-3-1 Hongo113-0033BunkyoTokyoJapan", "The Caltech Optical Observatories\nCalifornia Institute of Technology\n91125PasadenaCAUSA", "Observatoire de Genève\nUniversité de Genève\n51 Ch. des Maillettes1290VersoixSwitzerland", "INAF -Osservatorio di Astrofisica e Scienza dello Spazio di Bologna\nvia Gobetti 93/3I-40129BolognaItaly", "INAF -Osservatorio di Astrofisica e Scienza dello Spazio di Bologna\nvia Gobetti 93/3I-40129BolognaItaly", "INAF -Osservatorio di Astrofisica e Scienza dello Spazio di Bologna\nvia Gobetti 93/3I-40129BolognaItaly", "Space Telescope Science Institute\n3700 San Martin Drive21218BaltimoreMDUSA", "Instituto de Investigación Multidisciplinar en Ciencia y Tecnología\nUniversidad de La Serena\nRaúl Bitrán 1305, La SerenaChile", "Departamento de Astronomía\nUniversidad de La Serena\nAv. Juan Cisternas, La Serena1200NorteChile", "Centro de Astronomía (CITEVA)\nUniversidad de Antofagasta\nAvenida Angamos 601AntofagastaChile", "Dipartimento di Fisica e Astronomia\nUniversità di Bologna\nVia Gobetti 93/2 -I-40129BolognaItaly", "INAF -Osservatorio Astrofisico di Arcetri\nLargo E. Fermi 5I-50125FirenzeItaly", "Jet Propulsion Laboratory\nCalifornia Institute of Technology\n91109PasadenaCAUSA", "Instituto de Física y Astronomía\nUniversidad de Valparaíso\nAvda. Gran Bretaña 1111ValparaísoChile", "Cavendish Laboratory\nUniversity of Cambridge\n19 J. J. Thomson AveCB3 0HECambridgeUK", "Kavli Institute for Cosmology\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK", "Max-Planck-Institut für Astronomie\nKönigstuhl 17D-69117HeidelbergGermany", "Department of Astronomy\nCornell University\nSpace Sciences Building14853IthacaNYUSA", "California Institute of Technology\n1200 East California Boulevard249-17, 91125PasadenaMC, CAUSA", "Leiden Observatory\nLeiden University\nPO Box 95002300 RALeidenThe Netherlands" ]
[]
We present the ancillary data and basic physical measurements for the galaxies in the ALMA Large Program to Investigate C + at Early Times (ALPINE) survey − the first large multi-wavelength survey which aims at characterizing the gas and dust properties of 118 main-sequence galaxies at redshifts 4.4 < z < 5.9 via the measurement of [C II] emission at 158 µm (64% at > 3.5σ) and the surrounding 2 Faisst et al.far-infrared (FIR) continuum in conjunction with a wealth of optical and near-infrared data. We outline in detail the spectroscopic data and selection of the galaxies as well as the ground-and spacebased imaging products. In addition, we provide several basic measurements including stellar masses, star formation rates (SFR), rest-frame ultra-violet (UV) luminosities, UV continuum slopes (β), and absorption line redshifts, as well as Hα emission derived from Spitzer colors. We find that the ALPINE sample is representative of the 4 < z < 6 galaxy population selected by photometric methods and only slightly biased towards bluer colors (∆β ∼ 0.2). Using [C II] as tracer of the systemic redshift (confirmed for one galaxy at z = 4.5 out of 118 for which we obtained optical [O II]λ3727Å emission), we confirm red shifted Lyα emission and blue shifted absorption lines similar to findings at lower redshifts. By stacking the rest-frame UV spectra in the [C II] rest-frame we find that the absorption lines in galaxies with high specific SFR are more blue shifted, which could be indicative of stronger winds and outflows.
10.3847/1538-4365/ab7ccd
[ "https://arxiv.org/pdf/1912.01621v2.pdf" ]
208,617,478
1912.01621
1d326195b05f4291ac3ffd191c0fafb24bc8065b
The ALPINE-ALMA [CII] Survey: Multi-Wavelength Ancillary Data and Basic Physical Measurements March 16, 2020 13 Mar 2020 A L Faisst [email protected] IPAC M/C 314-6 California Institute of Technology 1200 East California Boulevard91125PasadenaCAUSA D Schaerer Observatoire de Genève Université de Genève 51 Ch. des Maillettes1290VersoixSwitzerland Institut de Recherche en Astrophysique et Planétologie − IRAP CNRS Université de Toulouse UPS-OMP 14, avenue E. BelinF31400ToulouseFrance B C Lemaux Department of Physics University of California One Shields Ave95616Davis, DavisCAUSA P A Oesch Observatoire de Genève Université de Genève 51 Ch. des Maillettes1290VersoixSwitzerland Y Fudamoto Observatoire de Genève Université de Genève 51 Ch. des Maillettes1290VersoixSwitzerland P Cassata Dipartimento di Fisica e Astronomia Università di Padova vicolo dell'Osservatorio3 I-35122PadovaItaly INAF Osservatorio Astronomico di Padova vicolo dell'Osservatorio 5I-35122PadovaItaly M Béthermin LAM (Laboratoire d'Astrophysique de Marseille) UMR 7326 Aix Marseille Université CNRS 13388MarseilleFrance P L Capak IPAC M/C 314-6 California Institute of Technology 1200 East California Boulevard91125PasadenaCAUSA The Cosmic Dawn Center University of Copenhagen Lyngbyvej 2DK-2100Vibenshuset, CopenhagenDenmark Niels Bohr Institute University of Copenhagen Lyngbyvej 2DK-2100CopenhagenDenmark O Le Fèvre LAM (Laboratoire d'Astrophysique de Marseille) UMR 7326 Aix Marseille Université CNRS 13388MarseilleFrance J D Silverman Kavli Institute for the Physics and Mathematics of the Universe The University of Tokyo 277-8583KashiwaJapan (Kavli IPMU Department of Astronomy School of Science The University of Tokyo 7-3-1 Hongo113-0033BunkyoTokyoJapan L Yan The Caltech Optical Observatories California Institute of Technology 91125PasadenaCAUSA M Ginolfi Observatoire de Genève Université de Genève 51 Ch. des Maillettes1290VersoixSwitzerland D Vergani INAF -Osservatorio di Astrofisica e Scienza dello Spazio di Bologna via Gobetti 93/3I-40129BolognaItaly G Zamorani INAF -Osservatorio di Astrofisica e Scienza dello Spazio di Bologna via Gobetti 93/3I-40129BolognaItaly E Zucca INAF -Osservatorio di Astrofisica e Scienza dello Spazio di Bologna via Gobetti 93/3I-40129BolognaItaly Andreas L Faisst Space Telescope Science Institute 3700 San Martin Drive21218BaltimoreMDUSA Instituto de Investigación Multidisciplinar en Ciencia y Tecnología Universidad de La Serena Raúl Bitrán 1305, La SerenaChile Departamento de Astronomía Universidad de La Serena Av. Juan Cisternas, La Serena1200NorteChile Centro de Astronomía (CITEVA) Universidad de Antofagasta Avenida Angamos 601AntofagastaChile Dipartimento di Fisica e Astronomia Università di Bologna Via Gobetti 93/2 -I-40129BolognaItaly INAF -Osservatorio Astrofisico di Arcetri Largo E. Fermi 5I-50125FirenzeItaly Jet Propulsion Laboratory California Institute of Technology 91109PasadenaCAUSA Instituto de Física y Astronomía Universidad de Valparaíso Avda. Gran Bretaña 1111ValparaísoChile Cavendish Laboratory University of Cambridge 19 J. J. Thomson AveCB3 0HECambridgeUK Kavli Institute for Cosmology University of Cambridge Madingley RoadCB3 0HACambridgeUK Max-Planck-Institut für Astronomie Königstuhl 17D-69117HeidelbergGermany Department of Astronomy Cornell University Space Sciences Building14853IthacaNYUSA California Institute of Technology 1200 East California Boulevard249-17, 91125PasadenaMC, CAUSA Leiden Observatory Leiden University PO Box 95002300 RALeidenThe Netherlands The ALPINE-ALMA [CII] Survey: Multi-Wavelength Ancillary Data and Basic Physical Measurements 727March 16, 2020 13 Mar 2020Submitted to ApJSDraft version Typeset using L A T E X twocolumn style in AASTeX63 Corresponding author:galaxies: evolution -galaxies: fundamental parameters -galaxies: ISM -galaxies: star formation -galaxies: photometry We present the ancillary data and basic physical measurements for the galaxies in the ALMA Large Program to Investigate C + at Early Times (ALPINE) survey − the first large multi-wavelength survey which aims at characterizing the gas and dust properties of 118 main-sequence galaxies at redshifts 4.4 < z < 5.9 via the measurement of [C II] emission at 158 µm (64% at > 3.5σ) and the surrounding 2 Faisst et al.far-infrared (FIR) continuum in conjunction with a wealth of optical and near-infrared data. We outline in detail the spectroscopic data and selection of the galaxies as well as the ground-and spacebased imaging products. In addition, we provide several basic measurements including stellar masses, star formation rates (SFR), rest-frame ultra-violet (UV) luminosities, UV continuum slopes (β), and absorption line redshifts, as well as Hα emission derived from Spitzer colors. We find that the ALPINE sample is representative of the 4 < z < 6 galaxy population selected by photometric methods and only slightly biased towards bluer colors (∆β ∼ 0.2). Using [C II] as tracer of the systemic redshift (confirmed for one galaxy at z = 4.5 out of 118 for which we obtained optical [O II]λ3727Å emission), we confirm red shifted Lyα emission and blue shifted absorption lines similar to findings at lower redshifts. By stacking the rest-frame UV spectra in the [C II] rest-frame we find that the absorption lines in galaxies with high specific SFR are more blue shifted, which could be indicative of stronger winds and outflows. INTRODUCTION The Early Growth Phase in Galaxy Evolution Galaxy evolution undergoes several important phases such as the ionization of neutral Hydrogen at redshifts z > 6 (also known as the Epoch of Reionizaton) as well as a time of highest cosmic star-formation rate (SFR) density at z ∼ 2 − 3. The transition phase at z = 4 − 6 (a time roughly 0.9 to 1.5 billion years after the Big Bang), often referred to as the early growth phase, is currently in focus of many studies. This time is of great interest for understanding galaxy evolution as it connects primordial galaxy formation during the epoch of reionization with mature galaxy growth at and after the peak of cosmic SFR density. During a time of only 600 Myrs, the cosmic stellar mass density in the universe increased by one order of magnitude (Caputi et al. 2011;Davidzon et al. 2017), galaxies underwent a critical morphological transformation to build up their disk and bulge structures (Gnedin et al. 1999;Bournaud et al. 2007;Agertz et al. 2009), and their interstellar medium (ISM) became enriched with metal from sub-solar to solar amounts (Ando et al. 2007;Faisst et al. 2016a), while at the same time the dust attenuation of the UV light significantly increased (Finkelstein et al. 2012;Bouwens et al. 2015;Fudamoto et al. 2017;Popping et al. 2017;Cullen et al. 2018;Ma et al. 2019;Yamanaka & Yamada 2019). Furthermore, the most massive of these galaxies may become the first quiescent galaxies already at z > 4 ( Glazebrook et al. 2017;Valentino et al. 2019;Tanaka et al. 2019;Stockmann et al. 2020;Faisst et al. 2019). All this put together, makes the early growth phase an important puzzle piece to be studied in order to decipher how galaxies formed and evolved to become the galaxies (either star forming or quiescent) that we observe in the local universe. It is evident from studies at lower redshift that multiwavelength observations are crucial for us to be able to form a coherent picture of galaxy evolution. To capture several important properties of galaxies, a panchromatic survey must comprise several spectroscopic and imaging datasets that cover a large fraction of the wavelength range of a galaxy's light emission, including (i) the restframe ultra-violet (UV) containing Lyα emission, as well as several absorption lines to study stellar winds and metallicity (Heckman et al. 1997;Maraston et al. 2009;Steidel et al. 2010;Faisst et al. 2016a), (ii) the restframe optical containing tracers of age (Balmer break) as well as important emission lines (e.g., Hα) to quantify the star-formation and gas metal properties (Kennicutt 1998;Kewley & Ellison 2008), and (iii) the far-infrared (FIR) continuum and several FIR emission lines (e.g,. [C II]λ158 µm or [N II]λ205 µm) that provide insights into the gas and dust properties of galaxies (De Looze et al. 2014;Pavesi et al. 2019). Fortunately, the early growth phase at redshifts z = 4 − 6 is at the same time the highest redshift epoch at which, using current technologies, such a panchromatic study can be carried out. The rest-frame UV part of the energy distribution at these redshifts has been probed in the past thanks to several large spectroscopic (Le Fèvre et al. 2015;Hasinger et al. 2018) and imaging (Capak et al. 2007;McCracken et al. 2012;Aihara et al. 2019) surveys from the ground as well as imaging surveys with the Hubble Space Telescope (HST, Grogin et al. 2011;Koekemoer et al. 2011;Scoville et al. 2007a). In addition, Hα has been accessed successfully through observations with the Spitzer Space Telescope (Stark et al. 2013;de Barros et al. 2014;Smit et al. 2014;Rasappu et al. 2016;Smit et al. 2016;Faisst et al. 2016aFaisst et al. , 2019Lam et al. 2019). However, the FIR of z > 4 galaxies has only been probed sparsely in the past in less than a dozen galaxies using the Atacama Large 1 10 100 1000 Hα [CII] gas, dust, dust obscured star formation hot gas and star formation stars, dust, metallicity, stellar winds Figure 1. ALPINE builds the corner stone of a panchromatic survey at z = 4 − 6. The diagram shows the multi-wavelength data products that are currently available for all the ALPINE galaxies. The currently covered parts of the spectrum are indicated in red. The numbers link to sections in this paper where the data products and their analysis are explained in detail. The spectrum sketch is based on a typical z = 5 galaxy (adapted from Harikane et al. 2018). (Sub-) Millimeter Array (ALMA, Riechers et al. 2014;Capak et al. 2015;Watson et al. 2015;Willott et al. 2015;Strandet et al. 2017;Carniani et al. 2018;Zavala et al. 2018b,a;Casey et al. 2019;Jin et al. 2019) as well as some as part of Herschel surveys in lensed and unlensed fields (e.g., Egami et al. 2010;Combes et al. 2012;Casey et al. 2012Casey et al. , 2014. Commonly targeted by observations with ALMA is singly ionized Carbon (C + ) at 158 µm, which is an important coolant for the gas in galaxies and is therefore broadly related to star formation activity and gas masses (Stacey et al. 1991;Carilli & Walter 2013;De Looze et al. 2014). The [C II] emission line is one of the strongest in the FIR and is in addition conveniently located in the ALMA Band 7 at redshifts z = 4 − 6 at one of the highest atmospheric transmissions compared to other FIR lines (see, e.g., Faisst et al. 2017). The origin of [C II] emission is still debated. In addition to photo-dissociation regions (PDRs) and the cold neutral medium, a significant fraction can also origin from ionized gas regions or CO-dark molecular clouds (Pineda et al. 2013;Vallini et al. 2015;Pavesi et al. 2016). Also, the increasing temperature of the Cosmic Microwave Background (CMB) has an effect on the relation between [C II] and star formation . Both potentially complicates the interpretation of [C II] as SFR indicator at high redshifts. Similar to Hα, [C II] traces the gas kinematics in a galaxy and is therefore an important component to quantify rotationand dispersion-dominated systems as well as outflows (Jones et al. 2017;Pavesi et al. 2018;Kohandel et al. 2019;Ginolfi et al. 2019). The FIR landscape has dramatically changed with the completion of the ALMA Large Program to Investigate C + at Early Times (ALPINE, #2017.1.00428.L). ALPINE is laying the ground work for the exploration of gas and dust properties in 118 main-sequence star forming galaxies in the early growth phase at 4.4 < z < 5.9 and herewith started the first panchromatic survey of its kind at these redshifts. ALPINE in a Nutshell In the following, we summarize the scope of the ALPINE survey, we refer to Le Fèvre et al. (2019) for a broader overview of the program. ALPINE is a 69 hour large ALMA program started in Cycle 5 in May 2018 and completed during Cycle 6 in February 2019. In total, 118 galaxies have been observed in Band 7 (covering [C II] emission at 158 µm and its nearby continuum) at a spatial resolution of < 1.0 and with integration times ∼ 30 minutes on-source depending on their predicted [C II] flux. The galaxies origin from two fields, namely the Cosmic Evolution Survey field (COSMOS, 105 galaxies, Scoville et al. 2007b) and the Extended Chandra Deep Field South (ECDFS, 13 galaxies, Giacconi et al. 2002). Due to gaps in the transition through the atmosphere, the galaxies are split in two different redshift ranges spanning 4.40 < z < 4.65 and 5.05 < z < 5.90 with medians of z = 4.5 and 5.5 and galaxy numbers of 67 and 51, respectively. All galaxies are spectroscopically confirmed by either Lyα emission or rest-UV absorption lines and are selected to be brighter than an absolute UV magnitude of M 1500 = −20.2. This limit is roughly equivalent to a SFR cut at 10 M yr −1 and corresponds roughly to a limiting luminosity in [C II] emission of L [CII] = 1.2 +1.9 −0.9 × 10 8 L (assuming the relation derived by De Looze et al. 2014). Assuming a 3.5σ detection limit, the [C II] detection rate is 64% and continuum emission is detected in 19% of the galaxies (see Figure 2). The main science goals enabled by ALPINE are diverse and cover many crucial research topics at high redshifts: − connecting [C II] to star-formation at high redshifts, − coherent study of the total SFR density at z > 4 including the contribution of dust-obscured star formation, − study of gas dynamics and merger statistics from [C II] kinematics and quantification of UV-faint companion galaxies, − study of gas fractions and dust properties at z > 4, − the first characterization of ISM properties using L FIR /L UV and [C II]/FIR continuum diagnostics for a large sample at z > 4, − quantifying outflows and feedback processes in z > 4 galaxies from [C II] line profiles. Note that ALPINE provides at the same time the equivalent of a blind-survey of approximately 25 squarearcminutes. This enables us to estimate the obscured fraction of star-formation (mostly below z = 4) by finding UV-faint galaxies with FIR continuum or [C II] emission. The serendipitous continuum sources and [C II] detections are discussed in detail in Bethermin et al. (2020) and Loiacono et al. (in prep.). A more detailed description of these science goals can be found in our survey overview paper . ALPINE is based on a rich set of ancillary data, which makes it the first panchromatic survey at these high redshifts including imaging and spectroscopic observations at FIR wavelengths (see Figure 1). The backbone for a successful selection of galaxies are rest-frame UV spectroscopic observations from the Keck telescope in Hawaii as well as the European Very Large Telescope (VLT) in Chile. These are complemented by ground-based imaging observations from rest-frame UV to optical, HST observations in the rest-frame UV, and Spitzer coverage above the Balmer break at rest-frame 4000Å. The latter is crucial for the robust measurement of stellar masses at these redshifts (e.g., Faisst et al. 2016b). For a survey overview of ALPINE see Le and for details on the data analysis see Bethermin et al. (2020). In this paper, we present these valuable ancillary data products and detail several basic measurements for the ALPINE galaxies. The outline of the paper is sketched in Figure 1. Specifically, in Section 2, we present the spectroscopic data and detail the spectroscopic selection of the ALPINE galaxies. In the same section, we also present stacked spectra and touch on velocity offsets between Lyα, [C II], and absorption line redshifts. Section 3 is devoted to the photometric data products, which include ground-and space-based photometry. In Section 4.1, we detail the derivation of sev-eral galaxy properties from the observed photometry. These include stellar masses, SFRs, UV luminosities, UV continuum slopes, as well as Hα emission derived from Spitzer colors. We conclude and summarize in Section 5. All presented data products are available in the online printed version of this paper 1 . The different catalogs and their columns are described in detail in the Appendix A. HST cutouts and rest-frame UV spectra for each of the ALPINE galaxies are shown in Appendix B. Throughout the paper we assume the ΛCDM cosmology with H 0 = 70 km s −1 Mpc −1 , Ω Λ = 0.70, and Ω m = 0.30. All magnitudes are given in the AB system (Oke 1974) and stellar masses and SFRs are normalized to a Chabrier (2003) initial mass function (IMF). SPECTROSCOPIC DATA AND SELECTION Spectroscopic selection of ALPINE galaxies The ALPINE survey is only possible due to a spectroscopic pre-selection of galaxies from large spectroscopic surveys on COSMOS and ECDFS. This is because the ALMA frequency bands are narrow (∼ 1000 km s −1 ), and in order to observe [C II] emission the redshift has to be known within a precision of ∼ 1000 km/s. The galaxy selection is refined to optimize the efficiency of the ALMA observations by creating groups of galaxies in spectral dimensions. Our sample also includes 7 galaxies that were previously observed with ALMA by Riechers et al. (2014) and Capak et al. (2015). These are HZ1, HZ2, HZ3, HZ4, HZ5, HZ6 /LBG-1, and HZ8, which correspond to the ALPINE galaxies DC 536534, DC 417567, DC 683613, DC 494057, DC 845652, DC 848185, and DC 873321, respectively. Furthermore, four galaxies from the VUDS survey (vc 5101288969, vc 5100822662, and vc 510786441 in COSMOS and ve 530029038 in ECDFS) are observed twice (resulting in a total number of 122 observations). The duplicate observations are used for quality assessment. Bethermin et al. (2020) describes the combination of these observations. The rest-frame UV spectroscopic data from which the ALPINE sample is selected combine various large surveys on the COSMOS and ECDFS fields. Out of the 105 ALPINE galaxies on the COSMOS field, 84 are obtained by the large DEIMOS spectroscopic survey (Capak et al. 2004;Mallery et al. 2012;Hasinger et al. 2018) et al. (2012) and Hasinger et al. (2018). ‡ Six of these galaxies are also observed as part of the Keck/DEIMOS survey (ref. 1). The corresponding number per selection from the Keck/DEIMOS program is given in square-brackets for those six galaxies. a Lyα emitters selected with NB711. b Lyα emitters selected with NB814. c Color-selected galaxies in B, g + , V , r + , and z ++ using the criteria from Ouchi et al. (2004);Capak et al. (2004Capak et al. ( , 2011Iwata et al. (2003); Hildebrandt et al. (2009). d Galaxies with a photometric redshift z > 4 with a probability of> 50% Vanzella et al. (2007Vanzella et al. ( , 2008; Balestra et al. (2010), (4) Malhotra et al. (2005); Rhoads et al. (2009) in Chile. In total 6 of the VUDS spectra are independently also observed as part of the Keck/DEIMOS survey (vc 5100559223, vc 5100822662, vc 5101218326, vc 5101244930, vc 5101288969, vc 510786441 ). The redshifts are consistent within 280 km s −1 and we do not find any systematic offsets between the two observations (see also Section 2.4.1). Out of the 13 galaxies in the ECDFS field, 11 are obtained from spectroscopic observations with VIMOS (9) GRAPES (HST grism, ECDFS) VLT VIMOS and FORS2 (ECDFS) X-ray (Chandra) LBG (color) Narrow-band (low-z) Narrow-band (high-z) 4.5 m excess photometric redshift VUDS (photo-z + LBG) Figure 3. Redshift distribution of ALPINE galaxies. Each bar shows the stacked number of different selections per bin (see Table 1 and description in text). The bins with galaxies from the ECDFS field are hatched. The left and right panels show galaxies in the two different redshift bins. VLT VIMOS and FORS2 (ECDFS) X-ray (Chandra) LBG (color) Narrow-band (low-z) Narrow-band (high-z) 4.5 m excess photometric redshift VUDS (photo-z + LBG) Parent (4 < z < 6) 9.5 10.0 10.5 11.0 11.5 12.0 log( L /L ) at rest-frame 1500 Å Table 1. The measurements on the parent sample in COSMOS at 4 < z < 6 is shown in light gray. The color-coding is the same as in Figure 3. The arrows show 1σ upper limits. The gray area denotes the M * UV , the knee of the UV luminosity function, which corresponds to −21.1 ± 0.15 (or log(νLν /L ) ∼ 10.77) at z = 5 (Bouwens et al. 2015). The derivation of the photometry is described in detail in Section 3. (2 2 ) at the VLT (Vanzella et al. 2007(Vanzella et al. , 2008Balestra et al. 2010), and 2 come from the HST grism survey GRAPES (Malhotra et al. 2005;Rhoads et al. 2009). The spectral resolution of the different dataset varies between R ∼ 100 (ECDFS/GRAPES grism), R ∼ 180 (ECDFS/VIMOS), R ∼ 230 (COSMOS/VUDS), R ∼ 660 (ECDFS/FORS2), and R ∼ 2500 (COS-MOS/DEIMOS). Biases towards dust-poor star-forming galaxies with strong rest-frame UV emission lines (such as Lyα) can be common in purely spectroscopically selected samples. To minimized such biases as much as possible, the spectroscopically observed galaxies have been pre-selected through a variety of different selection methods. The largest fraction of galaxies in ALPINE is drawn from the Keck/DEIMOS and VUDS surveys on the COS-MOS field. Both surveys include galaxies preselected in various ways, resulting in the most representative and inclusive spectroscopic high-redshift galaxy sample. Specifically, the VUDS survey combines predominantly a photometric redshift selection with a colorselected Lyman Break Galaxy (LBG) selection (Le Fèvre et al. 2015), known as the Lyman-break drop-out tech-nique (see, e.g., Steidel et al. 1996;Dickinson 1998). The Keck/DEIMOS survey (providing 71% of the total ALPINE sample) consists of galaxies that are selected by narrow-band surveys at z ∼ 4.5 (7%) and z ∼ 5.7 (27%), the drop-out technique (color selection) over the whole redshift range (49%), as well as purely by photometric redshifts (11%). In addition, 4 galaxies are selected by a 4.5 µm excess and one galaxy was preselected through X-ray emission using the Chandra observatory. On the ECDFS field, the galaxies are mostly color-selected. Table 1 summarizes the different selections and corresponding numbers of galaxies and provides a complete list of references. We also list the numbers of galaxies with Lyα emission (76%) and weak Lyα emission or Lyα absorption (∼ 24%). Note that the Keck/DEIMOS and VUDS samples have similar Lyα emission properties. However, note that above z = 5, the ALPINE sample is strongly dominated by narrow-band selected galaxies. Figure 3 shows the distribution of redshifts of the ALPINE galaxies in the COSMOS and ECDFS fields. The colored histogram bars show stacked numbers of galaxies that are preselected by the different methods discussed above. The bins with galaxies in the ECDFS field are hatched. The narrow-band selected galaxies are prominent at z ∼ 5.7 and represent the largest fraction of galaxies at z > 5 in ALPINE. On the other hand, the z < 5 sample consists mostly of color-selected galaxies. The VUDS galaxies are most represented at z < 5, while the DEIMOS spectra and the galaxies in ECDFS cover the whole redshift range. Figure 4 shows the distribution of observed magnitudes as well as rest-frame 1500Å and ∼ 4000Å luminosity of galaxies selected by the different methods. The photometry that is used is explained in detail in Section 3. The 1500Å rest-frame luminosity is derived from SED fitting (see Section 4.2 for details). The 4000Å rest-frame luminosity is derived directly from the Ul-traVISTA K s and VLT K v s magnitude for galaxies in the COSMOS and GOODS-S field, respectively. The magnitudes and luminosities are not corrected for dust attenuation. Note that the K-band is rest-frame 3000Å at the highest redshifts (z = 5.9), hence at these redshift older and dustier galaxies would be biased to lower luminosities. As expected for spectroscopically selected galaxies, the ALPINE sample covers the brighter part of the galaxy magnitude and luminosity distribution. The different selection methods on their own are distributed differently in this parameter space. Most noticeably, the z ∼ 5.7 narrow-band selected galaxies reside at the faintest luminosities, while the 4.5 µm continuum excess selected galaxies are among the brightest. The X-ray Chandra detected galaxy DC 845652 (green star) at z = 5.3 outshines all of the galaxies in UV luminosity. All in all, although naturally biased to the brightest galaxies, this diverse selection function makes ALPINE an exemplary panchromatic survey that enables the study of a representative high-z galaxy sample at UV, optical, and FIR wavelengths. Uniform calibration of spectra All of the rest-frame UV spectra discussed in Section 2.1 are relative flux corrected to remove sensitivity variations across the spectrograph as well as to correct atmospheric absorption features. However, not all of the spectra have been absolute flux calibrated, which is important to measure absolute quantities such as their . Examples of stacked ALPINE spectra. Panels 1a and 1b show stacked spectra at z < 5 (in COSMOS from DEIMOS observations and as part of the VUDS survey) and z > 5 (on COSMOS from DEIMOS and on ECDFS from VIMOS and FORS2 observations), respectively. The stacks are all normalized to the continuum between 1300Å and 1400Å and common emission and absorption features are indicated with gray bars. Note that the VUDS spectra have a lower native resolution (R ∼ 230) compared to the DEIMOS observations (R ∼ 2500), therefore the latter have been degraded in resolution using a 1-dimensional Gaussian window function for visual comparison. Panels 2a through 2e show stacks at z < 5 and z > 5 for the different datasets. The number of spectra included per wavelength is shown on the top of each panel. The uncertainty in flux is indicated by the light gray line. The y-axis scale is the same such that the continuum brightness can be compared. Lyα emission. Hence, we recalibrate the spectra using the Galactic extinction corrected total broad, intermediate, and narrow-band photometry of the ALPINE galaxies (see Section 3 for details on the photometry). It turns out that the absolute flux calibrated spectra are in excellent agreement (within better than 5% in flux) with our measured photometry and the recalibration is not necessary in these cases. As the spectra come from different surveys, we convert them to a common format during the recalibration procedure. To perform the absolute flux calibration, we convolve each of the spectra with the transmission functions of the various optical broad, intermediate, and narrowband filters that exist on the COSMOS and ECDFS fields, respectively. On average, we use 4 − 9 filters for galaxies at z < 5 and 2 − 4 at z > 5. If the filter extends further than the spectrum, we extrapolate the spectrum by its medium continuum value. If the filter extends significantly beyond the spectrum (> 50%), we do not consider the filter. We then compare the photom-etry obtained from the spectra to the total and Galactic extinction corrected photometry discussed in Section 3, which allows us to obtain an average correction factor for each spectrum. We found that a single number for this correction per galaxy is enough for the calibration as the spectra already have been relative flux calibrated. Since the uncalibrated spectra are mostly in units of counts, this correction is on the order of 10 −21 for most galaxies. Our recalibration corrects for slit-losses and seeing variations. We also scale the variance in order to conserve the S/N of the spectrum. The final precision of our calibration is around 5 − 10% in flux, which corresponds to the 1σ uncertainty in the photometry. Note that we do not consider undetected spectroscopic fluxes in this procedure, however, we use the constraints gained from the upper limits in the photometry for the calibration. Figure 5 shows three absolute calibrated spectra at z ∼ 4.53, z ∼ 4.56 (with weak Lyα), and z ∼ 5.68 to visualize our method. The filters that were used for the calibration are indicated in colors. Stacked Spectra: Overview over Rest-Frame UV Emission and Absorption Lines in ALPINE Galaxies Figures 6 and 7, show stacks of different spectra. In order to create median-stacks of the spectra, we resample the spectra to a common wavelength grid and stack them in rest-frame using their respective redshifts derived from [C II] or, if not available, from rest-UV absorption lines or Lyα emission. All stacks are subsequently binned to a resolution of 2Å for visual purposes to emphasize the UV absorption features. To obtain a per-pixel uncertainty from the sky background for each stack (visualize by the gray line), we simply combine the inverse variances in the individual spectra in quadrature. The latter are the original inverse variance that we adjusted to the new normalization described in Section 2.2. Panels 1a and 1b of Figure 6 compare the full stacked spectra of galaxies at z < 5 in COSMOS from observations with DEIMOS and as part of VUDS, as well as at z > 5 from observations in COSMOS from DEIMOS and in ECDFS from VIMOS and FORS2. In the former case, we adjust the resolution of the DEIMOS spectra (R ∼ 2500) to that of the VUDS observations (R ∼ 230) by applying a 1-dimensional Gaussian smoothing. The spectra are normalized to the median flux in the restframe wavelength range between 1300 and 1400Å before stacking. For stacks of galaxies at z > 5, the rest-frame wavelength reaches up to 1500Å, while for lower redshift stacks we show wavelengths up to restframe 1700Å. Several prominent spectral features are visible in the stacks in both redshift bins (indicated by gray bars). These include the Lyα emission line and N V at 1241Å and in addition UV absorption lines such as Si II at 1260Å, the Si III-O I-Si II complex at 1301Å, the two Si IV lines at 1398Å, as well as Si II, C IV, and He II at 1527Å, 1548Å, and 1640Å, respectively. Furthermore, we see indication of Fe II absorption at 1608Å in the COSMOS/DEIMOS spectra stack at z < 5. The depth of the UV absorption features are comparable for the different observations with the different instruments, verifying similar quality and little biases. However, note that the features in the ECDFS spectra are less pronounced due to the factor ∼ 6 smaller number of spectra contributing to the stacks compared to the DEIMOS stacks. Panels 2a through 2e show the stacks for variously selected datasets below and above z = 5. The spectra are not normalized before stacking in these cases to provide a comparison of the absolute flux values for the different redshifts and samples to the reader. The number of spectra per wavelength are shown on the top right for each panel. Note again that the number of high-redshift spectra drops towards redder wavelengths. This has to be kept in mind when analyzing the spectral features in the stacks. Emission and absorption lines are indicated as in the other panels. As expected, the stacked spectra at higher redshifts are fainter, but still significant UV absorption features are present (see also Faisst et al. 2016a;Pahl et al. 2019;Khusanova et al. 2019). Figure 7 shows stacked spectra in COSMOS observed with DEIMOS for the different selection categories (see also Table 1 and Figure 3). We split the LBG category in galaxies below and above z = 5. All the spectra are smoothed with a Savitzky-Golay filter with size of 2Å for visualization purposes. The total number of spectra per stack is indicated in the upper left corner. All panels are scaled the same way to emphasize differences in brightness. The X-ray detected galaxy at z = 5.3 is UV bright compared to the other stacks and shows strong N V emission with overlaid Si II absorption as well as broad C IV emission. LBGs (i.e., color-selected galaxies) are preferentially fainter but of similar continuum brightness as narrow-band selected galaxies at z ∼ 4.5. The latter show significant C II, Si IV, and C IV absorption. As expected, narrow-band selected galaxies at z ∼ 5.7 show strong Lyα emission and a faint continuum such that the S/N is too low to detect UV absorption features at great significance. The stack of galaxies selected by photometric redshifts shows to first order similar properties as the LBGs. The 4.5 µm-excess continuum selected galaxies are on average the continuum brightest galaxies and show significant Lyα emission as well as absorption features. Table 1. Emission and absorption features are indicated by gray bars and the number of spectra in the stack is shown on the upper right corner. We also show the X-ray detected galaxy (DC 845652 ) at z = 5.3, which shows strong and broad N V and C IV emission. The uncertainty in flux is indicated by the light gray line. Rest-UV Emission and Absorption Lines and Velocity Offsets Measurements We measure basic quantities from the individual restframe UV spectra. These include the redshift and equivalent width of Lyα emission as well as redshifts from various absorption lines. The Lyα redshift (z Lyα ) is based on the peak of the (asymmetric) Lyα emission to allow a direct comparison with models of Lyα radiative transfer (see, e.g., Hashimoto et al. 2015). The Lyα flux is measured by fit-ting a Gaussian to the line and for measuring the equivalent width (∼ f tot line /f continuum ) the continuum redward of the Lyα line is used. These measurements are explained in more detail in Cassata et al. (2020). The absorption redshifts are measured for each individual spectrum, if possible, using the lines Si II (1260.4Å), O I (1302.2Å) 3 , C II (1334.5Å), Si IV (1393.8Å) and Si IV (1402.8Å), Si II (1526.7Å), and C IV (1549.5Å) 4 . The first four are covered by observations in all galaxies, while the coverage of the latter depends on the redshift of the galaxy. Note that some of the above lines are predominantly formed in the ISM (low-ionization interstellar [IS] lines; Si II, O I, C II, Si II), while others are formed in stellar winds (highionization wind lines; Si IV or C IV) and therefore can display strong velocity shifts (e.g., Castor & Lamers 1979;Leitherer et al. 2011). To increase the S/N of our measurements, we use all the above lines to derive an absorption line redshift (referred to as z IS+wind ), but we compare the individual redshift from the IS (z IS ) and wind (z wind ) lines to investigate potential systematic differences. Before performing any measurements, we subtract a continuum model from each individual spectrum. The model is derived by fitting a 4 th -order polynomial to the spectrum, which is smoothed by a 5Å box kernel. We then fit the above absorption lines in five different rest-frame wavelength windows [ and the wind line C IV, respectively. The absorption lines can be significantly asymmetric due to stellar winds and the effect of optical depth. Fitting a single Gaussian to them could therefore bias the redshift measurements. Instead, we use the stacked spectrum of LBGs at z ∼ 3 from Shapley et al. (2006) as a template, which we cross-correlate to the observed data within the wavelength range of a given window by χ 2 minimization. We let the redshift vary within a velocity range of ±600 km s −1 (corresponding to roughly 0.01 in redshift) around a prior absorption redshift, which is obtained by a manual cross-correlation of the same template to all possible absorption lines at once using the interacting redshift-fitting tool SpecPro 5 (Masters & Capak 2011). We found that this approach significantly removes degeneracies in the fit and at the same time allows a visual inspection of all the spectra to flag the ones with low S/N where no reasonable fit can be obtained 6 . For each galaxy, the so obtained χ 2 (z) distribution is then converted into a probability density function p(z) for each of the windows. These are combined, by choosing the necessary absorption lines, to a total probability P (z) from . Stacked histograms of velocity offsets between redshifts derived from different spectral features. The number of galaxies and median of the distribution (including scatter) are indicated. Shown are the velocity offsets between Lyα emission and IS+wind absorption lines (top panel ), as well as between Lyα, IS+wind, and systemic redshift (middle and bottom panel). The latter two are in detail discussed in a forthcoming paper (Cassata et al. 2020). The average errors are on the order of ±100 km s −1 , which corresponds to the size of the bins. We do not find any significant biases introduced by the different selection methods (color coded as in previous figures). which the final absorption line redshifts (z IS+wind , z wind , or z IS ) are derived. The errors on these redshifts are derived by repeating this measurement 200 times, thereby perturbing the fluxes according to a Gaussian error distribution with σ defined by the average flux noise of the continuum. Typical uncertainties are on the order of ±100 km s −1 . As mentioned in Section 2.1, 6 galaxies in COS-MOS have been observed by the Keck/DEIMOS and VUDS spectroscopic surveys. Therefore there are two measurements for each of these galaxies. Specifically, for vc 5100559223, vc 5100822662, vc 5101218326, and vc 5101244930, the IS+wind redshift measurements agree within 200 km s −1 , 280 km s −1 , 70 km s −1 , and 110 km s −1 . These values are on the order of the measurement uncertainties. Note that while the VUDS slits are oriented east-west, the DEIMOS slits can be oriented north-south or in any other angle. This different orientation could also be responsible for the differences in velocity offsets. On the other hand, for vc 5101288969 and vc 510786441, we find significant differences of 1290 km s −1 and 1010 km s −1 . A close inspection of the spectra shows that these are very low in S/N. Also, both have low visual quality flags (−99 and 1, indicating not robust measurements are possible) and their redshifts are fit with less than 3 lines, hence should not be trusted. For all 6 spectra we decided to prefer the VUDS observations because of their slightly better S/N at a cost of lower resolution. Velocity Offsets with respect to [C II] FIR Redshifts The detection of [C II] by ALMA provides the systemic redshift of a galaxy. This enables us to study velocity offsets of rest-frame UV absorption lines and Lyα emission that will inform further about the properties of the ISM in these galaxies similarly to studies at lower redshifts using Hα and C II]λ1909 (e.g., Steidel et al. 2010;Marchi et al. 2019). Here we give an overview of the velocity properties and compare them for galaxies with high and low specific SFRs. In the following, we define the velocity difference for two redshifts (z 1 and z 2 ) as z 1 − z 2 ≡ ∆v 12 = c × ( 1+z1 1+z2 − 1) where c = 2.998 × 10 5 km s −1 . The measurement of the [C II] redshifts are detailed in Bethermin et al. (2020). They are defined as the peak of a Gaussian fit to the [C II] line with spectral resolution of 25 km s −1 . The uncertainty of the redshift measurements was es-timated by a Monte Carlo simulations with perturbed fluxes according to the error per spectral bin. The average uncertainty is roughly 50 − 60 km s −1 . For the absorption lines, we require that z IS+wind is measured from at least three absorption lines and we only show galaxies that have not been flagged by our visual inspection with SpecPro as unreliable (flag −99). The average intrinsic measurement error per galaxy is ±100 km s −1 . In relation to that, a systematic uncertainty of 0.5Å in the rest-frame wavelength of the absorption lines (e.g,. due to calibration issues) turns into a velocity shift of ∼ 120 km s −1 . Figure 8 shows stacked histograms of velocity differences. The number of galaxies used as well as the median of the distribution with scatter (not error on the median) are indicated as well. The upper panel compares the velocities measured from Lyα and the IS+wind absorption lines. We find a median offset on the order of 386 +257 −279 km s −1 , which is consistent with other measurements at the same redshifts (see, e.g., Faisst et al. 2016a;Pahl et al. 2019) as well as at z ∼ 2 − 3 . The center and bottom panels compare the IS+wind and Lyα redshifts to the systemic redshift (here defined as the [C II]λ158 µm redshift, Bethermin et al. 2020). For the former we find an offset of −227 +168 −206 km s −1 and for the latter we find 184 +201 −215 km s −1 . These negative and positive velocity offsets can be related in a simple physical model involving the resonant scattering of Lyα photons and outflowing gas in the outskirts of galaxies (see detailed discussion in Steidel et al. 2010). The redshifted Lyα emission line (with respect to systemic) can be explained by resonant scattering of the Lyα photons. Preferentially, red-shifted Lyα photons scattered from the back of the galaxy can make it unscattered through the intervening gas inside the galaxy along the line of sight. The blueshift of IS absorption may depend on the outflow velocity of the absorbing gas as well as its covering fraction (or optical depth) inside the galaxy along the line of sight towards the observer. For a more in-depth discussion we refer to a companion paper by Cassata et al. (2020). Overall, we do not see a significant dependence of the velocity differences on the various selection techniques (color-coded in the figure). Figure 9 compares the velocity offsets between IS (Si II, O I, C II, Si II) and wind (Si IV, C IV) lines. We require that at least three IS lines and one wind line is measured. In addition, only galaxies that pass our visual classification (i.e., have flags other than −99, see above) are used. Overall, we do not see any statistical difference between IS and wind lines, although there is a tail towards higher blueshifts in the case of wind lines. How- . Stacked spectra (in C II systemic redshift) in two bins of sSFR (red: < 4 Gyr −1 , blue: > 5 Gyr −1 ) for five wavelength regions covering prominent rest-UV absorption lines. The derivation of the sSFR for the ALPINE galaxies is detailed in Section 4.1. The average number of spectra in each bin is indicated together with the prominent absorption and emission lines. We note systematically stronger blue shifts of all absorption lines for the high sSFR stack. Particularly, note the strong blue-shift of the high-ionization wind lines. The C IV lines in the high sSFR bin also show indication of a more pronounced P-Cygni profile, indicative of strong stellar winds and outflows in high sSFR galaxies. The 1σ uncertainties of the stacked spectra is indicated by the shaded regions. ever, wind and outflows may be increased in galaxies with high and spatially dense star formation and young stellar populations. Therefore we would expect different velocity shifts for the absorption lines with respect to the systemic redshift for highly star-forming galaxies. In Figure 10, we investigate this picture by stacking galaxies at the extreme ends of the sSFR distribution (we refer to Section 4 for details on the measurement of the physical properties of our galaxies), namely low (< 4 Gyr −1 ) and high (> 5 Gyr −1 ) sSFR, in their corresponding rest-frames defined by the systemic redshift (i.e., [C II]λ158µm redshift). The sSFR is a good proxy of the star-formation density in a galaxy as well as the age of the current stellar population (see, e.g., Cowie et al. 2011). We show the stacked spectra in five wavelength regions covering prominent absorption lines for each sSFR bin. The vertical dashed lines show the different absorption lines in the [C II] rest-frame. First, we verify that the shifts between IS and wind lines are very similar for each sSFR bin (in concordance with Figure 8). However, intriguing is that in the low sSFR stack, all absorption lines agree well with the [C II] redshift, while in the high sSFR stack the lines are significantly blue shifted by 300 − 400 km s −1 . We also note that in the high sSFR stack, the C IV line shows a noticeable P-Cygni profile indicative of strong winds and outflows (Castor & Lamers 1979). These findings fit well into a picture of strong winds and outflows produced by the high star-formation in these galaxies, which is also in line with recent results obtained through the stacking of ALPINE [C II] spectra (Ginolfi et al. 2019 PDRs (Stacey et al. 1991;Gullberg et al. 2015;Vallini et al. 2015;Faisst et al. 2017). Moreover, recent work by Ginolfi et al. (2019) shows that [C II] emission is significantly affected by large-scale outflows caused by high star-formation in these galaxies. However, as shown by the same study, the outflows seem to be symmetric and therefore we do not expect them to significantly change the centroid of the [C II] emission line. During January 13-15, 2019, we were able to obtain a near-IR spectrum of one of our ALPINE galaxies (DC 881725 at z [C II] = 4.5777) using the Multi-Object Spectrometer For Infra-Red Exploration (MOS-FIRE, McLean et al. 2010McLean et al. , 2012 at the 10-meter Keck I telescope on Mauna Kea in Hawaii. The observations of a total on-source integration time of 24×3 min in K band (1.92 − 2.40 µm) were carried out under clear weather conditions with an excellent average seeing FWHM of 0.3 − 0.4 . We performed a standard data reduction using the MOSFIRE data reduction pipeline 7 (Version 2018). From the produced 2-dimensional spectrum and variance map, we extract the 1-dimensional spectrum at the spatial location of the galaxy using a weighted mean across ±3.5 spatial pixels (0.18 /px). We are able to detect the optical [O II] doublet (3727.09Å and 3729.88Å) at the spatial position of the galaxy at a level of > 5σ. Note that this is the first detection of optical [O II] in a galaxy with [C II] measurement from ALMA, which allows us for the first time to compare this two lines at these redshifts. In bottom panel of PHOTOMETRY FROM GROUND AND SPACE In this section, we summarize the ground-and spacebased photometric data that are available for the ALPINE galaxies in the COSMOS (105 galaxies) and ECDFS (13 galaxies) fields. Although these fields differ in survey depth, reduction methods, and number and type of photometric filters used, we find that their overall photometric measurements are comparable within 1σ limits after their conversion to total magnitudes and the correction for the specific biases of each survey. Therefore, we can treat them separately to first order for the matter of measuring various physical properties of the galaxies. The basis catalogs to which we match the 7 https://keck-datareductionpipelines.github.io/MosfireDRP/ ALPINE galaxies in the COSMOS and ECDFS field are the COSMOS2015 8 (Laigle et al. 2016) and the 3D-HST 9 (Brammer et al. 2012;Skelton et al. 2014) catalog, respectively. A summary of the different data available on the two fields including filter names, wavelengths, 3σ depths, and references to the measurements are given in Tables 2 and 3, respectively. In the following, we describe these data in more detail. Photometry on the ECDFS field The photometry for the galaxies on the ECDFS field is taken directly from the 3D-HST catalog, which provides ground-based observations as well as a wealth of data from HST imaging. The photometry (total fluxes and magnitudes) is corrected for Galactic extinction, PSF size as well as other biases, therefore no further correction are applied. The ALPINE galaxies are matched visually to the spatially closest 3D-HST counterpart using the HST WFC3/IR F 160W image as reference. The spectroscopic redshifts match the photometric redshifts within their uncertainty (∼ 0.1 − 0.2), ensuring that we identified the correct counterpart. The ground-based photometry available in ECDFS (including references) is listed in Table 2. Summarizing, this includes the U 38, b, v, R c , and I broad-band filters from the Wide Field Imager on the 2.2 meter MPG/ESO telescope, the U and R bands from VIMOS on the VLT, the near-IR filters J v , H v , and K v s from ISAAC on the VLT, J w and K w s data taken by WIRCam on the CFHT, as well as 14 intermediate-band filter from the Suprime-Cam on the Subaru telescope. For galaxies at z = 4.5 and 5.5, the Lyman-break falls roughly in the v and the R c -band and therefore the galaxies are expected to be only faintly (or not at all) visible in these and blueward filters. On the other hand, the galaxies are bright at observed near-IR wavelengths, i.e., filters red-ward of z-band (corresponding to roughly the F 850LP filter. Figure 12 shows the stacked F 850LP (z), J v (J) and K v s (K) magnitude distributions of the ECDFS ALPINE galaxies split in z < 5 (hatched blue) and z > 5 (hatched red). As expected, the latter sample occupies slightly fainter magnitudes. The space-based photometry includes the four Spitzer bands at 3.6 µm, 4.5 µm, 5.8 µm, and 8.0 µm. In addition, the public 3D-HST catalog includes a wealth of HST photometry. Specifically, it contains measurements in the ACS bands F 435W , F 606W , F 775W , F 814W , and F 850LP as well as in the WFC3/IR bands F 125W , and F 160W bands for all 13 ALPINE galaxies. 23 24 25 26 27 28 K-band magnitude COSMOS (z < 5) COSMOS (z > 5) ECDFS (z < 5) ECDFS (z > 5) Figure 12. Stacked histograms of the magnitude distribution for the ALPINE galaxies in the COSMOS (solid) and ECDFS (hatched) fields. The blue and red color-coding indicates galaxies at z < 5 and z > 5, respectively. The magnitudes (from left to right) correspond to z ++ , J, and Ks bands for COSMOS and F 850LP , J v , and K v s bands for ECDFS (see Tables 2 and 3 for more information on the filters). Only 10 galaxies have measurements in the WFC3/IR band F 140W . The HST photometry is measured on PSF-matched images. As described in Skelton et al. (2014), the Spitzer and ground-based photometry are measured using the MOPHONGO (Labbé et al. 2006;Wuyts et al. 2007;Whitaker et al. 2011), which uses a highresolution image (here the HST imaging) as spatial prior to estimate the contributions from neighboring blended sources in the lower resolution image. The different depths of these observations as well as references are listed in Table 2. A query of the Barbara A. Mikulski Archive for Space Telescopes (MAST 10 ) using the mastquery Python package 11 shows that in addition to the HST measurements contained in the 3D-HST catalog, four, ten, and two galaxies have coverage in the WFC3/IR bands F 098M , F 105W , and F 110W , respectively. None of the galaxies has ACS F 475W coverage. These additional data that are not published in the 3D-HST catalog come from various other observation programs in and around the ECDFS field. We subsequently measure this additional photometry for all ALPINE galaxies in ECDFS using SExtractor (version 2.19.5, Bertin & Arnouts 1996) in different aperture sizes (0.7 and 3 ) as well as auto magnitudes. For this, we first create a mosaic of all the HST pointings that overlap with the ALPINE galaxies using the AWS-drizzler 12 tool that is part of the grizli 13 Python package (Brammer, in prep.). We use a 0.06 pixel scale and all HST images are registered to Gaia (see Section 3.3). SExtractor is run with relative THRESH TYPE, and we set DETECT MINAREA, DETECT THRESH, DEBLEND MINCONT, and DEBLEND NTHRESH to 3, 1.5, 1.5, 0.001, and 64, respectively. If no object is detected above the threshold (1.5σ) within 0.7 (roughly the ground-based seeing) of the original ALPINE coordinates, we consider the galaxy as undetected in a given band and replace its flux by a 1σ limit that is computed from the RMS noise at the position of the galaxy. The photometry measured by SExtractor is subsequently corrected for galactic foreground extinction, which we assume to be constant for all galaxies in ECDSF at E(B − V ) = 0.0069 mag. Figure 13 summarizes the HST data available for the galaxies in the ECDFS field. The blue squares show the layout of all the HST pointings as of October 2019, with darker shades of blue indicating more observations. The ALPINE galaxies at z < 5 and z > 5 are indicated with orange circles and squares, respectively. Photometry on the COSMOS field Most of the ALPINE galaxies (105 out of 118) reside in the COSMOS field and we match them to the latest photometric measurements presented in the COS-MOS2015 catalog. The matching is again done on a galaxy-by-galaxy basis using the HST/ACS F 814W as well as UltraVISTA K s images as references. We also Figure 13. HST pointing footprints on the ECDFS field (blue) for different HST ACS and WFC3/IR filters. Darker colors mean more observations. The ALPINE galaxies are indicated by orange circles and squares for z < 5 and z > 5, respectively. match against the photometric redshifts given in the catalog in order to identify the correct counterpart 14 . In Table 3, we list all the photometric measurements available for the ALPINE galaxies on the COSMOS field. Summarizing, these include u * -band observations from MegaCam on CFHT, the B, V , r + , i + , z ++ as well as 12 intermediate-band and 2 narrow-band filters from the Suprime-Cam on Subaru, the Y HSC -band from the Hyper Suprime-Cam on Subaru as well as near-IR bands H w and K w s from WIRCam on CFHT and Y , J, H, and K s from VIRCAM on the VISTA telescope. In addition, the galaxies are covered by the four Spitzer channels from 3.6 µm to 8.0 µm from the SPLASH survey 15 Steinhardt et al. 2014;Laigle et al. 2016). As described in (Laigle et al. 2016), the Spitzer photometry is measured using IRACLEAN (Hsieh et al. 2012), which uses positional priors from higher resolution imaging (in this case the zY JKHK s detection χ 2 −image) to deblend the photometry. Contrary to the 3D-HST photometry catalog, the fluxes and magnitudes in the COSMOS2015 catalog are not total and not corrected for systematic biases and Galactic extinction. To perform these corrections, we follow the steps outlined in the appendix of Laigle et al. (2016). Specifically, we use the 3 diameter aperture magnitudes (M 3 ), which we correct for photometric (o i , see their equation 4) and systematic offsets (s f , see their table 3) by applying M tot,uncorr i,f = M 3 i,f + o i − s f ,(1) where i is the object identifier and f denotes the different filters. The total magnitudes are subsequently Figure 14. HST pointing footprints on the COSMOS field (blue) for different HST ACS and WFC3/IR filters. Darker colors mean more observations. The ALPINE galaxies are indicated by orange circles and squares for z < 5 and z > 5, respectively. corrected for Galactic extinction by applying M tot i,f = M tot,uncorr i,f − EBV i × F f ,(2) where EBV i is the Galactic extinction from the Schlegel et al. (1998) maps on COSMOS for each object as given in the catalog and F f are the extinction factors per filter given in table 3 of Laigle et al. (2016). Figure 12 shows the stacked magnitude distribution in z ++ (z), J, and K s (K) bands for the z < 5 (blue) and z > 5 (red) sub-samples. As expected, the high-redshift galaxies are fainter in all bands. In addition, we find the magnitude distributions between the galaxies in ECDFS and COSMOS to be similar. This indicates no major discrepancies in photometric, hence physical properties between the two samples. In terms of HST imaging, all galaxies except one are observed in ACS F 814W (Scoville et al. 2007a;Koekemoer et al. 2007). In addition to this, a MAST-search shows that several galaxies are covered by other observ-ing programs in the ACS bands F 435W (3), F 475W (5), F 606W (21), and F 850LP (5) as well as in the WFC3/IR bands, F 105W (11), F 110W (5), F 125W (16), F 140W (13), and F 160W (53) 16 . Note that the observations in F 160W primarily come from the CAN-DELS survey (covering the central part of COSMOS, Grogin et al. 2011;Koekemoer et al. 2011) as well as the "drift and shift" (DASH, Momcheva et al. 2017) survey. While the CANDELS imaging is deep (> 27.5 magnitudes at 3σ), the data from the DASH survey is much shallower (25.0 magnitudes at 3σ) and therefore only half of the galaxies are detected. Furthermore, the spatial sampling in the latter does not allow a detailed study of the structure of the galaxies. Figure 14 summarizes the HST pointings on the COSMOS field (blue). The CANDELS area in the center of the COSMOS field as well as the three DASH stripes are evident. The location of the ALPINE galaxies is indicated with circles (z < 5) and squares (z > 5). The photometry available is summarized in Table 3 including depths (where applicable) and references. In order to measure the photometry in these HST bands, we use SExtractor in the same way as described in Section 3.1. We first create a mosaic (0.06 pixel sizes and registered to Gaia) using the AWS-drizzler. We subsequently measure the HST photometry of on each of the images for all ALPINE galaxies with coverage using SExtractor in apertures (0.7 and 3 ) as well as auto magnitudes. If no object is detected above the set threshold level within 0.7 of the original ALPINE coordinates, we consider the galaxy as undetected in a given band. Similar to Equation 2, we correct the HST photometry for galactic foreground extinction using the Schlegel et al. (1998) extinction map. We compare these measurements to the ground-based photometry, by first performing a PSF matching by smoothing the original HST images with a Gaussian kernel with FWHM of 0.7 and measure the photometry in a 3 aperture (as used in the COSMOS2015 catalog). We also compare the ground-based (i + , Y , J, H) with the HST (F 814W , F 105W , F 125W , F 160W ) photometry in approximately matching filter bands. We find an overall agreement on average of 0.2 magnitudes (dark gray region) with an expected increase of scatter at fainter magnitudes. For the four brightest sources in H-band (< 23.4 AB), the ground-based measured photometry is systematically up to 0.5 magnitudes brighter in three out of four cases. Due to the low number of galaxies, it is difficult to investigate this statistically. A more detailed measurement of the HST photometry including deblending of specific sources such as merging or clumpy galaxies, will be provided in a forthcoming paper. Astrometric offsets between ALMA and ancillary data Astrometric accuracy is crucial in many cases. First, accurate spatial offsets of light emission from rest-frame UV, optical, and sub-mm wavelengths reveal the properties of the interstellar medium such as the location and interplay of stars, dust, and gas. Second, the robustness of the identification of sub-mm counterparts needs a high astrometric accuracy of all involved datasets. As described in Sections 3.1 and 3.2, the HST images produced for the ALPINE galaxies are all aligned to Gaia. Our tests show that the average offsets are less than 15 milli-arcseconds (mas) in both right-ascension and declination with a scatter of no more than 30 mas (G. Bram-mer, private communication). Unfortunately, the positional accuracy for current catalogs (such as 3D-HST or COSMOS2015 that are used here) are lower and we expect significant offsets between those astrometric solution and ALMA data products. In the following, we characterize these offsets. According to the technical handbook, ALMA observations are currently registered to the International Celestial Reference Frame (ICRF) to an accuracy better than ∼ 5 mas. The ICRF is based on hundreds of extragalactic radio sources such as quasars distributed over the whole sky. The positional accuracy ∆p of single observations can be estimated by ∆p = 70000 ν · B · σ ,(3) where ν is the observed frequency in GHz, B is the maximum baseline length in kilometers, and σ is the S/N at the peak of emission. For ALPINE (ν = 330 GHz and B = 0.2 km for C43-1) this leads to ∆p = 1060/σ. The calibrators are detected well above 50σ, which leads to an absolute positional accuracy of ∼ 20 mas or better. To check the astrometric alignment of the photometric catalogs used here, we make use of the Gaia DR2 catalog , which provides currently the most accurate absolute astrometry. As shown in , there are no significant offsets between this reference frame and the ICRF frame used by ALMA, hence this test directly reveals potential differences in astrometry between the 3D-HST and COSMOS2015 catalogs and our ALMA observations. Using the proper motion information of the stars from Gaia, we project their positions back in time to the year of calibration of the data products by to using the equations ∆α = cos (δ) · P α · (t ref − t gaia ), ∆δ = P δ · (t ref − t gaia ),(4) where α and δ denote the right-ascension and declination (and P α and P δ their proper motion), t gaia is the Gaia reference frame in years (here 2015.5), and t ref is the reference frame of the calibration of the catalogs. To increase accuracy, we only include stars with a proper motion in both coordinates of less than 5 mas yr −1 in the following. Note that no parallax motion is included in the above formulae, which would result in less than 5 mas yr −1 astrometric shifts. The COSMOS2015 catalog is calibrated to the Mega-Cam i−band data that was taken in 2004 (Laigle et al. 2016). For ECDFS, the catalog (3D-HST ) is calibrated to the same reference system as the CANDELS HST images, namely ground-based R−band data taken in 2001 (Koekemoer et al. 2011;Skelton et al. 2014). However, the exact year is not important as we have selected stars with a relatively slow proper motion. In order to select stars on the COSMOS field, we use the ACS/F814W SExtractor catalog (Scoville et al. 2007a) and select sources with a magnitude brighter than 23 AB and SExtractor star/galaxy classifier value of > 0.8. These stars are then matched to the COSMOS2015 catalog to obtain their position in that catalog. For ECDFS we extract stars directly from the 3D-HST catalog by selecting sources brighter than 23 AB in F160W with star flag = 1 and a SExtractor star/galaxy classifier value larger than 0.8. Other selections (e.g., different magnitude cuts) do not affect the following results. The Gaia stars are then matched to the star catalogs in COS-MOS and ECDFS to obtain the astrometric offsets. In total, 47 and 2724 Gaia stars are used in ECDFS and COSMOS, respectively. Figure 15 shows scatter plots and histograms comparing the position of the Gaia stars to the positions in the catalogs. We find significant systematic offsets in the astrometry in ECDFS of 98 mas in right-ascension and −256 mas in declination. These offsets are consistent with what was found in earlier studies (Dunlop et al. 2017;Franco et al. 2018;Whitaker et al. 2019). For COSMOS we only find a significant offset in rightascension of −64 mas. In addition to that, there is a significant scatter in the astrometry on the order of 100 mas in both coordinates in both fields. To compute the astrometric offset of individual ALPINE galaxies, we make use of the fact that the HST images are already aligned to the Gaia reference frame (see details in Sections 3.1 and 3.2). Specifically, we compute the offsets between the coordinates measured on the ACS/F814W images and the original coordinates from the 3D-HST or COSMOS2015 catalog. If no ACS F 814W image is available or if the galaxy is not detected (which happens for redshifts z > 5), we use the deepest image red of F 814W . If none is available (in four cases), we report the average offset as shown in Figure 15. Note that the coordinates given in the final ancillary data catalog are not corrected for these offsets. However, we give the offsets for each galaxy in the columns delta RA and delta DEC, which can be added to the original coordinates to obtain Gaia-corrected right-ascensions and declinations (see Appendix A). PHYSICAL PROPERTIES In this section, we detail measurements of various basic physical properties of the ALPINE galaxies that are based on their total, extinction corrected photometry described in Section 3. These include physical quantities from SED fitting such as stellar masses, SFRs, ages, and dust attenuation ( §4.1), and UV continuum slopes ( §4.3), as well as quantities directly derived from the photometry such as UV magnitudes and luminosities ( §4.2) and estimates of the Hα luminosity and equivalent width from Spitzer colors ( §4.4). 79594.9 22.9 1,7 † Depth is measured in 3 aperture for ground-based photometry. a Depths of the ultra-deep area are given in parenthesis. b Ancillary pointings, therefore depth varies depending on the specific observations. c Most of these data come from the CANDELS survey and their depth is HST/ACS F435W 4328.7 − b − F 475W 4792.3 − b − F 606W 5924.8 − b − F 814W 8058.2 29.2 3 F 850LP 9181.2 − b − HST/WFC3 F 098M 9877.4 − b − F 105W 10584.9 − b − F 110W 11623.8 − b − F 125W c 12516.3 27.6 b,c 4 F 140W 13969.4 − b − F 160W Stellar mass and SFRs from SED fitting Fitting Method For consistency and comparability with other studies on the COSMOS field, we choose the LePhare SED fitting code 17 (Arnouts et al. 1999;Ilbert et al. 2006) to derive stellar masses, SFRs, light-weighted stellar population ages, absolute magnitudes, optical dust reddening, and UV continuum slopes of the ALPINE galaxies. Importantly, stellar masses, SFR, and sSFR values are computed from the marginalized probability distribution functions over all the models and uncertainties are given in ±1σ of this distribution. We use a set of synthetic templates based on the Bruzual & Charlot (2003) stellar population library, which we tune to represent best galaxies at redshifts between 4 < z < 6. In detail, we use a series of different star-formation histories (SFH) based on exponentially declining (with τ = 0.1, 0.3, 1.0, and 3.0 Gyrs), delayed 18 (with τ = 0.1, 0.5, 1.0, and 3.0 Gyrs), and constant star formation. We add dust attenuation corresponding to a stellar E s (B − V) from 0 to 0.5 spaced in steps of 0.05 assuming a Calzetti et al. (2000) dust attenuation law. To account for metallicity dependence, we use a solar (Z ) and 0.2 Z metallicity. We also adopt a Chabrier (2003) IMF in the following. The model SEDs are generated for logarithmically spaced ages starting from 50 Myrs to the age of the universe at the redshift of each galaxy. To each SED, various rest-frame UV and optical emission lines are added following the description in Ilbert et al. (2009) by using common conversions outlined in Kennicutt (1998). Specifically, the UV luminosity at 2300Å is converted to a SFR using the relation SFR (M yr −1 ) = 1.4 × 10 −28 L ν (erg s −1 Hz −1 ) and subsequently trans- The models are fit to the photometry described in Section 3 (and listed in Tables 2 and 3). Specifically, for the galaxies in the ECDFS field we use the ground-based observations in U , R, J v , H v , and K v s , as well as the intermediate-bands IA427, IA505, IA527, IA574, IA624, IA679, IA738, IA767, and IA856, and all four Spitzer channels ch 1 , ch 2 , ch 3 , and 17 http://www.cfht.hawaii.edu/ ∼ arnouts/lephare.html 18 The delayed SFR is parameterzed as ψ(t) ∝ τ −2 t e −t/τ such that ψ(t) is maximal at t = τ . ch 4 . We also include observations in the HST filters F 435W , F 606W , F 775W , F 814W , F 850LP , F 125W , F 140W , and F 160W that are properly combined with the ground-based and Spitzer measurements in the 3D-HST catalog. For galaxies in the COSMOS field, we include the ground-based observations in u * , B, V , r + , i + , z ++ , Y , Y HSC , J, H, H w , K s , and K w s , as well as the intermediate-bands IA427, IA464, IA484, IA505, IA527, IA574, IA624, IA679, IA709, IA738, IA767, and IA827. We use all four Spitzer channels but no HST observations as only the one filter (F 814W ) exists for all galaxies. The fits are performed in flux density space (f ν in Jansky), which has several advantages compared to magnitude space. Specifically, it allows a proper statistical treatment of limits in the data of all bands. While in the case of magnitude limits an arbitrary significance level (e.g., 1σ, 3σ) has to be defined in order to use them in SED fitting, in flux density space no arbitrary choice needs to be made by the user as the limits manifest themselves only in the error bars of those bands where no flux is measured. This is important as slightly different levels of significance set for the limits can have profound effects on the output parameters, especially in cases where limits are imposed in astrophysically important parts of the rest-frame spectra (e.g., the Balmer break). In addition, some SED codes, including Le Phare, have difficulties dealing with limits (in the case of magnitudes) in a statistically consistent manner, and, instead, remove from consideration any models that slightly exceed these limits. Moving to flux density space for all fits alleviates these concerns. Several studies have found that the photometric errors are generally underestimated (e.g., Ilbert et al. 2006Ilbert et al. , 2009Ilbert et al. , 2013Skelton et al. 2014;Laigle et al. 2016). Artificially increasing the errors has been found to mitigate issues associated with poorly measured bands, poorly estimated zero points, or inhomogeneous methods of measuring photometry. To avoid artificially small errors that would dominate the χ 2 budget of a fit of a given galaxy, we follow the prescription of Laigle et al. (2016) and rescale the official flux density errors by a factor of 1.1. In addition, following Ilbert et al. (2009), we correct for underestimated photometric errors due to varying PSF sizes between optical and Spitzer imaging by adding in quadrature the following systematic errors to the photometric error budget prior to fitting: 0.01 mag for all ground-based UV/optical broad-band measurements; 0.05 mag for all ground-based UV/optical intermediatebands and near-IR broad-band measurements; 0.1 mag for Spitzer 3.6 µm and 4.5 µm and 0.3 mag for Spitzer 5.8 µm and 8.0 µm measurements. Figure 16. Comparison of stellar mass (left), SED-derived SFR (middle), and absolute rest-frame UV magnitude (right) measured by LePhare and HyperZ. The symbols are color-coded by the logarithmic difference in stellar population age (note that the scale is the same for all panels). The squares denote galaxies whose Spitzer photometry is blended with a nearby bright galaxy or star. The star-symbols denote mergers based on the classification in Le Fèvre et al. (2019). The 1-to-1 relation is shown as dashed line and a ±0.3 margin is shown by the gray band. The systematic differences in stellar masses derived by HyperZ are likely caused by different implementations of rest-frame optical emission lines, which affect most the measurement of stellar masses (based on rest-frame optical light). The large PSF sizes of the Spitzer observations result in a large risk of blended photometry. This can lead to an overestimation of flux and hence to an overestimation of stellar masses 19 . A cleaner and manual deblending of the Spitzer photometry (e.g., using a position prior from HST imaging) is possible and will be pursued in a forthcoming paper. In the following, we flag galaxies that have a bright companion galaxy within a Spitzer 4.5 µm PSF FWHM (2.5 ) based on the ground-based and HST imaging data. In total, one-third of the galaxies have contaminated Spitzer photometry to some degree. For 19% of the galaxies, the Spitzer photometry is severely contaminated and should not be trusted. We checked our simple flagging scheme against the contamination flags given in the 3D-HST catalog, which are based on the flux fraction that is overlapping with the galaxy for which the photometry is measured. For the 13 ALPINE galaxies in ECDFS, we find excellent agreement between both classifications. The Spitzer contamination flags are included in the ancillary data catalog (column spitzer cont). Systematic Uncertainties in Physical Properties from Modeling Assumptions Depending on their exact methods, different SED fitting codes may measure different physical properties even with the same photometry provided as input. In addition to more physical reasons (such as different assumptions on the stellar population models and implementation of emission lines), the different treatment of 19 Note that the stellar masses are primarily constrained through the Spitzer photometry that covers rest-frame wavelengths redward of the Balmer break. undetected fluxes (or fitting in magnitude space), varying minimization techniques and weightings, and different scaling of the error of the input photometry can contribute to these discrepancies. To investigate the amplitude of such differences, we compare the measurements from LePhare to a modified version of the HyperZ code (Bolzonella et al. 2000) that includes the effects of nebular emission (Schaerer & de Barros 2009), to estimate such systematic uncertainties in the fitted parameters. To minimize the degeneracy with other assumptions, we run HyperZ on the exact same photometry and with the exact same model SEDs (i.e., same metallicity, age, SFH, and dust attenuation law) as described in Section 4.1.1. Figure 16 compares the stellar mass (left), SED-derived SFR (middle), and absolute UV magnitude (right, see also Section 4.2) derived by the two codes. The symbols are color-coded by difference in stellar population age (positive values indicate younger ages derived by HyperZ). We find that the stellar masses derived by HyperZ are systematically smaller by 0.3−0.4 dex. Similarly, the SFRs are systematically larger by ∼ 0.3 dex. As expected, the absolute UV magnitudes are largely in agreement, as they are to first order independent of the physical parameters, and just represent a translation of the fitted UV flux. The differences in stellar mass and SFR are due to the effect of nebular emission, which results in younger ages and larger emission line corrections of the intrinsic rest-frame optical continuum (observed by the Spitzer broad-bands) in the HyperZ models, and therefore directly affects the stellar mass measurements (see, e.g., de Barros et al. 2014). This is confirmed by the fact that the stellar masses and SFR measured by the two codes agree well (within a factor of two) if the emission lines are turned off. The effect of nebular emission found by HyperZ may be overestimated for the ALPINE galaxies, e.g., if the emission lines are more strongly attenuated than the continuum (see Section 4.4). For consistency and comparability with other studies on the COSMOS field, we choose the LePhare fitting results as the default. Next to the systematic offset discussed above, we also find four galaxies (DC 472215, DC 503575, DC 722679, DC 790930 ) whose stellar mass measurements are significantly discrepant, by more than one order of magnitude, between the two codes. Specifically, they are fitted with an old, low-SFR, and massive galaxy template with HyperZ, while a young, high-SFR, and lowmass galaxy template is preferred by LePhare. Three of these outliers have significantly contaminated Spitzer photometry (indicated by the squares). As a conse-quence, the apparent Spitzer fluxes and stellar masses are overestimated and the optical colors are artificially reddened, which makes their stellar masses largely unreliable. Furthermore, we artificially increased the errors of the Spitzer photometry in our LePhare measurements (see Section 4.1.1), hence they have smaller weights, which might reduce the effect of photometric contamination on the fit. In addition, we investigate the effect of different dust reddening laws on the LePhare measurements. For this, we compare a Calzetti et al. reddening with a Small Magellanic Cloud (SMC, Prevot et al. 1984) reddening. The latter might be more suitable for metal poor lowmass galaxies. Running LePhare with the same settings but adopting a SMC reddening curve, we find only small changes in the stellar mass and SFR measurements. Specifically, for the former we find an average offset towards lower stellar masses in the case of SMC dust of 0.05 dex and a maximal offset of 0.2 dex. For latter we find similar offsets towards lower SFRs in the case of SMC dust. Finally, we note that the galaxy properties derived here are consistent within a factor of two with the ones published in the COSMOS2015 catalog (based on photometric redshifts). We conclude this from comparing galaxies with the same spectroscopic and photometric redshift within 0.1. 4.1.3. The ALPINE galaxies on the z = 5 main-sequence and [C II] fluxes Figure 17 summarizes the results of this section by showing the relation between stellar mass and SFR (the main-sequence) of our ALPINE galaxies together with the [C II] emission (in Jy km s −1 ) measured by ALMA (in color). The measurements are compared to all galaxies (with photometric redshifts) at 4 < z < 6 in the COS-MOS2015 catalog (blue points) as well as the mainsequence parameterization by Speagle et al. (2014) at z = 5 (gray band with ±0.3 dex width). The comparison to the photometrically selected COS-MOS parent sample indicates that the ALPINE sample is a fair representation of the overall population of star-forming z > 4 galaxies. The sample also includes at higher stellar masses galaxies that lie 2 − 3σ below the main-sequence. Note that two of these galaxies at log(M/M ) ∼ 10.3 and 10.7 have contaminated Spitzer photometry, and therefore their stellar masses are upper limits. Other two galaxies at log(M/M ) ∼ 10.6 and 10.9 do not show [C II] emission, which is expected if they are systems of low SFR below the main-sequence. The galaxy with [C II] detection at log(M/M ) ∼ 10.1 is classified as "extended dispersion dominated" by our [C II] morpho-kinematic classification (see Le Fèvre et al. 2019) and the HST/ACS imaging suggest a clumpy morphology. Although significantly below the mainsequence, [C II] is still detected in that galaxy, perhaps indicative of dust-obscured star-formation. The color-coding of the points suggests a [C II] emission increase along the star-forming main-sequence. Furthermore, the fraction of [C II] detected galaxies significantly drops below log(M/M ) ∼ 9.3 or a SFR of less than ∼ 10 M yr −1 . This could be due to the effect of metallicity on the [C II] line strength by either indirectly a higher ionization state or directly through lower Carbon abundance (e.g., Narayanan et al. 2017). The relationship between star formation and [C II] emission will be studied in more detail in a forthcoming paper (Schaerer et al. 2020). Note that there are galaxies with a SFR of less than ∼ 10 M yr −1 in our sample, contrary to our initial selection. We emphasize that the initial selection was based on the observed absolute UV magnitude and not on any property derived from SED fitting (such as SFR). This discrepancy is therefore expected within the uncertainty of measuring SFRs from SED fitting. Measurement of UV magnitudes and luminosities The UV luminosities and absolute UV magnitudes at rest-frame 1500Å (not dust corrected) are measured during the SED-fitting process with LePhare and are defined by the transmission curve of the GALEX FUV filter (∼ 1500Å). As shown in Section 4.1.2, the absolute UV magnitudes measured by LePhare and HyperZ are in very good agreement. We also compare these measurements to a more direct method by using the observed flux in a single filter that is closest to rest-frame 1500Å (Subaru z ++ filter at z < 5 and UltraVISTA Y or Y HSC band at z > 5 for galaxies in COSMOS and the HST filter F 850LP for all the galaxies in ECDFS). The left panel in Figure 18 shows a very good agreement between the two methods. The scatter is mainly due to the low S/N of the single-filter measurements of the second method (indicated by the large error bars). The scatter is enhanced for galaxies at z > 5 also due to the fact that the Ul-traVISTA Y -band observations are less deep than the z ++ observations used for z < 5 galaxies. In the following, we will use the more robust absolute UV magnitude from LePhare as they depend less on the S/N of single observations. The middle panel in Figure 18 shows the distribution of absolute UV magnitudes for galaxies in the COSMOS and ECDFS fields in two redshift bins as in the previous figures. The bulk of galaxies are between M UV of −22.7 and −20.2 (consistent with the faint absolute UV magnitude limit of the survey, see Section 1.2). One of the galaxies in ECDFS (CANDELS GOODSS 37 ) is significantly fainter (M UV = −19.2). This galaxy has been added to the sample to fill in an empty frequency window. The measurement of the absolute UV magnitude from single band and SED fit agree and the galaxy is compact and isolated (i.e., no contamination in the photometry). The fit to its photometry with LePhare suggests a dust-free, low-mass (log(M/M ) = 9.22) galaxy that is forming stars at a rate typical for the mainsequence (log(SFR/[M yr −1 ]) = 0.97). The right panel of Figure 18 compares the distribution of the UV luminosity of the ALPINE galaxies (stacked filled histogram for low and high redshift) to the same parent sample selected from the COSMOS2015 catalog and used in Figure 17 split in two redshift bins (empty histogram). As expected, the absolute UV magnitude cut applied for the selection of the ALPINE sample (see Density Parent (4 < z < 5) Parent (5 < z < 6) ALPINE (low-z) ALPINE (high-z) Figure 18. Left: Comparison of absolute UV magnitudes derived from LePhare and from one single filter close to rest-frame 1500Å. The scatter at fainter magnitudes is a S/N effect in the latter measurement. The gray stripe shows ±0.3 magnitudes around the 1-to-1 line. Middle: Distribution of absolute UV magnitudes of the whole ALPINE sample. Right: The UV luminosity distribution of ALPINE galaxies in relation to a parent sample from the COSMOS2015 catalog selected in the same redshift range. ALPINE galaxies occupy brighter luminosity, caused by the selection in MUV (see Section 2.1). Section 2.1) causes a bias towards the brighter end of the parent distribution. UV continuum slopes Method The UV continuum slope (β, defined as f λ ∝ λ β ) generally correlates with the attenuation of stellar light by dust and is therefore an important tool to study the dust properties of galaxies especially at high redshfits (e.g., Meurer et al. 1999;Bouwens et al. 2009;Finkelstein et al. 2012). The UV continuum slope of a galaxy can be derived by various methods. Here, we compute β from the best-fit SEDs derived by LePhare. Compared to deriving the slopes directly from the observed photometry by a linear fit, this approach results in less biased β measurements in the case of low S/N observations (e.g., Finkelstein et al. 2012;Barisic et al. 2017). This is particularly true for the ALPINE galaxies, whose restframe UV continuum is predominantly covered by relatively shallow ground-based imaging. Deep HST coverage by a sufficient number of bands of this wavelength range is only available for a small fraction of the galaxies. The β slopes are derived by a robust linear fit (to avoid the fit being affected by any absorption or emission lines) to the logarithmic slope of the best-fit LePhare SED in the wavelength range between 1300Å and 2300Å. To quantify uncertainties, we perturb the fluxes of each filter according to their individual errors assuming a Gaussian error distribution, refit the galaxies, and re-measure β from the resultant best-fit SED. For each galaxy we repeat this procedure 1000 times to produce a probability density function. The uncertainties for β quoted here are the 1σ percentiles of this distribution and are on average on the order of ∆β = 0.2 − 0.3. Systematic Uncertainties and Dependencies on Dust Attenuation In addition to photometric uncertainties, several model assumptions affect the measurement of β. We . Comparison of UV slopes and stellar mass. The offset panels show kernel density estimates of the β and stellar mass distribution. The ALPINE galaxies (large symbols) are split into their method of selection (see Section 2.1). We also show the data from our parent sample at 4 < z < 5 (gray dots). Statistically, our ALPINE peaks at ∼ 0.2 dex higher stellar masses and ∼ 0.3 bluer β. found that the two most important ones are the wavelength range over which β is fit and the assumed dust attenuation law. We choose the wavelength range over which β is measured to be consistent with the definition of several other studies (Calzetti et al. 1994;Meurer et al. 1999;Finkelstein et al. 2012;Bouwens et al. 2014;Barisic et al. 2017;Fudamoto et al. 2017). In the upper panel of Figure 19, we demonstrate how the β measured for our ALPINE galaxies would change if different wavelength ranges are used. First, we do not find any differences in our measurements compared to the definition by Calzetti et al. (1994), who use 10 discrete fitting windows between 1300Å and 2600Å to avoid strong absorption and emission lines (green diamonds). The other symbols show the comparison to different wavelength ranges and we notice significant offsets from our measurements. For example, defining β between 1600Å and 2200Å results in up to ∆β = 0.1 bluer slopes (orange squares). Using a significantly redder wavelength range, 1600Å to 2600Å, leads to 0.1 − 0.35 bluer slopes (blue circles) compared to our definition. Note that the offset varies as a function of β itself − specifically, differences are enhanced towards redder slopes. The second, more physically driven, quantity that affects the measurement of β is the assumed dust attenuation law. As described in Section 4.1.2, the choice of the dust attenuation law has a negligible affect on the stellar masses and SFRs. This is not the case for the β slopes as shown in the lower panels of Figure 19. The left panel compares β derived using Calzetti and SMC dust attenuation. We notice a consistent positive offset of up to ∆β = 0.3 for the reddest slopes. We compared the reduced χ 2 values output by LePhare for fits using a Calzetti and SMC dust attenuation in order to derive a preference for either of the dust attenuations. We find that the χ 2 values show insignificant differences, which lets us conclude that we are not able to distinguish between the different dust attenuations based on our SED fitting. Hence, we decided that the best way is to be agnostic about the dust attenuation and combine for each galaxy the two probability density functions P Calzetti (β) and P SMC (β) derived from our Monte Carlo approach (Section 4.3.1) assuming equal weighting to derive the median β and its 1σ uncertainties. In the lower left panel of Figure 19, the final combined β slopes are compared to the β derived assuming a Calzetti attenuation. The offset towards redder β is significantly reduced due to a narrower probability density functions assuming Calzetti dust (hence the average β are drawn to the Calzetti solution in most cases). The β Slopes of the ALPINE Galaxies in Context Figure 20 shows our β measurement (marginalized over both Calzetti and SMC dust) as a function of stellar mass split in the different methods of selection (Section 2.1). As the UV slope is mostly affected by the dust attenuation, the strong correlation between β and stellar mass is not surprising as more massive galaxies are expected to be more dusty. The z ∼ 5.5 narrow-band selected galaxies have statistically the bluest slopes, indicating their dust-poor nature. The other galaxies are spread out over the whole parameter space. We also show the data from our parent sample at 4 < z < 6 in gray and compare their β slope and stellar mass distribution to the ALPINE sample in the kernel density estimate plots. Note that the β slope distribution of ALPINE galaxies peaks at ∆β ∼ 0.2 bluer values than the parent sample at the same redshift. This is a minor bias (likely caused by our spectroscopic selection) that has to be kept in mind for future analyses. Measurement of Hα emission Rest-frame optical emission lines in z > 4 galaxies are out of reach of current spectrographs. Specifically, the Hα emission provides a good tool to study the star-formation properties of galaxies in more detail. Fortunately, in the redshift range 4 < z < 5, the Hα line falls in the Spitzer 3.6 µm filter, while the 4.5 µm filter lacks any strong emission lines. Therefore, the [3.6 µm]−[4.5 µm] color can be used to constrain the Hα line flux and its equivalent width (a proxy of recent stellar mass build up). This method leads to Hα emission properties that are statistically as accurate as derived from spectroscopic data (Faisst et al. 2016b). Several such measurements have been carried out in the past with success (Shim et al. 2011;Stark et al. 2013;de Barros et al. 2014;Smit et al. 2014;Mármol-Queraltó et al. 2016;Faisst et al. 2016b;Rasappu et al. 2016;Smit et al. 2016;Caputi et al. 2017;Faisst et al. 2019). About 55% of the ALPINE sample (66 galaxies) lie in this redshift range. To measure the Hα luminosity and equivalent widths, we follow the same technique as outlined in Faisst et al. (2019) (we refer to this paper for more technical details). In brief, this method makes an assumption on the rest-frame optical continuum to which emission lines are added in a consistent manner to reproduce the observed [3.6 µm]−[4.5 µm] colors of the galaxies. This approach is robust as it only depends on the slope of the rest-frame optical continuum, which is well defined and nearly independent of assumptions on age, metallicity, and star-formation history for galaxies younger than ∼ 1 Gyrs (mostly the case at z > 4). To describe the rest-frame optical continuum, we use several basis stellar population models based on the Bruzual & Charlot (2003) template library (see Table 4). For the dust correction of the Hα emission, we assume the stellar E s (B − V) values derived by LePhare, which we convert to nebular extinction factors Table 4). The f -factor (differential reddening between stellar continuum and nebular regions) has the strongest effect on the Hα luminosity measurements. by assuming an f -factor 20 of 0.44 as measured in local starburst galaxies (Calzetti et al. 2000). We also assume a Calzetti and SMC reddening law. Furthermore, we assume an [N II] to Hα ratio of 0.15, as expected for galaxies at log(M/M ) = 10 (Faisst et al. 2018), to correct the blending of the [N II] and Hα lines. In Figure 21, we show systematic uncertainties in the measurement of the Hα luminosity due to the assumptions in our model for the rest-frame optical continuum and the reddening law (models A through F , see Ta Hα luminosity (left y-axis) and Hα-derived SFR (right yaxis). The latter are derived from Hα assuming the conversion factor by Kennicutt (1998) for solar metallicity (dashed line and ±0.3 dex margin, gray). Only galaxies with noncontaminated Spitzer photometry are shown. The symbols are color-coded by stellar mass. The star-symbols denote mergers based on the classification in Le . The Hα-dependent quantities are derived assuming a Calzetti reddening law, constant SFH, solar metallicity, and f = 0.44 (see Figure 21 for effect of different assumptions). Above a SFR of ∼ 13 M yr −1 , the two SFRs are comparable. Below that threshold, the scatter in the Hα-derived SFRs increases due to low S/N of the 4.5 µm observations (see Faisst et al. 2019). Bottom: Comparison of Hα and [C II] luminosity. Shown are only galaxies with Hα measurements and non-contaminated Spitzer photometry. [C II] undetected galaxies (at < 3.5σ) are indicated with upper limits. The lines and dashed margins show the expected relation between Hα and [C II] derived combining the relations from Kennicutt (1998) andDe Looze et al. (2014) (for their entire sample and metal-poor dwarf galaxies, see text). Note that at lower [C II] luminosities (< 5 × 10 8 L ), the galaxies seem to be more consistent with the relation of local metal-poor dwarf galaxies. an SMC reddening law. The f -factor is the largest uncertainty in this measurement method and will have to be pinned down by future observations with the JWST. For now, the assumed f = 0.44 provides likely an upper limit on the Hα luminosities. As shown by the arrows in Figure 21, assuming an f -factor equal to unity (which is thought to be more likely based on observations at z ∼ 2) would decrease the Hα luminosities by up to 0.4 dex (0.8 dex) for a stellar dust reddening of 0.2 (0.4) magnitudes. The correction in case of an SMC reddening law are 0.1 − 0.2 dex less. We note that these factors also apply to Hα-derived SFRs and any other quantity that depends linearly on the Hα luminosity. The top panel of Figure 22 compares the SFRs derived from SED fitting (Section 4.1) to the Hα luminosity and Hα-derived SFRs for galaxies without contaminated Spitzer photometry. The latter is derived using the standard conversion factor given in Kennicutt (1998), SFR (M yr −1 ) = 4.5 × 10 −42 L Hα (erg s −1 ),(5) assuming solar metallicity and a Chabrier IMF. Assuming one-fifth solar metallicity, the inferred SFR is expected to be ∼ 0.2 dex lower (Ly et al. 2016). As shown in Faisst et al. (2019), the uncertainty of this conversion factor is negligible compared to the impact of the uncertain f -factor. The Hα luminosities in the ALPINE sample range from ∼ 10 41 L to ∼ 10 44 L assuming f = 0.44 and Calzetti et al. (2000) dust attenuation. The Hα-derived SFRs trace well the SED-derived SFRs above ∼ 13 M yr −1 . Below that value, we see a large scatter in Hα derived SFRs. which happens when the Hα emission becomes too faint to be measured reliably using the Spitzer broad bands. Specifically, this is the case roughly at log(M/M ) = 9.5, which corresponds to a 4.5 µm detection of less than 5σ (see figure 3 and appendix in Faisst et al. 2019). The lower panel of Figure 22 relates the Hα luminosity (here in units of solar luminosity) to the [C II] luminosity measured by ALMA (also solar luminosity). In addition, we show the expected relation between Hα and Figure 23. The rest-frame Hα EW of our ALPINE galaxies in context of local galaxies (gray cloud) and z = 4.5 galaxies from Faisst et al. (2019) (gray squares). The Hα EW is related to stellar mass and sSFR derived from SED fitting (color-coded). The ALPINE sample at 4 < z < 5 builds a representative subsample of the general galaxy population at z > 4, also in terms of Hα properties. Although with similar stellar masses as local galaxies, the high-redshift galaxies reside at significantly higher Hα EWs, which is naturally explained by their higher star formation. Note the two galaxies with particularly low sSFR (< 1 Gyr −1 , and consistently low Hα EWs of less than 30Å) that fall onto the massive end of the distribution of local galaxies. These galaxies are indicative for systems with evolved stellar populations at high redshifts with currently reduced star formation activity. we find a good agreement with the entire local sample. For lower [C II] luminosities, the Hα measurements fall below this relation and are instead more consistent, although with a large scatter, with the relation of metalpoor local dwarf galaxies. The majority of [C II] undetected galaxies align well with either relation, however, the uncertainty in the Hα measurements becomes substantial as galaxies fall below log(M/M ) = 9.5. Figure 23 shows the rest-frame Hα EW distribution of our ALPINE galaxies in the context of local galaxies (gray cloud) and other z ∼ 4.5 galaxies on the COS-MOS field (gray, Faisst et al. 2019). The Hα EW for the z > 4 galaxies is derived consistently assuming a constant SFH, Calzetti reddening law, and f = 0.44. The ALPINE galaxies cover well the parameter space of the other z ∼ 4.5 galaxies, hence build a representative sample also in terms of Hα properties. For a fixed stellar mass, the high-redshift galaxies have higher Hα EWs compared to local galaxies, which is expected from a galaxy evolution point of view as galaxies at higher redshift are highly star forming. Note that two galaxies in the ALPINE sample have similar Hα EW values as massive (log(M/M ) > 10.5) local galaxies. Consistently, also their sSFR are low (< 1 Gyr −1 ), which is indicative of them being systems with evolved stellar populations at high redshifts. SUMMARY AND CONCLUSIONS The early growth phase at redshifts z = 4 − 6 marks an important time in which galaxies build up their stellar mass, enrich in metals and dust, and change their structure to transform into galaxies at the peak of SFR density or thereafter. For a better understanding of this interesting galaxy population, a multi-wavelength survey is crucial. ALPINE comprises a valuable set of 118 galaxies at 4.4 < z < 5.9 with unprecedented ALMA data at ∼ 150 µm FIR wavelengths. Together with the ancillary data presented in this paper, it makes it the first large panchromatic survey to discover the formation and study the evolution of galaxies during the early growth phase. Summarizing, the science enabling corner stone datasets of ALPINE are: − unprecedented ALMA observations to study the dust, gas, and outflow properties of the largest sample of galaxies to-date at z > 4 (Bethermin et al. 2020), − consistently calibrated deep spectroscopic observations at rest-frame UV wavelengths ( §2) to study Lyα emission and absorption lines ( §2.4, 2.4.2), − coherent ground-based (and space-based in ECDFS) imaging data from the optical to near-IR ( §3) for the measurement of various properties from SED fitting methods ( §4.1), including stellar masses and SFRs ( §4.1.1), UV luminosities ( §4.2), and UV continuum slopes to study stellar dust attenuation ( §4.3) , − deep Spitzer imaging at 3.6 µm and 4.5 µm to measure Hα emission for 66 galaxies between 4 < z < 5 ( §4.4), − high-resolution HST/ACS imaging in F 814W for all galaxies and WFC3/IR imaging for a smaller fraction (less than 30% with deep F 160W data) to study their resolved structure in connection with FIR [C II] emission ( §3). The ALPINE sample is built upon several different selection methods (Section 2.1, Figure 3), hence contains a multitude of different spectroscopic properties. Because of the requirement for spectroscopic confirmation, the sample is slightly biased towards brighter UV magnitudes (Section 4.2, Figure 18) and blue UV continuum slopes (∆β ∼ 0.2) compared to the average 4 < z < 6 galaxy population (Section 4.3, Figure 20). Nonetheless, stellar masses and SFRs, derived from the wealth of ancillary data, show that the ALPINE sample is broadly representative of the 4 < z < 6 galaxy population. Figure 11). In general concordance with studies at z = 2−3 (using Hα to define the systemic redshift), we find that on average Lyα is red shifted (∼ 180 km s −1 ) with respect to the [C II] line, while and the absorption lines are blue shifted (∼ −230 km s −1 ). In Cassata et al. (2020), we perform a more detailed comparison to samples at lower redshifts and study the implication on the Lyα escape fraction in correlation with Lyα equivalent widths. Stacking the spectra in bins of sSFR, we find larger velocity offsets of absorption lines with respect to systemic for galaxies with high sSFRs, which is indicative of stronger winds and outflows in these galaxies (Section 2.4.2, Figure 10). This finding is in agreement with the recent work by Ginolfi et al. (2019), who show a broadening in the FIR [C II] profiles in ALPINE galaxies with high star formation. Statistically, the SFRs derived from Hα emission via the Kennicutt (1998) relation for galaxies between 4 < z < 5 agree well with the values derived from SED fitting, assuming a differential dust reddening factor of f = 0.44 (Section 4.4, upper panel of Figure 22). However, we observe a considerable scatter for fainter galaxies (log(M/M ) < 9.5) due to the lower S/N of the Spitzer observations. Thanks to the large sample size of ALPINE, we are able, for the first time, to compare the Hα luminosity to the [C II] luminosity (lower panel of Figure 22). Overall, we find Hα luminosities as expected from the local relation between L [CII] and SFR from De Looze et al. (2014) (using their fit to the entire sample). However, we find that at low [C II] luminosities (< 5 × 10 8 L ), the Hα luminosities are generally lower than what is predicted by that relation. Instead, the De Looze et al. relation derived from a sample of metal-poor dwarf galaxies is a better fit for those galaxies. This might suggest a more complex relation between SFR and [C II] luminosity driven by metallicity or other properties of the ISM. ALPINE is the beginning of a thorough exploration of galaxies at z > 4. It builds the foundation onto which future follow-up observations can build on. In fact, several follow-up programs are being granted, some of which are already on going. These include (i) additional HST WFC3/IR observations of interacting ALPINE galax-ies (PI: Faisst), (ii) follow up observations of [N II] at 205 µm with ALMA for 9 ALPINE galaxies (PI: Faisst), (iii) high spatial resolution (∼ 0.15 ) observations of the brightest ALPINE galaxies (PI: Ibar), and (iv) the follow up of four serendipitous objects at z > 4 with NOEMA (PI: Loiacono & Béthermin). In addition, several JWST proposals are in preparation. All ancillary data products (including catalogs, images, and spectra) will be made public accessible. In the Appendix A, we detail the layout of the catalogs including the measurements detailed in this paper. In Appendix B, we show HST cutouts in ACS F 814W and WFC3/IR F 160W bands as well as the rest-frame UV spectra of all individual ALPINE galaxies. This paper completes a series of three papers presenting the ALPINE survey (Le Fèvre et al. 2019) and the data processing . Acknowledgements: We would like to thank numerous people for the exchange of data without which the ALPINE ancillary data paper would not exist. Especially we would like to thank E. Vanzella for helping us gathering the spectra in the ECDFS field and O. Ilbert for useful discussions that improved the SED fitting results. We also thank the anonymous referee for the suggestions that improved this paper. This paper is based on data obtained with the ALMA Observatory, under Large Program 2017.1.00428.L. ALMA is a partnership of ESO (representing its member states), NSF(USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. This program receives funding from the CNRS national program Cosmology and Galaxies. This work is based on observations and archival data made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA along with archival data from the NASA/ESA Hubble Space Telescope. This research made also use of the NASA/IPAC Infrared Science Archive (IRSA), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. In parts based on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under ESO programme ID 179.A-2005 and on data products produced by TERAPIX and the Cambridge Astronomy Survey Unit on behalf of the UltraVISTA consortium. Based on data obtained with the European Southern Observatory Very Large Telescope, Paranal, Chile, under Large Program 185.A-0791, and made available by the VUDS team at the CESAM data center, Laboratoire d'Astrophysique de Marseille, France. This work is based on observations taken by the 3D-HST Treasury Program (GO 12177 and 12328) with the NASA/ESA HST, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Furthermore, this work is based on data from the W.M. Keck Observatory and the Canada-France-Hawaii Telescope, as well as collected at the Subaru Telescope and retrieved from the HSC data archive system, which is operated by the Subaru Telescope and Astronomy Data Center at the National Astronomical Observatory of Japan. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. Finally, we would also like to recognize the contributions from all of the members of the COSMOS Team who helped in obtaining and reducing the large amount of multiwavelength data that are now publicly available through IRSA at http://irsa.ipac.caltech.edu/Missions/cosmos.html. A.C., F.P., M.T., C.G., and F.L. acknowledge the support from grant PRIN MIUR 2017. G.C.J. acknowledges ERC Advanced Grant 695671 "QUENCH" and support by the Science and Technology Facilities Council (STFC). E.I. acknowledges partial support from FONDE- spitzer cont − Spitzer photometry contamination flag. Set to 0, 1, and 2 for no, slight, and heavy contamination, respectively. ewha medÅ Rest-frame Hα equivalent width assuming Calzetti et al. (2000) dust attenuation and f = 0.44 ewha lowÅ Lower 1σ limit of rest-frame Hα equivalent width ewha upÅ Upper 1σ limit of rest-frame Hα equivalent width log halum med erg s −1 Logarithmic Hα luminosity assuming Calzetti et al. (2000) dust attenuation and f = 0.44 log halum low erg s −1 Lower 1σ limit of Hα luminosity in log log halum up erg s −1 Upper 1σ limit of Hα luminosity in log log sfrha med M yr −1 Logarithmic SFR based on Hα luminosity. Derived assuming (Kennicutt 1998) (solar metallicity), Calzetti et al. (2000) dust attenuation, and f = 0.44 log sfrha low M yr −1 Lower 1σ limit of Hα based SFR in log log sfrha up M yr −1 Upper 1σ limit of Hα based SFR log a Note that NB1 and NB2 stand for the narrow-band selection of galaxies at z ∼ 4.5 and z ∼ 5.7, respectively. Excerpt of the column description of the photometry catalog for galaxies in the ECDFS field (Section 3.1). Wavelengths, depths, and references are given in Table 2. Continued on next page Column Unit Description B. ADDITIONAL FIGURES In the following, we show the imaging and spectroscopic data for the individual ALPINE galaxies. Figures B.1 through B.3 show 2 × 2 cutouts of the HST F814W band of each of the galaxies, sorted by increasing redshift. The redshift, stellar mass, and SFR is indicated. The dashed contours show −3σ levels and the solid contours show 3σ, 5σ, 10σ, 15σ, and 30σ levels. The cutouts are oriented such that north is up and east is to the left. Similarly, Figures B.4 and B.5 show the all the available HST F 160W data for the ALPINE galaxies as of October 2019. The cutout size and the drawn σ-levels are the same as in the previous figures. Note that the HST program DASH covers most of the galaxies, however, only a fraction is detected due to the low depth of these observations. Figures B.6 through B.10 show the rest-frame UV spectra for each ALPINE galaxy sorted by increasing redshift. The spectra are smoothed with a Savitzky-Golay filter of size 2Å. Prominent emission lines as well as individual absorption lines and absorption line complexes are indicated by the dark-red bars (compare to Section 2.4.2). For some of the galaxies in COSMOS a spectrum obtained by VUDS and Keck/DEIMOS is available. The names of these galaxies have an appended " v" or " d", respectively. Note their different spectral resolution. in this paper are indicated Spectroscopy (Keck/DEIMOS, VUDS, GOODS-S/VLT) - §2 Imaging (ground & HST; various programs) - §3 Spitzer imaging - §3 Hα from Spitzer colors - §4 Layout of Current Data Products for ALPINE Galaxies SED fitting - §4 (Stellar masses, SFR) ALMA Data reduction: Béthermin et al. (2019) Figure 2 . 2Signal-to-Noise (SNR) of the ALMA-detected sources in the ALPINE sample. The different histograms show the numbers for [C II] and continuum detections above 3.5σ. For more information, see Le Fèvre et al.(2019)andBethermin et al. (2020). based on the Ilbert et al. (2010) photo-z catalog. References: (1) Capak et al. (2004); Mallery et al. (2012); Hasinger et al. (2018), (2) Le Fèvre et al. (2015), (3) Figure 4 . 4Comparison of observed (i.e., not corrected for dust) z-band and K-band magnitudes (left) and luminosities (right) for different selections listed in Figure 5 . 5Absolute calibration of rest-frame UV spectra. Shown are three examples at z = 4.53, z = 4.56, and z = 5.68. The spectra are convolved by the filters and the photometry (open circles) is compared to the total and Galactic extinction corrected broad, intermediate, and narrow-band photometry from catalogs (filled circles) described in Section 3. Figure 6 6Figure 6. Examples of stacked ALPINE spectra. Panels 1a and 1b show stacked spectra at z < 5 (in COSMOS from DEIMOS observations and as part of the VUDS survey) and z > 5 (on COSMOS from DEIMOS and on ECDFS from VIMOS and FORS2 observations), respectively. The stacks are all normalized to the continuum between 1300Å and 1400Å and common emission and absorption features are indicated with gray bars. Note that the VUDS spectra have a lower native resolution (R ∼ 230) compared to the DEIMOS observations (R ∼ 2500), therefore the latter have been degraded in resolution using a 1-dimensional Gaussian window function for visual comparison. Panels 2a through 2e show stacks at z < 5 and z > 5 for the different datasets. The number of spectra included per wavelength is shown on the top of each panel. The uncertainty in flux is indicated by the light gray line. The y-axis scale is the same such that the continuum brightness can be compared. Figure 7 . 7Stacked spectra in COSMOS for each of the selections discussed in Section 2.1 and listed in Figure 8 8Figure 8. Stacked histograms of velocity offsets between redshifts derived from different spectral features. The number of galaxies and median of the distribution (including scatter) are indicated. Shown are the velocity offsets between Lyα emission and IS+wind absorption lines (top panel ), as well as between Lyα, IS+wind, and systemic redshift (middle and bottom panel). The latter two are in detail discussed in a forthcoming paper (Cassata et al. 2020). The average errors are on the order of ±100 km s −1 , which corresponds to the size of the bins. We do not find any significant biases introduced by the different selection methods (color coded as in previous figures). Figure 9 . 9Histogram of velocity offset with respect to systemic (defined by the [C II]λ158µm redshift) for IS (red; O I, C II, and Si II) and stellar wind affected absorption lines (blue; Si IV and C IV). The average errors are on the order of ∼ 200 km s −1 , which corresponds to the size of the bins. Figure 10 10Figure 10. Stacked spectra (in C II systemic redshift) in two bins of sSFR (red: < 4 Gyr −1 , blue: > 5 Gyr −1 ) for five wavelength regions covering prominent rest-UV absorption lines. The derivation of the sSFR for the ALPINE galaxies is detailed in Section 4.1. The average number of spectra in each bin is indicated together with the prominent absorption and emission lines. We note systematically stronger blue shifts of all absorption lines for the high sSFR stack. Particularly, note the strong blue-shift of the high-ionization wind lines. The C IV lines in the high sSFR bin also show indication of a more pronounced P-Cygni profile, indicative of strong stellar winds and outflows in high sSFR galaxies. The 1σ uncertainties of the stacked spectra is indicated by the shaded regions. Figure 11 , 11we show the final spectrum in the rest-frame of the [C II] emission. The width of ∼ 4Å includes both [O II] lines. The theoretical rest-frame wavelength of the doublet is indicated by the black arrows. The position of the line agrees perfectly with the [C II] redshift derived from ALMA, indicating that FIR [C II] and optical [O II] trace the same systemic redshift. In addition, the top panel of the Figure shows the [C II] FIR line for comparison of the central wavelength of the lines. Note that the actual width of the [C II] line is more than 100 times larger than the the one of the optical [O II] line. Figure 15 . 15Scatter diagrams and histograms of offsets between the Gaia reference frame and the COSMOS2015 (left) and 3D-HST (ECDFS, right) catalogs. The offsets are in the sense "Gaia − COSMOS" and "Gaia − ECDFS", respectively. For COSMOS we find only a systematic offsets in right-ascension direction of −64 mas. For ECDFS, the offsets are large in both directions. In addition to this, we measure a scatter in the astrometry of ∼ 100 mas for both fields in both coordinates. ( 1 ) 1Hildebrandt et al. (2006);Erben et al. (2005),(2)Nonino et al. (2009), (3) Wuyts et al. (2008); Retzlaff et al. (2010), (4) Hsieh et al. (2012), (5) Cardamone et al. (2010), (6) Giavalisco et al. (2004), (7) Koekemoer et al. (2011), (8) Grogin et al. (2011); Koekemoer et al. (2011), (9) Brammer et al. (2012); van Dokkum et al. (2013), (10) Ashby et al. (2013); Guo et al. (2013), (11)Dickinson et al. (2003) indicated. d Some of the F160W data comes from the DASH survey (depth given inparenthesis). References: (1) see Laigle et al. (2016), (2) Taniguchi et al. (2007, 2015) (3) Scoville et al. (2007b); Koekemoer et al. (2007), (4) Grogin et al. (2011); Koekemoer et al. (2011), (5) Momcheva et al. (2017), (6) Capak et al. (2012); Steinhardt et al. (2014), (7)Sanders et al. (2007) lated to an [O II] emission line flux using the relation SFR (M yr −1 ) = (1.4 ± 0.4) × 10 −41 L [OII] (erg s −1 ). Other emission lines (Lyα, [O III], Hβ, Hα) are derived by assuming specific ratios to [O II] that are calibrated by observations (see detailed description with references in Ilbert et al. 2009). Figure 17 . 17Relation between stellar mass and SFR (main-sequence) of our ALPINE galaxies compared to all COSMOS galaxies at 4 < z < 6 (blue points) and the main-sequence parameterization at z = 5 bySpeagle et al. (2014) (gray band with ±0.3 dex width). Galaxies with contaminated Spitzer photometry are marked with squares (their stellar mass is likely an upper limit) and mergers (classification by Le Fèvre et al. 2019) are shown as stars. The color denotes the [C II] flux in Jy km s −1 measured by ALPINE. Galaxies that are not detected at the 3.5σ level are shown with white face color. Figure 19 . 19Dependence of UV continuum slope β on the definition of the wavelength regions and assumption of dust attenuation law. The solid line shows the 1-to-1 relation and the dashed lines show different offsets. Top: Dependence of β on the adopted wavelength window (with respect to our choice, 1300Å to 2300Å). Bottom: Dependence of β on the assumed dust attenuation law. Using an SMC dust attenuation results in redder slopes compared to a Calzetti reddening law. Figure 20 20Figure 20. Comparison of UV slopes and stellar mass. The offset panels show kernel density estimates of the β and stellar mass distribution. The ALPINE galaxies (large symbols) are split into their method of selection (see Section 2.1). We also show the data from our parent sample at 4 < z < 5 (gray dots). Statistically, our ALPINE peaks at ∼ 0.2 dex higher stellar masses and ∼ 0.3 bluer β. Figure 21 . 21Effect of different assumptions of the rest-frame optical continuum, reddening law, and f -factor (arrows show absolute decrease in luminosity from f = 0.44 to 1.0 for different stellar dust attenuations and reddening laws) on the measurement of the Hα luminosity. The gray band shows the 1-to-1 relation with ±0.3 dex margin. The different models for the continuum are labeled in the same way as in 5 Figure 22 . 522ble 4), as well as the f -factor for E s (B − V) = 0.2 and 0.4. It is evident that different assumptions in metallicity and SFH have a negligible impact on the measured Hα luminosity. The choice of the reddening law matters as the Hα luminosity decreases by ∼ 0.3 dex for galaxies at high Hα luminosities (log(L Hα /L ) > 43.Top: Comparison of SED-derived SFRs to [C II] luminosity by combining Equation 5 with the linear relation between [C II] and SFR derived by De Looze et al. (2014) from local and low-redshift galaxy samples,log(SFR / [M yr −1 ]) = α × log(L [CII] / L ) + γ,(6)for different values of intercepts (γ) and slopes (α). Specifically, we are showing the relation for their entire sample (α = 1.01, γ = −6.99, blue hatched) and the metal-poor dwarf galaxies (α = 0.80, γ = −5.73, red hatched). For bright [C II] galaxies (L[CII] 5 × 10 8 L ) CYT through grant No. 1171710. The Cosmic Dawn Center (DAWN) is funded by the Danish National Research Foundation under grant No. 140. S.T. acknowledges support from the ERC Consolidator Grant funding scheme (project Context, grant No. 648179). LV acknowledges funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant agreement No. 746119. D.R. acknowledges support from the National Science Foundation under grant numbers AST-1614213 and AST-1910107 and from the Alexander von Humboldt Foundation through a Humboldt Research Fellowship for Experienced Researchers. Unique name for each galaxy in string format id 3dhst -Unique identification number in the 3D-HST catalog ra 3dhst degrees Right-ascension as given in 3D-HST catalog dec 3dhst degrees Declination as given in 3D-HST catalog Galactic extinction corrected total fluxes with 1σ uncertainty f f160w µJy HST/WFC3 F 160 flux and uncertainty e f160w µJy . at the Keck telescope in Hawaii. The remaining spectra on the COSMOS field are obtained from the VIMOS Ultra Deep Survey (VUDS, Le Fèvre et al. 2015; Tasca et al. 2017) at the VLT 1 http://www.astro.caltech.edu/ ∼ afaisst/ Table 1. Spectroscopy and selection of ALPINE galaxiesSurvey Selection Number Ref. COSMOS field (105 galaxies) Keck/DEIMOS † 84 1 narrow-band (z ∼ 4.5) a 6 narrow-band (z ∼ 5.7) b 23 LBG (color) c 41 pure photo-z d 9 4.5 µm excess 4 X-ray (Chandra) 1 with Lyα emission 66 weak Lyα emission or absorption 18 VUDS 21 2 photo-z + LBG 21 [narrow-band (z ∼ 4.5) 3] ‡ [narrow-band (z ∼ 5.7) 1] ‡ [LBG (color) 1] ‡ [4.5 µm excess 1] ‡ with Lyα emission 16 weak Lyα emission or absorption 5 ECDFS field (13 galaxies) VLT GOODS-S 11 3 primarily LBG (color) 11 total with Lyα emission 6 total without Lyα emission 5 HST/GRAPES 2 4 Grism (no a priori selection) 2 with Lyα emission 2 weak Lyα emission or absorption 0 † For a detailed description of the selection criteria, we refer to Mallery 1240Å,1280Å], [1280Å,1320Å], [1320Å,1350Å], [1370Å,1420Å], and [1500Å,1570Å]. For the separate fit of the IS and wind lines, we split the last window into two ranges, namely [1510Å,1540Å] and [1530Å,1570Å] to separate the IS line Si II ). 2.4.3. How well does [C II] trace systemic redshift? Comparison to optical [O II] emission The extended nature of [C II] may be indicative of its origin in the diffuse interstellar medium in addition to Sky [OII] in [CII] 158 m rest-frame Figure 11. Comparison of optical and FIR emission lines in the case of galaxy DC 881725. Top: [C II] emission line at 158 µm observed with ALMA. Bottom: The black line shows the optical [O II] doublet at 3727Å observed with the MOSFIRE spectrograph on Keck in the rest-frame of the [C II] emission. The theoretical rest-frame wavelength of the doublet is indicate by the two arrows. This shows that there is no significant velocity offset between the [C II] FIR and optical [O II] emission, hence verifies the validity of [C II] as tracer of the systemic redshift of this galaxy.0.00 1.00 2.00 3.00 4.00 5.00 6.00 flux (mJy/beam) [CII] 158 m FIR line 157.4 157.6 157.8 158.0 158.2 rest-frame wavelength ( m) 3720 3725 3730 3735 3740 rest-frame wavelength (Å) 0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 flux (electrons/sec) [OII] doublet (true) Table 2 . 2Photometry available for galaxies on the ECDFS field.Observatory/Instrument Filter Central λ 3σ depth Ref. (Å) (mag) Ground-based MPG-ESO/WFI U 38 3633.3 27.3 1 b 4571.2 26.6 1 v 5377.0 26.6 1 Rc 6536.3 26.9 1 I 9920.2 26.5 1 VLT/VIMOS U 3720.5 28.6 2 R 6449.7 27.8 2 VLT/ISAAC J v 12492.2 25.6 3 H v 16519.9 25.1 3 K v s 21638.3 25.0 3 CFHT/WIRCam Jw 12544.6 25.1 4 K w s 21590.4 24.5 4 Subaru/Suprime-Cam IA427 4263.4 25.7 5 IA445 4456.0 25.7 5 IA505 5062.5 25.8 5 IA527 5261.1 26.7 5 IA550 5512.0 26.0 5 IA574 5764.8 25.7 5 IA598 6000.0 26.6 5 IA624 6233.1 26.5 5 IA651 6502.0 26.7 5 IA679 6781.1 26.6 5 IA738 7371.0 26.5 5 IA767 7684.9 25.5 5 IA797 7981.0 25.2 5 IA856 8566.0 25.0 5 Space-based HST/ACS F435W 4328.7 29.1 6 F 606W 5924.8 29.1 (29.0) 6,7 F 775W 7704.8 28.4 6 F 814W 8058.2 28.9 7 F 850LP 9181.2 27.9 6 HST/WFC3 F 125W 12516.3 27.6 8 F 140W 13969.4 26.7 9 F 160W 15391.1 27.7 8 Spitzer /IRAC ch1 35634.3 ∼ 26.0 a 10 ch2 45110.1 ∼ 26.0 a 10 ch3 57593.4 24.4 11 ch4 79594.9 24.3 11 a The exposure time varies between 10 ks and 300 ks (corresponding to Table 3 . 3Photometry available on the COSMOS field.Observatory/Instrument Filter Central λ 3σ depth Ref. (Å) (mag) † Ground-based CFHT/MegaCam u * 3823.3 26.6 1 Subaru/Suprime-Cam B 4458.3 27.0 1,2 V 5477.8 26.2 1,2 r + 6288.7 26.5 1,2 i + 7683.9 26.2 1,2 z ++ 9105.7 25.9 1,2 IA427 4263.4 25.9 1,2 IA464 4635.1 25.9 1,2 IA484 4849.2 25.9 1,2 IA505 5062.5 25.7 1,2 IA527 5261.1 26.1 1,2 IA574 5764.8 25.5 1,2 IA624 6233.1 25.9 1,2 IA679 6781.1 25.4 1,2 IA709 7073.6 25.7 1,2 IA738 7361.6 25.6 1,2 IA767 7684.9 25.3 1,2 IA827 8244.5 25.2 1,2 N B711 7119.9 25.1 1,2 N B816 8149.4 25.2 1,2 Subaru/HSC YHSC 9791.4 24.4 1 CFHT/WIRCam Hw 16311.4 23.5 1 K w s 21590.4 23.4 1 VISTA/VIRCAM Y 10214.2 24.8 (25.3) a 1 J 12534.6 24.7 (24.9) a 1 H 16453.4 24.3 (24.6) a 1 Ks 21539.9 24.0 (24.7) a 1 Space-based Table 4 . 4List of basis stellar population models for the parameterization of the rest-frame optical continuum at 4 < z < 5 to derive Hα emission from Spitzer colors.Model SFH Metallicity Dust attenuation (Z = 0.02) A constant 0.02 Calzetti B constant 0.004 Calzetti C exp. declining a 0.01 Calzetti D constant 0.02 SMC E constant 0.004 SMC F exp. declining a 0.01 SMC a Assuming τ = 3 × 10 8 yrs. The FIR [C II] redshifts observed by ALMA allow us to set the systemic redshift of the galaxies in order to study velocity offsets of Lyα emission and several rest-frame UV absorption lines(Section 2.4). From one galaxy at z = 4.57 with optical [O II] measurements acquired from Keck/MOSFIRE, we show that the [O II] and FIR [C II] redshifts are in excellent agreement, hence the latter likely is a good tracer of the systemic redshift derived by optical emission lines at lower redshifts (Section 2.4.3, Table A A.1 -Continued from previous page Upper 1σ limit on sSFR in log M FUV mag Absolute rest-frame UV magnitude measured in the GALEX FUV filter (corresponding approximately to rest-frame 1500Å) M FUV low1sig mag Lower 1σ limit on absolute rest-frame UV magnitude M FUV high1sig mag Upper 1σ limit on absolute rest-frame UV magnitude UV continuum slopes (β) with different dust reddening (Section 4.3) Hα measurements from Spitzer colors (using Model A, see Section 4.4)Column Name Unit Description logsSFR higheff1sig yr −1 beta med calz − UV slope measured assuming Calzetti dust beta low1sig calz − Lower 1σ UV slope limit (Calzetti dust) beta high1sig calz − Upper 1σ UV slope limit (Calzetti dust) beta med smc − UV slope measured assuming SMC dust beta low1sig smc − Lower 1σ UV slope limit (SMC dust) beta high1sig smc − Upper 1σ UV slope limit (SMC dust) beta med comb − UV slope measured by marginalizing over Calzetti and SMC dust beta low1sig comb − Lower 1σ UV slope limit (Calzetti+SMC dust) beta high1sig comb − Upper 1σ UV slope limit (Calzetti+SMC dust) Table A .2. A Table A . A2 -Continued from previous pageGalactic extinction corrected total magnitudes with 1σ uncertainty Note: given are 1σ limits (and magnitude uncertainties are set to −1) if fluxes are smaller than 1σ flux uncertainties.Table A.3. Excerpt of the column description of the photometry catalog for galaxies in the COSMOS field (Section 3.2). Wavelengths, depths, and references are given in Table 3. Unique identification number in the COSMOS2015 catalog ra cosmos15 degrees Right-ascension as given in the COSMOS2015 catalog dec cosmos15 degrees Declination as given in the COSMOS2015 catalog Galactic extinction corrected total fluxes with 1σ uncertainty Ks FLUX APER3 µJy CFHT/WIRCam Ks-band flux and uncertainty Ks FLUXERR APER3 µJy . ... ... ... extinction corrected total magnitudes with 1σ uncertainty Note: given are 1σ limits (and magnitude uncertainties are set to −1) if fluxes are smaller than 1σ flux uncertainties. MAG mag CFHT/WIRCam Ks-band magnitude and uncertaintyColumn Name Unit Description ... ... ... mag f160w mag HST/WFC3 F 160 magnitude and error magerr f160w mag . ... ... ... Column Unit Description ALPINE ID - Unique name for each galaxy in string format id cosmos15 - Galactic Ks Ks MAGERR mag . ... ... ... One of these galaxies, ve 530029038, has also been observed by the VUDS survey. Here we refer to the O I absorption line complex consisting of Si III, C II, O I, and Si II. This absorption consists of two lines (1548.2Å and 1550.8Å) and here we give the average wavelength. 5 http://specpro.caltech.edu 6 The value of this visual flag is −99 if the S/N is too low to obtain a redshift, and 1 and 2 for reliable and very reliable redshift measurements, respectively. http://cosmos.astro.caltech.edu/page/photom 9 https://3dhst.research.yale.edu/Data.php https://mast.stsci.edu 11 https://github.com/gbrammer/mastquery 12 https://github.com/grizli-project/grizli-aws 13 https://github.com/gbrammer/grizli The photometric redshifts given in the COSMOS2015 catalog are consistent within 0.2 with our spectroscopic redshifts for more than 95% of all cases. 15 https://splash.caltech.edu There is also data in the F 098W filter, however, unfortunately no ALPINE galaxies are covered. The f -factor, f = Es(B − V)/En(B − V), describes the differential dust reddening between the stellar continuum and nebular regions. Its value is largely unknown at z > 2, but it is expected that f approaches a value closer to unity at higher redshiftsReddy et al. 2010;Kashino et al. 2013;Koyama et al. 2015;Valentino et al. 2015;Puglisi et al. 2016;Kashino et al. 2017;Faisst et al. 2019). Continued on next page Figure B.1. F 814W cutouts sorted by redshift (part 1). The dashed contours show −3σ levels and the solid contours show 3σ, 5σ, 10σ, 15σ, and 30σ levels. All cutouts are 2 on each side. Figure B.2. F 814W cutouts sorted by redshift (part 2). Figure B.4. F 160W cutouts sorted by redshift (part 1). The dashed contours show −3σ levels and the solid contours show 3σ, 5σ, 10σ, 15σ, and 30σ levels. All cutouts are 2 on each side. Figure B.6. Rest-frame UV spectra of all galaxies sorted by redshift (part 1). Figure B.7. Rest-frame UV spectra of all galaxies sorted by redshift (part 2). Figure B.8. Rest-frame UV spectra of all galaxies sorted by redshift (part 3). Figure B.9. Rest-frame UV spectra of all galaxies sorted by redshift (part 4). Figure B.10. Rest-frame UV spectra of all galaxies sorted by redshift (part 5). APPENDIXA. DESCRIPTION OF PUBLISHED DATA PRODUCTSThe data presented in this paper are summarized in three different catalogs.• The main catalog, which contains properties consistently measured for all the galaxies. These include general information (such as coordinates, redshifts, selection, morphological class), measurements performed on the spectra (such as Lyα redshift and properties as well as absorption line redshifts), measurements from SED fitting (including UV continuum slopes), and Hα line properties and SFRs.• The ECDFS photometry catalog, which contains all the Galactic extinction corrected total photometry (magnitude, fluxes, and uncertainties) of the galaxies in the ECDFS field. This catalog is based on the 3D-HST catalog.• The COSMOS photometry catalog, which contains all the Galactic extinction corrected total photometry (magnitude, fluxes, and uncertainties) of the galaxies in the COSMOS field. This catalog is based on the COSMOS2015 catalog.The following Tables A.1, A.3, and A.2 summarize the columns of each of these three catalogs. The catalogs can be downloaded in FITS format at http://www.astro.caltech.edu/ ∼ afaisst/ Note that the Tables A.2 and A.3 only show an excerpt of the description of the ECDFS and COSMOS photometry catalog. The full versions can be found at the link above. (2) pair-merger (major or minor); (3) extended dispersion dominated; (4) compact dispersion dominated; (5) too weak for assigning a class.Measurements on spectra (Section 2)has twin -A flag set to 1 of for a galaxy has been observed by Keck/DEIMOS and VUDS (two spectra available). If false, the flag is set to 0.z lya -Redshift determined from peak of Lyα emission (see details inCassata et al. 2020). Is -99 if no redshift measured.lya ewÅObserver-frame Lyα emission equivalent (see details inCassata et al. 2020).Is -99 if no equivalent width is measured.lya ew errÅ 1σ uncertainty on observer-frame Lyα emission equivalent (see details inCassata et al. 2020). Is -99 if no equivalent width is measured and −1 if no continuum measured (i.e., EW is upper limit).f lya erg/s/cm 2 Lyα emission flux (see details inCassata et al. 2020n lines wind used -Number of lines used for wind redshift measurement. We advice to generally only use galaxies with a value > 0 together with flag specpro > 0 for a conservative sample selection.Properties from SED fitting with LePhare (Sections 4.1 and 4.2)ID photcat -ID in the photometric catalogs. This is the 3D-HST catalog for galaxies in ECDFS and the COSMOS2015 catalog for galaxies in COSMOS.chi2χ 2 value given by the LePhare fit.Nband . 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[ "https://github.com/gbrammer/mastquery", "https://github.com/grizli-project/grizli-aws", "https://github.com/gbrammer/grizli" ]
[ "Sparsity information and regularization in the horseshoe and other shrinkage priors", "Sparsity information and regularization in the horseshoe and other shrinkage priors" ]
[ "Juho Piironen [email protected] \nDepartment of Computer Science\nHelsinki Institute for Information Technology\nHIIT\nAalto University\n\n", "Aki Vehtari [email protected] \nDepartment of Computer Science\nHelsinki Institute for Information Technology\nHIIT\nAalto University\n\n" ]
[ "Department of Computer Science\nHelsinki Institute for Information Technology\nHIIT\nAalto University\n", "Department of Computer Science\nHelsinki Institute for Information Technology\nHIIT\nAalto University\n" ]
[ "Electronic Journal of Statistics" ]
The horseshoe prior has proven to be a noteworthy alternative for sparse Bayesian estimation, but has previously suffered from two problems. First, there has been no systematic way of specifying a prior for the global shrinkage hyperparameter based on the prior information about the degree of sparsity in the parameter vector. Second, the horseshoe prior has the undesired property that there is no possibility of specifying separately information about sparsity and the amount of regularization for the largest coefficients, which can be problematic with weakly identified parameters, such as the logistic regression coefficients in the case of data separation. This paper proposes solutions to both of these problems. We introduce a concept of effective number of nonzero parameters, show an intuitive way of formulating the prior for the global hyperparameter based on the sparsity assumptions, and argue that the previous default choices are dubious based on their tendency to favor solutions with more unshrunk parameters than we typically expect a priori. Moreover, we introduce a generalization to the horseshoe prior, called the regularized horseshoe, that allows us to specify a minimum level of regularization to the largest values. We show that the new prior can be considered as the continuous counterpart of the spike-and-slab prior with a finite slab width, whereas the original horseshoe resembles the spike-and-slab with an infinitely wide slab. Numerical experiments on synthetic and real world data illustrate the benefit of both of these theoretical advances.MSC 2010 subject classifications: Primary 62F15.
10.1214/17-ejs1337si
null
54,584,958
1707.01694
440b77d462e6997396750c43bb5b50c67db2ea22
Sparsity information and regularization in the horseshoe and other shrinkage priors 2017 Juho Piironen [email protected] Department of Computer Science Helsinki Institute for Information Technology HIIT Aalto University Aki Vehtari [email protected] Department of Computer Science Helsinki Institute for Information Technology HIIT Aalto University Sparsity information and regularization in the horseshoe and other shrinkage priors Electronic Journal of Statistics 11201710.1214/17-EJS1337SIReceived June 2017.and phrases: Bayesian inferencesparse estimationshrinkage priorshorseshoe prior The horseshoe prior has proven to be a noteworthy alternative for sparse Bayesian estimation, but has previously suffered from two problems. First, there has been no systematic way of specifying a prior for the global shrinkage hyperparameter based on the prior information about the degree of sparsity in the parameter vector. Second, the horseshoe prior has the undesired property that there is no possibility of specifying separately information about sparsity and the amount of regularization for the largest coefficients, which can be problematic with weakly identified parameters, such as the logistic regression coefficients in the case of data separation. This paper proposes solutions to both of these problems. We introduce a concept of effective number of nonzero parameters, show an intuitive way of formulating the prior for the global hyperparameter based on the sparsity assumptions, and argue that the previous default choices are dubious based on their tendency to favor solutions with more unshrunk parameters than we typically expect a priori. Moreover, we introduce a generalization to the horseshoe prior, called the regularized horseshoe, that allows us to specify a minimum level of regularization to the largest values. We show that the new prior can be considered as the continuous counterpart of the spike-and-slab prior with a finite slab width, whereas the original horseshoe resembles the spike-and-slab with an infinitely wide slab. Numerical experiments on synthetic and real world data illustrate the benefit of both of these theoretical advances.MSC 2010 subject classifications: Primary 62F15. Introduction This paper deals with sparse Bayesian estimation and is an extension to our earlier work (Piironen and Vehtari, 2017a). We consider statistical models with a large number of parameters θ = (θ 1 , . . . , θ D ) but so that it is reasonable to assume that only some of them are far from zero. A typical example -and also the case we will mostly focus in this paper -is a regression or classification problem with a large number of predictor variables out of which we expect only a few to be relevant and therefore have a regression coefficient distinguishable from zero. A vast number of different estimators, both Bayesian and non-Bayesian, have been proposed for these problems. In the non-Bayesian literature the sparse problems are typically handled by Lasso (Tibshirani, 1996) or one of its generalizations (for an overview, see e.g., Hastie, Tibshirani and Wainwright, 2015). We focus on the probabilistic approach and carry out full Bayesian inference on the problem. Two prior choices dominate the Bayesian literature: two component discrete mixture priors known as the spike-and-slab (Mitchell and Beauchamp, 1988;George and McCulloch, 1993), and a variety of continuous shrinkage priors (see e.g., Polson and Scott, 2011, and references therein). The spike-and-slab prior is intuitively appealing as when the spike is taken to be a delta-spike in the origin, it is equivalent to Bayesian model averaging (BMA) (Hoeting et al., 1999) over the variable combinations, and often has good performance in practice. The disadvantages are that the results can be sensitive to prior choices (slab width and prior inclusion probability) and that the posterior inference can be computationally demanding with a large number of variables, due to the huge model space. The inference could be sped up by analytical approximations using either expectation propagation (EP) Suárez, 2010, 2015) or variational inference (VI) (Titsias and Lázaro-Gredilla, 2011), but this comes at the cost of a substantial increase in the amount of analytical work needed to derive the equations separately for each model and a more complex implementation. The continuous shrinkage priors on the other hand are easy to implement, provide convenient computation using generic sampling tools such as Stan (Stan Development Team, 2017), and can yield as good or better results. A particularly interesting example is the horseshoe prior Scott, 2009, 2010) θ j | λ j , τ ∼ N 0, τ 2 λ 2 j , λ j ∼ C + (0, 1) , j = 1, . . . , D, (1.1) which has shown comparable performance to the spike-and-slab prior in a variety of examples where a sparsifying prior on the model parameters θ j is desirable Scott, 2009, 2010;Polson and Scott, 2011). The horseshoe is one of the so called global-local shrinkage priors, meaning that there is a global hyperparameter τ that shrinks all the parameters towards zero, while the heavy-tailed half-Cauchy priors for the local hyperparameters λ j allow some θ j to escape the shrinkage (see Sec. 2.1 for more thorough discussion). Despite its good performance in many problems, the horseshoe prior has previously suffered from two shortcomings. First, there has been no consensus on how to carry out inference for the global hyperparameter τ which determines the overall sparsity in the parameter vector θ and therefore has a large impact on the results. We prefer full Bayesian inference (see Sec. 3.1) but the existing methodology has been lacking a systematic way of placing a prior for τ based on the information about the sparsity. Second, the horseshoe prior has the undesired property that the parameters far from zero are not regularized at all (see Sec. 2.3). While this is often considered as a key strength of the prior, it can be harmful especially when the parameters are only weakly identified by the data, for instance in the case of a flat likelihood due to separable data in logistic regression. We propose a solution to both of these problems. We introduce a concept of effective number of nonzero parameters m eff (Sec. 3.3), derive analytically its relationship between the global shrinkage parameter τ , and show an easy and intuitive way of formulating the prior for τ based on the prior information about the sparsity of θ. Based on these theoretical considerations, we argue that the previously proposed default priors are dubious based on the prior they impose on m eff , and that they yield good results only when τ (and therefore m eff ) is strongly identified by the data. Moreover, we introduce a generalization of the horseshoe prior, called the regularized horseshoe, that operates otherwise similarly as the original horseshoe but allows specifying the regularization to the coefficients that are far from zero (see Sec. 2.3). We show that the regularized horseshoe can be considered as the continuous counterpart of the spike-and-slab prior with a finite slab width, whereas the original horseshoe resembles the spikeand-slab with an infinitely wide slab. The benefit of both of these theoretical advances will be illustrated with examples on synthetic and real world data (Sec. 4). As a final remark, although we focus on the horseshoe in our discussion, we want to emphasize that both of these ideas could also be applied to other shrinkage priors, and several promising alternatives to the horseshoe have been proposed during the recent years (Bhattacharya et al., 2015;Zhang, Reich and Bondell, 2016;Ghosh, Li and Mitra, 2017). Horseshoe prior and its extension This section discusses the horseshoe and its connection to the spike-and-slab prior (Section 2.2). We also present an extension (Section 2.3) that both helps understanding the theoretical properties of the original horseshoe and -as will be demonstrated in Section 4 -robustifies the prior and improves its practical performance. Horseshoe prior for linear regression Consider the single output linear Gaussian regression model with several input variables, given by y i = β T x i + ε i , ε i ∼ N 0, σ 2 , i = 1, . . . , n , (2.1) where x is the D-dimensional vector of inputs, β contains the corresponding weights and σ 2 is the noise variance. The horseshoe prior is set for the regression coefficients β = (β 1 , . . . , β D ) β j | λ j , τ ∼ N 0, τ 2 λ 2 j , λ j ∼ C + (0, 1) , j = 1, . . . , D. (2.2) If an intercept term β 0 is included in model (2.1), we give it a relatively flat prior, because there is usually no reason to shrink it towards zero. As discussed in the introduction, the horseshoe prior has been shown to possess several desirable theoretical properties and good performance in practice Scott, 2009, 2010;Polson and Scott, 2011;Datta and Ghosh, 2013;van der Pas, Kleijn and van der Vaart, 2014). The intuition is the following: the global parameter τ pulls all the weights globally towards zero, while the thick half-Cauchy tails for the local scales λ j allow some of the weights to escape the shrinkage. Different levels of sparsity can be accommodated by changing the value of τ : with large τ all the variables have very diffuse priors with very little shrinkage, but letting τ → 0 will shrink all the weights β j to zero. The above can be formulated more formally as follows. Let X denote the n-by-D matrix of observed inputs and y the observed targets. The conditional posterior for the coefficients β given the hyperparameters and data D = (X, y) can be written as p(β | Λ, τ, σ 2 , D) = N β |β, Σ , β = τ 2 Λ τ 2 Λ + σ 2 (X T X) −1 −1β , Σ = (τ −2 Λ −1 + 1 σ 2 X T X) −1 , where Λ = diag λ 2 1 , . . . , λ 2 D andβ = (X T X) −1 X T y is the maximum likelihood solution (assuming the inverse exists). If the predictors are uncorrelated with zero mean and variances Var(x j ) = s 2 j , then X T X ≈ n diag s 2 1 , . . . , s 2 D , and we can approximateβ j = (1 − κ j )β j , (2.3) where κ j = 1 1 + nσ −2 τ 2 s 2 j λ 2 j (2.4) is the shrinkage factor for coefficient β j . The shrinkage factor describes how much coefficient β j is shrunk towards zero from the maximum likelihood solution (κ j = 1 meaning complete shrinkage and κ j = 0 no shrinkage). From (2.3) and (2.4) it is easy to verify thatβ → 0 as τ → 0, andβ →β as τ → ∞. The result (2.4) holds for any prior that can be written as a scale mixture of Gaussians like (2.2), regardless of the prior for λ j . The horseshoe employs independent half-Cauchy priors for all λ j , and for this choice one can show that, for fixed τ and σ, the shrinkage factor (2.4) follows the prior p(κ j | τ, σ) = 1 π a j (a 2 j − 1)κ j + 1 1 √ κ j 1 − κ j , (2.5) where a j = τ σ −1 √ n s j . When a j = 1, this reduces to Beta 1 2 , 1 2 which looks like a horseshoe, see Figure 1. Thus, a priori, we expect to see both relevant 0.0 0.5 1.0 0.0 0.5 1.0 κ j κ j Fig 1. The continuous curves show the densities for the shrinkage factor (2.4) for the horseshoe prior (2.2) when a j = τ σ −1 √ n s j = 1 (left) and when a j = 0.1 (right). The bars denote the corresponding point mass function for the spike-and-slab prior (2.7) with infinite slab width c → ∞, when π/(1 − π) = 1 (left) and π/(1 − π) = 0.1 (right). To aid visualization, the bars illustrating the point masses are scaled and show only the relative probability masses. (κ j = 0, no shrinkage) and irrelevant (κ j = 1, complete shrinkage) variables. By changing the value of τ , the prior for κ j places more mass either close to 0 or 1. For instance, choosing τ so that a j = 0.1 favors complete shrinkage (κ = 1) and thus we expect more coefficients to be close to zero a priori. Notice though that for a fixed τ , the sparsity assumptions will be dependent on the input dimension D, and to get around this issue, we need to consider the values of all the shrinkage factors κ j together. Using this idea, Section 3 discusses an intuitive way of designing a prior distribution for τ based on the assumptions about the number of nonzero components in β. Notice also that those variables which vary on larger scale s j are treated as more relevant a priori, which is the reason why we usually scale all the variables to have unit variance s 2 j = 1, unless the original scales really carry information about the relevances. Another way would be to use the original scales but adjust scales for the local parameters accordingly λ j ∼ C + 0, s −2 j . Spike-and-slab prior The spike-and-slab (Mitchell and Beauchamp, 1988;George and McCulloch, 1993) is a popular shrinkage prior that is often considered as the "gold standard" for sparse Bayesian estimation. The prior is often written as a two-component mixture of Gaussians β j | λ j , c, ε ∼ λ j N 0, c 2 + (1 − λ j ) N 0, ε 2 , λ j ∼ Ber(π), j = 1, . . . , D,(2.6) so that ε c and the indicator variable λ j ∈ {0, 1} denotes whether the coefficient β j is close to zero (comes from the "spike", λ j = 0) or nonzero (comes from the "slab", λ j = 1). Often we set ε = 0, that is, the spike is taken to be a delta spike at the origin δ 0 , although also ε > 0 could be used (George and McCulloch, 1993). The user then has to specify the values (or priors) for the slab width c and the prior inclusion probability π, which encodes the prior information about the sparsity of the coefficient vector β. Fixing c is probably the most common approach but by giving it a hyperprior, one can obtain a more heavy-tailed, such as Laplacian, slab (Johnstone and Silverman, 2004). With the choice ε = 0, the prior (2.6) can be written analogous to (2.2) as β j | λ j , c ∼ N 0, c 2 λ 2 j , λ j ∼ Ber(π), j = 1, . . . , D,(2.7) so instead of giving continuous priors for λ j s as in the horseshoe, here only two values (λ j = 0, 1) are allowed. Thus also the shrinkage factor κ j only has mass at κ j = 1 1+nσ −2 s 2 j c 2 and at κ j = 1, and the probabilities are π and 1−π, respectively. Letting c → ∞, all the mass is concentrated at the extremes κ j = 0 and κ j = 1, and the resemblance to the horseshoe becomes obvious, see Figure 1. Given the similarity of the shrinkage profiles between the horseshoe and spike-and-slab, it is not surprising that the two priors have shown comparable performance in a variety of experiments Scott, 2009, 2010;Polson and Scott, 2011). The next section discusses an extension of the horseshoe that closely resembles the spike-and-slab prior with a finite slab width c < ∞. Regularized horseshoe As discussed in Section 2.1, the horseshoe prior favors solutions β j ≈ 0 and β j ≈β j , and it can be shown that under certain conditions,β j →β j when |β j | → ∞ (Carvalho, Polson and Scott, 2010). While this guarantees that the strong signals will not be overshrunk -and is often considered to be one of the key assets of the prior -this property can also be harmful, especially when the parameters are weakly identified. An example of such case is the flat likelihood arising in logistic regression with separable data. As the horseshoe has Cauchy tails, in these problems it suffers basically from the same problems as the Cauchy prior, namely that the posterior means for the regression coefficients may vanish (Ghosh, Li and Mitra, 2017). Therefore it would be very useful to be able to control the amount of shrinkage for the largest coefficients, which in spike-and-slab prior (Sec. 2.2) is achieved by controlling the slab width. To guarantee that the prior always shrinks the coefficients at least by a small amount, we introduce the following regularized horseshoe prior β j | λ j , τ, c ∼ N 0, τ 2λ2 j ,λ 2 j = c 2 λ 2 j c 2 + τ 2 λ 2 j , λ j ∼ C + (0, 1) , j = 1, . . . , D,(2.8) where c > 0 is a constant that we assume is given for now. The intuition behind this definition is the following. When τ 2 λ 2 j c 2 , meaning the coefficient β j is close to zero, thenλ 2 j → λ 2 j and the prior (2.8) approaches the original horseshoe. However, when τ 2 λ 2 j c 2 , meaning the coefficient is far from zero, Shrinkage profiles as in Figure 1 but now for the regularized horseshoe (2.8) and spikeand-slab (2.7) with a finite slab width c = 1. In comparison to Figure 1, for both priors the first mode is shifted from κ j = 0 to κ j = 1/(1 + nσ −2 s 2 j c 2 ) (for the plots we have selected nσ −2 s 2 j = 10). thenλ 2 j → c 2 /τ 2 and the prior (2.8) approaches N 0, c 2 . Thus the prior will shrink the small signals as the horseshoe but will also regularize even the largest coefficients as a Gaussian slab with variance c 2 . Another way to see this is to notice that the conditional prior for β j can be factored as p(β j | λ j , τ, c) ∝ N 0, τ 2 λ 2 j N 0, c 2 ∝ N 0, τ 2λ2 j , (2.9) from which it is easy to see that depending on the relative magnitudes of τ 2 λ 2 j and c 2 , the prior operates (roughly) as the narrower one of the two factors. Therefore the role of N 0, c 2 is to "soft-truncate" the extreme tails of the horseshoe, thereby controlling the magnitude of the largest β j s. Letting c → ∞, we recover the original horseshoe. The shrinkage profile of the regularized horseshoe is illustrated in Figure 2 together with the spike-and-slab with the slab width c, which demonstrates the similarity of the two priors. Using c < ∞ has the advantage that it regularizes the parameters β j when they are weakly identified, and allows us also to specify our prior information about the maximum effect β j we expect to see. The benefit of the proposed approach is illustrated in Section 4.1.2. It must be noted that the shrinkage profile in Figure 2 does not have exactly the same shape as the original horseshoe shifted and scaled from interval (0, 1) to (b, 1) with b > 0, although it is very close to this. There is slightly more mass near the left hand side mode, although the difference is too small to be visible in Figure 2. It is possible to retain the exact shape of the horseshoe by defining the modified local parameterλ 2 j as λ 2 j = c 2 λ 2 j σ 2 ns 2 j + c 2 + τ 2 λ 2 j . (2.10) The details of this result are spelled out in Appendix A. The reason why we define the prior as (2.8) is that unless n or the slab width c 2 is very small, the term σ 2 ns 2 j is typically small compared to c 2 and leaving it out has little influence in practice. On the other hand, formulation (2.8) is simpler and has the nice interpretation as a product of the original horseshoe and the Gaussian slab (2.9). Thus we report the result (2.10) only for completeness. Our formulation requires choosing a value or prior for c. Unless substantial knowledge about the scale of the relevant coefficients exists, we generally recommend placing a prior for c instead of fixing it. Often a reasonable choice is c 2 ∼ Inv-Gamma(α, β), α = ν/2, β = νs 2 /2, (2.11) which translates to a Student-t ν (0, s 2 ) slab for the coefficients far from zero and is typically a good default choice for a weakly informative prior. Another motivation for using inverse-Gamma is that it has a heavy right tail accompanied by a light left tail thereby preventing much mass from accumulating near zero. This is natural as we do not want to shrink those coefficients heavily towards zero that are already deemed to be far from zero. Still, we emphasize that our approach is not limited to this choice and also other hyperpriors are possible. It would also be possible to use variable specific slab widths c i , but we do not explore this further in this paper and leave it for future investigation. Finally, we would also like to point out that the tail-cutting idea of Equation (2.9) could be used more generally with other priors when relevant. For instance, we expect our idea to be useful with the horseshoe+ prior of Bhadra et al. (2017) which also has Cauchy tails and therefore suffers from the same problem as the horseshoe. Hierarchical shrinkage In addition to the problems with vanishing means, earlier we have also reported sampling issues with the original horseshoe even in simple regression problems (Piironen and Vehtari, 2015). Technically speaking, the problem arises due to posterior having an extreme funnel shape which is challenging for Markov chain Monte Carlo (MCMC) methods. The problem was revealed with the help of the divergence diagnostics of the NUTS algorithm (Hoffman and Gelman, 2014;Betancourt and Girolami, 2015;Betancourt, 2017a,b) when fitting the models in Stan. The problem is related to the thick Cauchy tails of the prior, and to overcome the sampling issues, in our technical report we tentatively proposed replacing the half-Cauchy priors for the local parameters λ j in (2.2) with half-t priors with small degrees of freedom, such as ν = 3, and named this approach the "hierarchical shrinkage". With large enough ν, this seems to help with the sampling issues and remove the divergent transitions produced by NUTS, but the drawback is that the prior becomes less sparsifying. This is because when the tails of p(λ j ) are made slimmer, we need to increase the value for τ to accommodate large signals, and therefore the prior is not able to shrink the small coefficients efficiently towards zero. Another limitation of this approach is that in order to fight the problems arising from data separation in logistic regression (Sec. 2.3), we would also need to refrain from using half-Cauchy prior for the global parameter τ (which we might want to use, see Sec. 3) as this would also lead to Cauchy tails for p(β j ). The less sparsifying nature of the choice ν > 1 will be demonstrated in practice in Section 4.1.2, where we also show that the regularized horseshoe (Sec. 2.3) clearly outperforms this approach. Thus we no longer recommend increasing the local degrees of freedom, but instead of using the regularized horseshoe. The global shrinkage parameter This section discusses the prior choice for the global hyperparameter τ . We begin with a short note on why we prefer full Bayesian inference for τ over point estimation, and then go on to discuss how we propose to set up the prior p(τ ). Full Bayes versus point estimation In principle, one could use a plug-in estimate for τ , obtained either by crossvalidation or maximum marginal likelihood (sometimes referred to as "empirical Bayes"). The maximum marginal likelihood estimate has the drawback that it is always in danger of collapsing toτ = 0 if the parameter vector happens to be very sparse. Moreover, rather than being computationally convenient, this approach might actually complicate matters as the marginal likelihood is not analytically available for non-Gaussian likelihoods. While cross-validation avoids the latter problem and possibly also the first one, it is computationally less efficient than the full Bayesian solution and fails to account for the posterior uncertainty. For these reasons we recommend full Bayesian inference for τ , and focus on how to specify the prior distribution. Earlier approaches Carvalho, Polson and Scott (2009) also recommend full Bayesian inference for τ , and following Gelman (2006), they propose prior τ ∼ C + (0, 1),(3.1) whereas Polson and Scott (2011) recommend τ | σ ∼ C + 0, σ 2 . (3.2) If the target variable y is scaled to have marginal variance of one, unless the noise level σ is very small, both of these priors typically lead to quite similar posteriors. However, as we argue in Section 3.3, there is a theoretical justification for letting τ scale with σ. The main motivation for using a half-Cauchy prior for τ is that it evaluates to a finite positive value at the origin, yielding a proper posterior and allowing even complete shrinkage τ → 0, while still having a thick tail which can accommodate a wide range of values. For these reasons, C + 0, η 2 is a desirable choice when there are enough observations to let τ be identified by data. Still, we show that in several cases one can clearly benefit by choosing the scale η in a more careful manner than simply η = 1 or η = σ, because for most applications these choices place far to much mass for implausibly large values of τ . This point is discussed in Section 3.3. Moreover, the synthetic example in Section 4.1.1 shows that in some cases one could clearly benefit from even more informative prior. van der Pas, Kleijn and van der Vaart (2014) study the optimal selection of τ in model y i ∼ β i + ε i , ε i ∼ N 0, σ 2 , i = 1, . . . , n. (3.3) They prove that in such a setting, the optimal value (up to a log factor) in terms of mean squared error and posterior contraction rates in comparison to the true β * is τ * = p * n , (3.4) where p * denotes the number of nonzeros in the true coefficient vector β * (assuming such exists). Their proofs assume that n, p * → ∞ and p * = o(n). Model (3.3) corresponds to setting X = I and D = n in the usual regression model (2.1). It is unclear whether and how this result could be extended to a more general X, and how one should utilize this result when p * is unknown (as it usually is in practice). In section 3.3, we formulate our method of constructing the prior p(τ ) based on the prior information about p * , and show that if p * was known, our method would also give rise to result (3.4), but is more generally applicable. Effective number of nonzero coefficients Consider the prior distribution for the shrinkage factor of the jth regression coefficient for the linear Gaussian model, Eq. (2.5). The mean and variance can be shown to be E(κ j | τ, σ) = 1 1 + a j , (3.5) Var(κ j | τ, σ) = a j 2(1 + a j ) 2 , (3.6) where a j = τ σ −1 √ n s j as earlier. A given value for the global parameter τ can be understood intuitively via the prior distribution that it imposes on the effective number of coefficients distinguishable from zero (or effective number of nonzero coefficients, for short), which we define as m eff = D j=1 (1 − κ j ). (3.7) When the shrinkage factors κ j are close to 0 and 1 (as they typically are for the horseshoe prior), this quantity describes essentially how many active or unshrunk variables we have in the model. It serves therefore as a useful indicator of the effective model size. Using results (3.5) and (3.6), the mean and variance of m eff given τ and σ are given by E(m eff | τ, σ) = D j=1 a j 1 + a j (3.8) Var(m eff | τ, σ) = D j=1 a j 2(1 + a j ) 2 , (3.9) Let us now assume that, in addition of having a zero mean, each variable also has a unit variance s 2 j = 1. In this case the equations above simplify to E(m eff | τ, σ) = τ σ −1 √ n 1 + τ σ −1 √ n D, (3.10) Var(m eff | τ, σ) = τ σ −1 √ n 2(1 + τ σ −1 √ n) 2 D. (3.11) The expression for the mean (3.10) is helpful. First, from this expression it is evident that to keep our prior information about m eff consistent, τ must scale as σ/ √ n. Priors that fail to do so, such as (3.1), favor models of varying size depending on the noise level σ and the number of data points n. Second, if our prior guess for the number of relevant variables is p 0 , it is reasonable to choose the prior so that most of the prior mass is located near the value τ 0 = p 0 D − p 0 σ √ n , (3.12) which is obtained by solving equation E(m eff | τ, σ) = p 0 . Note that this is typically quite far from 1 or σ, which are used as scales for priors (3.1) and (3.2). For instance, if D = 1000 and n = 200, then prior guess p 0 = 5 gives about τ 0 = 3.6 · 10 −4 σ. To further develop the intuition about the connection between τ and m eff , it is helpful to visualize the prior imposed on m eff for different prior choices for τ . This is most conveniently done by drawing samples for m eff ; we first draw τ ∼ p(τ ) and λ 1 , . . . , λ D ∼ C + (0, 1), then compute the shrinkage factors κ 1 , . . . , κ D from (2.4), and finally m eff from (3.7). τ = τ 0 τ ∼ N + 0, τ 2 0 τ ∼ C + 0, τ 2 0 τ ∼ C + (0, 1) Fig 3. Histograms of prior draws for m eff (effective number of nonzero regression coefficients, Eq. (3.7)) imposed by different prior choices for τ , when the total number of input variables is D = 10 and D = 1000. τ 0 is computed from formula (3.12) assuming n = 100 observations with σ = 1 and p 0 = 5 as the prior guess for the number of relevant variables. Note the varying scales on the horizontal axes in the bottom row plots. Figure 3 shows histograms of prior draws for m eff for some different prior choices for τ , with total number of variables D = 10 and D = 1000, assuming n = 100 observations with σ = 1. The first three priors utilize the value τ 0 which is computed from (3.12) using p 0 = 5 as our hypothetical prior guess for the number of relevant variables. Fixing τ = τ 0 results in a nearly symmetric distribution around p 0 , while a half-normal prior with scale τ 0 yields a skewed distribution favoring solutions with m eff < p 0 but allowing larger values to also be accommodated. The half-Cauchy prior behaves similarly to the half-normal, but results in a distribution with a much thicker tail giving substantial mass also to values much larger than p 0 when D is large. Figure 3 also illustrates why prior τ ∼ C + (0, 1) is often a dubious choice: it places far too much mass on large values of τ , consequently favoring solutions with most of the coefficients unshrunk. Thus when only a small number of the variables are relevant -as we typically assume -this prior results in sensible inference only when τ is strongly identified by data. Notice also that, if we changed the value of σ or n, the first three priors for τ would still impose the same prior for m eff , but this is not true for τ ∼ C + (0, 1). This way, by studying the prior for m eff , one can easily choose the prior for τ based on the information about the number of nonzero parameters. Because the prior information can vary substantially for different problems and the results depend on the information carried by the data, there is no globally optimal prior for τ that works for every single problem. Some recommendations, however, will be given in Section 5 based on these theoretical considerations and experiments presented in Section 4. We conclude by pointing out a connection between our reference value (3.12) and the oracle result (3.4) for the simplified model (3.3). As pointed out in the last section, model (3.3) corresponds to setting X = I (which implies n = D and X T X = I) in the usual regression model (2.1). Using this fact and repeating the steps needed to arrive at (3.12), we get τ 0 = p 0 D − p 0 σ. (3.13) Suppose now that we select p 0 = p * , that is, our prior guess is oracle. Using the same assumptions as van der Pas, Kleijn and van der Vaart (2014), namely that n, p * → ∞ and p * = o(n), and additionally that σ = 1, we get τ 0 → p * /D = τ * . This result is natural, as it means it is optimal to choose τ so that the imposed prior for the effective number of nonzero coefficients m eff is centered at the true number of nonzeros p * . This further motivates why m eff is a useful quantity. Regularized horseshoe and other shrinkage priors As discussed in Section 2.3, when we set c < ∞, the shrinkage profile of the regularized horseshoe (2.8) becomes approximately equivalent to that of the horseshoe shifted from interval (0, 1) to (b j , 1), where b j = 1 1+nσ −2 s 2 j c 2 . Thus the shrinkage factor under the regularized horseshoe satisfies approximatelỹ κ j = (1 − b j )κ j + b j , where κ j denotes the shrinkage factor for the original horseshoe. From this we get 1 −κ j = (1 − b j )(1 − κ j ). Assuming further that all the variables have a unit variance s 2 j = 1 and thus b j = b = 1 1+nσ −2 c 2 , the effective model complexity under the regularized horseshoe satisfies m eff = (1 − b)m eff , where m eff is the effective number of nonzeros for the original horseshoe. Thus with a given τ , the effective complexity for the regularized horseshoe is always less than for the pure horseshoe, because those coefficients that are far from zero are still affected by the slab. Therefore we can naturally use result (3.12) also for the regularized horseshoe with p 0 as our prior guess for the number of coefficients far from zero, but remembering that those coefficients will experience the regularization by the slab. The concept of effective number of nonzeros could also be used with shrinkage priors other than the (regularized) horseshoe, as long as the prior can be written as a scale mixture of Gaussians like (2.2). Many such alternatives have been proposed, including the double-exponential or Laplace (Park and Casella, 2008), Dirichlet-Laplace (Bhattacharya et al., 2015), R-square induced Dirichlet decomposition (Zhang, Reich and Bondell, 2016), and horseshoe+ (Bhadra et al., 2017). For instance, the Dirichlet-Laplace prior has a Dirichlet concentration hyperparameter a that strongly affects the sparsity properties of the prior, and based on the experiments of Bhattacharya et al. (2015) has a substantial effect on the results. It would therefore be interesting the investigate a prior information calibrated selection of a using our framework. Depending on the prior, corresponding analytical results like (3.10) and (3.11) may or may not be available, but as long as one is able to sample both from p(τ ) and p(λ j ), it is always easy to draw samples from the prior distribution for m eff , and therefore investigate the effect of hyperprior or hyperparameter choice on the effective model complexity. It must be noted though, that for those prior for which the shrinkage factors κ are not near 0 or 1, the values of m eff can be more difficult to interpret. Non-Gaussian observation models When the observation model is non-Gaussian, the exact analysis from Section 3.3 is analytically intractable. We can, however, perform the analysis using a Gaussian approximation to the likelihood. Using the second order Taylor expansion for the log likelihood, the approximate posterior for the regression coefficients given the hyperparameters becomes p(β | Λ, τ, φ, D) ≈ N β |β, Σ , β = τ 2 Λ τ 2 Λ + (X TΣ −1 X) −1 −1β , Σ = (τ −2 Λ −1 + X TΣ −1 X) −1 , wherez = (z 1 , . . . ,z n ),Σ = diag(σ 2 1 , . . . ,σ 2 n ) andβ = (X TΣ −1 X) −1 X TΣ −1z (assuming the first inverse exists). Here φ denotes the possible dispersion parameter and (z i ,σ 2 i ) the location and variance for the ith Gaussian pseudoobservation. These are obtained from the first and second order derivatives of the log-likelihood terms L i (f i , φ) with respect to the linear predictor f i = β T x i at the posterior modef i =β T x i (Gelman et al., 2013, ch. 16.2) z i =f i − L i (f i , φ) L i (f i , φ) ,σ 2 i = − 1 L i (f i , φ) . The fact that some of the observations are more informative than othersmeaningσ 2 i is not constant -makes further simplification somewhat difficult. To proceed, we make the rough assumption that we can replace eachσ 2 i by a single variance termσ 2 . Assuming further that the covariates are uncorrelated with zero mean and variances Var(x j ) = s 2 j (as in Sec. 3.3), the posterior mean for the jth coefficient satisfiesβ j = (1 − κ j )β j with shrinkage factor given by κ j = 1 1 + nσ −2 τ 2 s 2 j λ 2 j . (3.14) The discussion in Section 3.3 therefore also approximately holds for the non-Gaussian observation model, except that σ 2 is replaced byσ 2 . Still, this leaves us with the question, which value to choose forσ 2 to exploit this result in practice? For the generalized linear models with y having a distribution in the exponential family with natural parameter θ and dispersion φ, the log likelihood for a single observation has the form (McCullagh and Nelder, 1989) Table 1 The pseudo variances for the most commonly used generalized linear models to be used as approximate plug-in values for σ 2 in equations of Section 3.3. In practice when necessary, we usually replace µ by the sample meanȳ. In the Gamma distribution, α denotes the shape parameter so that Var(y) = µ 2 /α. In the inverse Gaussian, λ is the shape parameter so that Var(y) = µ 3 /λ. See McCullagh and Nelder (1989). L = yθ − B(θ) A(φ) − C(y, φ),Model Link ∂µ/∂f V (µ) A(φ) Pseudo varianceσ 2 Gaussian Identity 1 1 σ 2 σ 2 Binomial Logit µ(1 − µ) µ(1 − µ) 1 µ −1 (1 − µ) −1 Poisson Log µ µ 1 µ −1 Gamma Inverse −µ 2 µ 2 α −1 µ −2 α −1 Inverse Gaussian Inverse squared −µ 3 /2 µ 3 λ −1 4µ −3 λ −1 for some specific functions A(·), B(·) and C(·). The pseudo variance for a given observation y is then (see Appendix B) σ 2 = − 1 L = A(φ) 1 V (µ) ∂µ ∂f 2 − (y − µ) ∂ ∂f 1 V (µ) ∂µ ∂f −1 , ( 3.15) where V (µ) = B (θ) is the variance function, µ = E(y) = B (θ) the expected value. The simplified expressions for the most commonly used generalized linear models with their canonical links are listed in Table 1. We observe thatσ 2 is a product between A(φ) and a term that in general depends on f , µ, and y, although for canonical links the dependence from y vanishes because the derivative of ( ∂µ ∂f )/V (µ) with respect to f is zero (McCullagh and Nelder, 1989, ch. 2). Thus it makes sense that for those non-Gaussian models that have a dispersion parameter φ (like Gamma and inverse Gaussian models), the pseudo variance and therefore also τ should scale with A(φ). For the non-constant multiplier of A(φ) in (3.15) we can use, for example, value obtained by setting µ equal to the sample mean ofȳ. This approach, although crude, seems to work reasonably well in practice. For instance, in binary classification, if we have the same number of observations from both classes, then µ = 0.5, yieldingσ 2 = 4 which was observed to give good results in our earlier study (Piironen and Vehtari, 2017a). More complex models Although we limit our discussion to generalized linear models, different authors have employed horseshoe prior in various other models. For instance, Faulkner and Minin (2017) consider trend filtering for modelling time series and use horseshoe as a sparsifying prior on the kth-order forward differences to express prior assumptions about the number of rapid changes in the underlying signal. As another example, Ghosh and Doshi-Velez (2017) use horseshoe prior over the weights in Bayesian neural networks to effectively turn off some of the nodes in the network. When the model gets more complicated, one cannot simply use the reference result (3.12) to guide the hyperprior choice because it is derived assuming the linear model (2.1). Unless similar results can easily be derived for the model of interest, we recommend a pragmatic approach of drawing from the prior for different values of τ and studying the effect on the sparsity (how many coefficients fall below a certain threshold) to get an idea of a reasonable range of values. Although this strategy may seem crude, we argue that it should still be better than not thinking about the prior at all. Moreover, our results (Sec. 4.2) suggest that with a weakly informative prior such as τ ∼ C + 0, τ 2 0 , having even a rough ballpark figure of the correct magnitude for τ 0 can already be a clear improvement compared to the simple τ ∼ C + (0, 1). Experiments This section illustrates the benefit of the theoretical advances on synthetic and real world data. All considered models were fitted using Stan 1 (codes in the supplementary material) with the default settings unless otherwise stated, running 4 chains, 2000 samples each, first halves discarded as warmup. Synthetic data Toy example We first illustrate the impact of the hyperprior choice p(τ ) with a toy example similar to the one discussed by van der Pas, Kleijn and van der Vaart (2014). Consider model (3.3), where each y i is generated by adding Gaussian noise with σ 2 = 1 to the corresponding signal β i . We generated 100 data realizations with n = 400 and the true β * having p * = 20 nonzero entries equal to A = 1, 2, . . . , 10 with the rest of the entries being zeros. We then computed the mean squared error (MSE) between the estimated posterior meanβ and the true β * assuming the pure horseshoe prior with hyperpriors τ ∼ C + (0, 1), τ ∼ C + 0, τ 2 0 and τ = τ 0 , where τ 0 is calculated from Equation (3.13) with the oracle prior guess p 0 = p * . Although the last choice reflects stronger prior information than we would typically expect to have in practice, the purpose of this setup is simply to demonstrate that one can substantially improve the inferences using our framework provided substantial prior knowledge exists. Notice though, that even when we set τ = τ 0 , we do not treat τ as completely fixed, because it depends on σ which is treated as an unknown parameter with vague prior p(σ 2 ) ∝ σ −2 . Figure 4 shows the MSE for the three hyperpriors for different values of A. For each prior, the error is largest close to A = √ 2 log 400 ≈ 3.5, which is called the "universal threshold" for this problem (Johnstone and Silverman, 2004; van der Pas, Kleijn and van der Vaart, 2014). Below this threshold the nonzero components in β are too small to be detected and are thus shrunk too heavily towards zero which introduces error. For A = 4 the informative τ = τ 0 actually yields the worst results due to this overshrinkage (see discussion below), but gives clearly superior results for larger A. The choice τ ∼ C + 0, τ 2 0 gives better results than τ ∼ C + (0, 1) but is clearly inferior to τ = τ 0 . Figure 5 illustrates the data y and the estimated coefficientsβ for one particular data realization when A = 4 and A = 6. In both cases the informative choice τ = τ 0 helps to shrink the zero components in β towards zero, but for A = 4 also overshrinks the nonzero components. The reason for the overshrinkage is that some observations y i that correspond to zero signal β i = 0 happen to have similar magnitude to the observations coming from a nonzero signal β i = A, and thus these irrelevant components "steal" from the limited budget for m eff . For this particular value of A the overshrinkage of the actual signals happens to be worse in terms of MSE than undershrinkage of the zero components, and thus one would get better results by setting p 0 to be slightly above the true p * (results not shown). For A = 6 the actual signals are large enough to be distinguished from zero, and the informative selection of τ yields substantially better estimate for β. Finally, Figure 6 illustrates the importance of scaling τ with the noise level σ. The top row shows one data realization and the posterior mean estimatesβ when the global parameter is fixed either to τ = p0 D−p0 or to τ = p0 D−p0 σ. For these data both yield essentially the same result, since the true noise variance σ 2 is one. However, when the observations are scaled by multiplying them by 0.1 (bottom row), the value for τ that does not scale with σ yields clearly worse results than in the first case, while the results for the latter value remain practically unchanged. What essentially happens is that when the observations are transformed to a smaller scale, then fixing τ = p0 n−p0 increases the prior expectation for m eff by the same factor (10 in this case) and thus it will favor solutions with many more coefficients far from zero. This small experiment has relevance also regarding the more general model (2.1) because in the ideal case of uncorrelated predictors with unit variance X T X = nI, we can think that we have a single observation of each β j with variance σ 2 /n. In practice these conditions are rarely met but the idea is still useful. Classification with separable data The purpose of this example is to illustrate the problem with the original horseshoe not controlling the magnitude of the largest coefficients and show how the regularized horseshoe can solve this issue. We generated n = 30 binary classification observations so that for the instances in the first class, the first two features were drawn from a Gaussian with mean 1 and scale 0.5, whereas in the other class the mean of the first two features was −1 (the data are visualized in Fig. 7). In addition, we generated 98 irrelevant features drawn from the standard Gaussian, so that the total input dimension was D = 100. We then fitted the standard logistic regression model with prior β 0 ∼ N 0, 5 2 for the intercept and four different priors for the regression coefficients β j : the original horseshoe, hierarchical shrinkage with ν = 3, and regularized horseshoe with slab scales c = 2 and c 2 ∼ Inv-Gamma(2, 8), the latter of which corresponds to a Studentt 4 0, 2 2 slab (see Sec. 2.3). In each case we used hyperprior τ ∼ t + 3 0, τ 2 0 (the conclusions of this example are not sensitive to this choice). For consistent sparsity assumptions, for the original and the regularized horseshoe, τ 0 was calculated from (3.12) using p 0 = 2, and for the hierarchical shrinkage by solving numerically E(m eff | τ = τ 0 ) = p 0 = 2. Figure 7 shows the scatter plots of the posterior draws for β 1 and β 2 (top row) as well as the medians and 80%-intervals for β 3 , . . . , β 100 (middle row), for the four different priors. Because the data are separable using only x 2 , this feature is a "solitary separator" and thus the mean for β 2 does not exist under the horseshoe prior due to its Cauchy tails (Ghosh, Li and Mitra, 2017). Although for β 1 the mean exist (x 1 is not a solitary separator), the posterior has substantial mass for very large values for this parameter also. Moreover, for the original horseshoe the NUTS produces almost 200 divergent transitions after the warmup showing clear problems with sampling. Using ν = 3 for the local parameters cuts down the tails and reduces the number of divergent transitions to a few, but still yields quite fat posterior tails for these two coefficients and results in much less shrinkage for the coefficients of the irrelevant features (middle row). Finally, the regularized horseshoe exhibits the most satisfactory performance cutting down the tails for β 1 and β 2 while still being able to shrink the irrelevant coefficients as well as the horseshoe. In this case, fixing the scale of the Gaussian slab to c = 2 results in too strong regularization for the relevant coefficients and generally we would prefer a less informative choice such as c ∼ Inv-Gamma(2, 8), but our purpose here was to demonstrate how easy it is to specify different levels of regularization for the largest coefficients using the new prior (2.8). The top row of Figure 7 clearly reveals the multimodality of the posterior for β 1 and β 2 . This does not produce marked problems in this case (e.g. MCMC convergence problems) but in general the multimodality can be an issue both for the original and the regularized horseshoe, and we will discuss this further in Sections 4.2 and 6. Bottom row: the observed data (red and green denoting the different classes) and the predictive class probabilities forỹ = 1 as a function of the first two inputs, given that the rest of the inputs are set to zero. First two columns denote the results for the pure horseshoe and hierarchical shrinkage, and the last two columns for the regularized horseshoe. Real world data -microarray cancer classification This section further illustrates the important concepts with some real world examples. We use the four microarray cancer classification datasets from our earlier paper (Piironen and Vehtari, 2017a). The datasets are summarized in Table 2 and can be found online. 2 For all the datasets we used the standard logistic regression model with a vague prior β 0 ∼ N 0, 10 2 for the intercept, and the original and regularized horseshoe priors for the regression coefficients to compare the differences between the two. For these problems, we reduced the number of draws per chain to 1000 to reduce the computation time. We first consider the Ovarian dataset as a representative example of how the prior choice p(τ ) and the use of the regularized horseshoe can affect the results. We fitted the model to the data with two hyperprior choices, τ ∼ C + (0, 1) and τ ∼ C + 0, τ 2 0 , where τ 0 is computed from (3.12) using p 0 = 3 as our prior Figure 8 shows prior and posterior draws for τ and m eff , and the absolute values of the posterior means for the regression coefficients, for the different prior configurations. The results for τ ∼ C + (0, 1) illustrate how weakly τ is identified by the data: there is very little difference between the prior and posterior samples for τ and consequently for m eff , and thus this "non-informative" Ovarian dataset : Histograms of prior (light gray) and posterior (dark gray) draws for τ (top row) and m eff (middle row), and absolute values of the posterior means for the regression coefficients |β j | (bottom row) imposed by different prior choices. The first two columns denote the results for the pure and regularized horseshoe with hyperprior τ ∼ C + (0, 1), and the last two columns the same but with τ ∼ C + 0, τ 2 0 , where τ 0 is computed from (3.12) with prior guess p 0 = 3. prior actually has a strong influence on the posterior. With the pure horseshoe this results in severe under-regularization and implausibly large magnitude for the regression coefficients. This happens because the effective model complexity is so large that the classes become separable, and therefore the coefficients are only weakly identified (as in the synthetic example in Sec. 4.1.2). With the regularized horseshoe, even with the complete separation, the magnitude of the coefficients remains sensible due to the regularization by the slab. Replacing the scale of the half-Cauchy hyperprior with τ 0 , reflecting a more sensible guess for the number of relevant variables, has a substantial effect on the posterior: the posterior mass for m eff becomes concentrated on much smaller values and the magnitude of the regression coefficients more sensible. For this hyperprior choice the difference between the original and regularized horseshoe is much less severe, but also in this case the largest coefficients are smaller for the regularized horseshoe. How this affects the predictive accuracy will be discussed in a moment. A potential explanation for why τ and therefore m eff are not strongly identified here is that there are a lot of correlations in the data. For instance, the predictor j = 1491 which appears relevant based on its regression coefficient, has an absolute correlation of at least |ρ| = 0.5 with 314 other variables, out of which 65 correlations exceed |ρ| = 0.7. This indicates that there are a lot of potentially relevant but redundant predictors in the data, and thus similar fit could be obtained by models with varying levels of sparsity. Another difficulty with correlating predictors is that they cause the posterior of the regression coefficients to become multimodal, which usually results in convergence issues. Figure 9 shows the posterior draws under the regularized horseshoe prior for the three coefficients with the largest absolute mean. All three coefficients have most of the draws near zero but some of the draws are far from zero, illustrating the multimodality. TheR-values (potential scale reduction factor, see e.g., Gelman et al., 2013, ch. 11) for these parameters were 1.09, 1.20 and 1.05 indicating problems with the convergence of the MCMC chains. Although these convergence issues do not seem to have a drastic negative impact on the predictive accuracy (see the discussion below), the multimodality dramatically reduces the sampling efficiency and is therefore a major practical and theoretical concern and deserves more attention in the future research (see also the discussion in Section 6). To investigate the effect of the prior choices on the prediction accuracy, we split each dataset into two halves, using one fifth of the data as a test set. All the results were then averaged over 50 such random splits into training and test sets. We carried out the tests for priors τ ∼ C + 0, τ 2 0 and τ ∼ N + 0, τ 2 0 with various τ 0 . The half-normal prior was included in the tests to investigate whether it sometimes could be beneficial to have a strong control over τ by using a short-tailed prior. To get a baseline for the comparisons, we also computed the prediction accuracies to Lasso with the regularization parameter tuned by 10fold cross-validation. The Lasso results were computed with the default settings of the R-package glmnet (Friedman, Hastie and Tibshirani, 2010). The top row of Figure 10 shows the effect of the prior choice on the test prediction accuracy, and the other two rows the computation time 3 and the fraction of divergent transitions produced by the NUTS after the warm-up. For the original horseshoe (solid lines) the results illustrate a clear benefit from using even a crude prior guess for the number of relevant variables p 0 : transforming any guess between p 0 = 1 and p 0 = n into a value of τ 0 using Equation (3.12) (shaded region) and using this as the scale for the half-Cauchy prior instead of τ 0 = 1 yields improved prediction accuracy and reduced computation time in all the datasets. The regularized horseshoe seems in general less sensitive to the prior choice for τ (Ovarian and Colon datasets), but in some cases clearly benefits from a more carefully chosen prior (Prostate and Leukemia datasets). For both priors, using half-Cauchy hyperprior for τ is clearly less sensitive to the prior guess τ 0 , and yields better results than the half-normal especially when τ 0 was chosen to be too small. In terms of the predictive accuracy, the regularized horseshoe performs overall comparably to the original horseshoe: for the Colon dataset it performs slightly better, for Leukemia slightly worse, and for Ovarian and Prostate about the same as the pure horseshoe. Only for the Colon and Leukemia datasets the difference is statistically significant (the errorbars are left out from the plot to avoid mess). In Leukemia dataset where the regularized horseshoe loses to the pure horseshoe the chosen prior for the slab width c is probably unnecessarily restrictive leading to too large regularization for the largest coefficients. It is evident that in this case using a looser prior for c could improve the results (because when c → ∞ we recover the original horseshoe), but we feel that these results are enough to convince the reader that a reasonably chosen hyperprior p(c) does not necessarily compromise and in some cases can even improve the accuracy compared to the pure horseshoe. On the other hand, the regularized horseshoe clearly improves the sampling robustness of the posterior. The regularized horseshoe produces only very few divergent transitions after the warm-up if any (the fraction is non-negligible only for the Prostate dataset with a poorly chosen global prior p(τ )), whereas for the original horseshoe the fraction varies between 1-30% which is really a lot. With this many divergences, there is always a concern about a biased inference and this cannot be taken too lightly (see the next Section for recommendations). Moreover, also the computation times are systematically either smaller or similar to the original horseshoe, which is due to better behaving posterior. For any reasonably selected prior (shaded region), the (regularized) horseshoe consistently outperforms Lasso in terms of predictive accuracy, but in some cases the difference is not very large. A clear advantage for Lasso, on the other hand, is that it is hugely faster, with computation time of less than a second for these problems, whereas even for the regularized horseshoe, the computation starts to get quite involved for the two biggest problems (around 30-60 minutes for Prostate and Leukemia datasets), which is the price for the full Bayesian inference. Recommendations Based on the theoretical considerations and the experimental results, instead of the original horseshoe, we recommend using the regularized horseshoe with a weakly informative prior on c, such as (2.11) with appropriately chosen scale s and degrees of freedom ν. As illustrated in Section 4, this does not compromise the predictive accuracy but greatly improves the sampling robustness of the prior and is resistant to problems originating from weak identifiability of the parameters. Still, it is worthwhile to compare the results to a model with a relatively loose prior on c using standard model assessment techniques (Vehtari and Ojanen, 2012;Vehtari, Gelman and Gabry, 2017) to get an indication if the slab width was chosen to be too restrictive. However, even if a very large value for c yields better predictive fit, if it produces a large number of divergent transitions, it cannot be recommended because in such cases the inference is likely to have a bias with an unknown magnitude (Betancourt, 2017a). In our experience, for the regularized horseshoe it is often possible to get rid off the divergences by tuning Stan's adaptation routine as explained by Betancourt (2017b). Also, instead of using the simple τ ∼ C + (0, 1), for linear regression we generally recommend a weakly informative default choice τ | σ ∼ C + 0, τ 2 0 , where τ 0 is computed from (3.12) using the prior guess p 0 for the number of relevant variables. For the other generalized linear models (GLMs), an approximately equivalent choice is obtained by replacing σ with a appropriate plug-in value as explained in Section. 3.5. Based on the results on the real world data, this choice seems to perform well unless p 0 is chosen to be much too large. However, the toy example shows that sometimes even better results can be obtained by more informative prior. Discussion This paper has discussed the use of the horseshoe prior for sparse Bayesian generalized linear models. We considered two methodological advances. First, we proposed a generalization of the horseshoe -called the regularized horseshoe -that operates otherwise similarly as the horseshoe but allows specifying the regularization to the coefficients that are far from zero. Second, we introduced a new concept -effective number of nonzero parameters -which is useful for guiding the hyperprior choice for the global shrinkage parameter. The experiments demonstrated the benefit of both of these approaches. The ability to regularize those parameters that are far from zero is useful especially when the parameters are only weakly identified by the data. As an example we discussed the logistic regression with data separation. Adding a small regularization to the largest coefficients ensures that the posterior mean will exist, leading to more reasonable parameter estimates and faster posterior exploration. The regularized horseshoe solves also the problems with the divergent transitions that have previously been an indication about problems in posterior simulation and possibly biased inference (Betancourt, 2017b). Regarding the hyperprior choice for the global shrinkage parameter, we argued that the previous default choices are often dubious based on their tendency to favor solutions with too many parameters unshrunk. The experiments show that for many datasets, one can obtain clear improvements -in terms of better parameter estimates, prediction accuracy and faster computation -by coming up even with a crude guess for the number of relevant variables and transforming this knowledge into a prior for τ using our proposed framework. Based on our results, for a reasonably selected global hyperprior, the (regularized) horseshoe outperforms Lasso in terms of predictive accuracy. A notable difference is that Lasso produces a truly sparse solution with exact zeros for some coefficients, whereas horseshoe does not. After fitting the full model, a truly sparse solution without losing predictive accuracy could be obtained using the projective variable selection (Piironen and Vehtari, 2017b) 4 . Although these advances improve the overall performance and applicability of the horseshoe prior to various problems, some challenges still persist. When using the horseshoe prior with correlating predictors, the major concern is always the multimodality of the posterior, which can lead to difficulties in sampling and especially to slow convergence of the MCMC. The experiments indicated that the regularized horseshoe can improve the sampling robustness but does not remove the multimodality. It must be noted though, that the multimodality is a direct consequence of the sparsifying prior assumption that favors solutions where only one in a group of correlating predictors would have a nonzero coefficient. Whether this assumption is reasonable in practice is a more fundamental question. It could make sense -both computationally and theoretically -to employ a horseshoe or other sparsifying prior on some transformed set of features -for instance principal components of the original predictors -instead of the original variables themselves. This would yield a unimodal posterior but the effects on the other aspects such as the predictive accuracy remain to be explored. We leave these ideas for future investigation. Appendix A: Derivation of the regularized horseshoe The regularized horseshoe from Section 2.3 can be derived by designing a new prior with shrinkage profile otherwise similar to the horseshoe, but so that instead of favoring κ j = 0 and κ j = 1, the prior favors κ j = b j and κ j = 1. This is achieved by defining the shrinkage factor for the new prior as κ j = (1 − b j )κ j + b j , (A.1) where κ j is the shrinkage factor for the original horseshoe. Nowκ j → 1 when κ j → 1, butκ j → b j when κ j → 0, so we shift the shrinkage profile of the horseshoe from interval (0, 1) to (b j , 1). By plugging in the expression for the shrinkage factor κ j = 1/(1+a 2 j λ 2 j ) where a 2 j = nσ −2 τ 2 s 2 j , with a straightforward manipulation we getκ j = 1 + b j a 2 j λ 2 j 1 + a 2 j λ 2 j . (A.2) We want to write this in the formκ j = 1/(1 + a 2 jλ 2 j ). Solvingλ 2 j from this equation yieldsλ 2 j = (1 − b j )λ 2 j 1 + b j a 2 j λ 2 j . (A.3) For convenience, we require that b j = 1 1+nσ −2 s 2 j c 2 which corresponds to shrinkage by Gaussian with variance c 2 . By plugging this into (A.3), after a few lines of straightforward algebra we are left with λ 2 j = c 2 λ 2 j σ 2 ns 2 j + c 2 + τ 2 λ 2 j . (A.4) Thus we have shown that the prior β j | λ j , τ ∼ N 0, τ 2λ2 j , λ j ∼ C + (0, 1), (A.5) withλ 2 j defined by (A.4) has the shrinkage profile of the horseshoe shifted from the interval (0, 1) to (b j , 1), where b j = 1 1+nσ −2 s 2 j c 2 . As discussed in Section 2.3, the term σ 2 ns 2 j is typically small compared to c 2 so by leaving this out from (A.5), we get the prior (2.8). Appendix B: Pseudo variance for Non-Gaussian observations As discussed in Section 3.5, assuming y has a distribution in the exponential family with natural parameter θ and dispersion φ, the log likelihood has the form We denote µ = B (θ) and V (µ) = B (θ), from which we can also deduce ∂µ ∂θ = V (µ). The first derivative is given by the chain rule ∂L ∂f = ∂L ∂θ ∂θ ∂µ L = yθ − B(θ) A(φ) − C(y i , φ),∂µ ∂f = y − µ A(φ) 1 V (µ) ∂µ ∂f . The second derivative is then ∂ 2 L ∂f 2 = 1 A(φ) (y − µ) ∂ ∂f 1 V (µ) ∂µ ∂f − 1 V (µ) ∂µ ∂f 2 . Thus the pseudo variance becomes The following shows the Stan code for the linear Gaussian model with the regularized horseshoe prior using a straightforward parametrization. In our experience this code works fine, but in Appendix C.2 we also provide another code using different parametrization (with which we generated our results). This is worth trying if the simple code has issues with sampling (produces divergences). σ 2 = − ∂ 2 L ∂f 2 −1 = A(φ) 1 V (µ) ∂µ ∂f 2 − (y − µ) ∂ ∂f 1 V (µ) In the code, both τ and λ j are given half-t priors with the degrees of freedom and the scale defined by the user (the scale can be adjusted only for τ ). Setting nu local = 1 corresponds to the horseshoe. nu global = 1 gives τ a half-Cauchy prior. The scale for τ is scale global*sigma, so if we want to set this to be τ 0 = p0 D−p0 σ √ n (Eq. (3.12)), we should set scale global = p0 (D−p0) √ n . The code assumes a Student-t slab for regularizing the largest coefficients, and the scale and degrees of freedom can be specified using slab scale and slab df arguments. # latent function values sigma = exp ( logsigma ); c = slab_scale * sqrt ( caux ); lambda_tilde = sqrt ( c^2 * square ( lambda ) ./ ( c^2 + tau^2* square ( lambda )) ); beta = z .* lambda_tilde * tau ; f = beta0 + x * beta ; } model { # half -t priors for lambdas and tau , and inverse -gamma for c^2 z ∼ normal (0 , 1); lambda ∼ student_t ( nu_local , 0 , 1); tau ∼ student_t ( nu_global , 0 , scale_global * sigma ); caux ∼ inv_gamma (0.5* slab_df , 0.5* slab_df ); beta0 ∼ normal (0 , scale_icept ); y ∼ normal (f , sigma ); } The code for the logistic regression model is very similar, we simply remove the lines related to the noise deviation sigma, and change the observation model and the type of the target variable data y. Notice also that now the scale for τ is simply scale global. Thus, to follow our recommendation, we set scale global = τ 0 = p0 D−p0 σ √ n (Eq. (3.12)), by using a suitable plug-in value for σ (Sec. 3.5). The lines that need to be changed (in addition to removing definitions related to sigma) are shown below. C.2. Alternative parametrization This parameterization was proposed by Peltola et al. (2014) (codes at https: //github.com/to-mi/stan-survival-shrinkage). In practice we have not observed problems with the code presented in Appendix C.1 but if it has issues with sampling, it is worth trying the following code (using which we ran our experiments). Below is the code for the Gaussian observation model. # latent function values sigma = exp ( logsigma ); lambda = aux1_local .* sqrt ( aux2_local ); tau = aux1_global * sqrt ( aux2_global ) * scale_global * sigma ; c = slab_scale * sqrt ( caux ); lambda_tilde = sqrt ( c^2 * square ( lambda ) ./ ( c^2 + tau^2* square ( lambda )) ); beta = z .* lambda_tilde * tau ; f = beta0 + x * beta ; } model { # half -t priors for lambdas and tau , and inverse -gamma for c^2 z ∼ normal (0 , 1); aux1_local ∼ normal (0 , 1); aux2_local ∼ inv_gamma (0.5* nu_local , 0.5* nu_local ); aux1_global ∼ normal (0 , 1); aux2_global ∼ inv_gamma (0.5* nu_global , 0.5* nu_global ); caux ∼ inv_gamma (0.5* slab_df , 0.5* slab_df ); beta0 ∼ normal (0 , scale_icept ); y ∼ normal (f , sigma ); } Again, the code for the logistic regression model is very similar. Below are the lines that need to be changed (in addition to removing the definitions related to sigma). Fig 2. Shrinkage profiles as in Figure 1 but now for the regularized horseshoe (2.8) and spikeand-slab (2.7) with a finite slab width c = 1. In comparison to Figure 1, for both priors the first mode is shifted from κ j = 0 to κ j = 1/(1 + nσ −2 s 2 j c 2 ) (for the plots we have selected nσ −2 s 2 j = 10). Fig 4 . 4Toy example: Mean squared error (MSE) between the estimated and the true coefficient vector of length n = 400 on average over 100 different data realizations. The true coefficient vector has p * = 20 elements with a nonzero value equal to A and the rest are zeros. Fig 5 .Fig 6 . 56Toy example: An example data realization y = (y 1 , . . . , yn) (gray crosses), posterior meanβ = (β 1 , . . . ,βn) (black dots) and the true signal β * (red lines) for A = 4 (top row) and A = 6 (bottom row). In both cases the oracle value for τ helps to shrink the zero components in β but also overshrinks the actual signals in the case A = 4. Toy example: Top row: one data realization y (gray crosses), the posterior mean estimatesβ (black dots) and the true signal β * (red lines) when A = 10 and the global parameter is fixed either to τ = p 0 Bottom row: Otherwise the same but now the scale of the observations is changed by multiplying them by 0.1. Fig 7 . 7Separable classification. Top row: draws for the regression coefficients of the two relevant features. Middle row: median and 80%-interval for the coefficients of the irrelevant features. Fig 8. Ovarian dataset : Histograms of prior (light gray) and posterior (dark gray) draws for τ (top row) and m eff (middle row), and absolute values of the posterior means for the regression coefficients |β j | (bottom row) imposed by different prior choices. The first two columns denote the results for the pure and regularized horseshoe with hyperprior τ ∼ C + (0, 1), and the last two columns the same but with τ ∼ C + 0, τ 2 0 , where τ 0 is computed from (3.12) with prior guess p 0 = 3. Fig 9 . 9Ovarian dataset : Histograms of the posterior draws under the regularized horseshoe prior for the three regression coefficients with the largest absolute mean. The histograms highlight the multimodality of the posterior. Fig 10 . 10Microarray datasets : Mean log predictive density (MLPD) on test data (top row), computation time (middle row), and the fraction of divergent transitions after warmup (bottom row) as a function of τ 0 for two hyperpriors: τ ∼ N + 0, τ 2 0 (red), and τ ∼ C + 0, τ 2 0 (yellow). Solid lines denote the pure horseshoe and dashed lines the regularized horseshoe with c 2 ∼ Inv-Gamma(2, 8). The shaded area denotes the values for τ 0 that correspond to sparsity guesses between p 0 = 1 and p 0 = n (Eq. (3.12)). The black dotted line in the top row plots shows the MLPD for Lasso. All the results are averaged over 50 random splits into training and test sets. for some specific functions A(·), B(·) and C(·). From the well- Table 2 2Summary of the real world microarray cancer datasets; number of predictor variables D and dataset size n.Dataset Type D n Ovarian Binary classif. 1536 54 Colon Binary classif. 2000 62 Prostate Binary classif. 5966 102 Leukemia (ALL-AML) Binary classif. 7129 72 guess for the number of relevant variables. For the regularized horseshoe we used hyperprior c 2 ∼ Inv-Gamma(2, 8) (as with the synthetic example, Sec. 4.1.2) which corresponds to a Student-t 4 0, 2 2 slab (see Sec. 2.3). data { int < lower =0 > n ; # number of observations int < lower =0 > d ; # number of predictors vector [ n ] y ; # outputs matrix [n , d ] x ; # inputs real < lower =0 > scale_icept ; # prior std for the intercept real < lower =0 > scale_global ; # scale for the half -t prior for tau real < lower =1 > nu_global ; # degrees of freedom for the half -t prior # for tau real < lower =1 > nu_local ; # degrees of freedom for the half -t priors # for lambdas real < lower =0 > slab_scale ; # slab scale for the regularized horseshoe real < lower =0 > slab_df ; # slab degrees of freedom for the regularized # horseshoe } parameters { real logsigma ; real beta0 ; vector [ d ] z ; real < lower =0 > tau ; # global shrinkage parameter vector < lower =0 >[ d ] lambda ; # local shrinkage parameter real < lower =0 > caux ; } transformed parameters { real < lower =0 > sigma ; # noise std vector < lower =0 >[ d ] lambda_tilde ; # ' truncated ' local shrinkage parameter real < lower =0 > c ; # slab scale vector [ d ] beta ; # regression coefficients vector [ n ] f ; data { ... int < lower =0 , upper =1 > y [ n ]; # outputs ... } transformed parameters { ... tau = aux1_global * sqrt ( aux2_global ) * scale_global ; ... } model { ... y ∼ b er no ul l i_ lo gi t ( f ); } data { int < lower =0 > n ; # number of observations int < lower =0 > d ; # number of predictors vector [ n ] y ; # outputs matrix [n , d ] x ; # inputs real < lower =0 > scale_icept ; # prior std for the intercept real < lower =0 > scale_global ; # scale for the half -t prior for tau real < lower =1 > nu_global ; # degrees of freedom for the half -t prior # for tau real < lower =1 > nu_local ; # degrees of freedom for the half -t priors # for lambdas real < lower =0 > slab_scale ; # slab scale for the regularized horseshoe real < lower =0 > slab_df ; # slab degrees of freedom for the regularized # horseshoe } transformed parameters { real < lower =0 > sigma ; # noise std real < lower =0 > tau ; # global shrinkage parameter vector < lower =0 >[ d ] lambda ; # local shrinkage parameter vector < lower =0 >[ d ] lambda_tilde ; # ' truncated ' local shrinkage parameter real < lower =0 > c ; # slab scale vector [ d ] beta ; # regression coefficients vector [ n ] f ;parameters { real logsigma ; real beta0 ; vector [ d ] z ; real < lower =0 > aux1_global ; real < lower =0 > aux2_global ; vector < lower =0 >[ d ] aux1_local ; vector < lower =0 >[ d ] aux2_local ; real < lower =0 > caux ; } data { ... int < lower =0 , upper =1 > y [ n ]; # outputs ... }transformed parameters { ... tau = aux1_global * sqrt ( aux2_global ) * scale_global ; ... } model { ... y ∼ b er no ul l i_ lo gi t ( f ); } The experiments with the original horseshoe on the real world data are taken from Piironen and Vehtari (2017a) and were run using Stan version 2.12.0, whereas the experiments with the regularized horseshoe are run using newer version 2.15.1 Colon, Prostate and Leukemia: http://featureselection.asu.edu/datasets.php; Ovarian data: request from the first author if needed. Wall time when the four chains are run in parallel using four cores. Code available at https://github.com/stan-dev/projpred. https://github.com/stan-dev/rstanarm AcknowledgmentsWe thank Andrew Gelman, Michael Betancourt and Daniel Simpson for helpful comments about the manuscript. We also acknowledge the computational resources provided by the Aalto Science-IT project.Appendix D: rstanarm codeThe horseshoe prior is implemented in the R-package rstanarm 5 , which contains a lot of precompiled Stan code and provides an easy-to-use interface to the most commonly used regression models. At the time of writing this, the package does not implement the regularized horseshoe but it will be added in the near future.Assuming the predictor matrix and the corresponding targets are loaded into variables x and y, the linear Gaussian model with the horseshoe prior can be fitted as follows:# use all available cores options ( mc . cores = parallel :: detectCores ()) # set up the prior , use hyperprior tau ∼ half -Cauchy (0 , tau0^2) D <-ncol ( x ) n <-nrow ( x ) p0 <-5 # prior guess for the number of relevant variables tau0 <-p0 /( D -p0 ) / sqrt ( n ) # rstanarm will scale this by sigma automatically prior_coeff <-hs ( df =1 , global_df =1 , global_scale = tau0 ) # fit the model fit <-stan_glm ( y ∼ x , family = gaussian () , data = data . frame ( I ( x ) , y ) , prior = prior_coeff )The other generalized linear models can be fitted in a similar manner. 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[ "https://github.com/stan-dev/projpred.", "https://github.com/stan-dev/rstanarm" ]
[ "\"NUDES? SHOULDN'T I CHARGE FOR THESE?\": MOTIVATIONS OF NEW SEXUAL CONTENT CREATORS ON ONLYFANS A PREPRINT", "\"NUDES? SHOULDN'T I CHARGE FOR THESE?\": MOTIVATIONS OF NEW SEXUAL CONTENT CREATORS ON ONLYFANS A PREPRINT" ]
[ "Vaughn Hamilton [email protected] \nMax Planck Institute for Software Systems Ananta Soneji\nMax Planck Institute for Software Systems\nArizona State University\nBoston University\n\n", "Allison Mcdonald \nMax Planck Institute for Software Systems Ananta Soneji\nMax Planck Institute for Software Systems\nArizona State University\nBoston University\n\n", "Elissa M Redmiles [email protected] \nMax Planck Institute for Software Systems Ananta Soneji\nMax Planck Institute for Software Systems\nArizona State University\nBoston University\n\n" ]
[ "Max Planck Institute for Software Systems Ananta Soneji\nMax Planck Institute for Software Systems\nArizona State University\nBoston University\n", "Max Planck Institute for Software Systems Ananta Soneji\nMax Planck Institute for Software Systems\nArizona State University\nBoston University\n", "Max Planck Institute for Software Systems Ananta Soneji\nMax Planck Institute for Software Systems\nArizona State University\nBoston University\n" ]
[]
With over 1.5 million content creators, OnlyFans is one of the fastest growing subscription-based social media platforms. The platform is primarily associated with sexual content. Thus, OnlyFans creators are uniquely positioned at the intersection of professional social media content creation and sex work. While the experiences and motivations of experienced sex workers to adopt OnlyFans have been studied, in this work we seek to understand the motivations of creators who had not previously done sex work. Through a qualitative interview study of 22 U.S.-based OnlyFans creators, we find that beyond the typical motivations for pursuing gig work (e.g., flexibility, autonomy), our participants were motivated by three key factors: (1) societal visibility and mainstream acceptance of OnlyFans; (2) platform design and affordances such as boundary setting with clients, privacy from the public, and content archives; and (3) the pandemic, as OnlyFans provided an enormous opportunity to overcome lockdown-related issues.
10.1145/3544548.3580730
[ "https://export.arxiv.org/pdf/2205.10425v2.pdf" ]
252,762,459
2205.10425
349a06d2de00b76eefdd870f5563a13e9291769b
"NUDES? SHOULDN'T I CHARGE FOR THESE?": MOTIVATIONS OF NEW SEXUAL CONTENT CREATORS ON ONLYFANS A PREPRINT 7 Oct 2022 October 10, 2022 Vaughn Hamilton [email protected] Max Planck Institute for Software Systems Ananta Soneji Max Planck Institute for Software Systems Arizona State University Boston University Allison Mcdonald Max Planck Institute for Software Systems Ananta Soneji Max Planck Institute for Software Systems Arizona State University Boston University Elissa M Redmiles [email protected] Max Planck Institute for Software Systems Ananta Soneji Max Planck Institute for Software Systems Arizona State University Boston University "NUDES? SHOULDN'T I CHARGE FOR THESE?": MOTIVATIONS OF NEW SEXUAL CONTENT CREATORS ON ONLYFANS A PREPRINT 7 Oct 2022 October 10, 2022 With over 1.5 million content creators, OnlyFans is one of the fastest growing subscription-based social media platforms. The platform is primarily associated with sexual content. Thus, OnlyFans creators are uniquely positioned at the intersection of professional social media content creation and sex work. While the experiences and motivations of experienced sex workers to adopt OnlyFans have been studied, in this work we seek to understand the motivations of creators who had not previously done sex work. Through a qualitative interview study of 22 U.S.-based OnlyFans creators, we find that beyond the typical motivations for pursuing gig work (e.g., flexibility, autonomy), our participants were motivated by three key factors: (1) societal visibility and mainstream acceptance of OnlyFans; (2) platform design and affordances such as boundary setting with clients, privacy from the public, and content archives; and (3) the pandemic, as OnlyFans provided an enormous opportunity to overcome lockdown-related issues. Introduction Social media platforms are innovating their business models to improve engagement of both viewers and content creators. To engage creators in developing high quality content that will drive viewership, platforms are increasingly facilitating methods for creators to be paid for their work through a variety of different monetization models Kopf [2020]. Some platforms, such as YouTube and TikTok, have created programs to pay creators in proportion to the views on their content, while others like Twitch, Patreon, and OnlyFans leverage a subscriber model in which fans pay creators directly through subscriptions, tips and gifts. The experiences of creators on these platforms have a deep impact on the success of the platform. Thus, to inform platform design and improve labor rights for a growing sector of gig workers-professional content creators-prior work has focused on the motivations of professional YouTube, Twitch, and Patreon creators, as well as the moderation and monetization schemes, harassment experiences, and subscriber motivations and relationships on these platforms, among other topics Wohn et al. [2019], Sheng and Kairam [2020], Johnson and , Taylor [2018], Seering et al. [2018], Törhönen et al. [2019], Regner [2021]. OnlyFans is among the newest subscription-based social media platforms. Launched in 2016, OnlyFans is an online subscription platform which creators use to earn money for their content. Users, also known as fans, subscribe to the "Nudes? Shouldn't I charge for these?": Motivations of New Sexual Content Creators on OnlyFans A PREPRINT content of these creators. OnlyFans creators can receive funds from their fans via monthly subscription fees, tips, paid private messages, and the pay-per-view feature Bonifacio and Wohn [2020], Bernstein [2019], Hall [2018]. OnlyFans is unique from its predecessors in that, while the platform advertises that it hosts a wide variety of content, it is primarily associated with adult content (i.e., sexual photos and videos that are protected by a paywall), rather than game streaming or artistic creation van der Nagel [2021a]. In contrast to other social media platforms (e.g., Twitch, YouTube, Patreon, Facebook), OnlyFans' terms of service are relaxed for adult content creation Bonifacio and Wohn [2020], Hall [2018]. Thus, OnlyFans is uniquely positioned between the spaces of digital sex work and subscriptionbased social media. Further, OnlyFans experienced rapid growth during the COVID-19 pandemic: revenue grew by over 500% Cooban [2021] and is now estimated to equal that of Twitch, despite having fewer subscribers. 1 Economists, psychologists, social scientists and researchers of computer-supported cooperative work have long sought to understand individual worker motivations for pursuing particular forms of labor Kanfer et al. [2017]. Most relevant to our particular study, prior work has examined what motivates professional creators on other social media platforms, people's choice to pursue various forms of gig work during the COVID-19 pandemic, as well as how sex workers adopt and navigate new commercial sexual platforms including OnlyFans Dunn et al. [2020], Cardoso and Scarcelli [2021], Hamilton et al. [Forthcoming], Vallas and Schor [2020], Matikainen [2015], Umar et al. [2021], Cano et al. [2021], Katta et al. [2020], Apouey et al. [2020], Dunn et al. [2020]. However, less is known about what draws people who have never previously worked in the sex industry to begin creating content on platforms like OnlyFans. Further, OnlyFans experienced an abrupt spike in popularity and visibility: after Beyoncé mentioned the site in a remix in April, OnlyFans reported "a 15 percent spike in traffic" in the subsequent 24 hours. 2 OnlyFans creators traverse the intersection of professional content creation-a growing form of gig work-and sex work. As a result, especially for those who were not previously working in the sex industry, joining OnlyFans to create sexual content can be uniquely stigmatizing. Our work aims to understand how those new to the commercial sex industry navigate content creation on OnlyFans and are motivated to overcome potential barriers to chose OnlyFans over other labor options. We find that celebrity hype around OnlyFans created by popular culture news such as Beyoncé mentioning the platform in her 2020 "Savage" Remix and creator Bella Thorne making $1 million in her first day on the platform Sanchez [2022], led the "cultural assimilation" of the platform, as one participant described it. This assimilation reduced the stigma of joining a platform associated with adult content and created significant interest among potential creators about how much money could be made. Further, the very design of the platform itself-which requires creators to advertise on mainstream social media platforms to drive traffic because there is no search or opportunity for creator discovery directly through OnlyFans-drastically raised the visibility of the platform, drawing in new creators. Beyond the hype and reduced stigma created by celebrity notoriety and the carefully curated visibility of the platform, creators were also motivated to join the platform by typical gig work motivations (e.g., money, flexibility, accessibility) and, similar to other fully-digital gig work, safety during the COVID-19 pandemic. However in many cases our participants considered creating on OnlyFans preferable to other gig work because of the platform's affordances: a lack of rating of creators, ease of blocking harassing subscribers, and autonomy over what content to create and how to construct their business model. Other factors that motivated creators were a desire to engage in sexual expression, the utility of the platform for curating their existing (non-commercial) sexual content, and the opportunity to leverage existing digital audiences or skills from other recreation or labor (e.g., cosplay, waitressing) in which they were already engaged. Finally, the pandemic resulted in both increased subscriber-side demand for digital entertainment and increased creatorside need for paid work due to job loss. We highlight how our participants intersect with and diverge from other workers in the spaces of professional content creation, the gig economy, and sex work. Related Work Here, we review prior literature on opportunities, risks, and challenges in digital content creation and gig work, as well as OnlyFans in particular. Content Creators Prior work has studied and investigated people's motivations to recreationally and professionally create digital content. Prior work finds that recreational creators have a variety of motivations, including a desire for self-expression, "Nudes? Shouldn't I charge for these?": Motivations of New Sexual Content Creators on OnlyFans A PREPRINT recognition, to promote ideas, to help or inspire others and/or to build community Brake [2014], Craig [2019], Omar and Dequan [2020], Markman [2012], Nardi et al. [2004], Leung [2009], Kopf [2020], Törhönen et al. [2019], Noonan [2018], Weber et al. [2021]. Creators may become professional creators -that is, monetize their creation, by happenstance or more intentionally, driven by entrepreneurial spirit, a desire to generate their own "media brands", to influence others, or to become famous Giardino [2021], Matikainen [2015], Freberg et al. [2011], De Veirman et al. [2020, Noonan [2018]. Prior work on motivations behind online video content creation on services such as YouTube and Twitch suggests that enjoyment and socialisation are more significant drivers for content creation than income and reputation Törhönen et al. [2019]. In contrast to platforms such as YouTube, creators on OnlyFans do not have the affordances to attract fans from the platform itself van der Nagel [2021b], Safaee [2021]. In order to succeed, creators must promote themselves and their content on other social media platforms Bonifacio et al. [2021]. Hence, OnlyFans creators fall in the category of influencers (internet micro-celebrities) as defined by prior work since they manage and engage with relatively large followings on social media platforms for self-branding and recognition Khamis et al. [2017], van der Nagel [2021b]. In this work, we explore how the motivations of OnlyFans creators intersect with and diverge from those of other content creators. Gig Workers The online "gig economy" is defined as a labor market where independent contracting happens through, via, and on digital platforms Woodcock and Graham [2019]. A digital gig economy platform is one that offers tools to bring together the supply and demand for labor Graham and Woodcock [2018]. The work performed on gig economy platforms is non-permanent, and lacks job security Woodcock and Graham [2019]. While gig work is often associated in the digital context with ridesharing and crowdwork, a far broader collection of workers engage in gig work. For example, scholars have shown that those working in the porn industry Safaee [2021], van der Nagel [2021a], Easterbrook-Smith [2022], Jones [2020], Butler [2020], professional content creators more generally Vallas and Schor [2020] and care workers who find employment via online marketplaces Hunt et al. [2019], Kasliwal [2020] are all gig workers. According to the 2021 Pew Research Center survey, 16% of Americans have engaged in and earned money through online gig work Center [2021-12-08]. Several studies have analyzed the factors that influence the growing adoption of gig economy platforms, finding that this work offers greater autonomy and flexibility to clients and workers compared to traditional formal labour Woodcock and Graham [2019], Wood et al. [2019], Graham et al. [2017], Lehdonvirta [2018]. Further, gig workers may be able to access more clients and thus greater income through platforms that connect them with clients from diverse industries and in some cases allow them to take on increasingly complex tasks as they gain more experience Wood et al. [2019], Graham et al. [2017]. Prior work finds differences in the motivations of part-vs. full-time workers, such as ride-hailing drivers: Rosenblat found that part-time ridehail drivers are mostly motivated by the flexibility of work offered by the platforms, whereas full-time drivers were motivated by having similar previous experience or lack of other job opportunities Rosenblat [2016]. Gig work platforms have low entry barriers along with flexible work hours and locations Rosenblat and Stark [2016]. As a result they may be more accessible to marginalized communities. Prior work has shown that gig work may offer greater ease of entry for disabled people and may provide necessary benefits such as greater control on when and how tasks are performed compared to traditional employment Harpur and Blanck [2020], Sannon [2021], Ali et al. [2011]. Gig work may reduce/avoid the need to disclose non-task-relevant disabilities, thereby reducing potential stigma and bias Harpur and Blanck [2020]. Across American gig work platforms, women gig workers make up approximately half of the workforce Stage [2022]. Prior work has focused on understanding platform engagement of women in the gig economy and exploring their challenges to economic opportunities and work-life balance Hunt et al. [2019], Raval and Pal [2019], Chaudhary [2020]. Through surveys and focus group discussions, Chaudhary finds that women are attracted to gig work because of flexible hours and easy management of other commitments, more incomegenerating potential and the opportunity to become potential breadwinners in the family Chaudhary [2020]. Although flexibility is the key motivator among women, they had to make difficult trade-offs between their time-use, income generation and caring roles Hunt et al. [2019]. In a recent study Ma et al. [2022], it was found that gig work platforms are gender-agnostic towards women's experiences and values and leave women vulnerable to bias and harassment by not enforcing anti-harassment policies in their design. This results in marginalization of workers as platforms discard the various social contexts that these workers are coming from Gray and Suri [2019]. While gig work platforms offer some preferential working conditions over fixed labour jobs, prior work also finds that gig workers suffer from low pay, crowded marketplaces, discrimination, employment insecurity, social isolation, overwork, financial precarity, and stressful and dangerous working conditions Graham and Woodcock [2018], Graham et al. [2017], Rosenblat and Stark [2016], Donovan et al. [2016], Seetharaman et al. [2021], Stewart and Stanford [2017]. Gig workers do not enjoy the benefits and protections provided by labor and employment laws as well as other bene-"Nudes? Shouldn't I charge for these?": Motivations of New Sexual Content Creators on OnlyFans A PREPRINT fits such as paid sick leave, health insurance and pensions Donovan et al. [2016], Doucette and Bradford [2019] and are forced to take on additional costs such as damage to tools, injury costs, and unpaid gaps between between paid gigs Zwick [2018], Vallas and Schor [2020]. While gig workers experience increased flexibility by using gig work platforms, workers are dependent on those platforms for payments and jobs. This creates significant vulnerability as the risk of losing platform access (i.e., being deplatformed) is ever present Vallas and Schor [2020], Pangrazio et al. [2021]. Further, prior work Rivera and Lee [2021] suggests through qualitative surveys and semi-structured interviews that gig economy platforms lack potential for career development. As a result, gig workers must transition out of those platforms to achieve their long-term career goals. Finally, recent works have also studied the impact of COVID-19 on the gig economy, finding increases in opportunities for fully-digital gig work, decreases in the number of in-person gig workers, and changes on the part of in-person gig work platforms, including both positive (employment benefits, including limited paid sick leave) and negative changes (restricting the flexibility and freedom of workers) Umar et al. [2021], Cano et al. [2021], Katta et al. [2020], Apouey et al. [2020], Dunn et al. [2020]. OnlyFans OnlyFans creators are independent contractors and earn their money through the platform's business model of subscription and tips Safaee [2021], much like other gig economy workers. In this work, we highlight similarities and differences between the motivations of those joining more typically studied gig work platforms such as those aforementioned and OnlyFans creators. Further, many of our participants had previously participated in other forms of gig work prior to joining OnlyFans and we report on their perspectives on the differences and similarities in their labor experiences. Between the stigma of studying sex work and OnlyFans being a relatively new phenomenon, the academic research on this platform is still in its infancy, despite an enormous body of media coverage regarding creators, fans, changes in OnlyFans terms and conditions, and celebrities on the platform Bernstein [2019-02-09], Shane [2021-05-18]. 3 As a result, much existing academic work on OnlyFans is presented in student capstone projects and theses rather than peer-reviewed papers. Offering empirical work on OnlyFans users broadly-including both creators and fans-Arañez Litam et al. recruited MTurk workers as well as university students who are OnlyFans users and compared their attitudes toward sex with those who were not on the platform, finding that they were similar in terms of attitudes towards sex Litam et al. [2022]. They additionally studied the demographics of OnlyFans users in their sample, but do not distinguish between fans and creators; they report that most of their participants were heterosexual, white, married men, consistent with prior work on the consumers of digital sexual content Rissel et al. [2017], Regnerus et al. [2016], Attwood [2005]. We answer their call to conduct qualitative research with Onlyfans users, concentrating specifically on creators new to content creation. As they propose, we directly examine the experiences of creating on the platform. Additional empirical prior work focuses on particular sub-populations of OnlyFans creators. Cardoso and Scarcelli carried out a qualitative research study with 20 creators, all young, white, Italian, cis women Cardoso and Scarcelli [2021]. They contextualise OnlyFans by explaining the rise in popularity of amateur and gonzo 4 porn genres. The study looked at the somatic experience of content creation and analysed the results by examining the kinds of corporeal production enacted by creators. Cardoso and Scarcelli begin to identify different categories of creators, distinguishing between creators who consider themselves full-time professionals vs. those who view their work more casually. Ryan studies male sex workers, finding that they use OnlyFans as an extension of their work as micro-celebrities on Instagram in their gay communities Ryan [2019]. Hamilton et al. investigated sex workers who pivoted from in-person to online work during the pandemic Hamilton et al. [Forthcoming], finding both new benefits and harms from this new modality. Our participants therefore form an additional hitherto unstudied population. Additionally, three smaller scale projects published as posters or theses empirically study OnlyFans creators. Ebersole interviewed eight undergraduate women who are OnlyFans creators, finding that they were working on OnlyFans for additional income; that some workers used the platform to express their sexuality; and that significant labour was also carried out off-platform and that creators connected in order to support their work, themselves and each other Ebersole [2022]. Uttarapong et al. also focused on this community aspect, interviewing 15 creators to find out how they used their personal networks for business and also for support Uttarapong et al. [2022]. Our work includes creators with vastly bigger audiences than their largest, who although not being celebrities, still utilized the intra-"Nudes? Shouldn't I charge for these?": Motivations of New Sexual Content Creators on OnlyFans A PREPRINT community practices described in their paper (among others) to great effect. Finally, Dominguez' thesis, a qualitative study (n=6) also studied creators' support networks Barroso Domínguez [2022]. In theoretical analyses, multiple scholars note how neoliberal discourses 5 set the scene for the rise of the self-employed porn performers on digital platforms and factor into the popularity of OnlyFans as a gig-work platform Easterbrook-Smith [2022], Ryan [2019]. They conclude that digital sex workers are more precarious than other gig workers because of the additional sex work stigma, related regulations (e.g., FOSTA-SESTA), and resulting structural implications (e.g., restricted banking access). Lastly, prior work suggests that other sex workers build communities online for activism, networking, peersupport, and sharing health and safety information Sanders et al. [2018], , Bernier et al. [2021], , Barakat and Redmiles [2022], Strohmayer et al. [2019]. Outside of the specific study of OnlyFans, there exists a large scholarship focused on many kinds of sex work including pornography and sex work that is digitally mediated. Whilst a full examination of all of these areas would be impossible within the scope of this paper, we highlight the findings of a subset of these works that are particularly relevant to our context. Much of the scholarship regarding online sex work involves the study of web-camming, which takes the form of one-to-one or one-to-many live-streaming shows that may involve stripping or sexual activities. Jones undertook a large-scale study of webcam workers Jones [2020], including particularly investigating racism Jones [2015] in the industry and the experiences of transmasculine Jones [2021] and fat Jones [2019] cam performers. Jones' work clearly shows that intersectionality affects both experience and earnings in digital sex work. Examining in-person work, Berg interviews more than 80 porn workers and, in analyzing these interviews through a Marxist-feminist labor-studies lens, finds her participants are undertaking "pleasurable resistance" to other kinds of labor Berg [2021]. Perhaps in an effort to capitalize on this sentiment, in analyzing the terms and conditions of webcam sites, Stegeman concluded that webcamming sites attempt to undermine the labor rights of performers by framing camming as "not-work" Stegeman [2021]. This dependence on both sex work (e.g., webcamming) platforms and on other platforms to engage in labor makes deplatforming (having an account banned or removed) a particularly onerous risk for digitally-mediated sex work. Prior work on digital marginalization of sex workers shows that platforms of all kinds regularly deplatform sex workers, taking away opportunities not only for business but for social participation and peer-support by censoring conversations around sex work and even sex education Barakat and Redmiles [2022], , Blunt and Wolf [2020]. Blunt et al. discuss how online sex-work platforms constantly update their algorithms and terms of services to sanitize online space, impacting sexual expression while simultaneously monetizing the data and content of online sex workers Blunt and Stardust [2021]. Finally, more broadly examining "sexual entrepreneurship," Rand describes digital sex work as absent from literature on digital labor Rand [2019] and argues for the inclusion of sex workers in academic and other conversations on gig work and digital labor. We answer this call and build on the above-mentioned prior work-focused on provision, benefits, and challenges of digitally-mediated sex work-by exploring the motivations of adults who do not have prior sex-work experience to create and monetize adult content as a form of gig work on a relatively mainstream subscription-based social media platform: OnlyFans. Methodology To understand the experiences and motivations of those new to professional sexual content creation to start a subscription service on OnlyFans, we conducted 22 semi-structured interviews with current and past OnlyFans creators in the United States during September and October 2021 during the global COVID-19 pandemic. Participant Recruitment We sought to recruit a broad range of U.S.-based content creators on OnlyFans. We intentionally avoided references to sexual content in our recruitment materials (see https://osf.io/ws54t/?view_only=3f553c40cfd940728f8e6329cd9a2795 for anonymized recruitment graphic) and interview protocols, in order to be inclusive of participants making any type of content. Despite this, all of our 22 participants did create sexual content on OnlyFans. We recruited our participants online by posting on social media and sending out recruitment flyers to the researchers' contacts to share in their networks Biernacki and Waldorf [1981], and in-person by posting flyers around several university campuses and cities (at coffee shops and grocery stores) in different parts of the U.S. "Nudes? Shouldn't I charge for these?": Motivations of New Sexual Content Creators on OnlyFans A PREPRINT The recruitment materials linked to a Qualtrics screening survey. OnlyFans does not publish demographic data about their creators Litam et al. [2022], and our goal in sampling was not to interview a representative sample of creators; rather we sought to hear from a diverse set of creators, including and especially from those with identities that might impact their experience of the platform. To do this, we intentionally set quotas for combinations of race, gender and age to the sign-up process in Qualtrics in an attempt to ensure that we recruited a diverse range of participants. We also used the survey to screen out those who had previously created content on OnlyFans but had stopped creating such content more than 18 months ago (as these creators would be less familiar with the current platform affordances and updates), and those who had previously engaged in sex work, as the focus of our work was to understand the motivations and experiences of those creators who were new to sex industry work (if that was the type of content they were creating on OnlyFans) as they may face unique barriers and challenges in their labor (see Section 2.3 for a summary of prior work on the experiences of those who have prior sex work experiences on OnlyFans). Qualified participants were redirected from the screening survey to anonymously self-schedule for an interview using a Calendly link. Participant Demographics. All of our participants were 18 years old or older, with median age of 29.5 years (σ = 7.01). Participants were invited to self-describe their gender and race/ethnicity in our screening survey. As such, our participants self reported their genders as: woman (12), man (3), non-binary (8), and three participants self-described as trans. 6 Participants who reported their race/ethnicity reported as: Black (3), Asian (2), Asian and white (1), Hispanic and white (2), and white alone (11). 9 out of 22 of our participants reported that they had a disability. Our participants were also asked their education status in the screening survey, reporting their highest level of education as a(n): Advanced degree (2), Bachelor's degree (4), Associate's degree (3), some college coursework but no degree (8), or a High School diploma (2). 10 of our participants reported in the survey that they had done gig work previously, on food delivery platforms (UberEats, PostMates, GrubHub, DoorDash), service provision websites (NextDoor, Craigslist, Facebook Marketplace), or ride-hailing platforms (Uber and Lyft). 6 participants had done other paid digital work such as tutoring, marketing, selling art, or writing reviews. Lastly, three of our participants described during the interview that they had participated in some kind of sex work prior: we had attempted to only interview those without such experience but can account for this by understanding that these participants could have either mis-read the screening survey or not perceived their work (stripping, phone sex, professional BDSM work respectively) as sex work. Our participants had spent a median of 15.5 months using OnlyFans; three of our participants had stopped using OnlyFans less than 6 months prior to the interview. Our participants reflect a wide range of income and subscriber numbers: small creators and creators with enormous reach. We asked the total income earned to date and found (in U.S. dollars) the minimum was $135 and the maximum $332, 000 with a median of $1, 237.84 (σ = $83, 974.24). The minimum number of subscribers reported (at the time of the survey) was 10 and the maximum 13, 000 with a median of 60 (σ = 4, 143.13). For seven of our participants, OnlyFans was their primary source of income. Lastly, our participants also reported their popularity ranking on the OnlyFans platform (a statistic provided by the platform); they provided the rank of their highest page if they had multiple pages. The most popular among our participants was in the top 0.03% of creators and least popular was in top 76% with a median popularity of 8.7% (σ = 24.82%). Interview Data Collection We used semi-structured interviewing methodology Bernard and Bernard [2013]. Interviewers first explained the goals of the study and answered any questions that the participants had before requesting to record the interview. At the start of the interview, participants verbally gave their consent to be interviewed and recorded and interviewers reconfirmed with each interviewee the screening survey information that was filled out by the participant. Participants were then asked background questions to understand the kind of work they did before joining OnlyFans. We then asked questions related to their experiences in those prior jobs and asked our participants to make comparisons with their experiences on OnlyFans (e.g., how do you find OnlyFans different or similar to these prior/current experiences?). We asked our participants questions about their initial exposure to OnlyFans and their motivations to start creating, publishing, and promoting content. The interview then queried what kinds of content the creators were making on OnlyFans, and the kinds of skills or interests they had to do this work. Finally, we asked our participants' platformrelated questions, which are out of scope for this paper. Participants were also invited to not answer any question that they did not want to answer. The relevant interview questions can be found at the following anonymous repository: https://osf.io/ws54t/?view_only=3f553c40cfd940728f8e6329cd9a2795. "Nudes? Shouldn't I charge for these?": Motivations of New Sexual Content Creators on OnlyFans A PREPRINT All the interviews were conducted in English via chat, voice, or video. Each interview lasted between 37 and 96 minutes, with an average interview length of 60.5 minutes. Participants were compensated with $50 via Amazon gift card or PayPal. All the interviews were then transcribed and any incidental identifying information was removed. Data Analysis We analyzed the interview data using an iterative open-coding process Corbin and Strauss [2014]. One author randomly selected five interview transcripts to identify common themes and create a thematic framework for the interview data. After creating the codebook, two of the authors coded another five transcripts. The coders achieved a Cohen Kappa of 0.75, which is considered a substantial inter-coder reliability score Landis and Koch [1977]. Subsequently, both the coders resolved any minor disagreements and refined the codebook, which was then used by one of the authors to code all the remaining transcripts. Given the qualitative nature of our results we report our findings primarily qualitatively, sparingly providing counts of overall themes (i.e., how many participants reported each motivation) only to highlight the prevalence of patterns in the interview data and provide transparency into data analysis and theme emergence. Moreover, we want to highlight that no inferences should be made about the prevalence of themes observed beyond the sample given the qualitative nature of our sample McDonald et al. [2019]. Ethics, Impacts & Research Justice Our work was approved by our institution's ethics review board. Since our participants are undertaking stigmatized work, we took extensive measures to protect participant data, anonymity, and privacy at every stage. Some examples include using platforms that are end-to-end encrypted for conducting virtual interviews (e.g., paid Webex), offering instructions on how to create an encrypted (ProtonMail) email address to organize an interview and receive reminders, and using a scheduling system (Calendly) that did not require personal information such as name from participants for scheduling. We offered participants multiple options for compensation: anonymous Amazon gift cards or PayPal payments (which could be sent to an encrypted email address they had created using our instructions). We took great care to protect the identities of participants and any unintended identifying information has been removed. In alignment with research justice principles, we employed a sex worker to transcribe the interviews and will send a copy of published research using this data to those participants who requested it MacNeil and Fernandez [2006], Bhalerao et al. [2022]. Lastly, as detailed further below in Section 3.5, we consulted OnlyFans creators on our research design and interview protocol and they voluntarily circulated the study recruitment materials in the networks of creators that they were members of; these creators were compensated for this labor. Broader Perspectives. Our hope for the positive impacts of this work is that adult content creators are taken seriously and considered part of the gig economy workforce. There have already been efforts to use research on gig workers in attempts to improve their working conditions and regulate employers and platforms in this area to their advantage (see the many papers in Section 2 for research on gig workers). However, the stigma of sex industry work often prevents adult content creators from being included in these efforts, which leaves them behind in terms of workers rights Butler [2020]. Sex worker and gig worker rights groups can use our research in conversation with policy makers to improve the working conditions for all content creators. We will provide plain-English, 1-page summaries of the results of this research project to be disseminated widely among the population studied. OnlyFans creators are already in community and overlap with sex worker communities, with whom we already have established relationships. Potential harms from the publication of this work include use of any negative experiences reported in our paper by anti-porn and anti-sex-work lobbying groups to advocate for OnlyFans banning adult content, as there has already been pressure to do van der Nagel [2021a]. Positionality The researchers are scholars of technology and sex work. We consulted with OnlyFans creators on the research design, interview protocol, and recruitment. However, those creators were not directly involved with the data analysis or writing of this paper. Additionally, there were participants with genders and ethnicities who are not reflected in the research or interview team, notably male and Black creators, since we are white/brown and American/british 7 /Indian, which may limit our capacity to fully understand and analyze the nuance of that data. "Nudes? Shouldn't I charge for these?": Motivations of New Sexual Content Creators on OnlyFans A PREPRINT Limitations Our study has several limitations. First, we conducted interviews only in English and only of U.S.-based creators, which limits participation to U.S. English-speakers. Future work should explore the motivations and experiences of OnlyFans creators in other geographies and contexts. Second, while we aimed to recruit only OnlyFans creators who had not done other forms of sex work previously, and used our screening questionnaire to screen for these participants, three of our participants had previously done sex work (see Participant Demographics above). Third, our interviews ask participants about their experiences conducting business in a competitive market. Thus, it is possible participants omitted relevant information about their business strategies. Results Overall, we find that our participants were motivated to join OnlyFans due to: 1) the visibility and acceptance of On-lyFans in society, which was created through a combination of celebrity hype, platform design, and peer conversation; 2) the potential earnings on OnlyFans, which they perceived as a (better) alternative to other forms of gig and service work; 3) a desire to engage in digital sexual expression; 4) already having existing content, audiences or skills that allowed them to quickly build a profitable OnlyFans page; and 5) pandemic factors such as increased flexible time, increased safety concerns about other forms of gig work, and loss of other forms of income due to economic impacts. Visible & Accepted: OnlyFans in Society OnlyFans gained significant visibility and mainstream acceptance, particularly starting in 2020. This came as a result of: 1) celebrity participation and conversation about OnlyFans, 2) peer conversation about the platform as a result of reduced stigma from (1), and 3) high visibility of the platform from both (1), (2) and OnlyFans' distinct design: it offers very limited creator discovery on the platform, pushing creators to promote their content and the platform itself across mainstream social media platforms. All 22 participants discussed this visibility or acceptance in some way. Our participants felt comfortable becoming OnlyFans content creators as a result of the societal pervasiveness of OnlyFans combined with broad participation from celebrities and peers, which decreased the stigma of creating on OnlyFans. For example, P11 explained that COVID-19 and the positioning of OnlyFans as "part of the gig economy" made people more comfortable with consuming and creating adult content through the platform. Thus, "every day regular people are associated with" OnlyFans, which made it feel like the platform had a "personalized feeling to it of like this isn't for porn, this is [a platform] for someone I just happen to know [to create adult content]." As a result, P11 explained, there is "less of a whorephobic 8 stigma" around creating content on OnlyFans. Because of the growth of OnlyFans and lessened stigma around the platform, P5 remarked that "in the future I think that there will be a lot more acceptance of sex work" in general, not just as part of OnlyFans. Celebrity Hype. Celebrities announcing on their social media and in the news that they were starting OnlyFans accounts also helped bring attention to the platform. Cardi B joined in August 2020 9 after, as aforementioned, Beyoncé mentioned the site in a remix in April causing, according to a spokesperson for OnlyFans, "a 15 percent spike in traffic" in the subsequent 24 hours. 10 It is important to note that clip sites (websites where adult performers can upload video content for clients to purchase) have existed for a long time, as have sites with other features that OnlyFans has, such as live streaming. However, OnlyFans' specific brand notoriety was important for our participants; 12 of our participants described OnlyFans as "compelling enough" to try out because of the promising, lucrative self-employment enabled by the platform. For example, P2 described OnlyFans as "becoming a mainstream term and semi-accepted (more so than other similar sites)" and P8 told us that the mention of OnlyFans by "internet celebrities...was definitely how it caught my attention first." However, it was not simply finding out about OnlyFans through celebrity mentions, but also imagining why celebrities joined the platform that inspired our participants: P8 continued "I thought it was cool, I liked the idea of these mostly women... reforming sexuality in their own way and being able to claim, reclaim something." Interestingly our participants P5, P10, and P21, were also motivated by the monetary hype created by celebrity earnings on the platform. The high earnings of several specific celebrities were referenced by our participants as a motivating factor to join. P5 discussed Bella Thorne joining OnlyFans, remarking "she's already fairly famous, but then she did "Nudes? Shouldn't I charge for these?": Motivations of New Sexual Content Creators on OnlyFans A PREPRINT one day of OnlyFans and she made one million dollars...that was pretty persuasive." Similarly, P10 "was swayed to do it [OnlyFans] after the Bella Thorne incident." Moreover, four of our participants (P1, P2, P10, and P15) read articles about how non-celebrities had increased their income by changing from regular jobs to OnlyFans work: "there's a nurse that started OnlyFans after the pandemic started and started making more money than being a nurse." (P1). P2 had "seen a couple posts about people being able to buy a house from OF earnings." Overall the celebrity allure, with the promise of money, was a common theme. P17 explains how these threads combined: "at first I was hearing that it was lucrative...[and then] I kept hearing about [OnlyFans everywhere]... [I] wanted to see what the hype is about." Peer Suggestion. Perhaps as a result of the reduced stigma created by celebrity discussion of and participation on OnlyFans, we observed that a significant number (12) of our participants' friends and family also directly discussed OnlyFans with them either as a conversational topic, or because they were creators themselves. P20 explained that they were motivated to join OnlyFans because a friend told them about it, saying "oh you should probably just start OnlyFans and [they] started sending me [clients]." Additionally, these current creators could serve as information bridges, helping new creators onboard. For example, P14 described how her sister encouraged her to join and collaborated with her: "Yeah my sister, she's like you have a great body why don't you just try OnlyFans? I was only going to do it for maybe just a couple of weeks just to help take care of things financially, but it lasted longer than that." P13 was similarly encouraged by the experiences of others: "I also had a friend who started doing it...so I started talking to her about it a little more and asked her how is it and she had nothing but good things to say so I was like okay I'll try it out." Platform Design. Finally, 16 of our participants described finding out about OnlyFans through other social media sites, where they saw friends or influencers posting about it. P8 describes how ubiquitous mentions of OnlyFans became mainstream on non-sexual content platforms: they "found out about [OnlyFans] through Twitter" and "had seen a few comments on it and it was trending at one point." Since OnlyFans does not have internal search functionality nor robust content discovery, creators have no choice but to use external advertising to drive clients to their accounts. In turn, this also amplifies the popularity of OnlyFans to other potential creators. P1 told us that "on every big tweet that blows up, there's usually someone's OnlyFans invite." Out of the 16 participants that described learning about OnlyFans through social media sites, more than half (9) mentioned Twitter. P12, who called the phenomenon of becoming aware of OnlyFans "cultural assimilation," added that "many people who I know and followed on Twitter were talking about and in some cases had OnlyFans accounts." One of the major reasons our participants mentioned Twitter more than any other mainstream social media sites is because Twitter allows consensually produced adult content provided the media within the Tweet is marked as sensitive. 11 OnlyFans as Work 18 of our participants mentioned money as a motivator to create on OnlyFans. While for three (P1, P6, P13) of our participants this was the only reason they created an OnlyFans-as P1 put it "money! Honestly, that's about it"-for the rest earning money was one of multiple reasons for joining the platform as a creator. Along with the known motivations-flexibility, autonomy, control, income-of gig workers to join gig economy platforms Rosenblat [2016], Chaudhary [2020], Wood et al. [2019], our participants perceived OnlyFans as better than other forms of gig work either due to the accessibility of the platform, more control, more income or replacement of lost income, enjoyment, ease of boundary setting, and physical and pandemic safety. It also allowed them to attend to increased responsibilities as carers or parents. Similarities to Other Work. Similar to other gig workers, 15 of our participants sought the autonomy to direct their own working hours on OnlyFans. P12, comparing it to other jobs, described it as "one of the better ones that I've had, just in terms of affording me flexibility." P14 summarized it thus: "I can sign in on my own times, my own hours... the fact that I could schedule my own time, I don't have a set time... with OnlyFans I can just show up." This autonomy was even described as pleasurable for some, with P10 explaining: "OnlyFans allows me to keep a schedule that I enjoy." Beyond the flexibility of being able to choose their own hours, there were other kinds of accessibility advantages that participants described. P4, who had multiple health diagnoses, had delivered food prior to working on OnlyFans; "Nudes? Shouldn't I charge for these?": Motivations of New Sexual Content Creators on OnlyFans A PREPRINT however: "I can still do OnlyFans even if my legs aren't working...because I can just kind of sit." Neurodivergence was also mentioned by two of our participants as a reason why OnlyFans was a more accessible workplace than regular jobs. For example, P21 explained that "due to the nature of my ADHD, my neurochemistry makes it very hard to maintain schedules, routines, consistency." P16 added "I have a lot of trouble working like regular jobs," describing their neurodivergence as "a bit of a barrier" to other kinds of work. Johnson has documented similar accessibility in digital patronage work for disabled creators on Twitch Johnson [2019]. The flexibility to attend to caring responsibilities also made OnlyFans a viable workplace for two (P4, P20) of our participants. Even though P20 had already been working from home, when their mother became ill, they lost that job due to workplace surveillance: "they're watching you, if you leave the computer they know that you left the computer so it was one of those things where I couldn't work and take care of her." Better than Other Gig Work. Whilst most of our participants did not make the kind of money they described hearing about in the discourse that motivated them to join (see Section 4.1), six of our participants (P2, P5, P7, P14, P19, P20) did experience increased earnings compared to other informal or gig work. For example, P2 described "making like 25 dollars for a night of pet-sitting where I had to stay at someone's house vs I can make that amount with 1-2 clip sales." Similarly, P5 felt that work on OnlyFans is "pretty easy... I do it maybe 10 minutes a day, it doesn't feel like a job." Further, OnlyFans creators control the price of the content and services they offer, not the platform. When asked to compare OnlyFans with other gig work, P2 went on to say: "I make more money on OF...And I can control the price more." Three participants also described OnlyFans as being better than other gig work not only in terms of earnings, but in terms of their power in the labor relationship. Prior work Kim et al. [2018], Raval and Dourish [2016] describes the additional emotional labor gig workers must expend in order to receive good reviews, which helps their algorithmic performance. However, on OnlyFans, creators are not reviewed. P2 described that because "there isn't a rating system like in so many app based gigs," on OnlyFans it was "easier to draw boundaries". Even in comparison to face-to-face work, P21 explained that OnlyFans offers more autonomy and thus: "allowed for me to establish my boundaries better, unlike the customer service jobs I've worked before where I can't talk back, I can't cut out a customer because it's up to the manager, whereas [on OnlyFans] I have the control to just block someone." P13 agreed, describing that in food service jobs they had been "hit on relentlessly," and that "the biggest difference" with doing OnlyFans compared to other work "has been respect." Finally, three (P5, P6, P7) of our participants preferred OnlyFans to other forms of gig work or service work because of the literal physical distance it offers from clients. P7 explains: "I definitely make more money on OnlyFans than I did with Uber and Lyft and I don't have to drive around all day and I feel safer because I'm not actually meeting with people. Before, strangers were getting into my car. It was always fine, but now there are no strangers that I'm seeing." Much like those who had done sex work prior to the pandemic Hamilton et al. [Forthcoming], our participants used OnlyFans to avoid face-to-face contact and the risks associated with meeting strangers in person during this time. Unique Challenges of OnlyFans. Although joining OnlyFans was lucrative and promising in the success stories our participants heard, many shared that overall OnlyFans earned them less money (6) and required more effort (17) than they anticipated. Moreover, while 13 of our participants reported exclusively positive experiences with the platform and found the work to be better than other types of gig work, the others shared both positive experiences and challenges. OnlyFans requires significant marketing and promotion (P1, P9, P20), offers no protection from chargebacks (P19), and is less socially acceptable than other gig work (P9, P21). Most notably, because OnlyFans does not match creators to clients like many other gig work platforms or offer internal creator listing or promotion features, creators must work to draw fans from other platforms. This contributes to the mainstreaming of OnlyFans but can also be one of the more challenging parts of being a creator. For P9, "promoting is definitely the most time consuming of all of it... people have to know it's out there in order to buy it." P20 said they find themselves doing OnlyFans work "all day every day... I spend hours hours hours, 17 hours on Telegram the past 24 hours, eight hours on Safari, most of that is OnlyFans, five hours on Twitter. Like it's just, it's obscene." Furthermore, our participants described multiple factors that could impact their success on the platform, including the level of explicitness of their content and their gender/racial identity. Some of our participants expressed a perception that there is a hierarchy of OnlyFans creators based on: the type of content (explicit content creators were perceived to be more successful), the type of fan base (celebrities/famous porn performers perceived to be more successful), the effort expended promoting the content off-platform, and lastly the sociodemographics of the creator. Regarding the latter, our participants specifically suggested that white women or those with an existing audience tend to have an easier time on OnlyFans, whereas fat, disabled and racialized performers as well as straight men might have greater struggles building a following on OnlyFans and making ends meet. P20 accurately summarizes the aforementioned collective observation by mentioning that "intersectionality is a bitch" and "race and body type has a lot to do with "Nudes? Shouldn't I charge for these?": Motivations of New Sexual Content Creators on OnlyFans A PREPRINT it (how much you earn on OnlyFans)". They ended their response by adding that "if you're disabled, if you're fat, if you're poor because being poor affects every stuff too, that affects your content production, that may even affect your ability to even engage your fans." Finally, the popularity of OnlyFans means that much of the sexual content market has moved to the platform. Even participants who want to use platforms other than OnlyFans felt they had no choice if they wanted to be successful. For example, P21 said, "OnlyFans is so big that whether you like it or not, if you do online [sex] work you're gonna have to use that site to be able to live off of your income." P12 shared that they don't like OnlyFans as a company, but stay on the platform because "it is where the audience is." Sexual Expression and Community Prior work suggests that more than 85% of adults create and share sexual content as text messages, images, and videos Stasko and Geller [2015]. Participants in our study were no exception. Prior to joining OnlyFans they were already creating and sharing sexual content; 14 of our participants described a desire to engage in sexual expression as a motivating factor for joining the platform. For these participants, OnlyFans afforded an opportunity to organize and monetize their content in a way that had not been available to them before. P3 described OnlyFans as a kind of sexual content archive, saying "it's a good way to archive all of my stuff and have it in one place." P3 further explained that the "general paywall" on OnlyFans offered both privacy and an opportunity to financially benefit from content creation they already engaged in, saying, "I've always been comfortable with showing myself off so [OnlyFans] was a good outlet for me to do that but also get a little benefit." P6 similarly described the situation of deciding to paywall their content: "I was giving my nudes away for free and I was like why? Shouldn't I charge for these? So that's how I joined." Finally P2, who enjoyed creating pinup content said that OnlyFans offered "a way to finally actually monetize it in a feasible way." OnlyFans also allowed participants to find a community of others who were open in their sexual expression. For example, P16 told us that OnlyFans "was a way for me to explore my sexuality a bit and it seemed like a community that like might be positive for me." P22 went further, saying "I envision a world where people are more comfortable with their bodies and with their own sexual expression, where the taboo of sex has been lifted... there's a part of me that wants to live those values, not just talk about them." Regarding colleagues on OnlyFans, P22 continued; "being in community with other creators has been such a frankly unexpected joy." Re-purposing Existing Content, Audiences & Skills 16 of our participants were drawn to join OnlyFans because they had existing content, audiences or skills that they could leverage to quickly develop a profitable OnlyFans page. As aforementioned, six of our participants were already creating sexual content recreationally. As a result, they had suitable content readily available to monetize and could make money instantly. P8 describes "my first 70 posts that I made were previous and old photos that I had" and that it was possible to "make money off of something that I was already doing." Three of our cosplayer participants (those who dress up, typically as characters from works of fiction) decided to make their OnlyFans content cosplay themed in order to leverage their existing audience. For example, P2 was "low key cosplay famous back in college" and knew that they could exploit this "(horny) fan base" on OnlyFans. P13 had a similar motivation: "I was already doing cosplay and I saw a lot of other cosplayers were having success with [OnlyFans] so I figured I would also try." Intersecting with a desire for sexual expression, P4 decided to use OnlyFans to showcase the "different ideas or things that I had in mind but I couldn't post on other socials" that don't allow "18+ cosplay content." Much like P2, prior to starting their OnlyFans, P4 "had people who showed interest in it [erotic cosplay content]" and therefore felt there could be financial potential of creating this material to sell on OnlyFans. Rouse and Salter note that cosplay was already highly sexualized and that cosplayers use a variety of monetization avenues, including OnlyFans Rouse and Salter [2021]. Another participant, P12, had already written a book about their sex life that had developed their "reputation as someone who had a lot of sex" (P12) and they opened an OnlyFans to give their followers "a look into this thing [their sex life] that they were clearly already interested in." Beyond repurposing existing digital audiences, six of our participants described how their prior job experiences (tutoring-P1, webcamming-P5, burlesque-P7, waitressing-P9, modelling-P10, P18) offered transferable skills for On-lyFans creative work. For example P9, who had been a waitress, explained that "waitressing is somewhat a show... you have to be very customer service friendly and you have to put on a little bit of an act to always be friendly because "Nudes? Shouldn't I charge for these?": Motivations of New Sexual Content Creators on OnlyFans A PREPRINT no one's always friendly 100 per cent of the time" concluding that "I guess working in the food industry and knowing how to please people is similar." P1, who was doing tutoring and OnlyFans at the same time explained: "it's just all kind of tailoring how you present yourself to these people. You know I'm directly pulling tricks that I do tutoring over to OnlyFans... I'm kind of practicing the way I talk on OnlyFans and tutoring and vice versa." Finally, four (P12, P14, P16, P19) of our participants described an intrinsic sense that they would gain pleasure or be good at "doing" OnlyFans. For example, P14 said simply "I knew that I would be good at it" and P16 similarly explained that OnlyFans: "just kind of seemed like something that I would enjoy doing." 4.5 Pandemic 12 participants described the COVID-19 pandemic as a motivator to joining OnlyFans. First, the pandemic decreased the appeal of in-person gig work (much like the decreased appeal of in-person sex work Hamilton et al. [Forthcoming]) due to the risk of catching COVID-19. For example, P7 explained that they had previously done ride-sharing work and had "stopped because of the pandemic. So I was not going to have people in the car. Even now, I won't go back until we are way past this." Similarly P10 said "I really needed work and I wanted to do something where I could work from home and not be exposed to Second, four (P5, P9, P10, P12) participants lost their jobs due to the pandemic and tried OnlyFans whilst waiting for unemployment benefits. P5 explained that "when COVID-19 struck I had no income for a little while before I started getting unemployment." These factors (risk of catching COVID-19, loss of job, and lack of work available) combined powerfully for P9 who described: "I was on unemployment after I lost my job for a little bit and then the benefits were coming to an end... I was a waitress before and I did DoorDash as well but I wasn't ready to go back to being a waitress... no one was going out to eat yet [because] it was still very not safe, so I need to make money somehow." Third, quarantines and lock-downs significantly changed how people spend their time. For example, as nearly all socialization and entertainment moved online, P22 described how "there was a time around the start of the pandemic where especially a lot of gay men were creating private Twitter accounts to share nude photos. [I decided that] I should be making money for this because my body is valuable and if I can support myself by creating content that is really exciting." While some people-marginalized groups in particular-experienced an erosion of free time during the pandemic Kantamneni [2020], on average Americans reported more leisure time of Labor Statistics [2020]. Correspondingly, P17 described having "nothing but free time" and asked "what's better than being at home in my room getting paid to be at home?" On the consumer side, increased time at home, boredom, and loneliness during the pandemic spurred growth in OnlyFans and increased pornography use in general Zattoni et al. [2021], Lau et al. [2021], Grubbs et al. [2022]. P3 summarized this effect, saying, "once the pandemic started and everyone was home [Only-Fans] definitely blew up." Discussion OnlyFans provides many of the same opportunities for creators that other gig work platforms do. For example, flexibility allowed disabled participants and those caring for dependents to build schedules that suited them better than those imposed by regular work. Similar to other content creators, our participants also found OnlyFans to offer a creative outlet for (sexual) self expression. Beyond these factors, our participants were also attracted to join OnlyFans for a unique set of reasons specific to the platform, namely the brand notoriety and market share directly caused by two factors: 1) the celebrity hype and general public discourse surrounding the platform, and 2) the lack of internal discoverability, which forces creators to do external promotion. We posit that these factors also drove an increase in mainstream general acceptance of OnlyFans as a platform and reduced the stigma of participating in it as a creator relative to other forms of sex work such as webcamming and traditional pornography. OnlyFans as "Playbour" As compared to other forms of gig work, the design of the OnlyFans platform has multiple factors that participants found preferable to other forms of gig work. First, whilst there is a black-box ranking system for creators to help convey how popular they are, OnlyFans has no direct rating system for creators. This is a distinction our participants found preferable to other kinds of gig work, which may subject them directly to client ratings and algorithmically dictate their ability to work based on those ratings Kim et al. [2018], Raval and Dourish [2016]. Second, OnlyFans allows for passive income opportunities. There is a hard limit to the number of jobs one may physically complete on Uber, Upwork, TaskRabbit and so on, which in turn limits one's possible income. On OnlyFans, the Motivations of New Sexual Content Creators on OnlyFans A PREPRINT same piece of content can be sold multiple times, offering the potential for greater financial return on the time/energy invested into it by the creator. The financial possibilities of OnlyFans over other types of gig work were a motivating factor for most of our participants despite the fact that many participants did not make the income they hoped. We note that there are several reasons why it is complex to study the actual per-hour return on work on OnlyFans: creators do many different kinds of work for their wage-including networking, promotion off-platform, and content creation itself-and different creators get different returns on their investments as a result of a variety of factors detailed in Section 4.2. However, we interviewed more than one creator who had achieved enough steady content sales to generate substantial income although this had taken enormous amounts of persistent, creative, driven labor over many months. Further research should investigate the effort-to-income ratio-and the role of intersectional marginalization in this financial return-on OnlyFans in order to better support creators. Third, the work itself could be enjoyable on OnlyFans. Whilst workers in the gig economy might find gig work preferable to regular work, finding the work pleasurable is relatively unusual. More than one of our participants described a sense of enjoyment in the work, in addition to those looking to fulfill their sexual expression needs by sharing erotic content. In Porn Work, Heather Berg calls the participation in studio porn "pleasurable resistance" against the kinds of poor labor conditions found in regular jobs Berg [2021]. Whilst some of our participants may not have joined OnlyFans if they had not needed the income, for many their content creation work was a pleasurable resistance against unemployment. Our participants are in many ways similar to other social media influencers Giardino [2021], Matikainen [2015], using sexual content creation as a kind of authentic self-expression and a vehicle for creativity. This is most similar to the concept of "playbour" Törhönen et al. [2019], where enjoyable or pleasurable activities become professionalized. Future research could similarly apply self-determination theory to study OnlyFans creators. OnlyFans also offers an unusual opportunity to monetize pleasurable activities that our participants were already doing: they could quickly re-purpose their existing digital audiences or sexual content that they had initially developed for recreation (e.g., while doing hobbies like cosplay or in romantic relationships). Platform features to protect privacy As aforementioned, a vast majority of adults make and share sexual content Stasko and Geller [2015]. However, a great deal of literature (see e.g., Hasinoff [2015], Stasko and Geller [2015], De Ridder [2019], Geeng et al. [2020]) focuses on the risks of creating and sharing sexual content. While Angela Jones, in her book Camming Jones [2020], describes some of her webcamming participants as fulfilling their exhibitionist natures by working on platforms that offer little-to-no privacy protections, the participants in our study wanted to share in a more boundaried and discreet way. Our participants found a sense of control and autonomy over their content by using OnlyFans to share it. Multiple platform features contributed to the level of comfort our participants had with sharing their intimate content on OnlyFans: affordances for member verification, the ability to moderate subscribers, and the lack of searchability on both the platform and the web. Further, the paywall-based monitization model of OnlyFans, in which fans expect to have to pay to view content vs. many webcamming platforms in which streams can be viewed for free and creators receive tips much like Twitch, created an additional sense of privacy. Platform features to support sexual expression In addition to the risks for creators in sharing their sexual content, there exist significant threats to platforms that host this content. As a result, many platforms chose to ban intimate content from being shared or eventually deplatform those who create and share such content due to pressures from funders or governments Tiidenberg and Van Der Nagel [2020]. OnlyFans requires its creators to identify themselves with both government ID and biometric identification. Content created with another person must be accompanied by the identification and written consent. Thus, OnlyFans' terms and conditions allow certain kinds of sexual content while requiring performer identification to inhibit illegal material. The performer identification to a certain extent immunizes the platform against the claim that there is illegal material on it, whilst the terms and conditions enable content that would likely get the creators deplatformed elsewhere. P21 described deplatforming thus: "everything you worked for is just gone and then you have to rebuild and try and get in contact with customers different ways and yeah. If you don't have a website it's just horrible." Deplatforming is the major income risk to adult content creators. The affordances, business practices, and terms of conditions of OnlyFans therefore combine to enable less personal and structural risk of deplatforming. There are still limitations to OnlyFans support for sexual expression. For example, the platform now bans particular types of content (such as content shot outside) which may restrict sexual expression among some potential users Cole [2021]. Despite these limitations, OnlyFans can serve as a valuable case study for understanding what precisely affords creators an opportunity for sexual expression and control and autonomy over that expression. Building on this work "Nudes? Shouldn't I charge for these?": Motivations of New Sexual Content Creators on OnlyFans A PREPRINT and on existing principles such as those expressed in the Sex Positive Social Media Manifesto Stardust et al. [2022], future work should explore additional avenues for supporting sexual expression on social media. This will help us better understand how to design consensual, safe social media platforms on which people may wish to share intimate content. Converging and diverging needs of creators Our work contributes to an emerging taxonomy of creators on platforms like OnlyFans, which includes professional creators (a mix of those with existing sex-work experience and new creators) who are focused primarily on earning income through OnlyFans and more recreational creators who are leveraging OnlyFans out of a desire to engage in sexual expression. As mentioned in Section 2, Cardoso and Scarcelli also note a distinction between self-identified "amateurs" and creators using OnlyFans in a "professional capacity" Cardoso and Scarcelli [2021]. Prior work finds that a majority of those already in the industry who started online sex work due to the pandemic intend to continue Hamilton et al. [Forthcoming]. Therefore, those formerly or simultaneously doing offline sex work also form a major cohort of OnlyFans creators. Prior work on webcamming also identified exhibitionism as a motivator for undertaking online sexwork Jones [2020]. In our work, 14 participants were motivated to join OnlyFans out of a desire to engage in sexual expression. Whilst 18 of our participants were motivated to join OnlyFans by the opportunity to monetize their content, others may have been willing to use the platform even if they did not earn income from it, because they sought a minimally stigmatized, convenient, discreet platform that permitted them to warehouse and socially share their erotic content. Continued development and refinement of this taxonomy is needed to improve the development of platforms for sexual content sharing as well as relevant policy (e.g., for protecting the rights of content creators). Conclusion Leveraging qualitative investigation through semi-structured interviews we examined the motivations and experiences of 22 U.S.-based content creators who joined OnlyFans without prior experience selling sexual content. Many of the motivations of our participants to create sexual content on OnlyFans overlap with those of other professional content creators (personal interests, self-branding, and recognition) and gig workers (flexibility, autonomy, control, and management of other commitments and responsibilities), thus supporting the need to include sexual content creators as part of research and conversation on digital labor and gig work. Uniquely however, our participants highlight the ability for their work on OnlyFans to be enjoyable-pleasureable labor-because of the opportunity the platform provides for sexual expression as well as the harms the platform aims to prevent through privacy-related affordances (e.g., lack of searchability, paywalls) and omission of many algorithmic management features (e.g., ratings) common to other gig work platforms. Our results also suggest that the celebritization and corresponding increased mainstream acceptance and lessened stigma of the platform as well as income loss and changes in how leisure time was spent created by the pandemic contributed to bringing our participants-and their fans-to the platform. We propose that OnlyFans uniquely profited off the simultaneous events of celebrity hype and the COVID-19 pandemic. It is unclear (and may be difficult to definitively determine) if the platform could have reached such saturation without either factor. OnlyFans has an estimated 2 million subscribers vs. Twitch's 7 million Bruner[2021]. Estimates by Statista (https://www.statista.com/) put both platforms' revenue around 2.5 billion for 2022. 2 https://www.thedailybeast.com/adult-site-onlyfans-experiences-big-beyonce-bump-following-savage-remix We have collected numerous additional media articles about OnlyFans here: https://osf.io/ws54t/?view_only=3f553c40cfd940728f8e632 4 Gonzo is a stylistic and aesthetic genre of pornography characterised by its anti-thesis to prior big-budget studio porn, with POV (point of view) camera work and pseudo-documentary style narrative. See e.g. Porn Studies Issue 3:4, introduced by Biasin and ZeccaBiasin and Zecca [2016]. Neoliberalism is the political and structural phenomena by which socialist policies and governance are systematically dismantled in favour of free-market individualism Bockman[2013]. We report gender in line with guidance from https://www.morgan-klaus.com/gender-guidelines.html. Authors prefer not to capitalize british as a decolonisation practice linda manyguns[2021]. https://www.theguardian.com/commentisfree/2010/jun/23/sex-workers-whorephobia 9 https://www.rollingstone.com/music/music-news/cardi-b-onlyfans-1043254/ 10 https://www.thedailybeast.com/adult-site-onlyfans-experiences-big-beyonce-bump-following-savage-remix https://help.twitter.com/en/rules-and-policies/media-policy "Nudes? Shouldn't I charge for these?": AcknowledgmentThe authors wish to thank Danielle Blunt and Maya Mundell for their contributions to recruitment and wish to thank Oshrat Ayalon, Hanna Barakat, Danielle Blunt, Kayla Booth, Angelica Goetzen, and Maya Mundell for their brainstorming and feedback on the interview protocol used in this work.JakeHall. rewarding good creators": Corporate social media discourse on monetization schemes for content creators. Susanne Kopf, Social Media+ Society. 642056305120969877Susanne Kopf. 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[ "Control-Aware Prediction Objectives for Autonomous Driving", "Control-Aware Prediction Objectives for Autonomous Driving" ]
[ "Rowan Mcallister ", "Blake Wulfe ", "Jean Mercat ", "Logan Ellis ", "Sergey Levine ", "Adrien Gaidon " ]
[]
[]
Autonomous vehicle software is typically structured as a modular pipeline of individual components (e.g., perception, prediction, and planning) to help separate concerns into interpretable sub-tasks. Even when end-to-end training is possible, each module has its own set of objectives used for safety assurance, sample efficiency, regularization, or interpretability. However, intermediate objectives do not always align with overall system performance. For example, optimizing the likelihood of a trajectory prediction module might focus more on easy-to-predict agents than safety-critical or rare behaviors (e.g., jaywalking). In this paper, we present controlaware prediction objectives (CAPOs), to evaluate the downstream effect of predictions on control without requiring the planner be differentiable. We propose two types of importance weights that weight the predictive likelihood: one using an attention model between agents, and another based on control variation when exchanging predicted trajectories for ground truth trajectories. Experimentally, we show our objectives improve overall system performance in suburban driving scenarios using the CARLA simulator.
10.1109/icra46639.2022.9811884
[ "https://arxiv.org/pdf/2204.13319v1.pdf" ]
248,426,803
2204.13319
963f1b388f5fadda0327b0accc5727ec108c1aa2
Control-Aware Prediction Objectives for Autonomous Driving Rowan Mcallister Blake Wulfe Jean Mercat Logan Ellis Sergey Levine Adrien Gaidon Control-Aware Prediction Objectives for Autonomous Driving Autonomous vehicle software is typically structured as a modular pipeline of individual components (e.g., perception, prediction, and planning) to help separate concerns into interpretable sub-tasks. Even when end-to-end training is possible, each module has its own set of objectives used for safety assurance, sample efficiency, regularization, or interpretability. However, intermediate objectives do not always align with overall system performance. For example, optimizing the likelihood of a trajectory prediction module might focus more on easy-to-predict agents than safety-critical or rare behaviors (e.g., jaywalking). In this paper, we present controlaware prediction objectives (CAPOs), to evaluate the downstream effect of predictions on control without requiring the planner be differentiable. We propose two types of importance weights that weight the predictive likelihood: one using an attention model between agents, and another based on control variation when exchanging predicted trajectories for ground truth trajectories. Experimentally, we show our objectives improve overall system performance in suburban driving scenarios using the CARLA simulator. I. INTRODUCTION Autonomous vehicles (AVs) must navigate busy roads using predictive models to anticipate what surrounding pedestrians and vehicles might do in order to plan safe trajectories around them. Safe operation requires such components be well calibrated, typically by minimizing some regression error on training data. However, not all errors made by prediction modules are equally important: some errors have minimal effect on downstream decisions, while some perceptual errors [24] and predictive errors [25] can have fatal outcomes. As no model is perfect, it is crucial to identify which prediction errors are safety-critical to ensure safety [19]. Whether trained independently or as part of multi-task end-to-end architectures [30], multi-agent trajectory forecasting models typically optimize prediction-specific objectives based on regressing recorded future trajectories by considering all agents equally important a priori. However, when considering the target control task of autonomous navigation, some predictions warrant more attention than others when deciding safe controls. Consequently, control-agnostic optimizing of prediction models may not result in improved downstream navigation performance due to limited data, model capacity, rare events, or computational constraints. Even with end-to-end training, multi-task objectives might not be aligned, thus resulting in performance degradation due to task interference [36]. In this work, we propose Control-Aware Prediction Objectives (CAPOs) to train prediction models that more ac- Fig. 1: A vehicle drives to the right while reacting to pedestrians with sample predicted trajectories shown in purple or pink. Our Control-Aware Prediction Objectives (CAPO) can learn to capture which predictions should have more influence on the vehicle's controls (cyan lines proportional to attention). Videos available at https://sites. google.com/view/control-aware-prediction curately reflect the relative effects of predictive errors on downstream control. Computing these downstream effects requires only forward passes without backpropagation between modules. This improves applicability with real-world AV planning and control systems, which might not be fully differentiable due to complex design constraints (e.g., verifiability, interpretability, comfort and safety constraints). Our method introduces importance-weighted prediction likelihood objectives using forward passes of the prediction model and planner. We investigate two weighting methods that can be trained with backpropagation. The first assigns weights based on control variations due to prediction changes. The second uses learned attention weights between agent predictions and AV controls. Using the CARLA simulator, we experimentally show that training prediction models with control-aware objectives leads to improved controller performance in complex multi-agent urban driving scenarios. Compared with existing prediction models, including prediction algorithms that treat everything as equal, we show that our new objective helps to avoid precisely those errors that would maximally influence downstream decisions. II. RELATED WORK Several related fields of study investigate objective-aware prediction metrics as we discuss here. A. Objective-Aware Prediction in Reinforcement learning Model-based reinforcement learning (MBRL) methods learn a dynamics model of an autonomous agent to predict which control decisions lead to states with higher objective rewards [7,22,21]. MBRL prediction is related to AV prediction, with the main difference being that AVs predict the trajectories of other human agents and not those of the autonomous agent. Nevertheless, several MBRL works have recently challenged the common assumption that the better a dynamics model's predictive accuracy, the better the downstream policy will maximize reward. For example, Lambert et al. [17] show task-agnostic loss functions used to train dynamics models are often uncorrelated with episode rewards, an issue termed "objective mismatch". Indeed, learned dynamics models need not be accurate everywhere in the state space, only in the areas that help maximize rewards [3]. Some RL works investigate training models insofar as they improve estimating the value function [11,2], policy gradient [14,1], or ability to reach a goal state [23]. Others optimize downstream policies directly using Bayesian optimization to search model parameters [3]. Work by Donti et al. [8] points out that in practice we often want a combination of training a model to optimize its likelihood as well as a downstream task term, although in the context of constrained optimisation. Similarly, Lambert et al. [17] correlate both metrics by increasing the weight in the loss of data points closer to data the optimal controller generates. Unfortunately for AV applications, such objective-aware prediction methods are often inapplicable for several reasons. First, MBRL assumes access to an objective reward function, reward samples, or goal state, but objective measures or goals of desirable driving are often difficult to define. By contrast, human-designed AV control systems are often preferable for verification and interpretability reasons. Second, a common assumption in RL is that the policy is either differentiable or stochastic (in the case of policy gradients), whereas realworld AV control systems often contain complex logic that is neither differentiable nor stochastic. Our work focuses instead on how to learn prediction given access to a safe, potentially non-differentiable controller. B. Map-Aware Prediction Metrics Map information can help incorporate prior knowledge into prediction metric design. For example, since the ego vehicle drives on the road, pedestrian forecasting errors could be given more weight on road surfaces than otherwise. Work by Shridhar et al. [32] use maps to help focus on potential collisions with other agents by generating a set of candidate ego trajectories along known lane tracks that any controller might follow. This method does not assume a particular downstream controller, but makes an educated guess as to what a reasonable controller might do. Another map-based prediction metric is Drivable Area Compliance (DAC) [5], which counts the proportion of model samples that exit the drivable area. A conceptual difference with our method is that our prediction metrics assume access to the specific downstream controller that will be used at test-time, improving test-time performance. Since controllers already consider road information, we circumvent the need to explicitly design prediction metrics around mapping information, which can incur additional hyperparameters, such as the relative costs of predicting if an agent will traverse either road / sidewalk / building. C. Control-Aware Perception Philion et al. [26] propose a control-aware 3D object perception metric called Planning KL-divergence (PKL) based on how perceptual errors cause distributional divergence in the ego's distribution of planned paths, compared to a planner with perfect observations, measured by the KL divergence. In contrast, our work focuses on prediction objectives, and additionally demonstrates how the new objective empirically affects online-control using a driving simulator. We also avoid KL distance losses in our work since this assumes non-trivial stochasticity in the controller or data, which is not always the case. Other works have also used KL policy distances to investigate how observations can be compressed while preserving human-like actions had they remained uncompressed [28]. Work by Piazzoni et al. [27] also investigates how perceptual metrics affect downstream planning but are specific to perception. III. PRELIMINARIES Here we formalize our notation and discuss some existing prediction metrics before presenting our own. A. Notation and Assumptions Let x ∈ X denote past trajectory information about all agents, used to make probabilistic predictionsŷ ∈ P Y about the future multi-agent trajectories y ∈ Y. Trajectories are predicted up to time horizon T , and y T denotes the future state at time T . As the intents of other agents are usually uncertain, we use a probabilistic prediction model q θ with trainable parameters θ to sample the motion of others:ŷ ∼ q θ (Y|x), denoting likelihoods as q θ (y|x) . = q θ (Y = y|x). If multiple samples are taken,ŷ k refers to the kth sample, and to single out the nth agent we overload notation usinĝ y n , and use y ego as the AV's future trajectory. Given such predictions, the AV controller π outputs ego controls u ∈ U to anticipate and avoid colliding with other agents' future trajectories: u = π(y). We assume our AV stack performs behavior prediction before control, a common assumption [31]. While conditioning behavior prediction on ego's intent provides more accurate prediction, for sake of simplicity we assume that other agents do not anticipate the AV's future, only the AV anticipates the other agents' future trajectories in order to avoid collisions. B. Common Prediction Metrics and Objectives Common prediction metrics in the literature and in prediction benchmarking challenges-including Argoverse Forecasting [5], Lyft Prediction [12], Waymo Open Motion [10], and AIODrive [35]-are summarized in Table I: TABLE I: Common prediction metrics in the literature. Metric Name Metric Equation Average Displacement Error (ADE) ||ŷ − y|| 2 Final Displacement Error (FDE) ||ŷ T − y T || 2 Minimum-ADE (minADE) min k∈[K] ||ŷ k − y|| 2 Minimum-FDE (minFDE) min k∈[K] ||ŷ k T − y T || 2 Miss Rate (MR) 1 K k 1[||ŷ k T − y T || 2 > α] Negative Log Likelihood (NLL): − log q θ (y|x) Most metrics compare the Euclidean distance between either the full predicted state-sequenceŷ (or final statê y T ) with the true sequence y (or final state y T ) an agent took, as recorded in data. Probabilistic models are typically trained to minimize the negative log likelihood (NLL) of the data. All such metrics are agnostic to road geometry and downstream planning, which implicitly assumes that all other agents' forecasts are equally relevant. For example, consider two pedestrians: one walking ahead of the ego vehicle and one behind. Assuming independent pedestrian motion, the NLL objective factorizes as: − log q θ (y ahead , y behind |x) = − log q θ (y ahead |x)−log q θ (y behind |x). Notice that this prediction metric is equally concerned with both y ahead and y behind . Intuitively, accurate prediction of the pedestrian ahead of the ego vehicle is more important for safe motion planning since the ego's planned path is more likely to intersect with y ahead than y behind . How can prediction metrics become "aware" that errors in predicting y ahead have greater downstream consequences than errors in y behind ? IV. CAPO: CONTROL-AWARE PREDICTION OBJECTIVES In this section we propose novel prediction loss functions that consider how predictions will be used downstream to improve predictive accuracy whenever prediction errors would cause a large change in control outputs. In Bayesian decision theory, a decision is evaluated as the expected utility of a decision u or controller π, integrating out any uncertainties [4]. In our case, it is the future trajectories of other agents that are unknown but can be probabilistically predicted according to a model with parameters θ. Following the literature on loss-calibrated variational inference [16,6,15], we define the gain of a decision or controller's value as a function of the model parameters θ that we wish to train. Gain x,y,π (θ) = utility(π, y,ŷ, x)q θ (ŷ|x)dŷ. (1) The choice of utility function in Eq. (1) is an open one, that is why we considered many possible input parmeter; it defines how desirable a course of actions would be given x andŷ. Alternatively, an existing metric like the NLL can simply be weighted without integration. In the next subsection we discuss some baseline choices for the utility or weight, and after, we propose two novel methods for computing these weights: a self-attention method and a counterfactual method. A. Baseline Objectives Most predictive metrics in the literature are agnostic to u and simply use a delta function to only score correct trajectory predictions, recovering the standard log likelihood metric: Gain x,y (θ) = δ(y −ŷ)q θ (ŷ|x)dŷ = q θ (y|x). However, we are interested in utilities that are a function of u in order to weight predictions by their downstream effect on the ego's control. For instance, we could score trajectory predictions based on the resultant ego controls π(ŷ) matching the ego's behavior under knowledge of the true future trajectories π(y): Gain x,y,π (θ) = δ(π(y) − π(ŷ))q θ (ŷ|x)dŷ. This integral is unfortunately intractable to derive or estimate, but softer utility functions can be used instead. One example is ||π(ŷ) − π(y)|| 1 , which we include as a baseline in Table II. Optimizing this controller output error guides the learning process towards predicting controller inputs (predicted trajectories) accurately, insofar as they result in the correct control. Any trajectory errors that do not induce a change in the AV's control are thus considered inconsequential and ignored. B. Attention-based CAPO We propose a GRU encoder-decoder architecture with an attention mechanism as introduced in [34]. Our method weights the agent predictions using attention factors between agents x and the AV's future trajectory y ego . The predictive model is a function parameterized by θ noted q θ : X → P Y×Yego . We note θ = {θ ego , θ agent } where θ ego is the set of parameters for the ego decoder only; X is the past observation space and P Y×Yego the probability spaces of future trajectories: P Yego for the ego and P Y for other agents. We use an architecture similar to [18,20] where we train a model with multi-head attention; the ego agent attends the other agents. The ego predictions are used as a proxy for the actual planner to compute the importance weights of other agents: The ego-attention blocks in figure Fig. 2 are heads of a multi-head attention mechanism. The computation performed by each head is given below: ego encoding . . . The attention vector is given by agent N encoding L k L v L q q 0 k 0 v 0 L k L v k n v n Q = q0 K =    k0 . . . kn    V =    v0 . . . vn    σ QK T √ dk V output encodedα = σ QK √ d k = [α 0 , ..., α N ],(2) where Q is the query matrix, K the key matrix, and σ the softmax operation that normalizes the attention vector (for details see [34]). The encoded output = αV is a weighted mean of the value vectors over the agents (including ego). Inspired by [20] the attention model produces outputs in the form of a sequence of Gaussian mixtures for each agent and is trained to minimize the NLL for all agents and the ego trajectory predictions. However, for our application, ego prediction is not the goal, it is only a proxy to compute the importance weights of the pedestrian contribution to the ego behavior. We propose to use attention coefficients α as importance factors in a weighted sum of per-human state prediction loss (as opposed to uniform weighting). Algorithm 1 summarizes how the model is trained with importance weighting. If multiple heads are used, the attention coefficients are averaged: w n = 1 H H h=1 α (h) n(3) The attention predictor imitates a planner that interacts with the other agents to avoid collision. The only way for the predictor to interact with the other agents is through attention. Therefore, as the model learns the correlations between the planner's trajectories and the agents trajectories, larger attention coefficients are given to the agents that cause larger reactions from the controller. It learns this offline and does not need access to the controller nor its gradient. Predicting jointly the ego trajectory and the other agents allows us to use the attention coefficients for concern weighting in a single run. The coefficient α 0 quantifies how independent the ego is from other agents. This method defines a concern about an agent but not about specific trajectories of that agent. It can define the concern without using the controller because it instead uses an offline-learned model that imitates the controller. Update model: θ ego ← θ ego + ∇ θego log q θ (y ego |x) θ agent ← θ agent + w(x)∇ θagent log q θ (y agent |x) Output: Predictive model q θ : X → P Y×Yego C. Counterfactual Action-Discrepancy CAPO Our second proposal can also be formulated as a reweighted maximization objective, where we weight the log likelihood of each agent's trajectory in a scene by its individual contribution to the ego's control decision. We do this by first enumerating through each agent in a scene, and computing counterfactual outputs from the AV's controller if every agent traversed their individual trajectory as recorded in the replay buffer, except for agent n. If we resample the trajectory that the nth agent might otherwise have taken, y k n ∼ q θ (Y n |x), we can compute the control output that would result:û k n = π({ŷ k n } ∪ y \ {y n }),(4) to compare against the control had no agent deviated from their recorded trajectories: u = π(y).(5) The difference in these two hypothetical controls corresponds to how much an individual agent affects the ego vehicle, and can represent the concern associated with predicting this particular agent in this particular instance accurately. If the model is probabilistic, then taking multiple samples (K > 1) helps ensure high importance even if the other agent only might cause a control deviation: w n = max k∈{1..K} ||u −û k n || 1 ,(6) which we use as weights for predictive model training: θ * = arg max θ N n=1 w n log q θ (y n |x). We summarize our counterfactual action discrepancy method in Algorithm 2. One benefit of this approach compared to the attention CAPO is that it is Update model: θ ← θ + w(u,û k n )∇ θ log q θ (y|x) Output: Predictive model q θ : X → P Y D. Summary of Objectives There are various choices for utilities, or weights for traditional module metrics. In Table II we summarize the several baselines methods, including NLL and our novel proposals. V. EXPERIMENTAL EVALUATION To evaluate our proposed method, we consider a representative scenario that is commonplace in autonomous driving: pedestrian trajectory prediction. The majority of pedestrian behaviors can safely be ignored by the AV's autonomy stack; however, in rare cases of pedestrian-ego interaction (e.g., road crossings), accurate prediction of pedestrian behavior Attention δ(y −ŷ) q θ (y|x) + q θ (yego|x) R2P2 Gainy δ(y −ŷ) q θ (y|x) R2P2 Gainπ1 ||π(y) − π(ŷ)||1 Eŷ [||π(y) − π(ŷ)||1] R2P2 Weight ∇ŷ ||∇ŷπ(ŷ)||1 Eŷ ||∇ŷπ(ŷ)||1 q θ (y|x) R2P2 Weight ∇y ||∇yπ(y)||1 ||∇yπ(y)||1q θ (y|x) Ours: R2P2 Weight π ||π(y)−π(ŷ)||1 Eŷ [||π(y)−π(ŷ)||1] q θ (y|x) R2P2 Weight πk max k ||π(y)−π(ŷ k )||1 max k ||π(y)−π(ŷ k )||1q θ (y|x) AttentionWeight α(x) α(x)q θ (yagent|x) + q θ (yego|x) becomes crucial in avoiding collisions. This sparsity of interaction showcases how predictive models may perform well with respect to traditional metrics (e.g., ADE) while still leading to suboptimal ego behavior when it matters most. Here, we first detail our experimental evaluation and implementation of the aforementioned scenario within the CARLA autonomous driving simulator [9]. We then compare results between our method and the various baselines discussed in Table II, where our experiments show that predictive models trained using our CAPO methods produce safe behavior with fewer collisions relative to other baselines. Fig. 4: Pedestrian Prediction Scenario. Pedestrians spawn on the sidewalk (yellow region) and the ego (red car) predicts the pedestrian trajectories within the next 3 seconds (green). Some pedestrians will cross the road at right angles (blue). Left: the planner predicts a collision with a crossing pedestrian and starts slowing (red ego prediction up to blue line but not further). Right: ego is safely passing the road segment where the pedestrian has already crossed. A. CARLA Scenario Design We implement our pedestrian prediction scenario in the CARLA simulator's Town01. A single ego vehicle is commanded to drive down a road that is adjacent to sidewalks which are populated with pedestrians. Occasionally, a pedestrian will cross the street and the ego agent must slow to avoid a collision when necessary. The ego vehicle adjusts its longitudinal control to balance the competing priorities of maintaining the current speed limit (45mph) while avoiding collisions with the pedestrians crossing the road (either their current or future-predicted distance on the road ahead). This balance is performed by an Intelligent Driver Model [33]. It uses predictions to estimate the closest collision distance and controls the vehicle to stop ahead of that point. Pedestrians spawn at random locations on the sidewalk and are then provided a long-range navigation goal that is also uniformly sampled from the sidewalk. When the long-range goal is reached, another is sampled to replace it. To induce pseudo-random motion, a short-range goal is also generated at each time step. This goal is generated by projecting a point 4m along the path to the long-range goal, starting at the pedestrian's location. The lateral offset β t+1 of the short-range goal is generated by sampling from a normal distribution centered about the previous lateral offset β t after it has been scaled down (to drive it towards the long-range goal): β t+1 = (1 − ε)β t + N (0, σ 2 ),(8) where σ is the variance of the noise, and ∈ [0, 1) is the commitment to the long-range goal. When on the sidewalk, pedestrians are programmed to walk at speeds sampled about 2m/s while navigating around other pedestrians to avoid collisions and, occasionally, will pause outside of shops. Each different kind of pedestrian is defined with various noise levels, commitment, and stopping chance. Pedestrians may also randomly decide to cross the road. The probability increases if their velocity vector points towards the road and increases greatly when the pedestrian is close to the road. While crossing, they travel at 2 m/s in the shortest path possible, i.e., perpendicular to the road direction. To increase task difficulty, the probability that pedestrians will cross the road is increased at test time. B. Compared models 1) Oracle distribution The pedestrian behavior is simulated with a known distribution at each time step, which is sampled to produce a trajectory. The trajectory distribution is approximated by sampling K = 5 trajectories for each pedestrian. The planner reacts to the trajectory that would cause the closest intersection with its desired path. This method is a perfect probabilistic predictor which is only accessible with simulated data. Its predictions are not biased towards sampling the most critical trajectories, and planning with relatively few samples can yield suboptimal results. 2) Gradient weighting Recent work has also investigated weighing prediction objectives by a measure of local sensitivity of downstream costs to individual predictions [13]. This method first learns a cost function to evaluate ego controls given predicted human trajectories, using inverse optimal control. The method then weights each prediction loss by the gradient magnitude of the cost outputs w.r.t. the prediction inputs. In our work, we consider using the controller directly, forming an analogous baseline: using gradients of the controller w.r.t. the predicted trajectory ||∇ŷπ(ŷ)|| 1 , or true trajectory ||∇ y π(y)|| 1 , included in Table II. While we do not assume differentiable controllers in our own methods, we nevertheless experiment using a differentiable controller to compare against this baseline method. 3) Attention weighting As presented in section IV-B. We train this model with our CAPO method (algorithm 1) and, as a baseline, we compare it with the unbiased prediction as the predictor [20] would produce. 4) Reparameterized Pushforward Policy (R2P2) we use the likelihood-based multi-agent prediction algorithm R2P2 [29] as baseline Gain y , and also use R2P2 as the base model for all other predictive models apart from the attention model. R2P2 is a autoregressive normalizing flow, capable of expressing multimodal agent trajectories, trained with NLL. We parameterize R2P2 to predict 30 steps with data at 10Hz, corresponding to a 3s prediction for all pedestrians. We train it with our CAPO method (algorithm 2) using K = 10 samples and we use K = 1 samples at test time. C. Metrics The table III presents our results for 100 sequences. We track the performance of the system (prediction and planner) with the success rate and the number of collisions. Three conditions may end a sequence: • Success: vehicles traverses 200m road without incident. • Collision: a pedestrian was hurt. • Time out: the car was too slow (> 60s). We also score efficiency and comfort indicators by average speed and average jerk respectively. Finally, we compute the average pedestrian trajectory prediction errors and their downstream effect on the planner with the average displacement error (ADE) and the Control Error equal to ||π(y)−π(ŷ)|| 1 . The control error measures the downstream effect of the prediction error on the ego's plans. VI. DISCUSSION The results in Table III show that while all methods do reasonably well, weighting predictive objectives by their downstream effect improves the downstream performance as illustrated by a low collision count and control error. While methods such as R2P2 Weight ∇ŷ assume a differentiable controller, we find this assumption does not need to be made, and our methods can work with any type of controller. While our methods did not score as well on the ADE metric of agents' trajectories, they did score best on the metrics that matters more: the control error, and success rate, thus mitigating error propagated downstream and improving the end task performance. We compared to objectives weighted by the planner's sensitivity ||∇ŷπ(ŷ)|| 1 , which is related to [13]. However, such methods assume the planner (or cost function) is differentiable, which is often not the case in real AV systems. Secondly, local sensitivity to a point prediction is not necessarily a good measure of relevance if the gradient is noisy or changes drastically over the full predictive distribution. While our experiments show encouraging results, testing with various setups and environments would be needed to give a clear best method. VII. CONCLUSIONS Modular autonomous systems (such as those commonly used in AVs) provide a number of advantages, but generally incur the disadvantage that individual components typically do not optimize for system-wide or downstream performance metrics directly. In this paper, we proposed weighted objectives for learning prediction models that account for the downstream objective without imposing stringent requirements on downstream components (such as end-to-end differentiability). These objectives weight the usual likelihood objective, either using attention weights derived from a behavior-cloned policy, or using the impact that substituting predicted trajectories for ground-truth trajectories has on planner output. Accounting for the downstream objective in this manner encourages prediction models to focus on what's important -either at the agent or individual trajectory level -and, as a result, improves system-wide performance, as we showed empirically in a pedestrian jaywalking scenario. A number of promising avenues exist for future research. First, control-aware objectives may provide out-of-domain generalization benefits by encouraging prediction models to focus on relevant aspects of the scene, and ignore spurious sources of information that are safe to ignore. Second, in this paper we focused on data collected from an expert. However, this requirement limits the applicability of the proposed metrics, and a broader coverage of the state-space resulting from the use of both expert and suboptimal data might improve the learned prediction models. Finally, although we focused in this paper on using control-aware weighting for optimizing prediction models. Our method might equally well be used to define a weighted metric for evaluating models in a validation setting where training-time access to the downstream planner is either not available or undesirable. Fig. 3 : 3Diagram of a self-attention head. Algorithm 1 1Attention CAPO Input: Controller: π : X → U 1: Record trajectory data D = {x, y} i 2: while training do Algorithm 2 2Counterfactual CAPO Input: Controller: π : X → U 1: Record trajectory data D = {x, y} i 2: while training do Fig. 2: Diagram of the attention model. All agent encoders and decoders share their weights. Encoders and decoders are GRUs. Attention is not used between agents.ego agent 1 . . . agent N Encoder ego Encoder agent . . . Encoder agent Ego-attention Ego-attention Ego-attention Decoder ego Decoder agent . . . Decoder agent prediction ego prediction agent 1 . . . prediction agent N TABLE II : IIComparison of utilities and weighted objectives.Method Utility or Weight Objective L(θ) Baselines: TABLE III : IIIScenario results, 100 episodes. Arrows indicate higher/lower preferred. Standard errors shown. Best, second.Predictive Model Success Rate ↑ Collisions ↓ Speed (m/s) ↑ Jerk (m/s −3 ) ↓ ADE (m) ↓ Control Error ↓ Baselines: R2P2 Gain y 89.0% 11 9.97 ±0.222 8.92 ±0.250 2.09 ±0.024 0.59 ±0.012 R2P2 Gain π1 85.0% 14 10.45 ±0.268 6.65 ±0.196 3.48 ±0.038 0.63 ±0.016 R2P2 Weight ∇ŷ 94.0% 4 9.53 ±0.216 8.21 ±0.140 1.98 ±0.024 0.60 ±0.012 R2P2 Weight ∇y 91.0% 9 9.74 ±0.216 8.74 ±0.184 2.00 ±0.025 0.60 ±0.011 Attention [20] 89.0% 11 13.79 ±0.214 4.48 ±0.147 2.61 ±0.050 0.63 ±0.026 Oracle distribution 98.0% 2 10.54 ±0.231 6.80 ±0.180 1.58 ±0.036 0.51 ±0.013 Our methods: R2P2 Weight π 93.0% 7 8.86 ±0.188 9.26 ±0.194 2.29 ±0.022 0.58 ±0.010 R2P2 Weight πk 99.0% 1 9.46 ±0.196 7.89 ±0.159 2.14 ±0.018 0.55 ±0.011 Attention Weight α 91.0% 9 14.36 ±0.217 4.22 ±0.154 2.58 ±0.053 0.64 ±0.024 Policy-aware model learning for policy gradient methods. 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[]
[ "Revisiting the Trade-off between Accuracy and Robustness via Weight Distribution of Filters", "Revisiting the Trade-off between Accuracy and Robustness via Weight Distribution of Filters" ]
[ "Xingxing Wei ", "Member, IEEEShiji Zhao " ]
[]
[]
Adversarial attacks have been proven to be potential threats to Deep Neural Networks (DNNs), and many methods are proposed to defend against adversarial attacks. However, while enhancing the robustness, the clean accuracy will decline to a certain extent, implying a trade-off existed between the accuracy and robustness. In this paper, we firstly empirically find an obvious distinction between standard and robust models in the filters' weight distribution of the same architecture, and then theoretically explain this phenomenon in terms of the gradient regularization, which shows this difference is an intrinsic property for DNNs, and thus a static network architecture is difficult to improve the accuracy and robustness at the same time. Secondly, based on this observation, we propose a sample-wise dynamic network architecture named Adversarial Weight-Varied Network (AW-Net), which focuses on dealing with clean and adversarial examples with a "divide and rule" weight strategy. The AW-Net dynamically adjusts network's weights based on regulation signals generated by an adversarial detector, which is directly influenced by the input sample. Benefiting from the dynamic network architecture, clean and adversarial examples can be processed with different network weights, which provides the potentiality to enhance the accuracy and robustness simultaneously. A series of experiments demonstrate that our AW-Net is architecture-friendly to handle both clean and adversarial examples and can achieve better trade-off performance than state-of-the-art robust models.
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[ "https://export.arxiv.org/pdf/2306.03430v1.pdf" ]
259,088,748
2306.03430
1281b56b9e16241307f98e1f6343591536d38ac1
Revisiting the Trade-off between Accuracy and Robustness via Weight Distribution of Filters Xingxing Wei Member, IEEEShiji Zhao Revisiting the Trade-off between Accuracy and Robustness via Weight Distribution of Filters 1Index Terms-Adversarial ExamplesAdversarial RobustnessDynamic Network StructureAccuracy-Robustness Trade-off ✦ Adversarial attacks have been proven to be potential threats to Deep Neural Networks (DNNs), and many methods are proposed to defend against adversarial attacks. However, while enhancing the robustness, the clean accuracy will decline to a certain extent, implying a trade-off existed between the accuracy and robustness. In this paper, we firstly empirically find an obvious distinction between standard and robust models in the filters' weight distribution of the same architecture, and then theoretically explain this phenomenon in terms of the gradient regularization, which shows this difference is an intrinsic property for DNNs, and thus a static network architecture is difficult to improve the accuracy and robustness at the same time. Secondly, based on this observation, we propose a sample-wise dynamic network architecture named Adversarial Weight-Varied Network (AW-Net), which focuses on dealing with clean and adversarial examples with a "divide and rule" weight strategy. The AW-Net dynamically adjusts network's weights based on regulation signals generated by an adversarial detector, which is directly influenced by the input sample. Benefiting from the dynamic network architecture, clean and adversarial examples can be processed with different network weights, which provides the potentiality to enhance the accuracy and robustness simultaneously. A series of experiments demonstrate that our AW-Net is architecture-friendly to handle both clean and adversarial examples and can achieve better trade-off performance than state-of-the-art robust models. INTRODUCTION D UE to the impressive performance, Deep Neural Networks (DNNs) have recently been applied in various areas. However, DNNs can be misled by adversarial examples produced by adding imperceptible adversarial perturbations on clean examples [34], which indicates that serious security issues exist in DNNs. To defend against adversarial examples, some researches are proposed to enhance the model's robustness. Adversarial Training [26] is proposed and proved to be an effective way to enhance the robustness of models. However, as a price of robustness, the clean performance will decline to a certain extent [35]. To balance the trade-off between the accuracy and robustness, various methods are presented from different views, such as advanced optimization algorithm [50], [53] or modified network architecture [8], [27], [52], and so on. As for the former one, TRADES [50] attempts to reduce the gap by utilizing a Kullback-Leibler (KL) divergence loss in adversarial training. MTARD [53] applies two teacher models to guide a student model to promote the accuracy and robustness by adversarial distillation. As for the latter one, MagNet [27] tries to learn the clean distribution and then distinguish adversarial examples to be not recognized. However, it is not robust to the adaptive attack proposed by [3]. AdvProp [47] uses an auxiliary Batch-Normalization (BN) layer to improve the robustness and can achieve good accuracy on clean examples. Furthermore, Liu et al. [25] propose multiple BN layers to control the domain-specific statistics for different attack methods. Although they show [42], [53], [54]. The figure shows filters' weight distribution exists an obvious difference for the standard and robust models. some positive effects, the trade-off still exists and needs to be further studied. To solve this problem, in this paper, we investigate the standard and robust models from the view of networks' weight distribution. In Figure 1, we observe that filters' weight distribution for the same network architecture between standard-trained and robust-trained models exist an obvious distinction, while the robust-trained models have the similar distribution. We analyse the underlying reason, and give a theoretical explanation for this phenomenon in terms of the gradient regularization of different optimization algorithms. The empirical and theoretical analysis indi- cate that a static network is challenging to eliminate the gap and is hard to improve both the accuracy and robustness because it is difficult to contain two groups of different weights within a static network. A reasonable solution is to adaptively change the network's weights based on the input samples, and thus utilize the optimal weights to process them, respectively. Inspired by the dynamic networks [23], [41], we design the Adversarial Weight-Varied Network (AW-Net) to perform a "divide and rule" weight strategy for the clean and adversarial example. Specifically, AW-Net contains two branches: a dynamic weight sub-network and an adversarial detector sub-network. The dynamic weight sub-network is in charge of changing the network's weights at the level of filters to process the clean and adversarial example, respectively. To handle different characteristic distributions of input samples and keep the model stable during training and testing, a MixBN [6], [25] structure is adapted into the sub-network. The adversarial detector sub-network is responsible for classifying the clean and adversarial example, and then produce the regulation signals to guide the weights' change in the dynamic weight sub-network. These two sub-networks are jointly trained in an end-toend adversarial training manner, and can interact with each other closely. With these careful designs, AW-Net has the potentiality to enhance the accuracy and robustness at the same. The flowchart is shown in Figure 2. We summarize our contributions as follows: • We empirically explore the difference existed between the standard-trained and robust-trained models versus the filters' weight distribution for the same network architecture, and theoretically explain this phenomenon in terms of the gradient regularization. We argue that this difference may be one reason for the poor trade-off between the accuracy and robustness because it is hard to contain two groups of different weights in a static network. • We propose a sample-wise dynamic network named Adversarial Weight-Varied Network (AW-Net), which dynamically adjusts the weights based on the input samples. For that, a dynamic weight sub-network at the filter level is designed, and a MixBN structure is used to handle the feature distributions for clean and adversarial examples. Meanwhile, we apply a multi-head adversarial detector sub-network to generate the regulation signal to guide the dynamic weights. Moreover, a joint training strategy is used to train AW-Net. • We extensively verify the effectiveness of AW-Net on two public datasets, and compare with stateof-the-art robust models against the white-box and black-box attack, respectively. The results show that our method achieve the best performance versus the trade-off between the accuracy and robustness. The rest of the paper is organized as follows: we introduce the related work in Section 2. Section 3 explores the differences for the filters' weight distribution. We present the AW-Net in Section 4. The experiments are conducted in Section 5, and the conclusion is given in Section 6. RELATED WORK Adversarial Attacks Szegedy et al. [34] demonstrate that imperceptible adversarial perturbations on inputs can mislead the prediction of DNNs. After that, lots of adversarial attack methods are proposed, such as Fast Gradient Sign Method (FGSM) [10], Projected Gradient Descent Attack (PGD) [26], Carilini and Wagner Attack (C&W) [4], and Jacobian-based Saliency Map Attack (JSMA) [29]. Generally speaking, adversarial attacks can be divided into white-box attacks [4], [10], [26] and black-box attacks [24], [43], [44], [45]. White-box attacks usually generate adversarial examples based on the gradients of the target models. While black-box attacks perform attacks via the transfer-based strategy or querybased strategy according to how much information of the target model can be obtained. In this paper, we propose a novel network architecture, and then test its robustness against the white-box and blackbox attacks, respectively. Adversarial Robustness To defend against adversarial attacks, lots of researches are proposed to enhance the robustness. Adversarial Training [26], [42] is a representative method among them. Madry et al. [26] formulate Adversarial Training as a min-max optimization. Further, Adversarial Robustness Distillation (ARD) methods [9], [53] use strong teacher models to guide the adversarial training process of small student models to enhance the robustness. Meanwhile, many pre-processing methods are applied to remove the effects of adversarial perturbations. For example, [2], [12] utilize the encoder structure to denoise the adversarial examples, which shows competitive performance against adversarial attacks. Image compression operations are also verified to be effective to deal with adversarial perturbations, such as JPEG compression [5] and DCT compression [1], etc. In addition, renormalizing the image from adversarial version to clean version [8], [30] can enhance the model robustness from the view of distribution. Shu et al. [30] use a batch normalization layer to achieve the clean feature by re-normalizing the adversarial feature with means and variances of clean features in adversarial training. Dong et al. [8] try to transform the adversarial feature into the clean distribution in object detection task. We can see that all the above methods aim to improve the robustness for a static network. In contrast, our method designs a dynamic network to adaptively adjust the weights to process the input sample. Trade-off between Accuracy and Robustness The Trade-off between accuracy and robustness has been widely studied [28], [39], [48], [50], [51], [53]. Zhang et al. [50] apply the prediction of clean examples to guide the training process toward adversarial examples (TRADES). Zhang et al. [51] enhance both accuracy and robustness by reweighting the training example. Stutz et al. [33] claim that manifold analysis can be helpful to achieve both accuracy and robustness. Yang et al. [48] argue that the trade-off can be mitigated by optimizing the locally lipschitz functions. Pang et al. [28] apply local equivariance to describe the ideal behavior of a robust model and facilitate the reconciliation between accuracy and robustness. Zhao et al. [53] apply multiple teacher models to improve the both accuracy and robustness of the student model. In this paper, we attempt to understand accuracy and robustness directly from the perspective of model weight distributions and try to mitigate the trade-off between model accuracy and robustness with the architecture of dynamic network weights. Dynamic Networks Different from static networks, dynamic networks usually change their architectures based on the input samples [37], [40]. The purpose of designing dynamic networks is to reduce the cost of network inference and utilize the most effective sub-networks to get the correct prediction. For example, Wang et al. [41] consider the filters as experts and select a part of them to activate some specific layers. Li et al. [23] propose the Dynamic Slimmable Network to divide the inputs into easy and hard samples and then design a dynamic network to predict samples' type. Veit et al. [37] and Wang et al. [40] design to dynamically skip the residual structure and reduce the calculation cost. Recently, the adversarial robustness of dynamic networks is evaluated [14], [17], [18]. [14] is the first work to show the vulnerability of dynamic networks. Hu et al. [18] argue that triple wins can be obtain for the accuracy, robustness, and efficiency via a multi-exit dynamic network. However, Hong et al. [17] show that the efficiency of this multi-exit network can be reduced by the proposed slowdown attack. Some methods named transductive defenses [38], [46] update the network's parameters in the evaluation process to dynamically defend against adversarial examples. We are different from the above works as follows: (1) we aim at defense rather than attacks like [14], [17]. (2) we focus on dynamically adjusting the network's weights rather than utilizing the multi-exit layers like [18]. The motivation of our method is based on the filter's weight observation in Figure 1, which has different defense mechanism with [18]. (3) During the testing phase, our method does not need to update the parameters, which is obviously different from transductive defenses [38], [46]. FILTER WEIGHT DISTRIBUTIONS Empirical Observation As mentioned before, although some models have shown the good adversarial robustness, their clean accuracy is always inferior to the standard models. To investigate the underlying reason, we pay attention to the network's weights. Because filters can be approximately considered as the independent units of feature extracting, so we compute the means and variances of the filters at different convolution layers, and then use them to evaluate the weight distributions. Figure 1 shows that the means' distribution between the standard and robust models varies greatly. To further explore this phenomenon, we directly illustrate the means and variances of a strong standard model and a strong robust model trained by [42] with the same network architecture in Figure 3. From the results, we can see that the filter's weight of the standard model shows a rough change, while the filter's weight of the robust model shows a smooth change. This contrast illustrates the obvious difference between the standard and robust model. The prominent means and variances of the standard model demonstrate that it is easy to capture more subtle image features, which can explain why the standard model is easily affected by adversarial samples with minor perturbations. As a comparison, the robust model shows smaller means and variances and is more robust to the adversarial perturbations. Different weight sizes between standard and robust models may directly lead to the sensitivity to clean and adversarial examples and can further influence the model's robustness. Theoretical Analysis Inspired by [31], we theoretically analyze the above performance from the perspective of model's optimization, which directly influences the weight distribution. We investigate the intrinsic mechanisms of robust models using adversarial training methods based on PDG-AT [26] and TRADES [50] following [31]. Firstly, we analyze the optimization loss of standard models and adversarial models during the training process. The optimization loss L adv (x, y) for adversarial training and the optimization loss L(x, y) for standard model training can be defined as follows: L adv (x, y) = 1 2 (L(x, y) + L(x + δ, y)),(1) where δ represents the adversarial perturbation with the maximum perturbation scale ϵ, and x and y refer to the input samples and true labels, respectively. By performing a first-order Taylor expansion on the loss L adv (x, y) for adversarial training, we can obtain: L adv (x, y) ≈ 1 2 (L(x, y) + L(x, y) + δ · ∂ x L(x, y)), = L(x, y) + 1 2 δ · ∂ x L(x, y),(2) where · denotes the matrix point multiplication operation, ∂ x L(x, y) represents the first-order partial derivative of L(x, y) with respect to x. Following [31], further expansion of ∂ x L(x, y) can be obtained as follows: δ · ∂ x L(x, y) = max ||δ||p≤ϵ | L(x + δ, y) − L(x, y) |, ≈ max ||δ||p≤ϵ | δ · ∂ x L(x, y) |, = ϵ || ∂ x L(x, y) || q ,(3) where p denotes the l p norm, || . || q is the dual norm of || . || p and can be defined as follows: || z || q = sup z T x || x || p ≤ 1 .(4) The regular term || ∂ x L(x, y) || q represents the result of regularization on the partial derivative of the optimization loss L(x, y) with respect to the input data x. Furthermore, by substituting the derivation result back into Eq. (2), we can obtain the L adv (x, y) for adversarial training as follows: L adv (x, y) = L(x, y) + ϵ 2 ||∂ x L(x, y)|| q .(5) Based on the above results, the difference between the two training methods is that the optimization loss L adv (x, y) of the adversarial training adds a gradient regularization constraint based on the optimization loss L(x, y) of the standard training, and the gradient regularization term constraint will directly affect the update of the model weight. Specifically, in the standard training process, the model weight w nat is updated based on the L(x, y) according to the gradient descent criterion: w nat = w nat − η∂ wnat L(x, y),(6) where η is the learning rate of the weight, ∂ wnat L(x, y) is the partial derivative of the L(x, y) relative to the weights w nat . During the adversarial training process, the weights w adv are updated as follows: w adv = w adv − η∂ w adv L adv (x, y).(7) In the initial state, the model weight w 0 is randomly initialized with the same means and variances. Based on this assumption, whether a model obtained by standard training or a model obtained by adversarial training, the model weight w 0 in the initial state has the same distribution. Since the update of the model weight w 0 depends on the gradient descent criterion, after the loss function guide the update, the expectations of the model weight are respectively as follows: E(w nat ) = E(w 0 ) − ηE(∂ w0 L(x, y)),(8)E(w adv ) = E(w 0 ) − ηE(∂ w0 L adv (x, y)).(9) Then the weight distribution difference between the standard training model and the adversarial training model can be derived as follows: E(∆ w ) = η || E(w adv ) − E(w nat ) ||, = η || E(∂ w0 L adv (x, y) − ∂ w0 L(x, y)) ||, = − ϵη 2 E(∂ w0 || ∂ x L(x, y) || q ).(10) The regular term || ∂ x L(x, y) || q can be used to measure the sensitivity of the input sample x in the model distribution space: when the value of || ∂ x L(x, y) || q is large, small perturbation in x will lead to obvious fluctuation in the prediction results of the model; When the value of || ∂ x L(x, y) || q is small, the prediction results of the model are not sensitive to small perturbation in x. After using it as the loss function for training optimization toward model weight w 0 , the partial derivative ∂ w0 || ∂ x L(x, y) || q of the model weight with respect to the regular term can optimize the sensitivity of the model weight toward x. In other words, after the constraints of the model weight, the final prediction does not change significantly when x is perturbed within perturbation scale ϵ. Therefore, the optimization process of the adversarial training model is constrained by the regularization term || ∂ x L(x, y) || q , the model is not obvious for the adversarial perturbation δ of the input x, which may be the inherent mechanism of strong robustness to adversarial examples; In contrast, the optimization of standard training is limited to the loss L(x, y), after gradient descent operation, the model weight w is only affected by the clean examples, and can achieve a pretty performance. Since the optimization process is not constrained by the regularization term || ∂ x L(x, y) || q , the standard training does not take into account the change of the model prediction results when x is slightly perturbed, which leads to weak robustness. Due to the term of gradient regularization, adversarial training and standard training will lead to different sensitivity for model weight to different types of samples, which directly leads to a significant gap between the robust model and the standard model. Based on the above observation and the theoretical analysis, we argue it is difficult to contain two groups of weights with different distributions in a static network. Inspired by dynamic networks, we consider to replace static networks with dynamic networks. A sample-wise dynamic network model can change the model weight according to the input samples, which provides the potentiality to solve the tradeoff between the accuracy and robustness. ADVERSARIAL WEIGHT-VARIED NETWORK Overview The pipeline of the proposed framework is shown in Figure 2. We propose a sample-wise dynamic Adversarial Weight-Varied Network (AW-Net) to enhance the accuracy and robustness. AW-Net contains two branchs: a dynamic weight sub-network and an adversarial detector subnetwork. The dynamic weight sub-network applies different filters' weights to adapt various inputs. Meanwhile, MixBN is used to handle the different distribution of clean and adversarial examples, respectively. The adversarial detector sub-network is used to generate regulation signals to adjust the filters' weights of dynamic weight sub-network. Dynamic Weight Sub-network The analysis in Section 3 inspires us to design a dynamic network composed of multiple AW-Net blocks which can dynamically adjust the filter weights. For the j-th block, we assume ψ j ∈ R c in j ×w in j ×h in j is the input feature with c in j channels, and ϕ j ∈ R c out j ×w out j ×h out j is the output feature with c out j channels. We utilize a residual structure to construct each block in AW-Net. Specifically, the j-th block is defined as follows: ϕ j = σ(F j (ψ j ) + ψ j ),(11) where σ(·) denotes the Relu layer. The symbol F j (·) contains the convolution layer W conv j ∈ R c in j ×kj ×kj ×c out j with kernel size k j and other nonlinear layers such as Batch-Normalization layer. W conv j consists of multiple filters K m ∈ R c in j ×kj ×kj , m = (1, 2, ..., c out j ), thus: W conv j = concat(K 1 , ..., K cout ),(12) where concat(·) is the concatenate operation along with the forth dimension. Without losing generality, different filters in convolution layer W conv j can be viewed as relatively independent units for extracting feature information and are similar to the individual "experts". From this point of view, different filters K m in our AW-Net block are supposed to have different importance when processing adversarial and clean samples. In order to make each filter play its speciality, we apply a filter-wise vector ω j = (ω 1 j , ..., ω cout j ) to give them different regulation signals as follows: W conv j = concat(ω 1 j · K 1 , ..., ω cout j · K cout ),(13) then we perform a convolution operation * to obtain the output feature: π j =Ŵ conv j * ψ j .(14) With the assistance of weight adjustment, the block can deal with adversarial and clean example in a distinctive way of feature processing. For the adversarial example, the filters' weights will have small variances and filters' weights will have large variances for clean examples, as the phenomenon shown in Figure 3. Batch Normalization [19] is an effective method to accelerate the training process, and is usually attached after the convolution layer. However, the existence of BN layer will eliminate the feature difference between adversarial and clean examples, and make their distributions tend to be the same. The weight adjustment mechanism toward filters will not take advantage of handling different characteristics and may negatively influence the performance of the network, leading to the catastrophic results that both distributions cannot be fitted well. BN mix = BN adv (π adv j ), ψ adv j ∝ x adv , BN nat (π nat j ), π nat j ∝ x nat .(15) In the testing process of the MixBN structure, we use type predictions P type (defined in the next section) from the adversarial detector sub-network as different BN's weights and use both Adv BN and Clean BN to deal with input features. The test process of MixBN can be denoted as follows: BN mix = (1 − P type )BN adv (π adv j ) + P type BN nat (π nat j ).(16) After the above operations, F j (·) is defined as: F j (ψ j ) = BN mix (Ŵ conv j * ψ j ).(17) Here, we use a two-layer residual structure, thus the j-th block is finally formulated as follows: ϕ j = σ(F j (σ(F j (ψ j ))) + ψ j ).(18) The prerequisite for an effective dynamic weight subnetwork is based on the excellent performance of regulation signals ω j and type predictions P type in the testing process. So how to design an effective adversarial detector subnetwork is needed to be solved. Adversarial Detector Sub-network We design a lightweight sample-wise adversarial detector sub-network, including a feature extractor and an adversarial weight regulator structure. The feature extractor is designed for getting semantic features from inputs and the following adversarial regulator structure is used to analyse the semantic features and generate guiding information toward dynamic weight sub-network. Here a backbone network is used as the feature extractor D to get the representative feature from origin images x. To facilitate subsequent processing for weight regulator, we get an one-dimension feature map ψ M ∈ R d after feature extractor D. The operation can be denoted as follows: ψ M = D(x).(19) Then the feature map ψ M is processed by the adversarial weight regulator, which has a type head to get the type predictions P type of input samples and signal heads to generate regulation signals ω j . For the design of type head structure, we use a fully connected layer W type ∈ R 2×d , and a softmax operation is applied to get the type predictions P type . The type head structure can be formulated as follows: P type = sof tmax(W type ψ M ).(20) For the design of signal head structure, in order to make full use of the entire information, individual signal heads are applied to generate regulation signals ω j for each jth block. Thus it is called as multi-head adversarial detetor sub-network. It can adjust the filter weights following Eq. (14) and allow every blocks to extract different semantic information of adversarial and clean examples at the filter level. Due to the instability of the neural network, directly using the feature map as regulation signals may lead to huge fluctuations whether in the training or testing process. Hence, a normalization operation toward the signal head are supposed to reduce the volatility. Here we use the tanh based function to make regulation signals more stable and can be formulated as follows: ω j = tanh(W j ψ M ) + 1 β ,(21) where W j ∈ R cout×d denotes the fully connected layer for the j-th block. β is a trade-off parameter to control the strength for guidance, if β is too high, the effect of weight adjustment will be too strong, and the network inference will be unstable. If β is too low, it will be difficult to distinguish adversarial and clean examples. All in all, with the favorable support of the adversarial detector sub-network, the dynamic weight sub-network can give full play to its ability to deal with adversarial and clean examples with different weights. Joint Adversarial Training Similar to the static networks, our AW-Net can also be trained using the existing various adversarial training algorithms based on the min-max formulation [26], such as TRADES [50], MART [42], and so on. However, to make full use of the dynamic characteristic in AW-Net, we choose Multi-teacher adversarial distillation (MTARD) [53] as the adversarial training algorithm to perform a joint training strategy. MTARD utilizes a clean teacher and a robust teacher to jointly train the network to achieve the balance between accuracy and robustness, which is highly compatible to our AW-Net. Specifically, we can apply these two teachers to train the different weights for the clean examples and adversarial examples, respectively. Clean examples are trained by the strong standard teacher T nat , and adversarial examples are trained by a strong adversarial teacher T adv , which is formulated as follow: L adv = τ 2 L KL (S τ (x adv ), T τ adv (x adv )),(22)L nat = τ 2 L KL (S τ (x nat ), T τ nat (x nat )),(23) where L nat and L adv denote adversarial distillation loss for clean and adversarial teacher, respectively, L KL denotes the Kullback-Leibler Divergence loss, S(·) denotes AW-Net, and S τ (·) denotes the tempered variant of S(·) with temperature τ [16]. Also, we design a classification loss to train the adversarial detector sub-network to get type predictions. The minimization loss of training the AW-Net is as follows: L total = L type (P type , y ′ ) + α 1 L adv + α 2 L nat ,(24) where L type denotes the type loss for the binary classification, y ′ is a label that can represent adversarial and clean types. α 1 and α 2 are the hy-parameters, which will be automatically adjusted in the training process as described in their original paper [53]. Therefore, we don't manually tune them in the experiment. More details about AW-Net For the dynamic weight sub-network, we replace the whole two-layer basic block with our AW-Net block mentioned in Section 4.2 based on the ResNet-18 [15], and the dimension of the final fully-connected layer adapts to the x's dimension of different applied dataset (e.g. CIFAR and Tiny-ImageNet) by average pooling operation. For the adversarial detector sub-network, an onedimensional vector ψ M of 512 size is obtained after the process of a lightweight feature extractor D. Then the feature vector ψ M is processed by the type head and the signal head mentioned in Section 4.3. After the process of type head W type , the feature vector ψ M transform into a 2-dimensional vector to predict the type predictions P type . For the signal head W j in j-th block, due to the two-layer block design in our setting, two individual signal heads W j are applied for two convolution layers in a block W conv j respectively and other setting is following the origin setting. EXPERIMENT Experiment Settings Datasets: We conduct the experiments on two public datasets, including CIFAR-10 [20], CIFAR-100 and Tiny-ImageNet [21]. For CIFAR-10 and CIFAR-100, the image size is 32 × 32, training set contains 50,000 images, and testing set contains 10,000 images. CIFAR-10 covers 10 classes while CIFAR-100 covers 100 classes. For Tiny-ImageNet dataset, the image size is 64 × 64. The training set contains 100,000 images, and testing set contains 10,000 images, covering 200 classes. Compared SOTA methods: Because our method proposes a novel network architecture, we compare with other state-of-the-art network architectures with similar parameter scales. To ensure fair comparisons, all the networks are trained with the same optimization algorithm. Specifically, for CIFAR-10 and CIFAR-100, the compared networks include ResNet-34 [15], ResNet-50 [15], WideResNet-34-8 [49], VGG-16-BN [32] and RepVGG-A2 [7]. All of them are trained by the same state-of-the-art adversarial distillation method [53] with multi-teachers. In addition, we also compare with a robust dynamic network [18]. Evaluation metric: Here we use the Weighted Robust Accuracy (W-Robust Acc) following [13] to comprehensively evaluate the accuracy and robustness, which is as follows: A w = γ nat A nat + γ adv A adv ,(25) where A w is the W-Roubst Acc computed from the clean accuracy A nat and the adversarial accuracy A adv . We set γ nat and γ adv both to 0.5. Implementation details: On CIFAR-10 and CIFAR-100 datasets, we use ResNet-18 as the basis to construct the dynamic weight sub-network, and then modify each residual block according to Section 4.2. Note that users can also adapt other networks like ResNet-34 and ResNet-50. The training optimizer is Stochastic Gradient Descent (SGD) optimizer with momentum 0.9 and weight decay 2e-4, and the initial learning rate is 0.1 for the dynamic weight network and 0.01 for the adversarial weight regulator, and the gradient of the feature extractor does not be updated. We train all the above models for 300 epochs, and the learning rate is divided by ten at the 215-th, 260-th, and 285th epoch, and the batch size is 128. We use PGD-10 (10 step PGD) to generate adversarial examples x adv with random start size 0.001 and step size 2/255, and the perturbation is bounded to the L ∞ norm ϵ = 8/255. β is set to 4 for CIFAR-10 and 1 for CIFAR-100, respectively. And τ is set to 1 for CIFAR-10 and 5 for CIFAR-100. The teacher models on CIFAR-10 and CIFAR-100 follow the setting in [53]. More Results about Filter Weight Distribution In order to better validate the weight difference between the robust model and the standard model, we validate against different network architectures. Here, in addition to the ResNet-18 network structures selected in Figure 3, we also chose three different network structures: including MobileNet-v2, ResNet-50, and WideResNet-34-8. The robust models are trained by MTARD [53]. The results are shown in Figure 4, Figure 5, and Figure 6. Also, we visualize the weight distribution for the above three models and the result is shown in Figure 7. The results show that the phenomenon about weight gap widely exists in different network structures but not just limited a particular network structure, which can effectively support our empirical judgment and theoretical analysis on the model weights' gap between the clean model and the robust model. Robustness Evaluation Robustness against White-box Attacks. We evaluate AW-Net and other comparison models against four white-box attack methods: FGSM [10], PGD sat [26], PGD trades [50], CW ∞ [4]. The step size of both two PGD attacks is 2/255 and total step is 20, the difference of two PGD attacks lies in the initialization. The step of CW ∞ is 30, and the perturbation of PGD and CW ∞ is bounded to the L ∞ norm ϵ = 8/255. Based on the results in Table 1, we find AW-Net can achieve the best W-Robust Acc compared with state-of-theart robust models with similar parameter scales. On CIFAR-100, AW-Net improves improves W-Robust Acc by 1.53% against PGD sat attack compared with other robust models. The relevant advantages also display in the evaluation on other dataset (CIFAR-10) and other attack (FGSM, PGD trades , and CW ∞ ). Robustness against Black-box Attacks. Similar to other works [22], [42], [50], we use the transfer-based attacks to evaluate the models' black-box robustness. For that, we conduct the white-box attacks on a surrogate model, and then evaluate the black-box robustness using these generated adversarial examples. Here, we select the strong robust model: WideResNet-70-16 [11] as surrogate model to generate adversarial examples. The used black-box attacks are PGD sat and CW ∞ . The results in Table 2 show that our AW-Net can outperform the state-of-the-art robust models and achieve better W-robust Acc. AW-Net improve W-robust Acc by 1.53% and 1.55% on CIFAR-10 against PGD sat attack and CW ∞ attack respectively. Meanwhile, AW-Net can achieve similar performance on CIFAR-100, which has an improvement of 0.83% against the CW ∞ attack. The robustness against blackbox attacks shows the effectiveness of our AW-Net. Ablation Study In order to verify the effectiveness of each component in our AW-Net, we conduct a set of ablation studies. Initially, we apply the adversarial detector sub-network to generate adversarial regulation signals and guide the dynamic weight network to distinguish clean and adversarial examples for different filters but without MixBN structure. Then, we explore the improvement brought by MixBN structure but without type predictions from the adversarial detector sub-network in the testing process, which only use the Adv BN. Furthermore, we use type predictions as MixBN weights, which is the final structure of our AW-Net. Meanwhile, to better show the effectiveness, we also report the performance of the baseline network: ResNet-18 as the comparison. The results are shown in Table 3. From the Table 3, we see that only using the dynamic weight cannot achieve the good performance. After adding the MixBN, the performance achieve a remarkable increase, which shows the necessity to use MixBN to encode different feature distributions for clean and adversarial examples. Meanwhile, when the MixBN is weighted with the type predictions, the performance has also been obviously improved, showing the effectiveness of the adversarial detector sub-network. As a comparison with the Baseline network, our dynamic network achieves a nearly 2% improvement, showing the effectiveness of our idea. 1 White-box robustness on CIFAR-10 and CIFAR-100 datasets. All the models are trained following the state-of-the-art adversarial training method MTARD [53]. All the results are the best checkpoints based on W-Robust Acc. Different Training Methods Here we try to explore the effect of different training algorithm on our AW-Net. We choose two different state-of-theart methods, MTARD [53] and TRADES [50], and visualize the performance and the training loss of AW-Net on CIFAR-10 and CIFAR-100 in Figure 8. The results show the model trained based on MTARD has better performances than TRADES, our model can achieve good performance with the help of MTARD. Experiment for β in AW-Net Here we explore the role of β in 14 used in our AW-Net. We select the AW-Net trained on CIFAR-10 and test the results using different β values. The results are shown in Table 4. The results show that the proper parameter settings has an influence on the final results. If β is too high, the network will fluctuate and lead to a lower performance. If β is too low, the adversarial and clean examples will be not recognized well. Finally, we choose the β as 4 in our setting for CIFAR-10 and CIFAR-100. Other Dynamic Networks Besides the static networks above, we here compare our AW-Net with a robust multi-exit dynamic weight network: RDI-ResNet-38 [18]. The evaluation is based on the PGD sat with step 20. The result is shown in Table 5, where shows that AW-Net has an obvious advantage compared with RDI-ResNet-38 [18]. The W-Robust Accuracy is improved from 63.54% to 71.00%, which demonstrates the effectiveness of the proposed dynamic network versus the varied-weights. Robustness on Tiny-ImageNet For Tiny-ImageNet dataset, we compare with the ResNet-34 [15]. Both our AW-Net and the baseline ResNet-34 are denotes the dynamic weight sub-network with the guidance of the adversarial detector; "MixBN" represents using the MixBN layer instead of single BN layer in the dynamic weight sub-network; "Type Head" denotes using the type predictions in the adversarial detector sub-network as the weights of MixBN. "Baseline" is the basis network structure to construct AW-Net. trained with state-of-the-art adversarial training algorithm: MTARD [53], the setting is also consistent. We train the above models for 50 epochs, and the learning rate is divided by 25-th and 40-th epoch, and the batch size is 64. β is set to 1 and τ is set to 5. The results on the Tiny-ImageNet are given in Table 6, showing that AW-Net can outperform ResNet-34 model when facing different attacks. Specifically, AW-Net improves W-Robust Acc by 4.49% against FGSM attack and improves W-Robust Acc by 5.18% against PGD sat attack. The results show that AW-Net can achieve better performance with both accuracy and robustness, and is still effective in more challenging dataset. Adaptive Attack According to the adaptive attack definition in [36], we augment the original objective function used in the PGD trades attack with our joint training loss (i.e., Eq. (14) in the original paper) to implement adaptive attacks. The comparison results of CIFAR-10 and CIFAR-100 under different attacks are shown in Table 7. Under the adaptive attack, we see that the Robust Accuracy of CIFAR-10 is 50.22%, which is almost the same with the W-Robust Accuracy under the original PGD trades attack (50.51%). In CIFAR-100 dataset, the W-Robust Accuracy under adaptive attack is even slightly higher than that under the original attack (26.21% vs 24.83%). These results verify the robustness of AW-Net against adaptive attacks. CONCLUSION In this paper, we investigated the state-of-the-art method and explored the reason of the accuracy and robustness of the model cannot be achieved simultaneously from the empirical and theoretical perspective. We argued that it may be caused by the gap in the filters' weight distribution. Based on the above observation, we proposed the Adversarial Weight-Varied Network (AW-Net) to deal with the clean and adversarial example with different filters' weights to selfadapt input samples. We used MixBN to fit the distributions of clean and adversarial examples, and applied an adversarial detector sub-network to distinguish the clean and adversarial sample. The generated regulation signals were applied to guide the adjustment of the dynamic network. A series of solid experiments proved that AW-Net could achieve state-of-the-art performance in W-Robust accuracy compared with other robust models on three datasets. Fig. 1 . 1The distribution versus the means of filters' weights in the second convolution layer in ResNet-18 on CIFAR-10 toward a standard model and three state-of-the-art robust models trained by Fig. 2 . 2The framework of our AW-Net. AW-Net includes two main branches: the dynamic weight sub-network composed of multiple AW-Net blocks and the adversarial detector sub-network to discriminate the clean and adversarial example and generate the regulation signals. In the training period, we utilize the MixBN (including Adv BN and Clean BN) to handle different feature distributions of clean and adversarial examples, respectively. In the testing period, Adv BN and Clean BN are weighted by the prediction of adversarial detector. Fig. 3 . 3The visualization of filters' means and variances for a strong standard model and a strong robust model trained by MART[42] on CIFAR-10. The left two figures are visualization of means and variances for the first layer in ResNet18; And the right two figures are visualization of means and variances for the second layer in ResNet 18. For simplicity, we only show these two layers, and the other layers also show the same phenomenon. Inspired by the gated batch normalization [25] , we design a MixBN, where a Clean BN BN nat is used to handle the features π nat j of clean examples x nat and Adv BN BN adv is used to handle the features π adv j of adversarial examples x adv . In the training process of the MixBN structure, the category of features can be obtained and we perform an operation of type selection, and separately train the Adv BN and Clean BN with adversarial examples x adv and clean examples x nat . The training process of MixBN can be denoted as follows: Fig. 4 .Fig. 5 . 45The visualization of filters' means and variances for a strong standard model and a strong robust model trained by MTARD [53] on CIFAR-10. The left two figures are visualization of means and variances for the first layer in MobileNet-v2; And the right two figures are visualization of means and variances for the second layer in MobileNet-v2. The visualization of filters' means and variances for a strong standard model and a strong robust model trained by MTARD [53] on CIFAR-10. The left two figures are visualization of means and variances for the first layer in ResNet-50; And the right two figures are visualization of means and variances for the second layer in ResNet-50. Fig. 6 .Fig. 7 . 67The visualization of filters' means and variances for a strong standard model and a strong robust model trained by MTARD [53] on CIFAR-10. The left two figures are visualization of means and variances for the first layer in WideResNet-34-8; And the right two figures are visualization of means and variances for the second layer in WideResNet-34-8. The distribution versus the means of filters' weights in the first and second convolution layer in MobileNet-v2, ResNet-50, and WideResNet-34-8 on CIFAR-10 toward standard models and robust models trained by[53]. Fig. 8 . 8The performance and training loss of AW-Net trained by TRADES and MTARD on CIFAR-10 (Top) and CIFAR-100 (Bottom), respectively. All the results are the best checkpoints based on W-Robust Accuracy. TABLE TABLE 2 2Black-box robustness on CIFAR-10 and CIFAR-100 dataset. All the models are trained following the same state-of-the-art adversarial training method MTARD[53]. All the results are the best checkpoints based on W-Robust Acc.CIFAR-10 CIFAR-100 TABLE 3 3Ablation study toward each component of AW-Net. "Dynamic Weight" TABLE 4 4The performance of AW-Net with different β. All the results are the best checkpoints based on Aw. The White-box Robustness of AW-Net and[18].Attack β value Anat A adv Aw PGDsat β = 1 90.30% 50.63% 70.47% β = 4 91.49% 50.44% 71.00% β = 5 92.04% 48.23% 70.14% CW∞ β = 1 90.30% 48.36% 69.33% β = 4 91.49% 47.65% 69.57% β = 5 92.04% 46.16% 69.10% TABLE 5 Attack model Anat A adv Aw PGDsat RDI-ResNet-38 [18] 83.79% 43.28% 63.54% AW-Net (ours) 91.49% 50.44% 71.00% TABLE 6 6White-box Robustness on Tiny-ImageNet. All the results are the best checkpoints based on Aw.Attack models Anat A adv Aw TABLE 7 Adaptive 7Attack toward AW-Net. 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[ "AN UPPER BOUND OF THE MINIMAL ASYMPTOTIC TRANSLATION LENGTH OF RIGHT-ANGLED ARTIN GROUPS ON EXTENSION GRAPHS", "AN UPPER BOUND OF THE MINIMAL ASYMPTOTIC TRANSLATION LENGTH OF RIGHT-ANGLED ARTIN GROUPS ON EXTENSION GRAPHS" ]
[ "Eon-Kyung ", "Sang-Jin Lee " ]
[]
[]
For the right-angled Artin group action on the extension graph, it is known that the minimal asymptotic translation length is bounded above by 2 provided that the defining graph has diameter at least 3. In this paper, we show that the same result holds without any assumption. This is done by exploring some graph theoretic properties of biconnected graphs, i.e. connected graphs whose complement is also connected.
null
[ "https://export.arxiv.org/pdf/2306.03326v1.pdf" ]
259,088,806
2306.03326
5ee32f5b38e4d56affe5f9096e4e3defd7494fb7
AN UPPER BOUND OF THE MINIMAL ASYMPTOTIC TRANSLATION LENGTH OF RIGHT-ANGLED ARTIN GROUPS ON EXTENSION GRAPHS 6 Jun 2023 Eon-Kyung Sang-Jin Lee AN UPPER BOUND OF THE MINIMAL ASYMPTOTIC TRANSLATION LENGTH OF RIGHT-ANGLED ARTIN GROUPS ON EXTENSION GRAPHS 6 Jun 2023right-angled Artin groupsextension graphstranslation lengthbiconnected graphs 2020 Mathematics Subject Classification: 20F3620F6520F6957M1557M60 For the right-angled Artin group action on the extension graph, it is known that the minimal asymptotic translation length is bounded above by 2 provided that the defining graph has diameter at least 3. In this paper, we show that the same result holds without any assumption. This is done by exploring some graph theoretic properties of biconnected graphs, i.e. connected graphs whose complement is also connected. Introduction For a simplicial graph Γ = (V (Γ), E(Γ)), let d Γ (v 1 , v 2 ) denote the graph metric on V (Γ), i.e. the length of the shortest path in Γ from v 1 to v 2 . Let diam(Γ) denote the diameter of Γ, i.e. diam(Γ) = max{d Γ (v 1 , v 2 ) : v 1 , v 2 ∈ V (Γ)}. LetΓ denote the complement graph of Γ, i.e. V (Γ) = V (Γ) and E(Γ) = {{v 1 , v 2 } ⊂ V (Γ) : v 1 = v 2 , {v 1 , v 2 } ∈ E(Γ)}. Γ is called nontrivial if |V (Γ)| 2. Γ is called biconnected if both Γ andΓ are connected. The right-angled Artin group A(Γ) on a finite simplicial graph Γ is the group generated by the vertices of Γ such that two generators commute if they are adjacent, hence A(Γ) = v ∈ V (Γ) : v i v j = v j v i if {v i , v j } ∈ E(Γ) . The extension graph Γ e of Γ is the graph such that the vertex set V (Γ e ) is the set of all elements of A(Γ) that are conjugate to a vertex of Γ, and two vertices v g 1 1 and v g 2 2 are adjacent in Γ e if and only if they commute as elements of A(Γ). (Here, v g denotes the conjugate g −1 vg.) Therefore V (Γ e ) = {v g : v ∈ V (Γ), g ∈ A(Γ)}, E(Γ e ) = { {v g 1 1 , v g 2 2 } : v g 1 1 , v g 2 2 ∈ V (Γ e ), v g 1 1 v g 2 2 = v g 2 2 v g 1 1 in A(Γ) }. Extension graphs are usually infinite and locally infinite. They are very useful in the study of right-angled Artin groups such as the embeddability problem between right-angled Artin groups [KK13,KK14a,LL16,LL18]. Let d e denote the graph metric d Γ e . It is known that (Γ e , d e ) is a quasi-tree, and hence a hyperbolic graph [KK13]. Throughout this paper, all graphs are simplicial, and assumed to be finite except for extension graphs. Definition 1.1. Suppose a group G acts on a connected metric space (X, d) by isometries from right. The asymptotic translation length of an element g ∈ G, denoted τ (X,d) (g) or τ (g), is defined by τ (g) = lim n→∞ d(xg n , x) n , where x ∈ X. This limit always exists, is independent of the choice of x, and satisfies τ (g n ) = |n|τ (g) and τ (h −1 gh) = τ (g) for all g, h ∈ G and n ∈ Z. If τ (g) > 0, g is called loxodromic. If {d(xg n , x)} ∞ n=1 is bounded, g is called elliptic. If τ (g) = 0 and {d(xg n , x)} ∞ n=1 is unbounded, g is called parabolic. The minimal asymptotic translation length of G for the action on (X, d), denoted L (X,d) (G), is defined by L (X,d) (G) = min{τ (X,d) (g) : g ∈ G, τ (X,d) (g) > 0}. The (minimal) asymptotic translation lengths have been studied for the actions of various geometric groups (see [LL07, GT11, KS19, BS20, Gen22, BSS23] for example). For the action of A(Γ) on (Γ e , d e ), Baik, Seo and Shin showed the following which is an analogue of the theorem of Bowditch [Bow08, Theorem 1.4]. Theorem 1.2 ([BSS23, Theorem A]). Let Γ be a nontrivial biconnected graph. For the action of A(Γ) on (Γ e , d e ), all loxodromic elements have rational asymptotic translation lengths. If Γ has girth at least 6 in addition, the asymptotic translation lengths have a common denominator. As in the above theorem, it is natural to require that "Γ is a nontrivial biconnected graph" when we consider the action of A(Γ) on Γ e , because Γ e is connected with infinite diameter if and only if Γ is nontrivial and biconnected [KK13,Lemma 3.5]. Observe that if |V (Γ)| ∈ {2, 3} then either Γ orΓ is disconnected. Hence the statement that "Γ is a nontrivial biconnected graph" implies |V (Γ)| 4. For the lower bound of the minimal asymptotic translation length of A(Γ) on (Γ e , d e ), it follows from a result of Kim and Koberda [KK14b,Lemma 33 ] that L (Γ e , de) (A(Γ)) 1 2|V (Γ)| 2 . This lower bound is improved to L (Γ e , de) (A(Γ)) 1 |V (Γ)|−2 in [LL22, Theorem 6.5]. For the upper bound, Baik, Seo and Shin showed the following. Theorem 1.3 ([BSS23, Theorem F]). Let Γ be a nontrivial biconnected graph. If diam(Γ) 3, then L (Γ e , de) (A(Γ)) 2. The graphs with diameter 2 form a large class of graphs. Therefore it would be natural to ask whether the same upper bound holds for these graphs. The following is the main result of this paper, which shows that the above upper bound holds without any assumption. If diam(Γ) is large, we obtain a better upper bound as follows. Theorem B (Theorem 4.5) Let Γ be a biconnected graph. If d = diam(Γ) 3, then L (Γ e , de) (A(Γ)) 2 d − 2 . Theorem B says that if d = diam(Γ) is large, then L (Γ e , de) (A(Γ)) is small. One may ask whether the converse holds, i.e. if L (Γ e , de) is small, then diam(Γ) is large. The following theorem shows that this is not the case. Theorem C (Theorem 4.6) For any n 4, there exists a biconnected graph Γ n such that |V (Γ n )| = n + 2, diam(Γ n ) = diam(Γ n ) = 2, L (Γ e n , de) (A(Γ n )) 2 n − 3 . Preliminaries Notations on graphs. For v ∈ V (Γ) and A ⊂ V (Γ), the stars St Γ (v) and St Γ (A) are defined as St Γ (v) = {u ∈ V (Γ) : d Γ (u, v) 1}, St Γ (A) = v∈A St Γ (v) = {u ∈ V (Γ) : d Γ (u, v) 1 for some v ∈ A}. For an edge e = {v 1 , v 2 }, the star St Γ ({v 1 , v 2 }) is also denoted by St Γ (e). A) = V (Γ). For A ⊂ V (Γ), Γ[A] denotes the subgraph of Γ induced by A, i.e. V (Γ[A]) = A, E(Γ[A]) = { {v 1 , v 2 } : v 1 , v 2 ∈ A, {v 1 , v 2 } ∈ E(Γ)}. A graph Γ 0 is called an induced subgraph of Γ if Γ 0 = Γ[A] for some A ⊂ V (Γ) . Notice the following. For graphs Γ 1 and Γ 2 , the disjoint union Γ 1 ⊔ Γ 2 is the graph such that V (Γ 1 ⊔ Γ 2 ) = V (Γ 1 ) ⊔ V (Γ 2 ), E(Γ 1 ⊔ Γ 2 ) = E(Γ 1 ) ⊔ E(Γ 2 ). The join Γ 1 * Γ 2 is the graph such that Γ 1 * Γ 2 = Γ 1 ⊔ Γ 2 , hence V (Γ 1 * Γ 2 ) = V (Γ 1 ) ⊔ V (Γ 2 ), E(Γ 1 * Γ 2 ) = E(Γ 1 ) ⊔ E(Γ 2 ) ⊔ { {v 1 , v 2 } : v 1 ∈ V (Γ 1 ), v 2 ∈ V (Γ 2 ) }. A graph is called a join if it is the join of two nonempty graphs. A subgraph that is a join is called a subjoin. The path graph P n = P n (v 1 , . . . , v n ) is the graph with V (P n ) = {v 1 , . . . , v n } and E(P n ) = {{v i , v i+1 } : 1 i n−1}. For example, P 4 (v 1 , v 2 , v 3 , v 4 ) looks like v 1 • v 2 • v 3 • v 4 • and its complement is P 4 (v 3 , v 1 , v 4 , v 2 ) . Notice that P n is biconnected for all n 4 and that P 4 is the only biconnected graph with four vertices. Right-angled Artin groups. An element of V (Γ) ±1 = V (Γ) ∪ V (Γ) −1 is called a letter. A word means a finite sequence of letters. Suppose that g ∈ A(Γ) is expressed as a word w. The word w is called reduced if w is a shortest word among all the words representing g. In this case, the length of w is called the word length of g. Definition 2.2 (support). For g ∈ A(Γ), the support of g, denoted supp(g), is the set of generators that appear in a reduced word representing g. It is known that supp(g) is well defined (see [HM95]), i.e. it does not depend on the choice of a reduced word representing g. Definition 2.3 (cyclically reduced). An element g ∈ A(Γ) is called cyclically reduced if it has the minimal word length in its conjugacy class. Notice that every positive word (i.e. a word composed of only letters in V (Γ)) is both reduced and cyclically reduced. The following equivalences are well known (see [KK14b,LL22] for example). Lemma 2.4. Let Γ be a nontrivial biconnected graph. For a cyclically reduced element g ∈ A(Γ) \ {1}, the following are equivalent: (i) g is loxodromic on (Γ e , d e ); (ii) supp(g) is not contained in a subjoin of Γ; (iii)Γ[supp(g)] is connected and supp(g) dominatesΓ. Biconnected graphs In this section we study biconnected graphs. Theorem 3.5 is the main result of this section. We begin with three easy lemmas. A vertex v of Γ is called a cut vertex of Γ if Γ \ v is disconnected. v 0 • • Γ 2 ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ • ( ( ( ( ( ( ( ( ( v 2 • • Γ 1 ❡❡ ❡❡ ❡❡ ❡❡ ❡ • (( (( (( (( ( • v 1 • Figure 1. Figure for Lemma 3.1. Lemma 3.1. Let Γ be a biconnected graph. If Γ has a cut vertex, then diam(Γ) 3. Proof. Let v 0 be a cut vertex of Γ. Then Γ \ v 0 is a disjoint union Γ 1 ⊔ Γ 1 for some nontrivial subgraphs Γ 1 and Γ 2 . (See Figure 1.) If d Γ (v 0 , v) 1 for all v ∈ V (Γ), then Γ = {v 0 } * (Γ \ v 0 ), which contradicts thatΓ is connected. Therefore there exists a vertex v 1 such that d Γ (v 0 , v 1 ) 2. We may assume v 1 ∈ Γ 1 . Choose a vertex v 2 of Γ 2 . Because v 0 is a cut vertex, each path from v 1 to v 2 must pass through v 0 . Therefore d Γ (v 1 , v 2 ) = d Γ (v 1 , v 0 ) + d Γ (v 0 , v 2 ) 2 + 1 = 3, hence diam(Γ) 3. Lemma 3.2. Let A and B be subsets of V (Γ) such that A ⊂ B ⊂ St Γ (A). If Γ[A] is connected then Γ[B] is also connected. Proof. Since A ⊂ B, the graph Γ[B] is obtained from the connected graph Γ[A] by adding some vertices and some edges. Since B ⊂ St Γ (A), each vertex in Γ[B] \ Γ[A] is adjacent to a vertex in Γ[A], hence Γ[B] is connected. Lemma 3.3. For any v ∈ V (Γ), St Γ (v) dominatesΓ. For any e ∈ E(Γ), St Γ (e) dominates Γ. Proof. For a vertex v, each u ∈ V (Γ) \ {v} is adjacent to v either in Γ or inΓ, hence V (Γ) = St Γ (v) ∪ StΓ(v) ⊂ StΓ(St Γ (v)). This means that St Γ (v) dominatesΓ. For an edge e = {v 1 , v 2 }, St Γ (e) dominatesΓ because St Γ (v 1 ) ⊂ St Γ (e) and St Γ (v 1 ) dominatesΓ. Lemma 3.4. Let Γ be a biconnected graph. If either diam(Γ) 3 or diam(Γ) 3, then there exists an edge e in Γ such that Γ[St Γ (e)] is biconnected. Proof. Since Γ[St Γ (e)] is connected for any edge e, it suffices to show thatΓ[St Γ (e)] is connected for some edge e of Γ. We use the following claim. Claim. Let v 1 , v 2 ∈ V (Γ). (i) If dΓ(v 1 , v 2 ) 3, then {v 1 , v 2 } dominates Γ, i.e. St Γ ({v 1 , v 2 }) = V (Γ). (ii) If d Γ (v 1 , v 2 ) 3, then {v 1 , v 2 } dominatesΓ, i.e. StΓ({v 1 , v 2 }) = V (Γ). Proof of Claim. Since (i) and (ii) are equivalent, we prove only (i). Suppose dΓ(v 1 , v 2 ) 3. We For each v ∈ V (Γ), either dΓ(v, v 1 ) 2 or dΓ(v, v 2 ) 2 because otherwise dΓ(v 1 , v 2 ) dΓ(v 1 , v) + dΓ(v, v 2 ) 1 + 1 = 2. Therefore either {v, v 1 } ∈ E(Γ) or {v, v 2 } ∈ E(Γ). This means that St Γ ({v 1 , v 2 }) = V (Γ). v 1 • v 2 • v 3 • v 4 • v 5 • v 6 • v 1 • v 2 • v 3 • v 4 • v 5 • v 6 • v 0 • ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ • v 1 • v 2 • v 3 • v 4 • v 5 ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✕ ✕ ✕ ✕ ✕ ✕ ✕ ✕ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍now assume diam(Γ) 3. Choose v 1 , v 4 ∈ V (Γ) such that d Γ (v 1 , v 4 ) = 3. Choose a path (v 1 , v 2 , v 3 , v 4 ) from v 1 to v 4 , and let e = {v 2 , v 3 } and A = {v 1 , v 2 , v 3 , v 4 }. Then Γ[A] is the path graph P 4 (v 1 , v 2 , v 3 , v 4 ), henceΓ[A] is the path graph P 4 (v 2 , v 4 , v 1 , v 3 ), which is connected. Notice that {v 1 , v 4 } ⊂ A ⊂ St Γ (e). Since d Γ (v 1 , v 4 ) 3, {v 1 , v 4 } dominatesΓ by the above claim, hence A also dominatesΓ. Therefore A ⊂ St Γ (e) ⊂ V (Γ) = StΓ(A). SinceΓ[A] is connected,Γ[St Γ (e)] is also connected (by Lemma 3.2). Using the above lemmas, we establish the following theorem. Example 3.6. Let Γ be the path graph P 6 in Figure 2 We use an induction on |V (Γ)|. If |V (Γ)| = 4, then Γ must be the path graph P 4 , hence we are done (by Lemma 3.4). We now assume |V (Γ)| 5. Choose a vertex v 0 , and let Γ 0 = Γ \ v 0 . By Lemma 3.1, if Γ 0 is disconnected then diam(Γ) 3, and if Γ 0 is disconnected then diam(Γ) 3. In either case, we are done (by Lemma 3.4). Therefore we may assume that Γ 0 is biconnected. • v 0 • v 1 • v 2 • v 3 ✎ ✎ ✎ ✎ e = e 0 St Γ 0 (e 0 ) By induction hypothesis, there exists an edge e 0 = {v 1 , v 2 } of Γ 0 such that Γ 0 [St Γ 0 (e 0 )] is biconnected. Notice the following: • St Γ (e 0 ) is either St Γ 0 (e 0 ) or St Γ 0 (e 0 ) ∪ {v 0 }; •Γ[St Γ 0 (e 0 )] = Γ 0 [St Γ 0 (e 0 )] = Γ 0 [St Γ 0 (e 0 )]; • Γ 0 [St Γ 0 (e 0 )] is connected because Γ 0 [St Γ 0 (e 0 )] is biconnected. Case 1. v 0 does not dominate St Γ 0 (e 0 ) in Γ. We will show that in this caseΓ[St Γ (e 0 )] is connected. Then we are done by taking e = e 0 . If v 0 is adjacent to neither v 1 nor v 2 in Γ, then St Γ (e 0 ) = St Γ 0 (e 0 ), hencē Figure 4, where v 4 ∈ St Γ 0 (e 0 ) in the left and v 4 ∈ St Γ 0 (e 0 ) in the right. Γ[St Γ (e 0 )] =Γ[St Γ 0 (e 0 )] = Γ 0 [St Γ 0 (e 0 )]. Since Γ 0 [St Γ 0 (e 0 )] is connected, we are done. If v 0 is adjacent to either v 1 or v 2 in Γ, then St Γ (e 0 ) = St Γ 0 (e 0 )∪ {v 0 }. Since v 0 does not dominate St Γ 0 (e 0 ) in Γ, there is a vertex v 3 ∈ St Γ 0 (e 0 ) that is not adjacent to v 0 in Γ (hence v 0 is adjacent to v 3 inΓ). SeeCase 2. v 0 dominates St Γ 0 (e 0 ) in Γ. Notice that St Γ (e 0 ) = St Γ 0 (e 0 ) ∪ {v 0 } ⊂ St Γ (v 0 ). There exists a vertex v 3 ∈ V (Γ)\ St Γ 0 (e 0 ) such that d Γ (v 0 , v 3 ) = 2 because otherwise d Γ (v 0 , v) 1 for all v ∈ V (Γ), hence Γ = {v 0 } * Γ 0 , which contradicts thatΓ is connected. Choose v 4 ∈ V (Γ) such that (v 0 , v 4 , v 3 ) is a path in Γ from v 0 to v 3 . See Let e = {v 0 , v 4 }. We will show thatΓ[St Γ (e)] is connected. Let A = St Γ 0 (e 0 ) ∪ {v 0 , v 3 , v 4 }, and observe the following. 3.3). Therefore Figure 4(a)) or {v 1 , v 4 } is an edge ofΓ (as in Figure 4(b)), the graph • v 0 ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ • v 4 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( • v 2 ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ • v 1 • v 3 ✗ ✗ ✗ ✗ ✗ e 0 ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ❡❡ ❡❡ ❡❡ ❡❡ ❡❡ ❡❡ ❡❡ ❡❡ ❡❡ e St Γ 0 (e 0 ) • v 0 ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ • v 1 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( • v 2 • v 4 • v 3 ✗ ✗ ✗ ✗ ✗ e 0 ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ e St Γ 0 (e 0 ) (a) v 4 ∈ St Γ 0 (e 0 ) (b) v 4 ∈ St Γ 0 (e 0 )A ⊂ St Γ (e) ⊂ V (Γ) = StΓ(A). • Since Γ 0 [St Γ 0 (e 0 )] is connected and since {v 1 , v 3 } and {v 3 , v 0 } are edges ofΓ, the graphΓ[St Γ 0 (e 0 ) ∪ {v 0 , v 3 }] is connected. Since either v 4 is contained in St Γ 0 (e 0 ) (as inΓ[St Γ 0 (e 0 ) ∪ {v 0 , v 3 , v 4 }] =Γ[A] is connected. Since A ⊂ St Γ (e) ⊂ StΓ(A) andΓ[A] is connected,Γ[St Γ (e)] is connected (by Lemma 3.2). The following proposition will be used in the proof of Theorem 4.6. Proposition 3.7. For each n 3, there exists a biconnected graph Λ n satisfying the following: (i) diam(Λ n ) = diam(Λ n ) = 2 and |V (Λ n )| = n + 2; (ii) Λ n contains a path graph P n as an induced subgraph in a way that V (P n ) dominates both Λ n and Λ n . Proof. If n = 3, let Λ 3 be the pentagon such that P 3 (v 1 , v 2 , v 3 ) is an induced subgraph of Λ 3 as in the left of Figure 5. It is straightforward to see that Λ 3 has the desired properties. If n 4, let Λ n be the graph obtained from P n = P n (v 1 , . . . , v n ) by adding two vertices x and y and by adding edges {x, v i } for i ∈ {2, . . . , n} and {y, v j } for j ∈ {1, 4, 5, . . . , n} as in the right of Figure 5 for the case of n = 7. It is straightforward to see that Λ n has the desired properties. Minimal asymptotic translation lengths In this section, we establish our results on the minimal asymptotic translation lengths of right-angled Artin groups on extension graphs. Let us abbreviate the translation length τ (Γ e ,de) (g) to τ e (g). For Lemma 4.1. Let {v 1 , v 2 } ∈ E(Γ) and g 1 , g 2 ∈ A(Γ) such that g 1 ∈ Z(v 1 ) and g 2 ∈ Z(v 2 ), and let g = g 1 g 2 . Then τ e (g) 2. v ∈ V (Γ), let Z(v) denote the centralizer of v in A(Γ), i.e. Z(v) = {g ∈ A(Γ) : gv = vg in A(Γ)}. It is well known that Z(v) is generated by St Γ (v). v 1 • v 2 • v 3 • x • y • v 1 • v 2 • v 3 • v 4 • v 5 • v 6 • v 7 • x • ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ✯ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ y • ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② q q q q q q q q q q q q q q q q q q q ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ Figure 5. Figures for Proposition 3.7 • v 1 • v 2 ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ • v g 1 ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ • v g 2 ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ • v g 2 1 ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ • v g 2 2 ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ • v g 3 1 ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ⑧ ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ Proof. Notice that {v h 1 , v h 2 } ∈ E(Γ e ) for any h ∈ A(Γ). In particular, {v g 2 1 , v g 2 2 } ∈ E(Γ e ). Since v g 2 1 = (v g 1 1 ) g 2 = v g 1 g 2 1 = v g 1 and v g 2 2 = v 2 , we have {v g 1 , v 2 } ∈ E(Γ e ). Now (v 1 , v 2 , v g 1 ) is a path in Γ e , hence (v 1 , v 2 , v g 1 , v g 2 , v g 2 1 , v g 2 2 , v g 3 1 , . . .) is also a path in Γ e . See Figure 6. Therefore d e (v 1 , v g n 1 ) 2n for all n 1, which implies τ e (g) 2. In the above lemma, the element g may not be loxodromic. The following lemma gives a sufficient condition under which a loxodromic element g with τ e (g) 2 exists. Proof. Let e = {v 1 , v 2 }, St Γ (v 1 ) = {v 1 , x 1 , . . . , x p } and St Γ (v 2 ) = {v 2 , y 1 , . . . , y q }. Let g 1 = v 1 x 1 · · · x p , g 2 = v 2 y 1 · · · y q , g = g 1 g 2 . Then g i ∈ Z(v i ) for i = 1, 2, hence τ e (g) 2 (by Lemma 4.1). Therefore it suffices to show that g is loxodromic. Since each g i is represented by a positive word on generators, g is cyclically reduced. Notice that supp(g) = supp(g 1 ) ∪ supp(g 2 ) = St Γ (v 1 ) ∪ St Γ (v 2 ) = St Γ (e). Since St Γ (e) dominatesΓ (by Lemma 3.3) andΓ[St Γ (e)] is connected by the hypothesis, g is loxodromic on Γ e (by Lemma 2.4). Combining the above lemma with Theorem 3.5 yields the following theorem. Proof. By Theorem 3.5, there exists an edge e = {v 1 , v 2 } in Γ such that Γ[St Γ (e)] is biconnected. By Lemma 4.2, there exists a loxodromic element g ∈ A(Γ) such that τ e (g) 2. In Theorem 4.5, we will obtain a better upper bound for the minimal asymptotic translation length for the case where diam(Γ) is large. For this, we first show the following lemma. Lemma 4.4. Let Γ be a nontrivial biconnected graph. Suppose that there exists an induced subgraph Λ of the complement graphΓ such that (i) Λ is connected and d = diam(Λ) 3; (ii) V (Λ) dominatesΓ. Then there exists a loxodromic element g ∈ A(Γ) such that τ e (g) 2 d−2 . Therefore Figure 7). Then each V k is nonempty and V (Λ) = V 0 ⊔ · · · ⊔ V d . Let V k = {v k1 , v k2 , . . . v kr k }. Define g k = v k1 · · · v kr k for 0 k d, g = g 0 g 1 · · · g d , h 0 = (g 0 g 1 · · · g d−2 )(g 0 g 1 · · · g d−3 ) · · · (g 0 g 1 g 2 )(g 0 g 1 ), L (Γ e ,de) (A(Γ)) 2 d − 2 . Proof. Since d = diam(Λ) 3, there are vertices v 0 , v d ∈ V (Λ) such that d Λ (v 0 , v d ) = d 3. For each 0 k d, let V k = {v ∈ V (Λ) : d Λ (v 0 , v) = k} (seeh d = (g d−1 g d )(g d−2 g d−1 g d ) · · · (g 3 · · · g d )(g 2 · · · g d ). Observe the following. • supp(g k ) = V k for each 0 k d, hence g i g j = g j g i if |i − j| 2; • g is cyclically reduced and supp(g) = V (Λ); • sinceΓ[supp(g)] = Λ is connected and supp(g) = V (Λ) dominatesΓ, g is loxodromic on (Γ e , d e ) (by Lemma 2.4); • g 0 , . . . , g d−2 ∈ Z(v d ), hence h 0 ∈ Z(v d ); • g 2 , g 3 , . . . , g d ∈ Z(v 0 ), hence A straightforward computation using the commutativity g i g j = g j g i for |i − j| 2 shows that h d ∈ Z(v 0 ). v 0 • V 0 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ • ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ • . . . ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ • | | | | | | | | | | | | | • V 1 • • . . . • • V 2 • v d • . . . • • V dh 0 h d = g d−2 . Since d Λ (v 0 , v d ) 3, {v 0 , v d } is an edge ofΛ and hence of Γ. Since h 0 ∈ Z(v d ), h d ∈ Z(v 0 ) and g d−2 = h 0 h d , we have τ e (g d−2 ) 2 (by Lemma 4.1). Therefore τ e (g) 2 d−2 . Since g is loxodromic, we are done. By putting Λ =Γ in the above lemma, we obtain the following theorem. From the above theorem, if d = diam(Γ) is large, then L (Γ e , de) (A(Γ)) is small. One may ask whether the converse holds, i.e. if L (Γ e , de) (A(Γ)) is small, then diam(Γ) is large. The following theorem shows that this is not the case. Theorem 4.6. For any n 4, there exists a biconnected graph Γ n such that |V (Γ n )| = n + 2, diam(Γ n ) = diam(Γ n ) = 2, L (Γ e n , de) (A(Γ n )) 2 n − 3 . Proof. Let Λ n be the biconnected graph mentioned in Proposition 3.7, and let Γ n = Λ n . Then diam(Γ n ) = diam(Γ n ) = 2; |V (Γ n )| = n + 2; P n is an induced subgraph of Γ n ; V (P n ) dominates Γ n . Since diam(P n ) = n − 1 3, there exists a loxodromic element g ∈ A(Γ n ) such that τ e (g) 2 diam(Pn)−2 = 2 n−3 (by Lemma 4.4). Therefore L (Γ e n ,de) (A(Γ n )) 2 n − 3 . For any nontrivial biconnected graph Γ, L (Γ e , de) (A(Γ)) 2. (i) For A ⊂ V (Γ),Γ[A] = Γ[A], i.e. the subgraph ofΓ induced by A is the same as the complement of the subgraph of Γ induced by A. (ii) If Γ 0 is an induced subgraph of Γ and B ⊂ V (Γ 0 ), then Γ[B] = Γ 0 [B], i.e. B induces the same subgraph in Γ and in Γ 0 . Figure 2 . 2Graphs for Example 3.6We first assume diam(Γ) 3. Choose v 1 , v 2 ∈ V (Γ) such that dΓ(v 1 , v 2 ) 3, and let e = {v 1 , v 2 }. By the above claim, St Γ (e) = V (Γ), henceΓ[St Γ (e)] =Γ and it is connected. Theorem 3 . 5 . 35For any nontrivial biconnected graph Γ, there exists an edge e of Γ such that Γ[St Γ (e)] is biconnected. Before proving the theorem, let us see examples. (a). If e = {v 1 , v 2 }, then Γ[St Γ (e)] is the path graphP 3 (v 1 , v 2 , v 3 ), which is not biconnected. If e = {v 2 , v 3 }, then Γ[St Γ (e)] is the path graph P 4 (v 1 , v 2 , v 3 , v 4 ), which is biconnected. It is easy to see that Γ[St Γ (e)] is biconnected if and only if e = {v i , v i+1 } for 2 i 4.Let Γ be the graph with 7 vertices inFigure 2(b). If e = {v 3 , v 4 }, then Γ[St Γ (e)] is the join {v 0 } * P 4 (v 2 , v 3 , v 4 , v 5 ), which is not biconnected. If e = {v 0 , v 6 }, then Γ[St Γ (e)] is the join {v 0 } * P 5 (v 2 , v 3 , v 4 , v 5 , v 6 ), which is not biconnected. It is easy to see that Γ[St Γ (e)] is biconnected if and only if e is {v 0 , v 2 } or {v 2 , v 3 }.Let Γ be the pentagon C 5 inFigure 2(c). Then Γ[St Γ (e)] is biconnected for any edge e. Figures 3 and 4 4are for the proof of Theorem 3.5, where the solid (resp. dotted) lines between two vertices indicate edges (resp. non-edges) in Γ.Proof of Theorem 3.5. Since Γ[St Γ (e)] is connected for any edge e, it suffices to show that Γ[St Γ (e)] is connected for some edge e. Figure 3 . 3Figure forCase 1 in the proof of Theorem 3.5 Figure 3 . 3SinceΓ[St Γ (e 0 )] is obtained from Γ 0 [St Γ 0 (e 0 )]by adding a vertex v 0 and adding at least one edge {v 0 , v 3 }, the graphΓ[St Γ (e 0 )] is connected, hence we are done. Figure 4 . 4Figures for Case 2 in the proof of Theorem 3.5 • A dominatesΓ because St Γ (e 0 ) ⊂ A and St Γ (e 0 ) dominatesΓ (by Lemma Figure 6 . 6Figure forLemma 4.1 Lemma 4 . 2 . 42Let Γ be a nontrivial biconnected graph. If there exists an edge e of Γ such that Γ[St Γ (e)] is biconnected, then there exists a loxodromic element g ∈ A(Γ) such that τ e (g) 2. Theorem 4. 3 . 3For any nontrivial biconnected graph Γ, L (Γ e ,de) (A(Γ)) 2. Figure 7 . 7Schematic diagram for the graph Λ in Lemma 4.4 Theorem 4 . 5 . 45Let Γ be a biconnected graph. If d = diam(Γ) 3, then L (Γ e , de) (A(Γ)) 2 d − 2 . Definition 2.1 (dominate). Let A and B be subsets of V (Γ). If B ⊂ St Γ (A) (i.e. each vertex v ∈ B is either contained in A or adjacent to a vertex in A), then we say that A dominates B in Γ. If A dominates V (Γ) in Γ, then we simply say that A dominates Γ.From the above definition, the statement that A dominatesΓ means StΓ( On the finiteness property of hyperbolic simplicial actions: the right-angled Artin groups and their extension graphs. H Baik, D Seo, H Shin, Geom. Dedicata. 21717H. Baik, D. Seo and H. Shin, On the finiteness property of hyperbolic simplicial actions: the right-angled Artin groups and their extension graphs, Geom. Dedicata 217(1) (2023) 7. Minimal Asymptotic Translation Lengths of Torelli Groups and Pure Braid Groups on the Curve Graph. H Baik, H Shin, Int. Math. Res. Notices. 202024H. Baik and H. Shin, Minimal Asymptotic Translation Lengths of Torelli Groups and Pure Braid Groups on the Curve Graph, Int. Math. Res. Notices 2020(24) (2020) 9974-9987. Tight geodesics in the curve complex. B H Bowditch, Invent. Math. 1712B.H. Bowditch, Tight geodesics in the curve complex, Invent. Math. 171(2) (2008) 281-300. Minimal pseudo-Anosov translation lengths on the complex of curves. V Gadre, C.-Y Tsai, Geom. Topol. 153V. Gadre and C.-Y. Tsai, Minimal pseudo-Anosov translation lengths on the complex of curves, Geom. Topol. 15(3) (2011) 1297-1312. Translation lengths in crossing and contact graphs of (quasi-)median graphs. A Genevois, arXiv:2209.06441A. Genevois, Translation lengths in crossing and contact graphs of (quasi-)median graphs, arXiv:2209.06441. Algorithms and geometry for graph products of groups. S Hermiller, J Meier, J. Algebra. 1711S. Hermiller and J. Meier, Algorithms and geometry for graph products of groups, J. Algebra 171(1) (1995) 230-257. Embedability between right-angled Artin groups. S.-H Kim, T Koberda, Geom. Topol. 171S.-h. Kim and T. Koberda, Embedability between right-angled Artin groups, Geom. Topol. 17(1) (2013) 493-530. An obstruction to embedding right-angled Artin groups in mapping class groups. S.-H Kim, T Koberda, Int. Math. Res. Not. 14S.-h. Kim and T. Koberda, An obstruction to embedding right-angled Artin groups in mapping class groups, Int. Math. Res. Not. 2014(14) (2014) 3912-3918. The geometry of the curve graph of a right-angled Artin group. S.-H Kim, T Koberda, Int. J. Algebra Comput. 242S.-h. Kim and T. Koberda, The geometry of the curve graph of a right-angled Artin group, Int. J. Algebra Comput. 24(2) (2014) 121-169. Small asymptotic translation lengths of pseudo-Anosov maps on the curve complex. E Kin, H Shin, Groups Geom. Dyn. 133E. Kin and H. Shin, Small asymptotic translation lengths of pseudo-Anosov maps on the curve complex, Groups Geom. Dyn. 13(3) (2019) 883-907. Translation numbers in a Garside group are rational with uniformly bounded denominators. E.-K Lee, S J Lee, J. Pure Appl. Algebra. 2113E.-K. Lee and S.J. Lee, Translation numbers in a Garside group are rational with uniformly bounded denominators, J. Pure Appl. Algebra 211(3) (2007) 732-743. Path lifting properties and embedding between RAAGs. E.-K Lee, S.-J Lee, J. Algebra. 448E.-K. Lee and S.-J. Lee, Path lifting properties and embedding between RAAGs, J. Algebra 448 (2016) 575-594. Embeddability of right-angled Artin groups on complements of trees. E.-K Lee, S.-J Lee, Int. J. Algebra Comput. 283E.-K. Lee and S.-J. Lee, Embeddability of right-angled Artin groups on complements of trees, Int. J. Algebra Comput. 28(3) (2018) 381-394. E.-K Lee, S.-J Lee, arXiv:2212.02708Acylindricity of the action of right-angled Artin groups on extension graphs. E.-K. Lee and S.-J. Lee, Acylindricity of the action of right-angled Artin groups on extension graphs, arXiv:2212.02708.
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[ "Field Output correction factors of small static field for IBA Razor NanoChamber", "Field Output correction factors of small static field for IBA Razor NanoChamber" ]
[ "D Mateus \nFaculdade de Ciências da\nUniversidade de Lisboa\nLisboaPortugal\n\nMercurius Health S.A\nLisboaPortugal\n\nFundação Champalimaud\nLisboaPortugal\n", "C Greco \nFundação Champalimaud\nLisboaPortugal\n", "L Peralta \nFaculdade de Ciências da\nUniversidade de Lisboa\nLisboaPortugal\n\nLaboratório de Instrumentação\n\n", "\nFísica Experimental de Partículas\nLisboaPortugal\n" ]
[ "Faculdade de Ciências da\nUniversidade de Lisboa\nLisboaPortugal", "Mercurius Health S.A\nLisboaPortugal", "Fundação Champalimaud\nLisboaPortugal", "Fundação Champalimaud\nLisboaPortugal", "Faculdade de Ciências da\nUniversidade de Lisboa\nLisboaPortugal", "Laboratório de Instrumentação\n", "Física Experimental de Partículas\nLisboaPortugal" ]
[]
Purpose:The goal of this work is to present results of field output factors (OF) using an IBA CC003 (Razor NanoChamber) and compared these results with PTW 60019 (MicroDiamond) and IBA Razor Diode. The experimental results for IBA CC003 were also compared with Monte Carlo (MC) Simulation, using Penelope and Ulysses programs. In addition, field output correction factors ( , , ) for IBA CC003 were derived with three different methods: 1) using PTW 60019 and IBA Razor as reference detectors; 2) comparison between MC and experimental measurements; and 3) using only MC.Material and Methods:The beam collimation included in this study were 1) square field size between 10x10 and 0.5x0.5 cm 2 defined by the MLC and jaws and 2) cones of different diameters. For IBA CC003 it was determined the polarity and ion collection efficiency correction factors in parallel and perpendicular orientation.Results:The results indicate 1) the variation of polarity effect with the field size is relevant for the determination of OF using IBA CC003, especially for parallel orientation; 2) there is no significant variation of the ion collection efficiency with the field size using IBA CC003 in parallel orientation; 3) OF differences between IBA CC003 and PTW 60019/IBA Razor and experimental and MC results increase with decreasing field size; The , , results indicate 1) using the first and second method, , , increase with decreasing field size, which can be related with the influence of the volume effect and 2) using the third method, , , decrease with decreasing field size, which can be explained by the perturbation effect. Conclusions: Our results demonstrate the need of applying , , for IBA CC003 for ≤ 1 cm, to compensate for volume averaging and perturbations effects.
null
[ "https://export.arxiv.org/pdf/2306.03636v1.pdf" ]
259,088,827
2306.03636
74ebc917b2f7f943d1a17a5ceebbffbeb0ba3366
Field Output correction factors of small static field for IBA Razor NanoChamber D Mateus Faculdade de Ciências da Universidade de Lisboa LisboaPortugal Mercurius Health S.A LisboaPortugal Fundação Champalimaud LisboaPortugal C Greco Fundação Champalimaud LisboaPortugal L Peralta Faculdade de Ciências da Universidade de Lisboa LisboaPortugal Laboratório de Instrumentação Física Experimental de Partículas LisboaPortugal Field Output correction factors of small static field for IBA Razor NanoChamber 1 Purpose:The goal of this work is to present results of field output factors (OF) using an IBA CC003 (Razor NanoChamber) and compared these results with PTW 60019 (MicroDiamond) and IBA Razor Diode. The experimental results for IBA CC003 were also compared with Monte Carlo (MC) Simulation, using Penelope and Ulysses programs. In addition, field output correction factors ( , , ) for IBA CC003 were derived with three different methods: 1) using PTW 60019 and IBA Razor as reference detectors; 2) comparison between MC and experimental measurements; and 3) using only MC.Material and Methods:The beam collimation included in this study were 1) square field size between 10x10 and 0.5x0.5 cm 2 defined by the MLC and jaws and 2) cones of different diameters. For IBA CC003 it was determined the polarity and ion collection efficiency correction factors in parallel and perpendicular orientation.Results:The results indicate 1) the variation of polarity effect with the field size is relevant for the determination of OF using IBA CC003, especially for parallel orientation; 2) there is no significant variation of the ion collection efficiency with the field size using IBA CC003 in parallel orientation; 3) OF differences between IBA CC003 and PTW 60019/IBA Razor and experimental and MC results increase with decreasing field size; The , , results indicate 1) using the first and second method, , , increase with decreasing field size, which can be related with the influence of the volume effect and 2) using the third method, , , decrease with decreasing field size, which can be explained by the perturbation effect. Conclusions: Our results demonstrate the need of applying , , for IBA CC003 for ≤ 1 cm, to compensate for volume averaging and perturbations effects. Introduction: In the last years, the use of small fields (less than 2 cm of diameter) in Radiotherapy has increased with the use of advanced treatment techniques, such as, stereotactic radiosurgery (SRS), stereotactic body radiotherapy (SBRT), intensity modulated radiotherapy (IMRT) and volumetric modulated radiotherapy (VMAT). The diagnosis of small lesions has also contributed to this need, due to the use of image techniques, such as, magnetic resonance imaging (MRI) and positron emission tomography (PET). The measurement of field output factors (OF) for MV small photon fields are subjected to large uncertainties, due to the challenges of small field dosimetry, because of the lack of electronic equilibrium, source occlusion and volume effect, and also the requirements of the detectors used in the measurements (IAEA TRS-483 2017, Aspradakis M et al. 2010, Das I et al. 2008. Thus, the small field dosimetry cannot be done with the conventional dosimeters and protocols (Andreo P et al. 2004). Alfonso et al (2008) [5] proposed a new formalism for small and non-standard field dosimetry, introducing a new concept of using correction factor , , which correlates the differences between the clinical field size and the machine-specific reference field size . The International Atomic Energy Agency (IAEA) in conjunction with American Association of Physicists in Medicine (AAPM) compiled the studies about small field dosimetry and developed the new formalism introduced by Alfonso and presented the TRS-483 Code of Practice (IAEA TRS-483 2017), which provides correction factors for different detectors and small field sizes. In the market there are different kinds of detectors to perform measurements of small fields, which present advantages and disadvantages for the measurements. However, the ionization chambers are known as the workhorse of reference dosimetry, thus it would be ideal performing the small field dosimetry using these kinds of detectors. The purpose of this study is to contribute with measurements for small field sizes with IBA Razor NanoChamber (IBA CC003) (Reggiori G et al. 2017, Razor NanoChamber 2017 using experimental data and Monte Carlo simulation for 6 MV and 10 MV with and without flattening filter beams. In addition, we seek to derive field output correction factors , , for IBA CC003 using three methods: 1) reference detectors (PTW 60019 and IBA Razor) with known output correction factors; 2) comparison between Monte Carlo simulation and experimental measurements; and 3) using only Monte Carlo to derive field output factors. Material & Methods: Accelerators and photon beams: The dosimetric measurements were performed in three Varian machines (Varian Medical Systems, Palo Alto, CA, USA): two TrueBeam (TrueBeam1 and TrueBeam2) and one Edge. The TrueBeam1 and Edge have a high definition MLC (central leafs with 0.25 cm of thickness) and TrueBeam2 has Millennium 120 MLC (central leafs with 0.5 cm of thickness). The nominal photon energies used were: 6 MV (with (6X) and without (6X-FFF) flattening filter) and 10 MV (with (10X) and without (10X-FFF) flattening filter). The dose-rate (MU/min) used was 600 MU/min and 800 MU/min for flattened and unflattened beams, respectively. The measurement geometry consisted of an isocentric setup with source-to-surface distance (SDD) of 90 cm and a depth of 10 cm with gantry at 0°. The beam collimation included in this study were 1) cones of different diameters (1.75, 1.5, 1.25, 1.0, 0.75 cm) with jaws opening of 5x5 cm 2 ; and 2) square field size of 10x10, 4x4, 3x3, 2x2, 1x1 and 0.5x0.5 cm 2 defined by the MLC with jaws positioning of 11x11, 5x5 and 4x4 cm 2 for square field size of 10x10, 4x4 and 3x3 cm 2 , respectively, and 3x3 cm 2 for the rest square field sizes; and 3) square field size of 10x10, 4x4, 3x3, 2x2, 1.5x1.5, 1x1, 0.8x0.8 and 0.5x0.5 cm 2 defined by the jaws. The 10x10 cm 2 field size was used as reference field size for the OF determination. Detectors: In this work, the detectors used were the ionization chamber IBA CC003, the microDiamond detector PTW 60019 and the diode IBA Razor. The IBA CC003 (Reggiori G et al. 2017, Razor NanoChamber 2017) is a microchamber (active volume of 3.0 mm 3 ), whose outer and inner electrodes are made of Shonka (C-552) plastic and graphite-EDM3, respectively. The outer electrode of Shonka, which corresponds to air equivalent plastic, has a diameter of 2.0 mm and a density of = 1.76 / 3 . The inner electrode of graphite has a diameter of 1.0 mm and a density of = 1.81 / 3 . The PTW60019 detector (PTW User Manual 2014), with an active volume of 0.004 mm 3 , consists of a single crystal intrinsic layer with thickness of 1 μm and a diameter of 2.2 mm. The IBA Razor diode (Razor Detector 2014) has an active volume of 0.6 mm in diameter and 20 μm of thickness. It is made with a n-type implant in p-type silicon and operates in photovoltaic mode. The IBA CC003 was placed in two setups, with the chamber stem parallel and perpendicular to the beam axis and operated with a bias voltage of 300 V. The polarity effect for IBA CC003 was evaluated for the 4 energies used and its dependence on the field size was investigated for the two orientations of the ionization chamber. The polarity correction factor was measured, for each field size, according to Eq. (1): = | +300 | + | −300 | 2 +300 (1) The ion collection efficiency was also investigated for the field sizes presented in this study. It was reported by different authors (Weinhous M S and Meli J A 1984, Hyun M A 2017, Zankowski C and Podgorsak E B 1998, Agostinelli S et al. 2008) that for very small volume ionization chambers, like IBA CC003, and high voltages the traditional two-voltage method is no longer adequate for definition of recombination effect ( ). Instead of that should be applied a method proposed by Agostinelli (Agostinelli S et al. 2008), which evaluates the variation of 1 vs. 1 (saturation curves) where is the collected charge and is the applied voltage. Jaffé plots (Zankowski C and Podgorsak E B 1998) were created from the saturation curves. The saturation value of the collected charge was calculated as the inverse of the interception for 1 → 0 of the linear regression curve. The linear fit was performed considering the linear part of the plot only (i.e. low voltages: 25, 50, 75, 100 ). Following this method, the ion collection efficiency correction factor was introduced as: = 300 (2) where 300 is the collected charge at the normal operating voltage of 300 V. The variable corrects for both recombination and excess charges and reduces to the conventional recombination factor if excess charges are null. The saturation curves were obtained for IBA CC003 for all studied beam collimation, energies, machines and orientation of the chamber in respect to the beam axis. The dependence of the ion collection efficiency on the field size was investigated and it was evaluated the need of applied ion collection efficiency correction factors for the determination of field output factors. The differences between ion collection efficiency and polarity effect determined in parallel and perpendicular orientations were investigated, through the evaluation of the ratio of these two factors acquired in both orientations. The PTW 60019 and IBA Razor were positioned with the stem parallel (parallel orientation) to the beam axis and operated with 0 V. Equivalent square small field size: The determination of the square small field size for each beam collimation and energy was performed with Gafchromic EBT3 films, due to its high spatial resolution. EBT3 films from lot 04022001 were used. For calibration curve purposes, 13 strips of 3.6x20.3 cm 2 were irradiated with a dose range between 2 Gy and 5.6 Gy with steps of 0.3 Gy and one strip was left unexposed. A field size of 10x10 cm 2 was used in order to expose them with homogeneous doses. For calibration and field size measurements procedures SDD of 90 cm was used and the films were covered with 10 cm thick piece of solid water. For each film measurement, 500 MU were delivered. An Epson Expression 10000XL (Seiko Epson Corporation, Nagano, Japan) flatbed scanner was used. The scanner was warmed up for at least 30 min before readings. Before acquisition and after long pauses, five empty scans were taken to stabilize the lamp. Scans were made in landscape orientation. Imagens were acquired in transmission mode using a glass compressor plate with 3 to 4 mm thick, over the films. Films were scanned in 48-bit RGB mode, with the colour correction tool not active, with 72 ppp of resolution, and saved as TIFF files. The calibration curves were created in OmniPro I'mRT software version 1.7 for each energy and applied for the exposed films with different beam collimation. The field dimensions in each direction (crossline and inline) were obtained from the OmniPro I'mRT software. Using equation (3) and (4), the equivalent square small field size was determined for square and circular small field sizes (IAEA TRS-483 2017), respectively, = √ (3) = √ = 1.77 (4) where A and B correspond to the crossline and inline dosimetric widths, defined by FWHM at the measurement depth and corresponds to the radius of the circular field defined by the points which on average, the dose level amounts to 50% of the maximum dose at the measurement depth. Monte Carlo The Monte Carlo simulations were made using the Penelope (NEA 2005) system, to simulate the physical interactions. For the geometry definition, and tracking and scoring of the particles, a homemade system called Ulysses was used (Peralta L and Louro A 2014). Phase-space files provided by Varian for TruBeam/Edge linear accelerator were used. These files were tailed on a plane located just upstream of the movable jaws and they were used as a radiation source. Downstream, it was created the geometry of the jaws X and Y, base plate, MLC (high definition and Millenium) and water phantom. The IBA CC003 was modeled according to the blueprints provided by the manufacturer, using three spheres, one inside the other (external: graphite; medium: shonka; inner: graphite) ( Figure 1). The simulation was performed for four types of geometry configurations: 1) only jaws; 2) high definition MLC (0.25 cm of thickness for central leafs at isocentre) named MLC1; 3) Millennium MLC (0.5 cm of thickness for central leafs at isocentre) named MLC2; and 4) Cones. The field sizes used were the same as the experimental measurements. To improve the statistics of the Monte Carlo results 1) the phase-space files were read several times until the defined total number of events was achieved (2x10 9 ) and 2) each program for each geometry configuration was run 20 times with different seeds, to minimize the uncertainty of the simulation result. Profiles and percentage depth doses (PDDs) Beams profiles in crossline and inline directions and PDDs were evaluated for small fields collimated by the MLC (3x3, 2x2 1x1 and 0.5x0.5 cm 2 ) for all energies presented in this study. The evaluations were done using the IBA CC003, PTW60019 and IBA Razor detectors and MC simulation. The measurements were performed in the water phantom using a scanning resolution of 0.2 mm for beam profiles and 0.5 mm for PDDs. Both detectors were placed in parallel orientation in respect to the beam axis. Profiles were acquired at isocentre with SSD of 90 cm and depth of 10 cm and PDD at SSD of 100 cm. FWHM and penumbra (distance between 80% and 20% dose) of beam profiles was evaluated and compared between the three detectors and MC simulation. Additionally, for the evaluation of the beam profiles and PDDs, Gamma analysis (Low D et al. 1998) was performed, using the passing criteria of 3%/3 mm and 2%/2 mm. Field Output factors For the IBA CC003 the field output factor was determined without correction factors, because this detector is not included in the TRS-483 list of field output correction factors. However, it was applied the polarity correction factor to the raw measurements, thus it was followed the given equation: The differences between field output factors determined in parallel and perpendicular orientations of IBA CC003 were investigated, through the evaluation of the ratio of the field output factors acquired in both orientations. For PTW60019 detector the field output factors were determined using the following equation: For the IBA Razor detector the field output factors were determined using the "Daisy-Chainning" method (IAEA TRS-483 2017, Griessbach I et al. 2005. For this method the ratio of the readings is measured between an intermediate and the reference field size using a suitable ionization chamber, and then measured the ratio of the readings between a clinical field size (defined by MLC, jaws or cones) and the intermediate field size using the diode detector. The ionization chamber used was the CC04 from IBA and the intermediate field size ( ) was 4x4 cm 2 . The field output factor used was given by: For each measurement performed using any detector, 200 MU were delivered and at least three measurements of collected charge were taken for each setup. The field output factors determined using MC simulation were calculated using the following equation: , , = , ,(8) here the , corresponds to the deposited energy in the simulated IBA CC003 for a given field size between 4x4 and 0.5x0.5 cm 2 for jaws and MLC and between 1.75 and 0.75 cm for cones and , corresponds to the deposited energy in the simulated IBA CC003 for the reference field size 10x10 cm 2 . The msr field was replaced for ref field because the 10x10 cm 2 reference field was used. Field Output Correction factors The field output correction factors were derived through three methods: 1) Reference detectors with known output correction factors, 2) Comparison between Monte Carlo simulation and experimental measurements, and 3) Only Monte Carlo Simulation. For the first method (Experimental method) it was assumed the PTW 60019 and the IBA Razor as the reference detectors and the field output correction factors for IBA CC003 was obtained for parallel and perpendicular orientation using the following equation: , , [IBA CC003] = ,, , , 003 (9) For the second method (Hybrid method) it was assumed the Monte Carlo results as a reference and the field output correction factors for IBA CC003 in parallel orientation was obtained using the following equation: where , and , were obtained taking the deposited energy from a sphere of water with a radius of 0.1 cm for the clinical fields and the reference field 10x10 cm 2 respectively. Uncertainties Measurement and Monte Carlo uncertainties were estimated following the recommendations of the Evaluation of measurement data -Guide to expression of uncertainty in measurement (BIPM 2008) and from IAEA publications TECDOC-158529 (IAEA 2008) and TRS-39830 (Andreo P et al. 2004). The field output uncertainties were calculated for experimental measurements and MC simulations for each machine, energy, beam collimation, detector and orientation of the chamber to respect the beam axis. An uncertainty budget was created and it was evaluated the different uncertainty sources which contribute to the measurement parameters. For the field output factor acquired by experimental measurements, the sources of uncertainty considered were 1) repeated measurements, 2) calibration certificate of electrometer and 3) resolution of electrometer. For the analysis of repeated measurement a Type A standard uncertainty was determined, in contrast, a Type B standard uncertainty was used to evaluate the influence of the calibration and resolution of the PTW Unidos electrometer. For , , , it was considered a combined uncertainty of 1% for field sizes larger than 1x1cm 2 and 2% for field sizes equal to or smaller than 1x1 cm 2 . For the field output factor acquired through MC simulation, the standard uncertainties of , and , were obtained through the standard deviation of the mean of the deposited energy on the simulated IBA CC003 and also 20 runs done for each geometric configuration and energy. The uncertainties for field output correction factors were also calculated using an uncertainty budget, for each energy, beam collimation (MLC, Jaws and Cones) and field size. Results and discussion: 3.1 Equivalent square small field sizes Data for the average of equivalent square field size between the three linear accelerators and corresponding expanded uncertainty are present in Table 1 and Table 2 for field sizes defined by MLC and jaws. In Table 3 it is presented the equivalent square field size and the associated uncertainty, for the cones acquired in the linear accelerator Edge. For the nominal ≥ 1 cm, the differences found are less than 0.5 mm, except for the energy 10X and the field size defined by the MLC, which presents a difference of 0.8 mm. For < 1 cm, the differences found are higher than 1.0 mm and can achieve 1.8 mm, especially for the smallest field size. Throughout this work, the field sizes will be indicated with nominal values, however, they will represent without exception the corresponding values. Profiles and PDD It is observed the differences of PDDs (between detectors and detectors and simulation) increase with the depth and the decrease of the field size, however the majority of the differences are in the buildup region. The PDDs for IBA CC003 present a slightly overestimated dose with increasing depths when compared with IBA Razor and PTW 60019, with the decreasing of the field size, as reported by Reggiori (Reggiori G et al. 2017). The Gamma Analysis show good results (> 90%) for field sizes above 1x1 cm 2 , for the two passing criteria and for the comparison between detectors, however between detectors and simulation only for the criteria 3%/3 mm, the Gamma results are above 90% (Table 4, 5, 6 and 7). For field size 0.5x0.5 cm 2 and evaluation between detectors, the worst Gamma results were between the detectors IBA CC003 and IBA Razor, maybe because Razor diode lead to an overestimation of the low energies (kV energy range) in the build-up region, and since PDD are normalized to , this effect induces the under response of the detector in depth. Gamma results between measurements and simulation for the field size 0.5x0.5 cm 2 were not satisfactory, because the number of particles, resulting from the phase-space files, is too few for this field size, which is something that is difficult to resolve because we are dependent on the limited number of particle existing in the phase-space file provided by Varian. Table 4 -Gamma analysis results for passing criteria of 3%/3 mm and 2%/2 mm for the PDD acquired with the three detectors and MC for 6X. Table 5 -Gamma analysis results for passing criteria of 3%/3 mm and 2%/2 mm for the PDD acquired with the three detectors and MC for 10X. Table 6 -Gamma analysis results for passing criteria of 3%/3 mm and 2%/2 mm for the PDD acquired with the three detectors and MC for 6X-FFF. Table 7 -Gamma analysis results for passing criteria of 3%/3 mm and 2%/2 mm for the PDD acquired with the three detectors and MC for 10X-FFF. No significant differences were found for FWHM and between the three detectors used, except for the smallest field size 0.5x0.5 cm 2 , which presents higher values for IBA CC003 compared to PTW60019 and IBA Razor (maximum difference of 0.8 mm). For the field size 1x1 cm 2 , TrueBeam2 presents higher values compared to Edge and TrueBeam1, which can be related to the thickness of the MLC leafs of the TrueBeam2, which are thicker than the ones of Edge and TrueBeam1. Comparing values between the three detectors and EBT3 film, it was found higher differences for the field size 0.5x0.5 cm 2 , but on average less than 0.5 mm. For the profiles, Gamma analysis show that differences between the detectors and detectors and simulation increase with the decreasing of the field size (Table 8, 9, 10 and 11). Only for the passing criteria of 3%/3 mm were the results satisfactory (> 90%) above 2x2 cm 2 field size and in some cases for the field size 1x1 cm 2 . For the evaluation between detectors, higher differences are between the detectors IBA CC003 and IBA Razor, which is explained by the difference of the active volumes of these two detectors. For the evaluation between detectors and simulation, the Gamma analysis results for profiles are not as satisfactory as for PDD, possibly because the profiles present much less points to evaluate. This is a big disadvantage for the small field especially on the penumbra, because even two points considered very close can be far enough to have a large difference in % dose, and this influences the Gamma analysis result. Table 8 -Gamma analysis results for passing criteria of 3%/3 mm and 2%/2 mm for the Profiles acquired with the three detectors and MC for 6X. Table 9 -Gamma analysis results for passing criteria of 3%/3 mm and 2%/2 mm for the Profiles acquired with the three detectors and MC for 10X. Polarity correction factor and Ion collection efficiency of IBA CC003 A strong variation of polarity effect with the field size is observed for all the energies presented in this study in the parallel orientation of the chamber (Figure 2). The polarity effect increases continuously with the decrease of the field size, which is also observed by Looe (Looe et al. 2018) and Reggiori (Reggiori G et al. 2017). Although, for perpendicular orientation of the chamber, the field size dependence is not as relevant, as reported by Looe (Looe et al. 2018) and Gul (Gul et al. 2020). A difference of around 10% and 1.5% was found between the reference field size and the smallest field size for all energies, for parallel and perpendicular orientation of the chamber, respectively. of the cases, above ( ) and can achieve 0.01 for the smallest field size of 0.5x0.5 cm 2 . Small differences in the polarity correction factor were found between the different linear accelerators (<1%). At reference field size, the polarity correction factor for parallel orientation of the chamber was 1.004±0.002, 1.013±0.002, 0.999±0.002 and 1.009±0.002 for 6X, 10X, 6X-FFF and 10X-FFF, respectively. And for perpendicular orientation were 1.012±0.002, 1.019±0.002, 1.008±0.002 and 1.017±0.002 for 6X, 10X, 6X-FFF and 10X-FFF, respectively. Analysis of the ion collection efficiency correction factor with field size (Figure 3) showed no significant variation for the parallel orientation of the chamber, the differences between the reference field size and the small field sizes are less than 0.5%. However for the perpendicular orientation of the chamber it is possible to observe a significant difference between the reference field size and the smallest field size (0.5x0.5 cm 2 ) defined by the jaws for all energies presented in this study, which is around 3%, but this behavior is not observed for the smallest field size defined by the MLC (difference less than 0.5%). Analyzing the ratio between the ion collection efficiency correction factor acquired in both directions, there were differences less than 1%, except for the smallest field size defined by the jaws, where and [ ] , considering = 2. Small differences in the ion collection efficiency correction factor were found between the different linear accelerators (<0.5%). At reference field size, the ion collection efficiency correction factor for parallel Assuming the condition a) b) c) d) orientation of the chamber was 1.001±0.006, 1.003±0.008, 1.003±0.006 and 1.004±0.006 for 6X, 10X, 6X-FFF and 10X-FFF, respectively. For perpendicular orientation were 1.003±0.006, 1.004±0.008, 1.005±0.006 and 1.005±0.006 for 6X, 10X, 6X-FFF and 10X-FFF, respectively. It was decided not to apply the ion collection efficiency correction factor to determine the field output factor for both orientations of the IBA CC003, because no significant dependence was observed as function of the field size. Field Output factors No significant differences were observed for the OF acquired with IBA CC003 between measurements and Monte Carlo simulation down to the field size 2x2 cm 2 defined by the Jaws and MLC, however from field size 1x1 cm 2 the field output factor differ significantly between measurements and simulation within 95% confidence limits, because for the IBA CC003. The same happens for the measurements of OF between parallel and perpendicular orientation of the IBA CC003. For the field sizes defined by Cones, the differences appear for the Cone 7.5 ( = 0.66 cm) when it is compared measurements versus simulation, and from the Cone 12.5 ( = 1.11 cm), when it is compared the measurements between parallel and perpendicular orientation of IBA CC003 (Figure 4, 5 and 6). The measurements OF presented lower values compared to the Monte Carlo OF, because of the volume effect which appears for < 1.0 cm, since the dose measured is underestimated for these field sizes. Small differences in the field output factors were found between the different linear accelerators for all the field sizes (< 1%), except for the field sizes defined by jaws below the field size 0.8x0.8 cm 2 . It is also observed that for the field sizes defined by the Jaws, the TrueBeam2 presents lower field output values than the others machines for field sizes below 1x1 cm 2 . This effect is understood due to the field size dimension defined by the Jaws of TrueBeam2 being smaller compared to TrueBeam1 and Edge. It was found expanded uncertainties less than 1% for measured OF and between 2-3% for simulated OF. Field output correction Factors Analyzing the IBA CC003 field output correction factors obtained through the first (Figure 7 and 8) and second (Figure 9) method, it seems that increases with the decreasing of the field size, which is related to the influence of the volume effect of the IBA CC003 for < 1 cm. , , obtained with IBA CC003 in parallel orientation (Figure 7) presents a more linear behavior compared with the IBA CC003 in perpendicular orientation (Figure 8). Comparing the results of the IBA CC003 field output correction factors using PTW 60019 and Razor as reference detectors, it is observed that for all field sizes defined by the MLC and for field sizes defined by the Jaws and Cones, for ≥ 0.8 cm, there is no significant differences in the , , obtained with the two detectors used as reference (less than 1.0%). However, for the smallest field size defined by the Cones (Cone 0.75) and by the Jaws (0.5x0.5 cm 2 ), higher a) b) c) differences are observed: maximum differences of around 1.5 % for field size defined by the Cones and 5.5 % for the field size defined by the jaws. The , , results obtained using the second method (Figure 9), present higher values compared to the first method, which could be related with the limited number of particles existing in the file provided by Varian that limits the number of particles that reach the detector. We have tried to solve this problem, reading the file several times and running the program for each beam configuration 20 times with different seeds, to minimize the uncertainty of the simulation result. Analyzing the IBA CC003 , , obtained through the third method ( Figure 10), it is observed that the field output correction factors decrease with the decreasing of the field size. In this case there is no influence of the volume effect because the volume of the simulated IBA CC003 is the same as the water sphere. Therefore, the effect of perturbation of the particles is more predominant, which can explain the decrease of field output correction factors with the field size. The expanded uncertainties of , , for each method were: 1) around 1% -3% for MLC and Jaws and 2% -4% for Cones, for the experimental method. For the field sizes down of 1 cm the field output correction factors present an expanded uncertainty higher than the others field sizes, because it was assumed a combined uncertainty of 2% for field sizes less than 1 cm and 1% for field sizes higher than 1 cm for the PTW 60019 and Razor field output correction factors; 2) around 2% -4% for all the beam collimation for hybrid method; and 3) around 2% -6% for all the beam collimation for MC method. Conclusions: The IBA CC003 is a viable option for relative dosimetry for small field sizes (until 1x1 cm 2 ). A significant improvement in spatial resolution compared to standard ionization chambers. For very small fields, however, the solid detectors (such as IBA Razor and PTW 60019) provide better results in terms of spatial resolution, which lead to reduced penumbra (around 1-2 mm less) and limited volume average effect. With respect to the field output factors, IBA CC003 can be used for small field sizes, but for field sizes ≤ 1 cm it is necessary to apply field output correction factor to compensate for volume averaging and perturbation effects. The parallel orientation of the IBA CC003 seems to present more stable behavior when compared to the perpendicular orientation, because in perpendicular orientation the IBA CC003 can lead to stem effect issues, which were observed especially for field sizes below to 2x2 cm 2 . For the experimental determination of the field output factors using IBA CC003, it is important to take into account the polarity effect, especially for the parallel orientation of the ionization chamber, because the polarity effect increases with the decreasing of the field size. However, the ion collection efficiency is not as relevant, because the variation of ion collection efficiency a) b) c) d) with the field size is not significant. There is a difference between experimental and MC , , , which increases with the decreasing of the field. The major weakness of this simulation is the number of particles that reach the simulated ionization chamber, directly proportional to the field size, because the program is dependent on a limited particle file (phase-space file) provided by Varian. The results for , , demonstrated the need of field output correction factors for equivalent square field sizes equal or less than 1 cm, for both orientations of the IBA CC003 with respect to the beam axis. Acknowledgment: We acknowledge Varian for the supply of the phase-space files of TrueBeam accelerator and IBA Dosimetry and Sociedade Avanço for putting at our disposal the Razor NanoChamber and Razor diode. One of the authors (Mateus D) acknowledges the financial support of Mercurius Health. We also thank Ashley Rose Peralta for the review of the English text. References Figure 1 - 1Dimensions of simulated IBA CC003. For the third method (MC method) the field output correction factors were derived according to the following equation: Figure 2 - 2Average polarity correction factor between the three linear accelerator for field sizes defined by the Jaws, MLC and Cones in parallel and perpendicular orientation of the IBA CC003: a) 6X, b) 10X, c) 6X-FFF and d) 10X-FFF. is higher than an expanded uncertainty of [ ] Figure 3 - 3Average ion collection efficiency correction factor between the three linear accelerator for field sizes defined by the Jaws, MLC and Cones in parallel and perpendicular orientation of the IBA CC003: a) 6X, b) 10X, c) 6X-FFF and d) 10X-FFF. Figure 4 -Figure 5 -Figure 6 - 456OF for the beam collimation defined by the Jaws acquired through measurements (in parallel and perpendicular orientation of IBA CC003) and Monte Carlo simulation: a) 6X, b) 10X, c) 6X-FFF and d) 10X-FFF. OF for the beam collimation defined by the MLC acquired through measurements (in parallel and perpendicular orientation of IBA CC003) and Monte Carlo simulation: a) 6X, b) 10X, c) 6X-FFF and d) 10X-FFF. OF for the beam collimation defined by the Cones acquired through measurements (in parallel and perpendicular orientation of IBA CC003) and Monte Carlo simulation: a) 6X, b) 6X-FFF and c) 10X-FFF. types of beam collimation and beam energy. The value of , , Figure 7 7the beam collimation acquired through the Experimental Method using IBA CC003 in parallel orientation: a) 6X, b) 10X, c) 6X-FFF and d) 10X-FFF. the beam collimation acquired through the Experimental Method using IBA CC003 in perpendicular orientation: a) 6X, b) 10X, c) 6X-FFF and d) 10X-FFF. the beam collimation acquired through the Hybrid Method: a) 6X, b) 10X, c) 6X-FFF and d) 10X-FFF. the beam collimation acquired through the Monte Carlo Method: a) 6X, b) 10X, c) 6X-FFF and d) 10X-FFF. Table 1 - 1Equivalent square field size ( ) for field sizes defined by the MLC. Table 2 - 2Equivalent square field size ( ) for field sizes defined by the Jaws. Table 3 - 3Equivalent square field size ( ) for field sizes defined by the Cones. Table 10 - 10Gamma analysis results for passing criteria of 3%/3 mm and 2%/2 mm for the Profiles acquired with the three detectors and MC for 6X-FFF.Table 11 -Gamma analysis results for passing criteria of 3%/3 mm and 2%/2 mm for the Profiles acquired with the three detectors and MC for 10X-FFF. : - :Andreo P et al. 2004 IAEA TRS-398 protocol: Absorbed dose determination in external beam radiotherapy: An international code of practice for dosimetry based on standards of absorbed dose to water, International Atomic Energy Agency (IAEA) -IAEA TRS-483 2017 Dosimetry of small static fields used in external beam radiotherapy -An international code of practice for reference and relative dose determination, International Atomic Energy Agency (IAEA) -Aspradakis M et al. 2010 IPEM Report 103: Small field MV photon dosimetry, International Atomic Energy Agency (IAEA) 310 -Das I et al. 2008 Small fields: Nonequilibrium radiation dosimetry Med. 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Phys. 45 5608-5621 -Gul A et al. 2020 Feasibility study of using Stereotactic Field Diode for field output factors measurements and evaluating three new detectors for small field relative dosimetry of 6 and 10MV photon beams J. Appl Clin Med Phys. 21:11, 23-26 -BIPM Joint Committee for Guides in Metrology 2008 Evaluation of measurement data -Guide to the expression of uncertainty in measurement JCGM 100:2008 GUM 1995 with minor corrections Sevres: BIPM -IAEA 2008 Measurement Uncertainties, IAEA-TECDOC-1585 Vienna: IAEA -NEA 2005 Penelope-2014: A code System for Monte Carlo Simulation of Electron and Photon Transport NEA -Peralta L and Louro A 2014 AlfaMC: A fast alpha particle transport Monte Carlo code Nucl. Instr. and Meth. in Phys. Res. A 737 163-169 -Belosi M F et al. 2014 Monte Carlo simulation of TrueBeam flattening-filter-free beams using phase-space files: Comparison with experimental data Med. 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[ "RELATIVISTIC PHASE SPACE DIFFUSION OF COMPACT OBJECT BINARIES IN STELLAR CLUSTERS AND HIERARCHICAL TRIPLES", "RELATIVISTIC PHASE SPACE DIFFUSION OF COMPACT OBJECT BINARIES IN STELLAR CLUSTERS AND HIERARCHICAL TRIPLES" ]
[ "Chris Hamilton ", "Roman R Rafikov " ]
[]
[]
The LIGO/Virgo detections of compact object mergers have posed a challenge for theories of binary evolution and coalescence. One promising avenue for producing mergers dynamically is through secular eccentricity oscillations driven by an external perturber, be it a tertiary companion (as in the Lidov-Kozai (LK) mechanism) or the tidal field of the stellar cluster in which the binary orbits. The simplest theoretical models of these oscillations use a 'doubly-averaged' (DA) approximation, averaging both over the binary's internal Keplerian orbit and its 'outer' barycentric orbit relative to the perturber. However, DA theories do not account for fluctuations of the perturbing torque on the outer orbital timescale, which are known to increase a binary's eccentricity beyond the maximum DA value, potentially accelerating mergers. Here we reconsider the impact of these short-timescale fluctuations in the test-particle quadrupolar limit for binaries perturbed by arbitrary spherical cluster potentials (including LK as a special case), in particular including 1pN general relativistic (GR) apsidal precession of the internal orbit. Focusing on the behavior of the binary orbital elements around peak eccentricity, we discover a new effect, relativistic phase space diffusion (RPSD), in which a binary can jump to a completely new dynamical trajectory on an outer orbital timescale, violating the approximate conservation of DA integrals of motion. RPSD arises from an interplay between secular behavior at extremely high eccentricity, short-timescale fluctuations, and rapid GR precession, and can change the subsequent secular evolution dramatically. This effect occurs even in hierarchical triples, but has not been uncovered until now.
null
[ "https://export.arxiv.org/pdf/2306.03703v1.pdf" ]
259,088,905
2306.03703
8ad4e42724587b97d71cc42c906bd8840d0ddb0f
RELATIVISTIC PHASE SPACE DIFFUSION OF COMPACT OBJECT BINARIES IN STELLAR CLUSTERS AND HIERARCHICAL TRIPLES June 7, 2023 Chris Hamilton Roman R Rafikov RELATIVISTIC PHASE SPACE DIFFUSION OF COMPACT OBJECT BINARIES IN STELLAR CLUSTERS AND HIERARCHICAL TRIPLES June 7, 2023Draft version June 7, 2023Draft version Preprint typeset using L A T E X style emulateapj v. 12/16/11 The LIGO/Virgo detections of compact object mergers have posed a challenge for theories of binary evolution and coalescence. One promising avenue for producing mergers dynamically is through secular eccentricity oscillations driven by an external perturber, be it a tertiary companion (as in the Lidov-Kozai (LK) mechanism) or the tidal field of the stellar cluster in which the binary orbits. The simplest theoretical models of these oscillations use a 'doubly-averaged' (DA) approximation, averaging both over the binary's internal Keplerian orbit and its 'outer' barycentric orbit relative to the perturber. However, DA theories do not account for fluctuations of the perturbing torque on the outer orbital timescale, which are known to increase a binary's eccentricity beyond the maximum DA value, potentially accelerating mergers. Here we reconsider the impact of these short-timescale fluctuations in the test-particle quadrupolar limit for binaries perturbed by arbitrary spherical cluster potentials (including LK as a special case), in particular including 1pN general relativistic (GR) apsidal precession of the internal orbit. Focusing on the behavior of the binary orbital elements around peak eccentricity, we discover a new effect, relativistic phase space diffusion (RPSD), in which a binary can jump to a completely new dynamical trajectory on an outer orbital timescale, violating the approximate conservation of DA integrals of motion. RPSD arises from an interplay between secular behavior at extremely high eccentricity, short-timescale fluctuations, and rapid GR precession, and can change the subsequent secular evolution dramatically. This effect occurs even in hierarchical triples, but has not been uncovered until now. INTRODUCTION The compact object -black hole (BH) and/or neutron star (NS) -binary mergers discovered by the LIGO/Virgo collaboration in recent years (Abbott et al. 2021) have reinvigorated the detailed study of secular evolution of binaries in external tidal fields. The most famous scenario of this kind is a hierarchical triple, in which the compact object binary in question is orbited by a bound tertiary companion. In this case the tertiary companion can drive secular oscillations of the binary orbital elements -known as Lidov-Kozai (LK) oscillations (Lidov 1962;Kozai 1962) -on timescales much longer than either orbital period. Alternatively, very similar secular evolution arises if one considers a binary perturbed not by a tertiary point mass but by the global tidal field of the stellar cluster in which it resides (Hamilton & Rafikov 2019b,c,a;Bub & Petrovich 2020). In either scenario, the key idea is that secular tidal forcing can increase a binary's eccentricity dramatically. Provided this forcing is strong enough to overcome 1pN general relativistic (GR) apsidal precession -which acts to decrease the maximum eccentricity (Wen 2003;Hamilton & Rafikov 2021) -the binary can thereby achieve a small pericenter distance, allowing it to radiate gravitational waves (GWs) efficiently and thus rapidly merge. Similar mechanisms have been invoked to explain the origin of other exotic objects, such as Type 1a supernovae (Katz & Dong 2012), blue stragglers (Leigh et al. 2018), hot Jupiters (Fabrycky & Tremaine 2007), and so on. Regardless of the particular system under consideration, the same two key questions always arise: (i) under what circumstances can a binary reach extremely high eccentricity on an astrophysically relevant timescale? and (ii) how does the binary behave when it reaches such extreme eccentricities? Question (i) is straightforward to answer using secular theories, the simplest of which involve truncating the perturbing potential at quadrupole order, taking the 'test particle' approximation and then 'double-averaging' (DA) -that is, averaging the dynamics over both the binary's 'inner' Keplerian orbit and over its 'outer' barycentric motion around the potential. In Hamilton & Rafikov (2019b,c); Hamilton & Rafikov (2021) -hereafter Papers I, II and III respectively -we developed the most comprehensive such DA theory to date, capable of describing the secular evolution of any binary perturbed by any fixed axisymmetric potential (in the test particle, quadrupole limit), accounting for 1pN GR precession of the inner orbit. In this theory, which includes the LK scenario as a special case, the binary's maximum eccentricity e max can be calculated (semi-)analytically as a function of the initial conditions. As a result one can easily determine the region of parameter space that leads to extremely high e max . In the case of spherical cluster potentials (including the Keplerian LK case) which we will focus on exclusively throughout this paper, one always finds that e max is limited by the initial relative inclination i 0 between the binary's inner and outer orbital planes: e max ≤ e lim ≡ (1 − Θ) 1/2 ,(1) where Θ ≡ (1 − e 2 0 ) cos 2 i 0 , and e 0 is the initial eccentricity. Hence, a necessary (but not always sufficient) part of the answer to question (i) is that Θ ≪ 1. However, DA theories often do not provide an accurate answer to question (ii). That is because DA theory ignores a component of the torque that fluctuates on the timescale of the outer orbit, and normally washes out to zero upon averaging over that timescale. This becomes problematic at extremely high eccentricity, when the relative changes in the binary's (very small) angular momentum due to this fluctuating torque can become O(1). As a result, the DA theory can fail to capture the dynamics in detail. A more accurate (if more cumbersome) description is provided by the singly-averaged (SA) theory, in which one only averages over the binary's inner Keplerian orbit, and hence captures fully the fluctuations in the orbital elements on the outer orbital timescale. In particular, these short-timescale fluctuations 4 (sometimes called 'SA fluctuations') can increase a binary's maximum eccentricity beyond e max (Ivanov et al. 2005). Because of this they can be of great significance when predicting LK-driven merger rates of black hole (BH) or neutron star (NS) binaries, blue straggler formation rates, white dwarf collision rates, and so on (e.g. Katz & Dong 2012;Antonini & Perets 2012;Bode & Wegg 2014;Antonini et al. 2014;Antognini et al. 2014;Luo et al. 2016;Grishin et al. 2018;Lei et al. 2018;Lei 2019). Faced with this assessment, one might decide simply to abandon DA theory altogether and only work with the SA equations of motion. Alternatively one might choose to forego all averaging, and instead to integrate the 'Nbody' equations of motion directly. There are three main objections to these approaches. First, numerical integration of the SA or N-body equations is prohibitively expensive if one wants to evolve millions of binary initial conditions, as done in e.g. Hamilton & Rafikov (2019a). Second, SA and N-body approaches necessarily demand more initial data, inflating the parameter space. Third, whatever one gains though brute-force computation, one also often sacrifices in terms of analytical and physical insight. Instead, our approach will be to understand the SA, at high eccentricity in an approximate analytical fashion, guided by the DA theory and by numerical integrations where appropriate. We will restrict ourselves to studying the test particle quadrupole limit, and we will include GR precession of the inner orbit but we will ignore GW emission (though see Hamilton & Rafikov 2022). In particular, by examining in detail the numerical solutions to the SA equations of motion, we have encountered an important phenomenon which we call relativistic phase space diffusion (RPSD). This phenomenon must have been present in many authors' direct numerical integrations of the LK problem, but has never been explicitly discussed before. Since RPSD is the key result of this paper, let us now motivate our study with an example of it. 1.1. Example of relativistic phase space diffusion It is well known that when a binary is driven secularly to very high eccentricity, then even in the DA approximation, its orbital elements (i.e. its eccentricity, argument of pericenter, and longitude of ascending node) exhibit O(1) fractional changes on the timescale (see Paper III): t min ∼ j min t sec ,(3) where j min ≡ (1−e 2 max ) 1/2 is the minimum dimensionless angular momentum, e max is the maximum eccentricity, and t sec is the secular timescale. Clearly, for very large eccentricities, t min ≪ t sec . The central result of this paper is that in the SA approximation, if t min is so short as to be comparable to or smaller than the outer orbital period T ϕ , and GR apsidal precession is included, then the binary can very quickly 'jump' to a new phase space orbit, such that the DA approximation would fail completely to track the next secular cycle. Said differently, RPSD drives abrupt shifts of the binary's approximate DA integrals of motion, just like GW emission or stellar encounters could, but in their absence. To illustrate the RPSD phenomenon, we present Figures 1 and 2. To create these Figures we considered a NS-NS binary with semimajor axis a = 50 AU, orbiting in (and tidally perturbed by) a spherical Hernquist potential of mass M = 10 7 M ⊙ and radial scale b = 1 pc, which constitutes a simple model of a nuclear stellar cluster. The outer orbit's azimuthal period is T ϕ = 0.064 Myr. (The full set of initial conditions used here will be described in §3.1.1 after we have introduced our notation more fully). In each Figure, we integrate the (testparticle, quadrupole) SA equations of motion for roughly one secular period t sec . We ran this integration seven times, each time with identical initial conditions except that we give the binary's outer orbit a different initial radial phase. We also integrated the DA equations of motion for the same initial conditions (in the DA case the outer orbital phase information is irrelevant, since we have averaged over the outer motion). The only difference between Figures 1 and 2 is that in Figure 2, we switched on the effect of 1pN GR apsidal precession. This is a weak effect which only becomes important at very high eccentricity (the dimensionless strength of GR here is ϵ GR ∼ 10 −3 -see §2.2.1). We repeat that we do not include GW emission in any of our calculations in this paper. In both Figures we plot the SA eccentricity, log 10 (1 − e), for each run (different colored lines) at three stages of the evolution: (a) the very earliest stages near t = 0, (b) around the eccentricity peak, and (c) the latest stages near t = t sec . The thick dashed blue line in each panel is the DA solution. In Figure 1, we see that although the different SA realizations track each other very closely (at least in log space) in panel (a), around the eccentricity peak in panel (b) they differ quite considerably. Some runs do not even reach the DA maximum eccentricity log 10 (1 − e max ) ≈ −4.75, while some runs achieve an extremely high eccentricity, log 10 (1 − e max ) ≈ −5.65. These SA fluctuations at high-e can have a major effect on merger timescales, as is well known (Grishin et al. 2018). Nevertheless, upon emerging from the eccentricity peak, the different runs converge together (see panel (c)), closely tracking the 'underlying' DA solution. Now we compare this with Figure 2. The early evolution (compare Figures 1a and 2a) is essentially identical with and without GR, as expected away from highe. Next, comparing Figures 1b and 2b, we see that the maximum eccentricity reached by a given binary is very slightly diminished by the inclusion of GR (the green line now reaches 1 − e max ≈ −5.6 rather than −5.65), as we would expect given that the binary resides in the weak GR regime (Paper III), but otherwise the evolution is almost indistinguishable from the case without GR. However, by the time the DA eccentricity has returned to its minimum around 6.8 Myr (panel (c)) most of the SA trajectories have diverged significantly both from the DA solution and from each other. They will subsequently follow entirely different secular evolutionary tracks, many of which will be badly approximated by the original DA solution. As we will see in §4, over the course of many secular cycles this trajectory divergence results in a 'diffusion' of the value of the DA Hamiltonian, which is an approximate integral of motion around which each SA solution would normally fluctuate (as in Figure 1), and will be defined properly in §2. This diffusion behavior breaks down at sufficiently high eccentricity provided GR is switched on, so we call it RPSD. However, by using the term 'diffusion' we do not mean to suggest that the system's integrals of motion follow any simple Brownian walk or that their evolution can be described by a diffusion equation. In fact, the 'diffusion' we have found does not seem to obey any simple statistical behavior, as we will see in §4.3.3. We note here that Luo et al. (2016) also found that short-timescale fluctuations can cause the mean trajectory of a binary to drift gradually away from the DA prediction, even in the test-particle quadrupole LK problem, and derived a 'corrected DA Hamiltonian' to account for this deviation 5 . However, this effect is distinct from RPSD, for several reasons: (i) it has nothing to do with GR precession; (ii) it does not require extremely high eccentricity behavior, and (iii) it occurs due to an accumulation of nonlinear quadrupolar perturbations, which are normally averaged out in the standard LK theory, over many secular periods (see Tremaine 2023; these nonlinear effects are too small to be discernible in Figure 1). On the contrary, RPSD only occurs when GR is included, and happens on a much shorter timescale, requiring just one (very high) eccentricity peak. We discuss this comparison further in §5.3. 1.2. Plan for the rest of this paper The rest of this paper is structured as follows. In §2 we briefly recap some key results from Papers I-III and establish our notation. In §3 we provide several numerical examples that illustrate the phenomenology of shorttimescale fluctuations when GR is not included, particularly with regard to high eccentricity behavior. We then proceed to explain the observed behavior quantitatively, and derive an approximate expression for the magnitude of angular momentum fluctuations at high e. In §4 we switch on GR precession and give several numerical examples of systems exhibiting RPSD. We then analyse this phenomenon more quantitatively and offer a physical explanation for it. In §5 we consider the astrophysical importance of RPSD, and discuss our results in the context of the existing LK literature. We summarise in §6. DYNAMICAL FRAMEWORK In this section we recap the basic formalism for describing a binary perturbed by quadrupole-order tides, including 1pN GR precession. For more details see Papers I-III. Inner and outer orbits Consider a binary with component masses m 1 and m 2 , inside some spherically symmetric host system (the 'cluster') whose potential is Φ(R). The binary's barycentre R g (t) is assumed to move as a test particle in this potential: d 2 R g /dt 2 = −∇Φ(R g ). We call this the 'outer orbit', and assume it is confined to the (X, Y ) plane of a fixed inertial Cartesian coordinate system (X, Y, Z). Meanwhile, the binary's internal Keplerian orbital motion (the 'inner' orbit) is described by orbital elements: semi-major axis a, eccentricity e, inclination i, longitude of the ascending node Ω, argument of pericenter ω, and mean anomaly M . Here i is measured relative to the outer orbital (X, Y ) plane, and Ω is measured relative to the fixed X axis. For a spherical potential (which includes the LK case of a Keplerian potential), we take (X, Y ) to be the plane of the outer orbit. An alternative description of this inner orbit, which will be more convenient for our Hamiltonian formalism, is provided by introducing the Delaunay actions I = (L, J, J z ), with L = G(m 1 + m 2 )a, J = L √ 1 − e 2 , and J z = J cos i, and their conjugate angles ψ = (M, ω, Ω). Sometimes we will find it useful to refer to the dimensionless versions of these variables: j ≡ J/L = (1 − e 2 ) 1/2 ,(4)j z ≡ J z /L = (1 − e 2 ) 1/2 cos i,(5)Θ ≡ j 2 z = (1 − e 2 ) cos 2 i.(6) Obviously j is just the dimensionless angular momentum. Moreover, e and j must obey 0 ≤ e ≤ e lim ≡ √ 1 − Θ, Θ 1/2 ≤ j ≤ 1,(7) to be physically meaningful for a given Θ. 2.2. Dynamical equations Let us ignore GR precession for now, so the only forces that the binary feels are the internal two-body Keplerian attraction and the tidal perturbation from the cluster. Let these forces be encoded in the Hamiltonian Figure 1, except with 1pN GR apsidal precession switched on. The fact that the trajectories have diverged by t ∼ tsec is a manifestation of relativistic phase space diffusion (RPSD), which is the central result of this paper. H(ψ, I, t). Then the Delaunay variables evolve according to Hamilton's equations of motion: dψ dt = ∂H ∂I , dI dt = − ∂H ∂ψ .(8) After averaging over the binary's inner orbit, H = H 0 + H 1 where H 0 (I) = −µ 2 /(2L 2 ) (with µ ≡ G(m 1 + m 2 )) is just the Keplerian energy. The 'singly-averaged' (SA), test-particle quadrupole tidal Hamiltonian is then 6 (Paper I): H 1,SA (ψ, I, t) = 1 2 αβ Φ αβ ⟨r α r β ⟩ M .(9) The averages ⟨r α r β ⟩ M are given explicitly in terms of orbital elements, expressible through (ψ, I), in Appendix A of Paper I. The singly-averaged Hamiltonian H 1,SA (ψ, I, t) ends up being a function of the variables 6 In Papers I-II we referred to H 1,SA as ⟨H 1 ⟩ M . We will stick to the H 1,SA notation in what follows. An analogous statement holds for the upcoming H 1,DA and H GR . J, J z , ω, Ω and the time t (through the dependence of Φ αβ on R g (t)). The SA equations of motion follow by differentiating (9) according to (8) -these are given explicitly in Appendix A (equations (A1)-(A4)). In particular, since by averaging over the inner orbit eliminates the angle M , the conjugate action L = √ µa is conserved, and so the binary's semi-major axis a is constant under SA dynamics. If we further average the Hamiltonian (9) over the outer orbital motion R g (t) (i.e. over the orbital torus or annulus -see Paper I), the resulting doubly-averaged (DA) perturbing tidal Hamiltonian is H 1,DA = 1 2 αβ Φ αβ ⟨r α r β ⟩ M = A 8µ 2 × J −2 (J 2 − 3ΓJ 2 z )(5L 2 − 3J 2 ) − 15Γ(J 2 − J 2 z )(L 2 − J 2 ) cos 2ω .(10) Here A and Γ are constants (see §6 of Paper I) that depend on the potential and outer orbit; A measures the strength of the tidal potential and has units of (frequency) 2 , whereas Γ is dimensionless. In the Keplerian (LK) limit we find Γ = 1 and A = GM/[2a 3 g (1 − e 2 g ) 3/2 ], where a g and e g are respectively the semimajor axis and eccentricity of the outer orbit and M is the perturber mass. Being time-independent, H 1,DA is a constant of motion in the DA approximation. Moreover, the DA Hamiltonian (10) does not depend on the longitude of ascending node Ω, meaning that in the DA approximation, J z is also a constant of motion. The nontrivial DA equations of evolution of ω, J and Ω arising from (10) are given in equations (12)-(14) of Paper III. 1pN GR precession Next we wish to we include the effects of 1pN GR apsidal precession on the binary's inner orbit. Whether we use SA or DA theory, we can achieve this by adding to our Hamiltonian a term H GR = − AL 3 8µ 2 × ϵ GR J = − Aa 2 8 × ϵ GR j ,(11) where the strength of the precession is measured by the dimensionless parameter (see Papers II & III) ϵ GR ≡ 24G 2 (m 1 + m 2 ) 2 c 2 Aa 4 (12) = 0.258 × A * 0.5 −1 M 10 5 M ⊙ −1 b pc 3 × m 1 + m 2 M ⊙ 2 a 20 AU −4 .(13) In the numerical estimate (13) we have assumed a spherical cluster of mass M and scale radius b, and A * ≡ A/(GM/b 3 ) -see Paper I. The inclusion of GR precession obviously affects the equation of motion for dω/dt (whether we work in the SA or DA approximation) by adding an extra term CLϵ GR /J 2 . The physical effect of GR precession is typically to quench the cluster tidedriven eccentricity oscillations, as we explored in detail in Paper III (see also Miller & Hamilton 2002;Fabrycky & Tremaine 2007;Bode & Wegg 2014). As we showed there, there are typically no large e oscillations in the 'strong GR' regime, defined by ϵ GR ≳ ϵ strong ≡ 3(1+5Γ). Integrals of motion We know from Paper III that in the DA approximation, i.e. under the dynamics prescribed by the total DA Hamiltonian H DA ≡ H 1,DA + H GR , there are two independent integrals of motion. We could take these to be the value of the Hamiltonian H DA and the z-component of angular momentum J z , but for the purposes of this paper it will be most useful to take them to be j z ≡ J z /L and D, defined as (see equation (19) of Paper III): D ≡ e 2 1 + 10Γ 1 − 5Γ sin 2 i sin 2 ω − ϵ GR 3(1 − 5Γ) √ 1 − e 2 .(14) In the SA approximation, i.e. under the dynamics prescribed by the total SA Hamiltonian H SA ≡ H 1,SA + H GR , the quantities j z (t), D(t) are not precise integrals of motion 7 . Nevertheless, they can be usually be regarded as adiabatic invariants, i.e. quantities which fluctuate on the timescale of the outer orbital period but are approximately conserved upon averaging over this period, provided it is sufficiently short. In other words, under normal circumstances a binary's j z and D values simply fluctuate around the underlying DA solution corresponding to one of the level curves in the characteristic (ω, e) phase space -see Paper II. This allows one to consider short-timescale fluctuations as a perturbation on top of a dominant secular effect (Ivanov et al. 2005;Luo et al. 2016). On the other hand, in §4 we will see that for non-zero ϵ GR and very high eccentricities, this perturbative assumption can break downfor instance, the time-averaged value of D can change dramatically and abruptly, reflective of a violation of the adiabatic invariance condition, and this is what gives rise to RPSD. A note on phase space morphology The phase space trajectories described by the DA Hamiltonian (10) fall into two categories -librating trajectories, which loop around fixed points at ω = ±π/2 in the (ω, j) phase plane, and circulating trajectories, which traverse all ω ∈ (−π, π). Whether a binary's trajectory will librate or circulate depends on its initial orbital elements as well as the value of Γ. In Paper II we showed how to determine the family to which a binary's phase space trajectory belongs, explained how this affects the resulting minimum/maximum eccentricity achievable, etc. We also explained there how altering Γ changes the phase space morphology, and hence the relative importance of librating versus circulating trajectories. In particular, Γ = 0, ±1/5 turn out to be critical values, such that e.g. systems with Γ > 1/5 have a qualitatively different phase space morphology to those with 0 < Γ < 1/5, a result which has strong implications for the types of cluster which are able to easily excite high eccentricity behavior (Hamilton & Rafikov 2019a;Hamilton & Rafikov 2022). Moreover, In Paper III we extended these results to include the effect of GR precession, which also alters the phase space morphology. However, for the purposes of studying short-timescale fluctuations at extremely high eccentricity, it is largely irrelevant whether a binary is on a librating or circulating trajectory, or which Γ regime it happens to be in. This is because the details of the high eccentricity fluctuations depend predominantly on very short short-timescale (∼ T ϕ ) torquing that the binary experiences as e → 1, and not on the averaged behavior over the rest of the secular cycle. For instance, the numerical examples we present in this paper all happen to be for librating trajectories, but we have found qualitatively indistinguishable results for circulating examples. Similarly, we do show examples with both Γ > 1/5 and 0 < Γ < 1/5 in this paper, but the distinction between these two Γ regimes is not central to our discussion, so we mostly neglect to mention it from now on. SHORT-TIMESCALE FLUCTUATIONS AND HIGH ECCENTRICITY BEHAVIOR In this section we first provide a qualitative discussion of several numerical examples which demonstrate the phenomenology of short-timescale fluctuations ( §3.1). Crucially, for simplicity and in order to cleanly separate certain physical effects, we do not include GR precession in any of these examples (GR precession will be added in §4). In §3.2 we provide a quantitative analysis of the behavior we have uncovered. Finally in §3.3 we derive an approximate expression for the magnitude of angular momentum fluctuations at highest DA eccentricity, which gives us the maximum achievable e. Figure 3 we give an example of a binary that undergoes significant short-timescale fluctuations at high eccentricity. This figure is very rich in information and exhibits several interesting features that we wish to explore throughout the paper. We will also see several other figures with this or similar structure. It is therefore worth describing the structure of Figure 3 in detail. At the very top of the figure (top line of text) we provide the values of 6 input parameters (Φ, M, b, r p , r a , ϕ 0 ) that define the perturbing potential as well as the outer orbit's initial conditions. In this case (which is the same setup as in Figure 1) we are considering a binary in a Hernquist potential Φ(r) = −GM/(b+r), with total mass M = 10 7 M ⊙ and scale radius b = 1pc. Since this potential is spherical, the shape of the outer orbit is determined by two numbers: its pericenter distance r p = 0.7b, its apocenter r a = 1.4b. Unless otherwise specified, in this paper we always initiate the outer orbit at t = 0 from (R, ϕ) = (r p , ϕ 0 ) withφ > 0, where ϕ is the outer orbit's azimuthal angle relative to the X axis (see Paper I). In the second line of text we list 7 input parameters (m 1 , m 2 , a 0 , e 0 , i 0 , ω 0 , Ω 0 ) that concern the binary's inner orbit; the subscript '0' denotes initial values. In this example we are considering a NS-NS binary (m 1 = m 2 = 1.4M ⊙ ) with initial semimajor axis a 0 = 50AU. Note also that the initial inclination i 0 is chosen close to 90 • , which is necessary to achieve very large eccentricities starting from e 0 ≪ 1 (since this requires Θ ≪ 1). In the third and final line of text at the top of the figure, we list 5 important quantities that follow from the choices of 13 input parameters above: Γ, Θ, the inner orbital period T in = 2π µ/a 3 , the outer orbit's radial period T R , and its azimuthal period T ϕ . In this instance we have a Γ value of 0.326 and Θ = 2.1 × 10 −5 , which allows e max to become extremely high. Lastly, we emphasise that we have artificially switched off GR precession in this example. Thus we set ϵ GR = 0 in the equations of motion and in evaluating D, which is equivalent to taking the speed of light c → ∞. This choice is also indicated in the third line of text. GR precession will be incorporated in §4, allowing for direct comparison with the results of this section. Now we move on to the figure proper. In panels (a) and (b) we display the trajectory of the outer orbit through the (X, Y ) plane, integrated using GALPY (Bovy 2015). In both panels we show the trajectory from t = 0 to t = T R (cyan line), and from t = T R to t = 2T R (yellow line). In black we show the entire trajectory traced up to time t = 0.1t sec (panel (a)) and t = 0.5t sec (panel (b)), where t sec is the period of secular oscillations, computed using equation (33) of Paper II. In total we integrated the outer orbit R g (t) until t = 4.5t sec . We then fed the resulting Φ αβ (R g (t)) time series into the SA equations of motion (A1)-(A4) and integrated them numerically. In panels (c), (d) and (e) we compare the numerical integrations of the SA equations of motion for e, ω, Ω (green curves) against the prediction of DA theory (blue dashed curves). We also show the results of direct 'N-body' integration 8 (red dotted curves). In panel (c) we see that the binary reaches extremely high eccentricity, with the DA result 1 − e DA reaching a minimum at ≈ 10 −4.8 . In this panel we already see that the maximum eccentricity reached in the SA approximation can be rather different from the DA value, and changes from one eccentricity peak to the next -near the second peak, around 10 Myr, 1 − e SA plunges to ∼ 10 −5.4 . Another striking feature of panels (d) and (e) is the step-like jumps in ω and Ω that occur near maximum eccentricity in both SA and DA integrations. These are just what we expect from our investigation in Appendix C of Paper III. In panels (f) and (g) we show the evolution of the quantities D (equation (14)) and j z (equation (5)). In the DA approximation, D and j z are integrals of motion -hence the blue dashed DA result is simply a straight horizontal line. We see that the SA result oscillates around the constant DA value in both cases, with an envelope that has period t sec for j z and t sec /2 for D. In panel (f) there is in fact a small offset between D DA and the mean value of D SA , which is due to an initial phase offset of the outer orbit (Luo et al. 2016;Grishin et al. 2018). We also notice a characteristic behavior which is that fluctuations in D are minimised around the eccentricity peak, while fluctuations in j z are maximised there. In the right hand column, in panels (i)-(m) we simply reproduce panels (c)-(g), except we zoom in on the sharp eccentricity peak at around 10.28 Myr. At the top of this column we have panel (h), which shows the outer orbital radius R(t) during this high-eccentricity episode. In addition, in each panel (h)-(m) we shade in light blue the region t(j min ) − t min < t < t(j min ) + t min .(15) Here t(j min ) is the time corresponding to the minimum of j DA (i.e. peak DA eccentricity), namely when j DA = j min , and t min is the time taken for j DA to change from j min to √ 2j min -see §4.2 of Paper III. In panel (h) we also indicate the value of the ratio 2t min /T ϕ , which will turn out to be very important when we switch on GR in §4. In this particular case we see that 2t min /T ϕ = 0.41, so that most of the interesting (very highest eccentricity) behavior happens on a timescale shorter than an outer azimuthal period. From panels (h)-(m) we see that the N-body and SA integrations agree very well even at very high eccentric- -Example of a binary that undergoes significant short-timescale fluctuations at very high eccentricity (the same one we used in Figure 1). The details of the plot are explained fully in §3.1. Note the initial inclination i 0 = 90.3 • ities, giving us confidence that the SA approximation is a good one here 9 . However, the SA prediction differs markedly from the DA prediction at very high eccentricity. In particular, from panel (i) we see that 1 − e DA becomes ≈ 10 −4.6 at its minimum, while 1 − e SA reaches a significantly smaller value still, ≈ 10 −5.4 . Panels (j) and (k) reveal that the large jumps in ω and Ω both happen on a timescale ∼ 2t min . In panels (l) and (m) we show how the integrals of motion j z and D fluctuate around the maximum eccentricity. | j( )| 2 n φ n R R ± φ 2 R ± φ 2 φ ± R(r) In panels (n)-(q) we again show the time series of R and 1 − e around the second eccentricity peak (although over a wider timespan), as well as the differences δj(t) ≡ j SA (t) − j DA (t), δω(t) ≡ ω SA (t) − ω DA (t),(16) between the results of SA and DA integration. The vertical dotted magenta line in panels (o) and (p) corresponds to t = t 0.99 , which is the time when e DA first reaches 0.99. From panel (p) we see that δj fluctuates in a complex but near-periodic manner, with period ∼ 5T R . The fluctuations themselves are not perfectly centered around zero; before the eccentricity peak, the mean value of δj is slightly negative, whereas afterwards it is slightly positive. The blue and cyan bars in panel (p) correspond to simple approximations to the amplitude of δj at peak eccentricity -see §3.3. Meanwhile, from panel (q) we see that the fluctuation δω is negligible until the very highest eccentricities are reached, where there is a sharp pulse before it decays to zero again. This pulse is approximately antisymmetric in time around t = t(j min ). The pulse episode lasts for ∼ 2T R . Finally, in panel (r) we show the power spectrum of δj fluctuations, which is the square of the Fourier transform δj(ν) ≡ dt exp(iνt)δj(t). We calculated this Fourier transform numerically using the δj(t) data from panel (p). We see that the signal is concentrated at frequencies ν = n 1 Ω R + n 2 Ω ϕ for certain pairs of integers n 1 , n 2 , where Ω i ≡ 2π/T i is the outer orbital frequency. We discuss these power spectra briefly in §3.3 and §4.3.3. 3.1.2. Changing the initial inclination to i0 = 93.3 • In Figure 4 we run the same calculation as in Figure 3, except we take i 0 = 93.3 • rather than 90.3 • . 10 The main effect of this choice is to reduce the maximum eccentricity significantly, so that 1 − e max ≈ 10 −2.7 . As a result, DA evolution near maximum eccentricity is slower than in Figure 3, while the outer orbit is unchanged; hence we find 2t min /T ϕ = 4.55 in this case. The qualitative fluctuating behavior of D, j z and δj is quite similar between the two figures, although in Figure 4 many more fluctuations fit into the 'blue stripe' surrounding maximum eccentricity. The fluctuations δω are very different: the brief 'pulse' that lasted for only ∼ 2T R in Figure 3q has been replaced with a much broader signal with a slightly smaller amplitude. 9 The SA approximation can itself break down at extremely high eccentricity -see §5.2 for an example and a rough criterion. 10 To keep the plots clean, we do not show any N-body information for this example, or the example in §3.1.3. However, we have checked that N-body and SA integrations give indistinguishable results in each case. An example in the Plummer potential The phenomenology reported above is rather characteristic of binaries orbiting in cusped potentials, and will be analysed more quantitatively in §3.2. First, however, we perform the same calculation except this time with a cored potential, namely the Plummer sphere. In Figure 5 we provide an example of a binary exhibiting short-timescale fluctuations in a cored potential. All input parameters are the same as in Figure 3 except we change the potential from Hernquist to Plummer, so Φ = −GM/ √ b 2 + r 2 (we again use M = 10 7 M ⊙ and b = 1pc). The change of potential means that we now have Γ = 0.194 < 1/5, which leads to a rather large D DA value of around 17.6 (equation (14)). Noteworthy in this case is the morphology of the fluctuations of j z,SA near highest eccentricity, around the DA value j z,DA = − √ Θ = −0.0045 (panel (m)). In this case, small oscillations on the timescale ∼ T R are superimposed upon a larger 'carrier signal' oscillation, which has amplitude ∼ 0.006 and its own period ∼ 4T R . In this case these fluctuations can actually change the sign of j z , which corresponds to the SA binary inclination i temporarily passing through 90 • , a so-called 'orbital flip' -see Naoz (2016); Grishin et al. (2018). In Figure 6 we show the inclination i(t) explicltly for SA and DA integrations; we see that near peak eccentricity the DA approximation fails entirely to capture the flip behavior. A similar morphology is exhibited by the δj time series (panel (p)). Again the fluctuations δω (panel (q)) are negligible until the very highest eccentricities are reached, where there is a sharp, negative pulse of maximum amplitude ∼ 0.2π, that lasts for ∼ 2T R in total before decaying back to zero. Dependence on phase angles It is also worth emphasising here the dependence of these results on the choice of various phase angles, namely the initial radial phase of the outer orbit, the initial azimuthal angle of the outer orbit ϕ, and the initial choice of longitude of ascending node Ω of the inner orbit 11 . These choices feed into the solutions of the SA equations of motion (A1)-(A4), though of course they do not affect the DA solutions. By inspecting numerical examples without GR precession (such as that in Figure 1, as well as several others not shown here), we found that the choice of these phase angles can significantly affect e.g. the maximum value of e SA that is reached by a binary, but that the qualitative behavior is very similar from one realization to the next, and the averaged values of D and j z are conserved 12 . However, this ceases to be true when GR precession is included, as we saw in Figure 2 (see also §4). Precisely, we aim to understand the characteristic behaviors of δj and δω around the eccentricity peak and to understand the envelopes of D and j z fluctuations over secular timescales. We will address the separate problem of determining the amplitude of fluctuations δj around the peak eccentricity in §3.3. | j( )| 2 n φ n R R ± φ 2 R ± φ 2 φ ± R(r)| j( )| 2 n φ n R R ± φ 2 R ± φ 2 φ ± R (r) Plummer, Notation To achieve these aims we must first introduce a clean, precise notation to describe fluctuations. Let us define the vector w ≡ [ω, J, Ω, J z ]. Then the 'SA solution' w SA (t) ≡ [ω SA (t), J SA (t), Ω SA (t), J z,SA (t)],(17) is found by self-consistently integrating the SA equations (A1)-(A4), which are the Hamilton equations result- ing from H SA (ω SA , J SA , Ω SA , J z,SA , t) ≡ H SA (w SA , t). Meanwhile the 'DA solution' w DA (t) ≡ [ω DA (t), J DA (t), Ω DA (t), J z,DA ],(18) is found by self-consistently integrating the DA equations of motion, which are the Hamilton equations for H DA (ω DA , J DA , J z,DA ) ≡ H DA (w DA ). Consistent with equation (16) we formally define: δw(t) ≡ w SA (t) − w DA (t).(19) Next, we will also find it useful to define a 'fluctuating Hamiltonian': ∆H(w, t) ≡ H SA (w, t) − H DA (w),(20) which is written out explicitly in Appendix B. Using ∆H, we can define four new quantities ∆w(w, t) ≡ [∆ω(w, t), ∆J(w, t), ∆Ω(w, t), ∆J z (w, t)],(21) as the solution to the equations of motion d∆w(w, t) dt ≡ ∂ ∂J , − ∂ ∂ω , ∂ ∂J z , − ∂ ∂Ω ∆H(w, t). (22) As an example, the partial derivative ∂∆H/∂ω is given explicitly for spherical potentials in equation (B5). Note that equation (22) is defined for arbitrary arguments w. In Figure 7 we plot the time series of d∆w(w, t)/dt for w = w DA and w = w SA using the data from the first 8 Myr of evolution from Figure 3. Repeating the same exercise for other examples gives plots that look qualitatively the same as Figure 7; we will refer to this Figure several times in the upcoming discussion. Now we must bear in mind that in general, δw(t) ̸ = ∆w(w SA , t) ̸ = ∆w(w DA , t).(23) In other words, if we feed a numerical result w(t) = w SA (t) into equation (22) and integrate forwards in time, we do not in general reproduce the 'SA minus DA' solution (19); for example: δω(t) ≡ ω SA (t) − ω DA (t) ̸ = ∆ω(w SA , t) ≡ t 0 dt ′ ∂∆H(w SA (t ′ ), t ′ ) ∂J SA .(24) Similarly, δJ ̸ = ∆J and δΩ ̸ = ∆Ω. The exception is J z , for which there is no DA evolution, so that δJ z (t) = ∆J z (w SA , t). We feel it is important to make this distinction since it affects perturbative calculations -for example, Luo et al. (2016) implicitly used δw(t) = ∆w(w DA , t) when calculating the cumulative impact of short-timescale fluctuations over many secular cycles, by 'freezing' the DA elements on the timescale of the outer orbit. See §5.3 for more. Nevertheless, for our purposes it is normally sufficient to approximate δw(t) ≈ ∆w(w SA (t), t) ≈ ∆w(w DA (t), t).(25) To demonstrate this, we took the quantities from Figure 7 and integrated them forwards in time using (22). We plot the result in Figure 8 on top of the 'true' fluctuation δw(t), shown with black lines. We see that the approximation (25) is very accurate in this case (we also confirmed this with other examples not shown here). Scaling of fluctuations at high eccentricity in spherical cluster potentials Having established the approximation (25), we can use the equations of motion (22) to gain a better understanding of the behavior of fluctuating quantities δw in To do this we take derivatives of ∆H as given in (B4) (which is valid for spherical potentials only) and then take the high eccentricity limit 13 L 2 ≫ J 2 ≳ J 2 z . As a result we find the following scalings: d dt δj ∝ J 0 , d dt δj z ∝ J 0 , d dt δω ∝ J −1 , d dt δΩ ∝ J −1 .(26) Thus, we expect the fluctuations δj, δj z to be independent of j as e → 1, i.e. as j 2 , j 2 z → 0. In other words, as the binary approaches maximum eccentricity we do not expect any sharp peak in δj(t) or δj z (t), but we do expect a spike in δω(t) and δΩ(t). Such behavior is precisely what we found in panels (m), (p) and (q) of Figures 3-5, and is also exhibited clearly in Figure 8. Envelope of fluctuations in D and jz 13 Note we are not assuming J 2 → J 2 z , i.e. we are assuming nothing about the ratio Jz/J ≡ cos i other than that it is ∈ (−1, 1). This means that the results (26) hold regardless of the complicated behavior of the inclination near high-e as seen in e.g. Figure 6. In panels (a)-(d), black solid lines show the evolution of δw ≡ w SA − w DA using the data from Figure 3 during the time interval t ∈ (3, 3.32)Myr (the first peak in DA eccentricity occurs at t ≈ 3.34Myr). Cyan dashed lines and red dotted lines show the approximations ∆w(w SA , t) and ∆w(w SA , t) respectively. In panels (e)-(h) we show the same data using a logarithmic scale. In the numerical examples above, fluctuations in j z,SA consisted of rapid oscillations on the timescales ∼ T ϕ , T R (reflective of the torque fluctuating on the outer orbital timescale), modulated by an envelope with period t sec . Similarly, D SA oscillated on the timescale ∼ T ϕ , modulated by an envelope with period t sec /2. We now explain each of these envelope behaviors in turn. First, we consider the envelope of j z,SA fluctuations. It is clear from the numerical examples that the amplitude of this envelope is largest around the eccentricity peak (minimum j), and smallest around the eccentricity minimum (maximum j). This is easily explained by evaluating the torque formula dj z,SA /dt using equations (22) and (B6), and examining the scaling with j (which is well illustrated by the example in Figure 7d). The envelope of j z fluctuations simply reflects the amplitude of the fluctuating torque. Second, we address the fluctuations in D SA . In this case the amplitude of the envelope exhibits minima at times corresponding to both j DA = j max and j DA = j min , and maxima in-between. To see why, we differentiate (27). The green and orange dotted lines show contributions fromḊ 1 (equation (28)) andḊ 2 (equation (29)) respectively. The amplitude of both terms is smallest at the extrema of e DA . (14) with ϵ GR set to zero: dD dt =Ḋ 1 +Ḋ 2 (27) whereḊ 1 = 2 D(ϵ GR = 0) e de dt = − 2j D(ϵ GR = 0) 1 − j 2 dj dt ,(28)D 2 = 10Γe 2 1 − 5Γ d dt sin 2 i sin 2 ω .(29) In Figure 9 we plot dD/dt following equation (27), again over the first 8 Myr of evolution from Figure 3. With green and orange dotted lines we overplot the contributions coming fromḊ 1 andḊ 2 respectively. Like with j z,SA , the envelope of fluctuations in D SA reflects the envelope of dD/dt. In particular, the amplitudes of botḣ D 1 andḊ 2 are minimized at the eccentricity extrema, and maximized in-between. A much more detailed discussion of dD/dt at high eccentricity is postponed to §4.3.1. 3.3. Characteristic amplitude of δj fluctuations Perhaps the most important consequence of shorttimescale fluctuations is that they enhance the value of e max when e gets very large, which can lead to e.g. more rapid compact object binary mergers (Grishin et al. 2018). With this in mind, we wish to estimate (δj) max , which we define to be the absolute value of the maximum fluctuation δj in the vicinity of maximum eccentricity. Unfortunately, a given binary does not have a single value of (δj) max , because the precise details of the fluctuating behavior differ from one secular eccentricity peak to the next. (We already saw something closely related to this in Figure 1b, where binaries which approached the same secular eccentricity peak with different outer orbital phases ended up exhibiting very different behavior around e max ). Nevertheless, our goal in this section will be to estimate the characteristic size of such fluctuations and then demonstrate numerically that our estimate is a reasonable one. For simplicity we will again assume that Φ is spherically symmetric. Then to evaluate the torque at high eccentricity we can use (B5), which by (22) and (25) is a good approximation to −dδj/dt if we evaluate it using DA quantities. The maximum eccentricity as predicted by the DA theory is e DA = e max ≈ 1, and it always occurs either at ω DA = ±π/2 or at ω DA = 0. Let the corresponding minimum inclination be i min . Evaluating (B5) at these (assumed fixed) DA values, we find dδJ dt ω=±π/2 = 5 4 a 2 cos i min × 2f − (R) sin[2(ϕ − Ω)],(30) or the same thing with an additional minus sign if evaluating at ω = 0. Note that the function f − (R), defined in equation (B2), depends on the instantaneous value of the outer orbital radius R(t). Finally, one can check that for a Keplerian potential Φ = −GM/R we recover equation (B4) of Ivanov et al. (2005). Next we use the fact that in DA theory i min does not vary from one eccentricity peak to the next, and we assume that Ω DA is stationary on the timescale T ϕ . (This assumption, along with that of stationary ω DA and e DA on the timescale T ϕ , breaks down whenever t min ≲ T ϕ -see Figures 3 and 5, as well as Appendix C of Paper III. Nevertheless, these assumptions are good enough in order to get for a simple estimate of (δj) max which is all we need here). Placing the maximum DA eccentricity at t = 0 without loss of generality, we set Ω = Ω(0). Then the only time dependence in equation (30) comes from R(t) and ϕ(t). Furthermore, f − (R) < 0 for all R in sensible cluster potentials 14 . As a result, the sign of the torque at highest eccentricity (equation (30)) is dictated entirely by the instantaneous value of the phase angle 2(ϕ − Ω). The fluctuation (δj) max is therefore accumulated over a quarter period in azimuth, say from ϕ(t 1 ) − Ω = 0 to ϕ(t 2 ) − Ω = π/2, after which the torque changes sign. Integrating (30) over time we find (δj) max = 5 4 a 3 G(m 1 + m 2 ) 1/2 cos i min F (r p , r a )(32)=10 −4 × √ Θ/j min 1 m 1 + m 2 M ⊙ −1/2 a 10AU 3/2 × F * 0.8 M 10 5 M ⊙ 1/2 b pc −3/2 ,(33) 14 To see this, suppose the cluster has density profile ρ(r). From Poisson's equation ∇ 2 Φ = 4πGρ it is straightforward to show that ∂ 2 Φ ∂R 2 − 1 R ∂Φ ∂R = R ∂ ∂R GM(R) R 3 ,(31) where M(R) = R 0 4πr 2 drρ(r) is the mass enclosed inside a sphere of radius R. In particular, for any model in which ρ is a monotonically decreasing function of radius, the expression (31) is negative for all R. where all the details of the potential and outer orbit have been absorbed by the function F (r p , r a ) = t2 t1 dt ∂ 2 Φ ∂R 2 − 1 R ∂Φ ∂R sin[2(ϕ − Ω)],(34) and in the numerical estimate (33) we defined the dimensionless number F * ≡ GM b 3 −1/2 F.(35) 3.3.1. Parameter range where SA effects are important The problem with equations (32) and (34) as they stand is that we do not know precisely which quarterperiod in ϕ will provide the dominant fluctuation, because this would require knowledge of R(t) and ϕ(t). Thus, we cannot evaluate (34) directly for an arbitrary outer orbit -and even if we could, we would not expect the resulting (δj) max to be exactly correct because of the non-stationarity of ω, j, Ω. However, we can make a very rough estimate of the importance of these fluctuations if we note that F is normally of the same order of magnitude as the azimuthal frequency of the outer orbit, 2π/T ϕ . Then using cos i min ∼ 1 and the weak GR result j 2 min ∼ Θ ∼ cos 2 i 0 , we get (δj) max j min ∼ 1 | cos i 0 | T in T ϕ .(36) This equation is useful for estimating whether the effect of short-timescale fluctuations can drastically change j at very high eccentricity. In particular, it tells us that SA effects are crucial for binaries with initial inclinations in the range | cos i 0 | ≲ T in T ϕ .(37) We can also relate the estimate (36) to the ratio t min /T ϕ , which will be a very important parameter in our discussion of RPSD (see §4). We know from Paper II that t sec ∼ T 2 ϕ /T in . Then the time spent near highest eccentricity (equation (3)) is on the order of t min ∼ j min t sec ∼ j min T 2 ϕ /T in . Using this to eliminate T in from the right hand side of (36) gives (δj) max j min ∼ T ϕ t min .(38) Thus short-timescale fluctuations are important in precisely those regions of parameter space where t min is comparable to or smaller than T ϕ . 15 Circular outer orbits The simplest (and practically speaking, only) way to proceed more quantitatively than this is to estimate F by imagining that the binary is on a circular outer orbit 15 Unfortunately, this is also the regime in which the calculation we have performed in this section is not really valid, since this relied on our freezing the DA quantities while we calculated the fluctuations -see the discussion below (30). See §5.3 for more. with radius R. Then F = F circ (R) where F circ (R) = 2Ω circ ∂ ln Ω circ ∂ ln R ,(39) where Ω circ (R) = [R −1 ∂Φ/∂R] 1/2 is the angular frequency of a circular orbit of radius R. In the LK case of Keplerian potentials, the result arising from (32) with the circular approximation (39) was already derived by Ivanov et al. (2005). In Figure 10 we plot the dimensionless number F * circ ≡ (GM/b 3 ) −1/2 F circ as a function of R/b for circular outer orbits in various spherically symmetric cluster potentials with scale radius b. For reference we also plot F * circ for the Kepler potential Φ = −GM/R. We see that in the cored (Plummer and isochrone) models F * circ has a maximum value of order unity which is realised when R ∼ b, and that it falls sharply to zero towards the centre of the cluster. For centrally cusped potentials (Hernquist and NFW) we again have F * circ ∼ 1 at intermediate radii R ∼ b, but F * circ diverges towards the centre as ∼ R −1/2 , typically reaching F * circ ∼ 10 at the smallest sensible radii. At very large radii R ≫ b, the isochrone, Plummer and Hernquist potentials tend toward Keplerian behavior, F * circ ∼ R −3/2 . (The logarithm in the NFW potential means it never quite becomes Keplerian at these radii). From Figure 10 we learn that (i) the magnitude of short-timescale angular momentum fluctuations is roughly independent of potential type for R ≳ b, (ii) short-timescale fluctuations are significantly larger in cusped potentials than in cored potentials for R < b, and (iii) very large values of F * can be reached at small radii in the Kepler potential. Further discussion and examples Although it is strictly valid for circular outer orbits only, Figure 10 can also teach us something about noncircular outer orbits. For instance, in cusped potentials, the scaling of F * circ with R suggests that as long as the outer orbit is not too eccentric, a decent approx-imation to the dominant short-timescale fluctuation can be found by employing the circular approximation with F circ (equation (39)) evaluated at R = r p . In cored potentials this is no longer true because of the turnover in F * circ at R ∼ b. Then, for example, for orbits with r a ≲ b the dominant j fluctuations clearly arise around apocenter passage, since this is where F * circ is largest and this is where the outer orbit spends the most time. However, outer orbits in cored potentials with r a ≲ b tend to have Γ < 1/5 (Paper I), so they tend not to reach such high eccentricities anyway 16 , and besides, the values of F * circ never exceeds ∼ 1 regardless of R for these potentials. Hence, for an order of magnitude estimate we may choose simply to evaluate (δj) max using equation (39) with R = r p , regardless of the type of potential or outer orbit. In panel (p) of Figures 3-5 we show as 'error bars' the values of ±(δj) p (cyan) and ±(δj) a (yellow), which are calculated by evaluating ±(δj) max (equation (32)) using the circular approximation (39) at R = r p and R = r a respectively. We see that the circular approximation gives a reasonable estimate of the amplitude of fluctuations δj. One important caveat here is that while the δj(t) behavior is often rather regular up to e DA ≲ 0.99, it often becomes rather irregular in the immediate vicinity of the eccentricity peak, as can be seen in Figures 3-5. This is because of the rapid evolution, ω, Ω and i when j ≈ j min (see the light blue shaded bands in panels (j) and (k) of each of those Figures, as well as the inclination plot in Figure 6) which introduces a significant phase dependence into the detailed fluctuation behavior. Of course, since (32) was derived by assuming stationary ω, Ω, i, j, it necessarily fails to capture this irregular behavior. Finally, we attempted to capture the complicated behavior of δj(t) near the eccentricity peak for non-circular outer orbits, and thereby move beyond the circular approximation, by isolating the contribution of individual Fourier modes δj(ν) (panel (r) of Figures 3-5). However, even in the cases where a single dominant Fourier mode can be extracted (such as the ν = 2Ω ϕ − Ω R mode in Figure 5r), our lack of knowledge of the outer orbital phase as high eccentricity was approached made it nearimpossible to predict in detail the SA behavior e.g. in Figure 5o. THE EFFECT OF GR PRECESSION In the previous section we gained insight into how short-timescale fluctuations in the tidal torque affect high eccentricity behavior, but if our results are to be relevant for studying dynamical compact object merger channels, then it is vital that we also account for 1pN GR precession. As we will see in this section, including GR precession can change the picture significantly. To begin with, in §4.1 we rerun the numerical calculations from §3.1 except this time with GR precession switched on, and simply describe the altered phenomenology. In §4.2 we compare the GR and non-GR calculations in the special case of the LK problem. The main new result that arises in each of these subsections is RPSD, which stems from non-conservation of the approximate integral of motion D in the SA approximation. In §4.3 we offer a physical explanation for this new phenomenon, explain the criteria for its existence, and attempt a rough statistical analysis. Numerical examples with GR precession Fiducial Hernquist example In Figure 11 we rerun the calculation from Figure 3, except we switch on the GR precession term (with strength ϵ GR = 0.00107) in the SA and DA equations of motion. We now discuss Figure 11 in some detail. The structure of panels (a)-(m) is identical to those of Figure 3, except that we have dispensed with N-body integration since it is prohibitively computationally expensive (to capture accurately the fast GR precession during eccentricity peaks tends to require an extremely tiny timestep). Comparing panels (a)-(m) with those of Figure 3 we immediately notice several qualitative differences. Whereas in Figure 3 the DA and SA predictions for log 10 (1 − e) agreed almost perfectly except at extremely high eccentricity, now in Figure 11 (with GR precession switched on) they disagree manifestly after the second eccentricity maximum. Moreover, while the period of secular oscillations is fixed in the DA case, the SA secular period changes from one eccentricity peak to the next. By the time of the third eccentricity peak the DA and SA curves in panels (c)-(e) are completely out of sync, as we intimated would happen back in §1.1, when we were discussing Figure 2. Crucially, from panel (f) we see that D SA no longer fluctuates around D DA indefinitely like it did in Figure 3f, but rather exhibits discrete jumps during very high eccentricity episodes. In panel (l) we zoom in on the D SA behavior around the second eccentricity peak. We see that D SA jumps from ≈ −1.05 to ≈ −0.8 and that this jump happens on the timescale ∼ 2t min (the width of the blue shaded band). This jump in the approximate integral of motion D means that the binary has jumped to a new constant-Hamiltonian contour in the DA (ω, e) phase space (Papers II-III), as we confirm in Figure 12. Since this behavior depends crucially on the presence of finite GR precession we choose to call it 'relativistic phase space diffusion' (RPSD). Furthermore, each phase space contour has its own secular period, i.e. the secular period t sec depends on D (see §2.6 of Paper II, and especially Figure 3 of that paper). Hence it is unsurprising that a jump in D SA leads to a modified SA secular period compared to the fixed DA period 17 . Meanwhile, we observe in panel (g) that the envelope of j z,SA fluctuations does undergo abrupt, although very modest, changes that coincide with the second and third eccentricity peaks. Thus, the adiabatic invariance of j z,SA is also slightly broken, but its time average is much Figure 3, except we switch on GR precession. This causes the time-averaged value of D SA to diffuse away from D DA , with discrete jumps occurring during episodes of extremely high eccentricity. We call this relativistic phase space diffusion ('RPSD'). better preserved than is the time average of D SA . To investigate the RPSD behavior further, we next ran the integration for a much longer time, 20t sec . In panels (n) and (o) of Figure 11 we plot 1 − e and D respectively as functions of time over this entire duration. The same picture holds in that D SA is roughly static between eccentricity peaks, but often makes a discernible jump during a peak. These jumps seem to have no preferred sign. 4.1.2. Changing the initial inclination to 93.3 • RPSD does not occur in Figure 13, in which we keep the initial conditions exactly the same as Figure 11 except that we change i 0 from 90.3 • to 93.3 • . (In other words, we rerun the same calculation as in Figure 4 except with GR switched on). Note that this case has 2t min /T ϕ = 4.55, as opposed to Figure 11 which had 2t min /T ϕ = 0.42. An example in the Plummer potential In Figure 14 we give another example of RPSD, this time in the Plummer potential. This example is identical to that in Figure 5 except that we switched on GR precession, and zoomed in on the first eccentricity peak (10) and (11)), and the black dashed line gives the maximum achievable DA eccentricity e lim ≡ √ 1 − Θ, all calculated at t = 0. The binary's initial phase space location is shown with a green circle, and this of course coincides with the blue dashed contour on which the DA solution lives indefinitely. The binary's final SA location is shown with a red circle. We see that the SA trajectory jumps to a new 'parent' DA contour (i.e. off the blue dashed DA contour) during the second high eccentricity episode. in the right column rather than the second. Panel (l) shows that this first eccentricity peak coincides with a large jump in D. As a result, the subsequent SA evolution is entirely different from the original DA prediction. Dependence on phase angles Analogous to the discussion in §3.1.4, we have also run several more numerical calculations identical to those shown in this section but varying the outer orbit's initial radial phase angle and the initial value of ϕ − Ω. As expected from Figure 2, we find that RPSD is highly phase-dependent, meaning that SA simulations run with slightly different initial conditions can have dramatically different outcomes. This suggests we should attempt a statistical analysis -see §4.3.3. An example in the Lidov-Kozai limit It is important to note that RPSD is not exclusive to the non-Keplerian potentials that we have investigated so far, but is present in the Lidov-Kozai problem as well, at the test-particle quadrupole level (though to our knowledge, nobody has mentioned it explicitly). To demonstrate this, in Figure 15 we show a calculation with exactly the same initial condition as in Figure 3, except that we change the potential to the Keplerian one, Φ = −GM/R. In other words, we are now investigating the classic test particle quadrupole Lidov-Kozai problem, relevant to e.g. a NS-NS binary orbiting a SMBH (e.g. Antonini & Perets 2012;Hamers 2018;Bub & Petrovich 2020), except that we have GR precession switched off. In panels (a) and (b) we simply see the outer orbital ellipse, which has semimajor axis a g = (r a + r p )/2 = 1.05b and eccentricity e g = (r a − r p )/(r a + r p ) = 0.33. Panels (c)-(m) exhibit behavior which is qualitatively similar to that in Figure 3. Since 2t min /T ϕ = 0.15, the highest eccentricity episode lasts for significantly less than one outer orbital period. Despite this, the SA result tracks the secular (DA) result indefinitely. (We have also checked that the SA result shown here is indistinguishable from that found by direct N-body integration). Next, in Figure 16 we perform the same calculation as in Figure 15, but now with GR precession switched on. This causes significant and repeated diffusion of D SA around the secular eccentricity peaks, meaning the binary ultimately follows a completely different SA trajectory compared to the one we would have predicted had we only used DA theory. This confirms that RPSD is present as a phenomenon in LK theory at the testparticle quadrupole level as long as 1pN GR precession is included. Physical interpretation of RPSD and quantitative analysis The aim of this section is to provide some physical understanding of RPSD and to attempt a rough quantitative analysis of this phenomenon. Mechanism behind RPSD To understand why RPSD occurs, we take note of two pieces of empirical evidence from the above examples (which we confirmed in several additional numerical experiments not shown here): • When GR precession is switched off, there is no RPSD. • RPSD can occur when 2t min /T ϕ ≲ 1 but we do not observe it to occur when 2t min /T ϕ ≫ 1. Taking these strands of evidence together will allow us to identify the physical mechanism behind RPSD, as we now explain. Whether we consider SA or DA dynamics, at eccentricities far from unity, significant changes in the orbital elements occur only on secular timescales, i.e. timescales much longer than T ϕ . Of course, D (equation (14)) is a function of these orbital elements. Thus at eccentricities far from unity, D SA (t) invariably exhibits relatively small and rapid (timescale ∼ T ϕ ) oscillations around the constant value D DA . However, at the highest eccentricities e → 1, significant changes in orbital elements can occur on timescales much shorter than t sec . This is true even in DA theory: for instance, in Appendix C of Paper III (which concerned high-e behavior in the limit of weak, but nonzero, GR precession) we saw several numerical examples of binaries whose ω DA value is turned through ∼ 90 • or more on the timescale 2t min . Now, when 2t min /T ϕ ≫ 1 this evolution is still slow from the point of view of the outer orbit. But in the regime 2t min /T ϕ ≲ 1, the DA theory essentially predicts its own demise, since it tells us that O(1) relative changes in the orbital elements occur on the timescale of the outer orbit, contrary to the fundamental assumption of the DA approximation, namely outer orbit-averaging (Paper I). In that case, it is possible that the fluctuations of SA theory may no longer be considered small, and would no longer oscillate rapidly around the DA orbital elements while those DA elements change significantly. As an example, consider e.g. Figure 11j, for which 2t min /T ϕ = 0.42. In this case ω DA undergoes a large 'swing' on the timescale ∼ 2t min near peak DA eccentricity. This time range is so short that ω SA does not have a chance to fluctuate around ω DA during it. By contrast, consider Figure 13j, for which 2t min /T ϕ = 4.55. In this instance ω DA undergoes a swing of very similar magnitude but on a much longer timescale, allowing ω SA to perform multiple small fluctuations around ω DA while the 'swing' is in progress (i.e. within the blue range). Now let us incorporate GR precession into the discussion. We know (see Figures 3-5 and 15) that when GR is switched off, the aforementioned short timescale fluctuations do not affect D SA very dramatically -instead, D SA just oscillates around the constant value D DA indefinitely. Indeed, in the absence of GR, dD SA /dt is minimized at very high eccentricity, as we already showed for a particular example in Figure 9. When we do include GR precession, dD SA /dt gets an additional contribution arising from the last term in equation (14) Figure 11 except we use the Plummer potential -in other words, the same as Figure 5 except we switch on GR precession. In this case, after a few secular periods the DA prediction does not even qualitatively describe the SA dynamics. (27) we now have dD dt =Ḋ 1 +Ḋ 2 +Ḋ GR ,(40) withḊ 1 ,Ḋ 2 still given by (28)-(29), anḋ D GR ≡ ϵ GR 3(1 − 5Γ) 1 j 2 dj dt .(41) However, the extra term (41) cannot be directly responsible for RPSD, because when integrated across an eccentricity peak it gives zero. (Equivalently, the third term in (14) is unchanged by passing through an eccentricity peak, since it only depends on the value of j). Nevertheless, when coupled with the lack of timescale separation 2t min /T ϕ ≲ 1, GR is responsible for RPSD, as we will now demonstrate. In Figure 17a we plot dD/dt (black line) using data from the first 8 Myr of evolution from Figure 11 (c.f. Figure 9). The contributions fromḊ 1 andḊ 2 (equations (28)-(29)) are overplotted with green and orange dotted lines respectively, while the termḊ GR (equation (41)) is shown with a red dashed line. We see that contrary to Figure 3, except we use the Kepler potential (so we are studying the classic test-particle quadrupole Lidov-Kozai mechanism). GR precession is switched off here. Figure 9, there is a large spike in the signal at maximum eccentricity, around t = 3.35 Myr (recall that t = 6.8 Myr is an eccentricity minimum). In Figure 18 we zoom in on this eccentricity peak, and find thatḊ 1 (green dotted line) contributes negligibly to dD/dt at this time. Since we just argued above that the GR term (red dashed line) cannot be responsible for RPSD, the total change in D that occurs across the eccentricity peak must come from the integral over the orange curve, i.e. it must arise due toḊ 2 , which is proportional to d(sin 2 i sin 2 ω)/dt. Let us now dig even more deeply into this term. We write it asḊ Figure 15 except with GR precession switched on. This Figure shows that RPSD is present even in the test-particle quadrupole LK problem as long as GR is included. 2 =Ḋ 2a +Ḋ 2b +Ḋ 2c ,(42)whereḊ 2a ≡ 10Γe 2 1 − 5Γ sin 2i di dt sin 2 ω(43)D 2b ≡ 10Γe 2 1 − 5Γ sin 2 i sin 2ω dω dt −ω GR ,(44)D 2c ≡ 10Γe 2 1 − 5Γ sin 2 i sin 2ωω GR ,(45)andω GR = AL 5 ϵ GR 8µ 2 J 2 = Aϵ GR T in 16πj 2 ,(46) is the direct contribution of GR precession (see equation (A1)). In Figure 18b we replot the orange dotted curve from Figure 20, but this time we also plot its component parts,Ḋ 2a ,Ḋ 2b andḊ 2c . It is very striking that both contributionsḊ 2a andḊ 2b have large amplitudes (they would individually give rise to fluctuations |∆D| ∼ 5 on a timescale T ϕ ), but that they almost perfectly cancel one another. In fact, by using equations (A1), (A2) and (A4) it is possible to show that in the high-eccentricity limit L ≫ J ≳ J z (the same limit we took in §3.2.2), the combinationḊ 2a +Ḋ 2b → 0. 18 As a result, at high 18 More precisely, one findṡ D 2a +Ḋ 2b = αβ Φ αβ (t)Q αβ (w SA ),(47) and each of the Q αβ (w SA ) individually vanish in the high eccentricity limit j, jz → 0, assuming nothing about ω, Ω or the ratio cos i ≡ jz/j. For instance, without any approximations we get (40)) for the first 8 Myr of evolution from Figure 11; in other words, this is the counterpart of Figure 9 with GR included. The green and orange dotted lines again show the contributions fromḊ 1 andḊ 2 (equations (28)-(29)), whereas the red dashed line shows (41); the black solid line is the total dD/dt. Figure 17, but zoomed in around the first eccentricity peak. In panel (b) we ignore the red, green and black lines from panel (a), leaving only the contribution from the termḊ 2 (orange), which we decompose into its constituent parts (43)-(45). The termsḊ 2a andḊ 2b have large amplitudes but cancel eachother almost perfectly at high eccentricity. eccentricity the orange dotted line is comprised entirely of contributions from termḊ 2c (equation (45)), i.e. the part driven directly by GR precession. Qzz ∝ (1 − j 2 )j sin 2 2i sin 2ω sin 2 ω,(48)T φ dD/dt (b)Ḋ 2 D 2ȧ D 2ḃ D 2c In Figures 19-20 we repeat the exercise of Figures 17-18, except using the data from Figure 13, i.e. the example with 2t min /T ϕ = 4.55 which does not exhibit RPSD. This time there is no spike in any contribution to dD/dt at high eccentricity. The contributionsḊ 2a andḊ 2b cancel each other again as they must, but this time there is no significant contribution fromḊ 2c either. From these figures, the root cause of RPSD finally becomes clear. In the absence of sufficiently rapid GR precession at high eccentricity, there is near perfect synchronicity between the evolution of the inclination i SA and that of the argument of pericenter ω SA built into the SA equations of motion. As a result, the factor sin 2 i sin 2 ω is essentially a constant during a high eccentricity episode. But when GR gets switched on, it acts only on ω SA and not (directly) on i SA -instead i SA can which obviously tends to zero at high eccentricity. The other Q αβ are more complicated but all tend to zero in the limit j, jz → 0. only 'react' indirectly to the changes in ω DA . Thus there is a GR-driven lack of synchronization between ω DA and i DA on the timescale T ϕ , meaning sin 2 i sin 2 ω is not precisely constant. This 'phase-lag' between i and ω is instigated as the binary enters the eccentricity peak around j ∼ j min , since this is where GR is most effective, and it is driven for a time ∼ 2t min . If 2t min /T ϕ ≫ 1 then the lack of synchronization will be negligible and there will be no RPSD. But if 2t min /T ϕ ≲ 1, the GR-driven evolution of ω DA gives rise to RPSD before the value of i DA can catch up. In terms of D, like we saw in Figure 17l, the value of D SA has no time to oscillate back and forth around its 'parent' D DA value before the high eccentricity episode is over that upon emerging from the high-eccentricity blue stripe it 'settles' on a new parent D DA value (compare this with Figure 13l). Equivalently, the binary 'jumps' to a new contour in the (ω, e) phase space while at high e (still at fixed Γ, Θ and ϵ GR ), aided by the fact that contours at high e are bunched so closely together ( Figure 12). Criteria for producing significant RPSD Using the above argument, we can estimate the 'jump' that D SA sustains across the eccentricity peak, and in so doing find rough criteria for this jump to be significant. We know that the jump in D SA is equal to the time integral of (45) across the eccentricity peak, i.e. to the area under the black curve in Figure 18: D jump = 10Γ 1 − 5Γ dt e 2 sin 2 i sin 2ωω GR ,(49) whereω GR is given in equation (46). Here everything should have an 'SA' suffix, and the width of the integration domain should be several t min , centered on the eccentricity peak. To get a characteristic value for the integral, we first approximate e ≈ 1. We also know that sin 2 i sin 2ω will be oscillating in a complicated way as the binary passes through the eccentricity peak, but as long as 2t min /T ϕ ≲ 1 these oscillations will not average out to zero. To order of magnitude, we therefore replace dt sin 2 i sin 2ω j −2 → 0.1t min j −2 min for a very rough estimate. This gives |D jump | ∼ 0.1 × 10Γ 1 − 5Γ ϵ GR T in At min 16πj 2 min ,(50) though the crudeness of our approxmations means that the prefactor 0.1 is chosen somewhat arbitrarily. If we now recall that A ∼ 4π 2 /T 2 ϕ (Paper I), and that t min ∼ Figure 19, but zoomed in around the first eccentricity peak. Panel (b) showsḊ 2 (orange), and its constituent parts (43)-(45). j min t sec ∼ j min T 2 ϕ /T in (Paper II), we find T φ dD/dt (b)Ḋ 2 D 2ȧ D 2ḃ D 2c|D jump | ∼ 0.1 × 20Γ 1 − 5Γ ϵ GR 2t min T ϕ −1 T ϕ T in .(51) For the examples of RPSD shown in Figures 11,14 and 16,equation (51) returns the values |D jump | ∼ 0.8, 5 and 0.3 respectively 19 . Finally, we return to the assumption that 2t min /T ϕ ≲ 1. Assuming weak GR so that j min ∼ Θ 1/2 , this requirement may be recast as: Θ ≲ T in T ϕ 2 (52) ≲ 5 × 10 −5 × m 1 + m 2 2.8M ⊙ −1 a 50AU 3 × M 10 7 M ⊙ 1 R 1pc −3 ,(53) where we assumed a binary on a circular outer orbit of radius R in a spherical cluster of mass M, i.e. we put 19 Note that we use the values of t min calculated at t = 0, before the binary has undergone any RPSD. Once it has shifted to a new parent D DA value, this value will change (since both the minimum angular momentum j min and the secular period tsec will have changed). T 2 ϕ ∼ GM/R 3 . For binaries that are not initially very eccentric, Θ ∼ cos 2 i 0 , meaning that that RPSD operates in the inclination window: | cos i 0 | ≲ 0.007 × m 1 + m 2 2.8M ⊙ −1/2 a 50AU 3/2 × M 10 7 M ⊙ 1/2 R 1pc −3/2 .(54) Throughout this paper we have considered clusters with mass 10 7 M ⊙ , binaries with the mass of a NS-NS binary, m 1 = m 2 = 1.4M ⊙ , and outer orbits with semimajor axis a g = 1pc. Putting R ∼ a g with these numbers into equation (54) Moreover, the requirement that 2t min /T ϕ ≲ 1 is essentially the same as the requirement that short timescale fluctuations are important at high eccentricity, (δj) max ≳ j min -see equation (38). This leads us to the important conclusion that in situations with GR precession included, if short-timescale fluctuations are dominating the high-eccentricity evolution then some level of RPSD is inevitable. We discuss the astrophysical implications of RPSD further in §5.1. In conclusion, RPSD occurs if (54) is satisfied; if so, the resulting jumps in D SA have a characteristic size (51). Obviously, if ϵ GR = 0 then there is no RPSD (i.e. D jump = 0 regardless of initial inclination). Statistical analysis of RPSD We know from §1.1 and §4.1.4 that the details of the erratic high eccentricity behavior in the presence of GR precession are highly dependent on the outer orbital phase, i.e. on the values of R and ϕ − Ω as the binary approaches the eccentricity peak. This makes it very difficult to predict the behavior of D SA precisely for a given set of initial conditions. Thus, a natural next step is to investigate the statistics of jumps in D SA , by creating an ensemble of D jump values for the same binary on the same outer orbit, but set off with different initial values of R and ϕ − Ω. To do this, it is necessary to first define more precisely what we mean by a 'jump in D'. This immediately raises some technical issues: (i) the value of D SA is never actually fixed, and (ii) a steady-state approximation of D SA clearly breaks down as e SA approaches unity. We therefore choose to study time-averages of D SA before and after the eccentricity peak, taken over time intervals that do not include the peak itself, i.e. sufficiently far from the peak for the averaged value to be meaningful 20 . In particular, before the first eccentricity peak this guarantees that the time average ⟨D SA ⟩ coincides with the 'parent' value D DA . We then define the jump in D across the peak to be D jump ≡ ⟨D SA ⟩ after − ⟨D SA ⟩ before .(55) In order to probe the statistics of D jump , we first considered an ensemble of 10 4 systems with exactly the same setup and initial conditions as in Figure 11, except in each case we drew a random initial value for ϕ − Ω (uniformly in (0, 2π)) and a random initial radial phase (i.e. for fixed r p , r a we chose at random the initial radius R 0 , correctly weighted by the time spent at each radius, i.e. dN ∝ dR 0 /v R (r p , r a )). We stopped each integration at t ≈ t sec , i.e. after one full eccentricity oscillation, meaning D jump was well-established according to equation (55). The results of this exercise are shown in Figure 21a,b. In panel (a) of this Figure we present a scatter plot of the values of |D jump | against the minimum j SA value that was achieved in the corresponding random realization. With a vertical cyan line we show the DA prediction j DA,min , and with a horizontal orange line we show the characteristic estimate (51). In panel (b) we marginalize over j SA,min in order to produce a histogram of D jump values, or rather two histograms, one for positive D jump (red) and one for negative D jump (blue). From these panels it is clear that there is no very simple dependence of D jump on j SA,min , nor is there necessarily any symmetry between the distribution of positive and negative D jump values, nor does the distribution converge to any simple form. In fact, the value of |D jump | varies over several orders of magnitude, and equation (51) merely provides an estimate (though often not a particularly accurate one) of its maximum value. On the other hand, the upper limit of the envelope of D jump values is fairly well-fit by a power law |D jump | ∝ j −1.5 SA,min (see the black solid line in panel (a)). In the remaining panels of Figure 21 we repeat this exercise for several other examples known to exhibit RPSD. In panels (c)-(d) and (e)-(f) respectively we change the potential to the Plummer potential (i.e. taking the initial conditions from Figure 14) and the Kepler potential (taking initial conditions from Figure 16). Again we see that the |D jump | upper envelopes are fairly well-fit by power laws |D jump | ∝ j −1.5 SA,min , but aside from this no robust trend that can be pulled out. Although the histograms of D jump values in the Kepler case may seem like they have a promising symmetry (panel (f)), this also proves unreliable. To show this, we ran two further examples in the Kepler potential, this time for nearly circular outer orbits. Precisely, in panels (g)-(h) and (i)-(j) we use the same initial conditions as in Figure 16 except we change the outer orbit's peri/apocenter from (r p /b, r a /b) = (0.7, 1.4) to (0.7, 0.702) and (1.4, 1.402) respectively. In these cases, the tracks through (j SA,min , D jump ) space approximately follow one-dimensional curves (panels (g) 20 In practice, it is sufficient to average D over several outer orbital periods during a time range corresponding to e SA / ∈ (0.99, 1). and (i)) rather than a broad two-dimensional distribution, due to the fact that we have removed a degree of freedom (the initial radial phase) by making the outer orbit near-circular. Despite this, there seems to be no simple trend in the distribution of D jump values (panels (h) and (j)). In fact, we plotted several such figures for several different sets of initial conditions, cluster potentials, and so on, tried increasing the number of random realizations substantially, and experimented with different ways of binning the values of D jump , but we were unable to uncover any striking insights. We also tried taking the Fourier transform of δj(t) near the eccentricity peak and isolating the important frequencies that contribute to the signal, hoping that way to gain insight into RPSD (the same exercise that we performed in panel (r) of Figures 3-5). Naturally, we found that these Fourier spectra are concentrated at frequencies n 1 Ω R + n 2 Ω ϕ for pairs of integers n 1 , n 2 (here Ω i ≡ 2π/T i is the outer orbital frequency). Unfortunately, there were typically several important frequencies at play simultaneously, especially for outer orbits that were far from circular, and we did not extract anything useful from this effort. DISCUSSION Astrophysical relevance of RPSD Since we have uncovered a new effect in this paper the obvious question is: how relevant is it to astrophysical systems? Let us assume that our compact object binary satisfies ϵ GR ≪ ϵ weak and hence reaches very high eccentricity j min ∼ Θ 1/2 (Paper III). Assume also that its initial eccentricity is not so large. Then the important necessary condition for significant RPSD to occur is equation (54). In Hamilton & Rafikov (2019a) we considered compact object mergers driven by spherical cluster tides, so we will use that as a test case. There, 10 7 M ⊙ was the upper limit on sensible cluster masses; m 1 = m 2 = 1.4M ⊙ was the lower limit on compact object masses; and 50AU was the upper limit on any sensible distribution of (still rather soft) binaries. Since cos i 0 is distributed uniformly ∈ (0, 1) for isotropically oriented binaries, we conclude that RPSD would have affected much less than 1% of our sample. Moreover, this fraction will be even smaller when we consider e.g. BH-BH binaries with m 1 = m 2 = 30M ⊙ . Thus, we do not expect RPSD to be important for the bulk populations of binary mergers that we considered in Hamilton & Rafikov (2019a). Nevertheless, it is easy to find numerical examples of compact object binaries orbiting in clusters for which RPSD is a contributing effect (Rasskazov & Rafikov, in prep.). In these cases the analytic description of secularly-driven inspirals developed in Hamilton & Rafikov (2022) breaks down completely, and the merger timescale can be wildly different (either longer or shorter) from that used in Hamilton & Rafikov (2019a). We speculate that RPSD may be important for certain exotic phenomena that involve even more extreme eccentricities, such as head-on collisions of white dwarfs in triple systems (Katz & Dong 2012). Note that RPSD occurs even for circular outer orbits and for equal mass binaries, i.e. in triples where the octupole contributions to the potential are very small, which are not usually considered promising for producing high merger rates. However, further analysis of this possibility will require incorporating the effects GW emission, which we have neglected throughout this paper. This is left as an avenue for future work. Breakdown of the SA approximation In this paper we have considered the SA dynamics of binaries driven by cluster tides, particularly at very high eccentricity. We have implicitly assumed that these equations are accurate. It is worth noting, however, that even the SA approximation can itself break down, in situations where any of (J, ω, J z , Ω) evolve significantly on the timescale of the inner orbital period (Antonini et al. 2014). Such a situation is shown in Figure 22. In this figure we consider a soft NS-NS binary with T in ∼ 600 yr, that reaches 1 − e ∼ 10 −4 in the Hernquist potential. (We do not switch on GR precession in this example, to make it clear that the breakdown of the SA approximation has nothing to do with GR effects). The blue band in the right hand panels spans 2t min while the red band spans 2T in . We see that, by the time of the 7th eccentricity peak, the N-body and SA results differ significantly, with 1 − e differing by up to ∼ 0.5 dex. Note that the binary component masses are equal in this example, m 1 = m 2 , so any octupole terms in the tidal force ought to be zero, meaning any corrections are of hexadecapole or higher order. Nevertheless, we checked that the disagreement is really due to a breakdown in the orbit-averaging approximation, rather than these higher order terms, by running an N-body simulation with only the perturbing potential truncated at quadrupole order. The agreement with the full (untruncated) N-body simulation was excellent. The SA approximation is based on the assumption that T in is small compared to any other timescale of interest. Thus, the breakdown of the SA approximation occurs if the binary orbital elements, particularly ω and Ω, change sufficiently rapidly near peak eccentricity, which occurs if t min is not sufficiently large compared to T in . For instance, in Figure 22, ω changes by ∼ π on a timescale of ∼ 5000 yr. Given T in ≈ 600 yr, this corresponds to a ratė ω ∼ 20 • per inner orbital period. Such a large value is really an indication that the orbit is not truly Keplerian, and thus that the Keplerian orbital elements themselves are not particularly well-defined; it is perhaps unsurprising that in this regime even the SA approximation breaks down. Relation to LK literature The question of how short-timescale fluctuations affect high eccentricity evolution in LK theory was first addressed by Ivanov et al. (2005), who estimated the amplitude of the angular momentum fluctuations experienced near maximum eccentricity by a binary undergoing LK oscillations. Ivanov et al. (2005)'s result and other results very similar to it have since been used extensively for modelling hierarchical triples (e.g. Katz & Dong 2012;Bode & Wegg 2014;Antognini et al. 2014;Silsbee & Tremaine 2017;Grishin et al. 2018). Their high-e fluctuating torque formula is a special case of our equation (30). To evaluate this formula, Ivanov et al. (2005) took the outer orbit to be circular; hence our analysis in §3.3.2 encompasses theirs as a special case. On the other hand, in a Keplerian potential the circular outer orbit approximation is not necessary; Haim & Katz (2018) generalized the result of Ivanov et al. (2005) to eccentric outer orbits. No such generalization is possible for arbitrary spherical cluster potentials. More recently, as we mentioned in §1.1, Luo et al. (2016) took a perturbative approach to the SA LK problem for arbitrary inner and outer eccentricities. They showed that short-timescale fluctuations captured by the SA equations of motion can accumulate over many secular periods, resulting in secular evolution that does not resemble the original DA prediction. In other words, the time-averaged SA solution does not agree with the DA solution, but instead diverges from it gradually (see Tremaine 2023 for a more rigorous mathematical treatment of this phenomenon). This divergence occurs in a predictable way, and Luo et al. (2016) were able to derive a 'corrected DA' Hamiltonian (essentially a ponderomotive potential, see Tremaine 2023) that accurately reproduces the time-averaged SA dynamics. Although Luo et al. (2016) only considered the LK problem up to quadrupolar order in the tidal expansion, their results were generalized to arbitrary (octupole, hexadecapole, ...) order by Lei et al. (2018), and then further extended to include fluctuations on the inner orbital timescale ∼ T in by Lei (2019). Moreover, Grishin et al. (2018) applied the formalism of Luo et al. (2016) to high eccentricity behavior. Assuming a circular outer orbit they calculated the new maximum eccentricity arising from Luo et al. (2016)'s 'corrected' secular theory, as well as the magnitude of angular momentum fluctuations at highest eccentricity. Though we have not done so here, in the special case of circular outer orbits the results of Luo et al. (2016) and Grishin et al. (2018) could be trivially extended to arbitrary axisymmetric cluster potentials of the sort considered in this paper. However, in the key papers mentioned above (Ivanov et al. 2005;Luo et al. 2016;Grishin et al. 2018), GR precession was not directly included when calculating the fluctuating behavior at high eccentricity. Those authors also all implicitly assumed the timescale separation 2t min /T ϕ ≫ 1, allowing them to freeze the time-averaged values of (ω, J, Ω, J z ) on the timescale T ϕ while they calculated the fluctuations. Our work is different in that we have included GR precession and, crucially, investigated systems with 2t min /T ϕ ≲ 1. In this case, the RPSD effect that we have uncovered means that SA dynamics do not converge to the original DA prediction on average, just as was found by Luo et al. (2016) in the non-GR LK theory. However, unlike Luo et al. (2016)'s discovery, RPSD depends critically on the strength of GR precession and also happens very rapidly (on a timescale t min ≲ 2T ϕ ) rather than accumulating over many secular periods. It is also worth contrasting the behavior found in this paper with that from Hamilton & Rafikov (2022). In that case, we found that the secular behavior of binary orbital elements changed over time due to bursts of GW emission at eccentricity peaks, in a purely DA framework -we did not include short-timescale fluctuations. By contrast, in the present paper we have found secular behavior that changes over time due to bursts of RPSD at high eccentricity, which is driven by short-timescale fluctuations and GR precession, but we have not included -Example of SA and N-body integrations disagreeing at high eccentricity. In the right column we zoom in to the seventh eccentricity peak, where the two approaches for calculating 1 − e disagree by as much as ∼ 0.5 dex. The blue band in the right hand panels spans 2t min while the red band spans 2T in . Note that we started the outer orbit at apocenter, rather than pericenter, in this example. GW emission. Thus, although we motivated our study by using parameters typical of compact object binaries, in our case the binaries will never merge since there is no dissipation of energy. Future work focusing on dynamical merger channels might investigate the interplay between RPSD and GW emission, combining the results of Hamilton & Rafikov (2022) with the present paper. Finally, we have considered only quadrupole terms in the tidal potential, i.e. we have considered an SA problem whose DA counterpart is completely integrable. The octupole terms (see Appendix E of Paper I) are very small in most cases we consider, since a/R is very small in applications to binaries in clusters (e.g. Hamilton & Rafikov 2019b). In fact, in all the numerical examples shown in this paper the octupole terms are identically zero, since we always took the binary components to have equal masses m 1 = m 2 . In LK theory, octupole and higher-order terms are expected to become important when the outer orbit is significantly eccentric and the component masses are not equal, and can lead to non-integrability and hence chaos, both in the SA and DA approximations (Li et al. 2015). We have shown that in the SA approximation, quasi-chaotic phase space behavior can arise even at the pure test particle quadrupole level via RPSD, provided 1PN GR precession is included. Indeed, perhaps the reason that quadrupole-level RPSD has not been mentioned before in the LK literature is that the majority of numerical integrations of the LK equations with GR do include octupolar or higher order terms and/or relax the test-particle approximation, and any complicated behavior that results is then attributed to these effects. SUMMARY In this paper we have investigated the role of shorttimescale fluctuations upon (test particle quadrupole) tide-driven evolution of binary systems, particularly with regard to their high eccentricity behavior. Our main results can be summarized as follows. • We analyzed the behavior of fluctuations of binary orbital elements in the singly-averaged (SA) approximation, in the absence of 1pN GR apsidal precession. In particular, we derived an expression for the magnitude of angular momentum fluctuations at high eccentricity for binaries orbiting in arbitrary spherically symmetric cluster potentials. Roughly, these fluctuations are comparable in magnitude to the minimum angular momentum predicted by doubly-averaged (DA) theory whenever the cosine of the initial inclination is comparable to or smaller than the ratio of inner and outer orbital periods (equation (36)). • We then investigated the high eccentricity SA behavior including 1pN GR precession, and found that the evolution can be dramatically different from the case without GR. This can be true even in the weak GR regime (Paper III), where GR makes negligible difference to the DA dynamics. In particular, relativistic phase space diffusion (RPSD) may kick the binary to a new phase space contour on the timescale of the outer orbit, potentially leading to quasi-chaotic evolution and extreme eccentricities, and a full breakdown of the naive DA theory. The rough criterion for RPSD to occur is essentially the same as for angular momentum fluctuations to be comparable to the DA minimum angular momentum (equation (54)), with the additional requirement that GR precession be switched on. The size of the typical is proportional to the dimensionless strength of GR precession ϵ GR . • RPSD likely affects only a very small fraction of binaries in population synthesis studies, but may be a crucial ingredient for e.g. head-on collisions of white dwarfs. Fig. 1 .Fig. 2 . 12-Example of eccentricity evolution for a NS-NS binary orbiting a Hernquist potential with outer azimuthal period T ϕ = 0.064 Myr, for seven different values of the outer orbit's initial radial phase. Panels (a) and (c) show the beginning (t ≈ 0) and end (t ≈ tsec) of the calculation, respectively, while panel (b) focuses on the eccentricity peak around t ≈ tsec/2. The DA solution (independent of radial phase) is shown with a blue dashed line. -As in 3. 1 . 1Three numerical examples 3.1.1. Fiducial example in the Hernquist potential, i0 = 90.3 • In Fig. 3.-Example of a binary that undergoes significant short-timescale fluctuations at very high eccentricity (the same one we used in Figure 1). The details of the plot are explained fully in §3.1. Note the initial inclination i 0 = 90.3 • As inFigure 3except we change i 0 to 93.3 • . As a result emax is decreased and 2t min becomes significantly larger than T ϕ . As inFigure 3, except this time we use the Plummer potential. Plotting the inclination i = arccos(jz/j) around the first eccentricity peak fromFigure 5. The SA result reveals that near peak eccentricity, the binary experiences orbital 'flips' whereby its inclination crosses 90 • . Fig. 7 .Fig. 8 . 78-Time series of d∆w/dt following the definition(22), for the first 8 Myr of evolution fromFigure 3(the first peak in DA eccentricity occurs at t ≈ 3.34Myr; the subsequent eccentricity minimum is at t ≈ 6.8Myr). Cyan dashed lines and red dotted lines show d∆w(w SA , t)/dt and d∆w(w SA , t)dt respectively. -Justifying the approximation δw(t) ≈ ∆w(w SA , t) ≈ ∆w(w DA , t). Fig. 9 . 9-Time series of dD/dt (black solid line) for the first 8 Myr of evolution from Figure 3, following equation Fig. 10 . 10-Plot of the dimensionless function F * circ ≡ F circ (R)/(GM/b 3 ) 1/2 , where F is defined in equation(39), for four spherical potentials with scale radius b, as well as the Kepler potential Φ = −GM/r. Fig . 11.-As in Fig . 12.-Illustration of RPSD in phase space. We plot the trajectory of the binary fromFigure 11through the (ω, e) phase space, over the first ∼ 2tsec. We show the DA trajectory in blue and the SA trajectory in green. The grey lines are contours of constant DA Hamiltonian (equations Fig . 13.-As inFigure 11except we take i 0 = 93.3 • -in other words, the same asFigure 4except we switch on GR precession. There is no RPSD in this case. Fig . 14.-Fiducial Plummer example including GR precession. As in Fig. 15.-As in Figure 3, except we use the Kepler potential (so we are studying the classic test-particle quadrupole Lidov-Kozai mechanism). GR precession is switched off here. Fig . 16.-As in Fig . 17.-Time series of dD/dt (equation Fig. 18 . 18-Panel (a) is the same as in Fig . 19.-As inFigure 17, but using data fromFigure 13, which does not exhibit RPSD. Fig. 20 . 20-Panel (a) is as in gives | cos i 0 | < 0.007, which corresponds approximately to i 0 ∈ (89.6 • , 90.4 • ). This is concomitant with what we found numerically; all examples that exhibited RPSD had i 0 = 90.3 • , while the example for which there was no RPSD (Figure 13) had i 0 = 93.3 • . Fig. 21 . 21-In panels (a)-(b) we rerun the first secular period fromFigure 11, except for 10 4 randomly drawn outer orbital phases, i.e. values of ϕ − Ω and R. Panel (a) shows a scatter plot of |D jump | values from each run against the minimum j value achieved in the SA calculation. Panel (b) shows a histogram of the |D jump | values from panel (a). Note the initial DA value of D here is D DA = −1.036. Panels (c)-(d) are the same except for the Plummer potential (c.f.Figure 14), with D DA = 17.624. Panels (e)-(f) are for the Kepler potential (c.f.Figure 16), with D DA = −0.371. Panels (g)-(h) are again for the Kepler potential as inFigure 16except we change the outer orbit from (rp/b, ra/b) = (0.7, 1.4) to (rp/b, ra/b) = (0.7, 0.702). Panels (i)-(j) are the same except this time we change the outer orbit to (rp/b, ra/b) = (1.4, 1.402). Fig. 22.-Example of SA and N-body integrations disagreeing at high eccentricity. In the right column we zoom in to the seventh eccentricity peak, where the two approaches for calculating 1 − e disagree by as much as ∼ 0.5 dex. The blue band in the right hand panels spans 2t min while the red band spans 2T in . Note that we started the outer orbit at apocenter, rather than pericenter, in this example. Throughout this paper, we use the term 'short-timescale fluctuations' to mean those fluctuations that arise in SA theory when compared with DA theory. We do not consider other fluctuations that might occur on short timescales, e.g. flyby encounters with passing stars. This effect, which seems to have been first investigated at arbitrary eccentricities and inclinations byBrown (1936), was recently put on a solid mathematical footing byTremaine (2023). To be clear, in the SA approximation the value of D is found by evaluating the right hand side of (14) using the SA orbital elements. The 'N-body' integration involves directly integrating the exact binary equations of motion in the presence of the full, smooth timedependent field Φ(Rg(t)) -without any tidal approximationusing REBOUND(Rein & Liu 2012). .2. Analysis of fluctuating behavior We now wish to explain more quantitatively the behavior that we observed in the previous two subsections.11 In fact, the SA equations for binaries perturbed by spherically symmetric clusters only depend on the difference ϕ−Ω, rather than on ϕ and Ω individually -see equation (B4).12 Apart from the gradual drift away from the DA solution that occurs over many secular timescales(Luo et al. 2016), which is distinct from RPSD, as discussed in § §1.1 and 5.3. Of course, the Γ < 1/5 numerical examples shown in this paper do reach very high e (Figures 5, (14)), but that is because we have purposely chosen a rather special set of initial conditions in order to make this happen. For instance, inFigure 11, making D less negative while keeping jz -and therefore Θ -fixed moves the binary closer to the separatrix in the (D, Θ) phase space (seeFigure 3cof Paper II), which accounts for the increase in the period of the subsequent secular oscillation. (Strictly speaking, tsec is not the exact DA secular period when we include GR precession, as it is calculated assuming no GR (equation (33) of Paper II), but for ϵ GR ≪ 3(1 + 5Γ) -the 'weak-tomoderate GR regime', see Paper III -it is a very good approximation.) We thank Ulrich Sperhake and Bence Kocsis for their careful scrutiny of an earlier version of this work, and Martin Pessah for helpful comments on §5.2. This work was supported by a grant from the Simons Foundation (816048, CH), as well as the STFC grant ST/T00049X/1 and Ambrose Monell Foundation.APPENDIXSINGLY-AVERAGED EQUATIONS OF MOTION IN THE TEST-PARTICLE, QUADRUPOLE LIMIT FOR ARBITRARY CLUSTER POTENTIALSThe full SA Hamiltonian is H SA = H 1,SA + H GR where H GR is given by(11)and H 1,SA is given by(9). The corresponding SA equations of motion are dω dt = ∂H 1,SA ∂J + ∂H GR ∂J =[L 2 /(8J 4 µ 2 )]× (Φ xx + Φ yy ) − J 5 (3 + 5 cos 2ω) − 5JJ 2 z L 2 (1 − cos 2ω) + (Φ xx − Φ yy ) − J 5 (3 + 5 cos 2ω) cos 2Ω + 5JJ 2 z L 2 (1 − cos 2ω) cos 2Ω + 5J 2 J z (J 2 + L 2 ) sin 2Ω sin 2ωcos 2Ω sin 2ω + 2JJ z cos 2ω sin 2Ω + Φ zz 0.5(J 2 − J 2 z ) sin 2ω + Φ xy JJ z cos 2ω cos 2Ω − 0.5(J 2 + J 2 z ) sin 2ω sin 2Ω + Φ xz J 1 − J 2 z /J 2 (J cos 2ω cos Ω − J z sin 2ω sin Ω) + Φ yz J 1 − J 2 z /J 2 (J cos 2ω sin Ω + J z sin 2ω cos Ω) ,cos Ω sin 2ω + J z (−3J 2 + 5L 2 + 5(J 2 − L 2 ) cos 2ω) sin Ω) .In the case of axisymmetric cluster potentials, it is straightforward to recover the DA equations from the SA equations (A2)-(A3) by replacing the time-dependent quantities Φ αβ with their time-averages Φ αβ , and using the identities Φ xx = Φ yy and Φ xy = Φ xz = Φ yz = 0 (see equations (33)-(36) of Paper I).FLUCTUATING HAMILTONIANBy subtracting the DA Hamiltonian from the SA Hamiltonian, assuming them to be functions of the same variables (J, ω, ...), and using Φ xx = Φ yy and Φ xy = Φ xz = Φ yz = 0, we get an expression for the 'fluctuating Hamiltonian' (equation(20)):(Note that the term involving ϵ GR has disappeared, since it is the same in SA and DA theory). Equation (B1) holds for binaries in arbitrary axisymmetric potentials. We can simplify matters significantly if we restrict ourselves to spherical potentials Φ(r) = Φ( √ R 2 + Z 2 ). Let us defineand assume without loss of generality that R g is confined to Z = 0. Then it is easy to show (see equations (33)-(36) of Paper I) that:(Note that we have dropped the 'g' subscript for ease of notation). If we also define ∆f ± ≡ f ± − f ± where f ± is the annulus-averaged value of f ± then the fluctuating Hamiltonian can be written concisely asAs explained in §3.2, equations for the evolution of fluctuations in orbital elements be derived from the fluctuating Hamiltonian ∆H(J, ω, ..) by taking its partial derivatives (equation(22)). In particular, we haveAssuming this to be a good approximation to −dδj/dt at high e, as we do in §3.3, it constitutes a generalisation of equation (B4) ofIvanov et al. (2005). The result ofIvanov et al. (2005)is recovered if one assumes the outer orbit to be circular (so ∆f ± = 0), the perturbing potential to be Keplerian, and evaluates (B5) at ω = ±π/2 and e → 1. We also have ∂∆H ∂Ω = L 2 4J 2 µ 2 f − 10JJ z (J 2 − L 2 ) sin 2ω cos[2(ϕ − Ω)] − ((J 2 − J 2 z )(3J 2 − 5L 2 ) + 5(J 2 + J 2 z )(J 2 − L 2 ) cos 2ω) sin[2(ϕ − Ω)] ,which coincides precisely with (minus) the right hand side of (A4) if one assumes Φ to be spherical. This is as it must be, since in DA dynamics J z is perfectly conserved, and so dJ z /dt = dδJ z /dt, as we argued below equation(24). . R Abbott, T Abbott, S Abraham, The Astrophysical journal letters. 9137Abbott, R., Abbott, T., Abraham, S., et al. 2021, The Astrophysical journal letters, 913, L7 . J M Antognini, B J Shappee, T A Thompson, P Amaro-Seoane, Monthly Notices of the Royal Astronomical Society. 4391079Antognini, J. M., Shappee, B. J., Thompson, T. A., & Amaro-Seoane, P. 2014, Monthly Notices of the Royal Astronomical Society, 439, 1079 . 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[]
[ "Dance Generation by Sound Symbolic Words", "Dance Generation by Sound Symbolic Words" ]
[ "Miki Okamura \nUniversity of Tsukuba\nJapan\n\nNARUYA KONDO\nUniversity of Tsukuba\nJapan\n\nTATSUKI FUSHIMI\nUniversity of Tsukuba\nJapan\n\nYOICHI OCHIAI\nMAKI SAKAMOTO\nThe University of Electro-Communications\nJapan\n\nUniversity of Tsukuba\nJapan\n" ]
[ "University of Tsukuba\nJapan", "NARUYA KONDO\nUniversity of Tsukuba\nJapan", "TATSUKI FUSHIMI\nUniversity of Tsukuba\nJapan", "YOICHI OCHIAI\nMAKI SAKAMOTO\nThe University of Electro-Communications\nJapan", "University of Tsukuba\nJapan" ]
[ "Conference acronym 'XX" ]
Fig. 1. The dance motion generated by our model: By providing a creative onomatopoeia (in this case, "Bulber") as input, we can generate a dance that suits it. The input onomatopoeia is embedded in terms of the intensity of 43 adjective expressions by the Sakamoto system[14]and is input into a deep learning-based dance generation model called FACT[36], which generates the dance. A zero vector is input during times when no onomatopoeia is provided.This study introduces a novel approach to generate dance motions using onomatopoeia as input, with the aim of enhancing creativity and diversity in dance generation. Unlike text and music, onomatopoeia conveys rhythm and meaning through abstract word expressions without constraints on expression and without need for specialized knowledge. We adapt the AI Choreographer framework and employ the Sakamoto system, a feature extraction method for onomatopoeia focusing on phonemes and syllables. Additionally, we present a new dataset of 40 onomatopoeia-dance motion pairs collected through a user survey. Our results demonstrate that the proposed method enables more intuitive dance generation and can create dance motions using sound-symbolic words from a variety of languages, including those without onomatopoeia. This highlights the potential for
null
[ "https://export.arxiv.org/pdf/2306.03646v1.pdf" ]
259,088,916
2306.03646
816695b07383dc65bb7bc22e2d167efd018ee4be
Dance Generation by Sound Symbolic Words June 03-05. 2018 Miki Okamura University of Tsukuba Japan NARUYA KONDO University of Tsukuba Japan TATSUKI FUSHIMI University of Tsukuba Japan YOICHI OCHIAI MAKI SAKAMOTO The University of Electro-Communications Japan University of Tsukuba Japan Dance Generation by Sound Symbolic Words Conference acronym 'XX Woodstock, NYJune 03-05. 2018ACM ISBN 978-1-4503-XXXX-X/18/06. . . $15.00 https://doi.org/XXXXXXX.XXXXXXX diverse dance creation across different languages and cultures, accessible to a wider audience. Qualitative samples from our model can be found at:CCS Concepts: • Computing methodologies → Animation; Machine learning Additional Key Words and Phrases: motion generationmachine learningdatasetssound symbolic wordsonomatopoeia ACM Reference Format: Miki OkamuraNaruya KondoTatsuki FushimiMaki Sakamotoand Yoichi Ochiai 2023 Dance Generation by Sound Symbolic Words In Woodstock '18: ACM Symposium on Neural Gaze DetectionJune 03-052018WoodstockNY ACMNew YorkNYUSA10 pages https://doiorg/XXXXXXXXXXXXXX Fig. 1. The dance motion generated by our model: By providing a creative onomatopoeia (in this case, "Bulber") as input, we can generate a dance that suits it. The input onomatopoeia is embedded in terms of the intensity of 43 adjective expressions by the Sakamoto system[14]and is input into a deep learning-based dance generation model called FACT[36], which generates the dance. A zero vector is input during times when no onomatopoeia is provided.This study introduces a novel approach to generate dance motions using onomatopoeia as input, with the aim of enhancing creativity and diversity in dance generation. Unlike text and music, onomatopoeia conveys rhythm and meaning through abstract word expressions without constraints on expression and without need for specialized knowledge. We adapt the AI Choreographer framework and employ the Sakamoto system, a feature extraction method for onomatopoeia focusing on phonemes and syllables. Additionally, we present a new dataset of 40 onomatopoeia-dance motion pairs collected through a user survey. Our results demonstrate that the proposed method enables more intuitive dance generation and can create dance motions using sound-symbolic words from a variety of languages, including those without onomatopoeia. This highlights the potential for INTRODUCTION Dance is a universal form of expression that goes beyond the limits of language, culture, and geography. As an essential aspect of human history, dance has continually evolved with advancements in technology. One such development is the incorporation of computer graphics and animation in the automatic generation of dance. This combination of technology and creativity has the potential to offer new perspectives, support the creation of dance, and reduce effort. Furthermore, it contributes to developing diverse movements for interactive characters in video games and virtual reality applications. With the progress of technology, machine learning techniques have been introduced to the field of dance generation. Traditional textbased dance generation methods can only handle expressions that exist in language, making it challenging to incorporate new sounds and sensations not represented in existing language. Consequently, these methods face difficulties in generating unique dance motions that go beyond the constraints of language and culture, limiting creativity and diversity. On the other hand, music-based dance generation methods, while rich in rhythm, may not effectively convey meaning. Moreover, these approaches are not easily accessible to users who cannot create music themselves. We propose the utilization of onomatopoeia as a effective and innovative input source for a machine learning model aimed at generating dance motion. Onomatopoeia possesses the unique ability to convey both rhythm and meaning through abstract word sense expressions, making it an ideal candidate for this purpose. Its versatile nature allows it to be employed in various contexts, such as describing the texture of a meal, the touch of a cat, the speed of a car, or the ambiance of a comic strip, emphasizing its broad applicability and potential for enhancing the dance generation process. Building upon the framework of an existing study named AI Choreographer [36], we adapt this method to incorporate onomatopoeia as input for generating dance motion. In contrast to text and music inputs, we utilize the Sakamoto system [14] for extracting features from onomatopoeia. The system is a deterministic feature extraction method focusing on phonemes and syllables, representing onomatopoeia through 43 numerical values corresponding to the strength of adjective expressions. This system can be used with any language containing phonemes, as it is not influenced by linguistic meaning. Combining the Sakamoto system with a deep generative model is a novel approach, as there are few examples of such implementation in the literature. We collected 44 pairs of onomatopoeia and dance motions through a user survey and trained the model on this dataset. The results of our study indicate that the proposed model successfully generates dance motions that correspond with the input onomatopoeia. Furthermore, we demonstrate that the model can generate dances using sound-symbolic words from various languages, including those without onomatopoeia, allowing for more intuitive dance generation. However, it must be acknowledged that the quality of the generated motions is not on par with the latest state-of-the-art dance generation methods. Future research and development can focus on model improvements, dataset expansion, and the development of interactive applications for onomatopoeia-based dance motion generation. This paper's contributions are as follows: • We introduce a novel approach using onomatopoeia as input for dance motion generation. • We adapt the AI Choreographer framework and employ the Sakamoto system for feature extraction from onomatopoeia. • We present a new dataset of 44 onomatopoeia-dance motion pairs collected from a user survey. • Our study demonstrates the successful generation of dance motions using sound-symbolic words from various languages, including those without onomatopoeia. • The results indicate potential for intuitive and diverse dance creation across different languages and cultures. RELATED WORK 2.1 Human Motion Generation The problem of human motion generation has been extensively studied in the fields of computer vision, graphics, and robotics. Early approaches focused on employing statistical models, such as kernel-based probability distribution [6,7,19,43], and motion matching techniques [25] to synthesize motion. However, these methods tended to abstract away motion details and were primarily restricted to simple domains like locomotion. The advent of deep learning has led to a surge of interest in exploring the applicability of neural networks for generating 3D motion by training on large-scale motion capture datasets. Various network architectures have been investigated, including CNNs [26,27], GANs [23], RBMs [50], RNNs [3,8,9,15,18,20,30,56,57], and Transformers [2,5,37,58]. While these approaches have demonstrated the capability to generate more diverse and realistic human motions than earlier methods, they still face challenges in capturing the physical laws governing human movement. MDM [52] is a motion generation method using diffusion models [24], which has become capable of producing quite human-like movements. These motion generation models, while skilled at producing common human actions, are not well-suited for generating expressive and creative motions. Generating human motion conditioned on various inputs has also become an active area of research. Studies have explored conditioning on joystick control [37], class labels [22,41], text descriptions [42,64], and seed motions [16,26,44,63]. These methods enable motion generation under various control conditions. However, unlike existing researches, our study focuses on generating motions conditioned by sound-symbolic words, which is the first attempt of its kind. Dance Generation The challenging task of generating dance motions that are stylistically faithful to input music has attracted the attention of many researchers. Early approaches in this domain followed a motion retrieval paradigm [17,34,39] or used optimization-based approaches [51] to generate 2D pose skeletons from conditioning audio. However, these methods often resulted in unrealistic choreographies that lacked the complexity of human dances. More recent works have shifted towards synthesizing motion from scratch by training on large datasets. Various modeling approaches have been proposed like LSTMs [4,31,49,59,65], GANs [21,33,48], convolutional sequence-to-sequence models [1,62], normalizing flow [55], transformer [28,29,32,35,36,47] and diffusion models [53]. Among these methods, recent developments like AI Choreographer [36] and EDGE [53] have started to generate diverse and human-like dances based on music inputs. However, because these methods require music for creating dance, they are not suitable for those who simply want to generate dances without providing music. Moreover, it has been challenging to casually edit input data or output input output or function [60] Sound Symbolic Words Image retrieval Sakamoto et al. (2017) [45] Sound Symbolic Words Material retrieval [12] Sound Symbolic Words Medical conditions Table 1. Overview of related works. We have summarized studies that are closely related to our research, involving dance as the output and sound symbolic words as the input. Currently, we are exploring new combinations within these fields. dances, preventing anyone from intuitively generating dances with ease. While EDGE [53] has implemented editing features to simplify dance generation, enabling partial masking and generating under certain constraints, it can be difficult to master without some dance experience. On the other hand, we propose using sound-symbolic words as the input, allowing anyone to easily generate dances by merely entering intuitive words. Leveraging Onomatopoeia Representation In recent years, there has been a growing interest in research on utilizing onomatopoeic representations for applications such as search and prediction. Onomatopoeic words can represent abstract concepts such as the texture and feeling of an object or the rhythm of a sound, which are unique features not present in other forms of expression like language or music. Consequently, onomatopoeia can be considered an ideal medium for handling information in a more sensory-oriented manner. Several studies have explored the use of onomatopoeia for data visualization and search, such as [11-13, 45, 46, 60]. For instance, [60] proposes an image retrieval system that uses sound-symbolic words to represent surface texture, while [13] develops a color design support system that quantifies images using onomatopoeia to recommend suitable colors for intuitive and ambiguous design images. [45] introduces a method for selecting material samples using sensory vocabulary, effectively exploring the tactile perceptual space. [11] presents a product search system that uses onomatopoeia to express rich product textures and rank related images accordingly. [12] constructs a system that proposes metaphors corresponding to the onomatopoeia expressing medical conditions, enhancing the quality of medical interviews. [46] develops a system for estimating multidimensional personality traits from a single sound-symbolic word, contributing to the prediction of personality evaluations and the efficiency of communication. Furthermore, [61] proposes a deep convolutional neural network (DCNN)-based computer vision method that generates onomatopoeic expressions of material textures from input images, demonstrating human-like texture representation through onomatopoeia. In addition, [38] introduces a game system that incorporates onomatopoeia as a game controller, comparing and evaluating the user experience with traditional controllers and confirming an improved sense of enjoyment and novelty. While there are several search and generation methods using onomatopoeia, to the best of our knowledge, there have been no attempts at content creation utilizing onomatopoeia. This study presents the first endeavor in onomatopoeia-based generation, aiming to extract the great potential of onomatopoeia as a form of expression. PILOT STUDY AND DATA COLLECTION 3.1 Questionnaire survey To confirm the relationship between sound symbols a.k.a onomatopoeia and dance choreography and the possibility of generating dance movements with onomatopoeia, we conducted a questionnaire on dance and onomatopoeia among experienced dancers before and after the data collection experiment.In preliminary survey, participants answered the following questions: "Do you use onomatopoeia during dance practice? The options for whether or not they use onomatopoeia were "use frequently," "use often," "occasional use," and "I don't use it". After the annotation experiment, participants were asked if it was easy for them to annotate the onomatopoeia to the dance choreography. The options were "fairly easy," "easy," "somewhat easy, " "normal, " "somewhat difficult, " "difficult, " and "fairly difficult. Finally, respondents answered free-form questions about their impressions and findings of the annotations. Questionnaire results. In the questionnaire asking whether they use onomatopoeia during dance practice, 62.5% answered "often" and 32.3% answered "yes," and in the questionnaire asking about the degree of difficulty of annotation, 47.1% answered "somewhat difficult" and 23.5% answered "difficult. The results indicated that although they may use onomatopoeia spontaneously during dance practice, it is difficult to imagine onomatopoeia for each movement Data collection In order to construct a dataset of dance and onomatopoeia to be used for training data and to explore the relationship between dance and onomatopoeia, we conducted a subject experiment in which we collected data on what kind of onomatopoeia was recalled in response to what kind of choreography and asked participants to describe their impressions and findings. Japanese speakers aged 18-69 with dance experience were asked to annotate the onomatopoeia they recall in response to dance videos. Data on onomatopoeia for dance was collected by selecting silent dance videos mainly in the genre of the subjects' experience and A questionnaire that asked respondents if they use onomatopoeia during dance practice in the pre-survey. Many respondents answered that they use onomatopoeia "often" or "often. Because one person did not respond, the result is the result of 15 respondents. Fig. 3. Post questionnaire asked whether it was easy to imagine onomatopoeia for the dance choreography. Many responded "somewhat difficult" or "difficult". adding onomatopoeia to the dance choreography as subtitles using Youtube Studio's subtitle creation function. Dance videos from the front viewpoint of Advanced Dance in the AIST Dance Video Database [54] were used to generate a variety of dance motions. In the annotations, the following instructions were given: "Annotate the onomatopoeia that came to you intuitively rather than carefully, and divide the dance movements into as many smaller pieces as possible to make the onomatopoeia. Separate onomatopoeia and dance movements as finely as possible. Annotate the choreography with onomatopoeia that comes to mind freely. You are free to create your own onomatopoeia, not limited to existing common vocabulary. " During the one-hour experiment, the participants were asked to annotate one to five dance videos, depending on their progress. This subject experiment yielded the annotation data for a total of 44 dance videos in 10 dance genres. SYSTEM In this study, we collected annotation data for AIST Dance Database [54], using the method described in section 3. The onomatopoeia collected in the subject experiments was quantified using sound symbol estimation system which we call 'Sakamoto System'. The quantified onomatopoeia were arranged in a time series and converted into a series vector to be used during training. AIST++, which is threedimensional motion data corresponding to the AIST Dance Database, and the series vectors were trained on the FACT model, and the motion was output. The following is a detailed description of the methods and techniques used to create the datasets treated as training data. Preparation and pre-processing of training data 4.1.1 Preparation. In preprocessing, the collected onomatopoeia is converted into a 43-dimensional series vector corresponding to the dance motion and made into sequence data. First, all the annotated onomatopoeia described in the caption file created using the subtitle editor of the video service Youtube in the subject experiment of the 3 chapter are extracted, and the impression of the onomatopoeia is quantified by Sakamoto System. The system converted each onomatopoeia into a 43-dimensional numerical vector, and created an onomatopoeia quantification dictionary that describes the 43dimensional numerical impression of each onomatopoeia. Next, an array of 43 dimensions per index, each with 0 elements, is created, with the series length of the onomatopoeia series vector being the series length of the music corresponding to the dance motion in the previous research [36] implementation. Next, the onomatopoeia of the input is quantified from the numeric dictionary, and the time (in seconds) when each onomatopoeia is given as a subtitle is multiplied by fps=60 to obtain the number of frames, and the features of the corresponding index number of the series vector are replaced. For example, if the onomatopoeia "kuru-kuru" is annotated between 2.0 and 4.0 seconds, the index numbers from 120 to 240 of the series vector are replaced with a 43-dimensional numeric vector representing the "kuru-kuru" impression. The sample of the onomatopoeia series vector is the following array where = 43 is the 43-dimensional feature of the adjective pair calculated by Sakamoto System, is the number of frames corresponding to the motion video The number of frames is the number of frames corresponding to the motion video. [14]: Embedding sound symbolism. To quantify onomatopoeia, we use Sakamoto System for estimating fine impressions of onomatopoeia. Based on the phonetic symbolism of the Japanese language, the System quantifies the image of the input onomatopoeia using a scale of 43 adjective pairs related to mainly visual and tactile perception, for example light -dark, warm -cold, sharp -mild, and heavy -light. Sakamoto System The system is able to estimate the wealth of information conveyed by not only existing onomatopoeia but also newly created onomatopoeia, and to distinguish between a variety of highly similar onomatopoeia. For this reason, in the subject experiments in this study, participants were asked to freely respond to the onomatopoeia they imagined from the action. Since the system used in this study does not quantify the impression of onomatopoeia in real time, the onomatopoeia collected in the subject experiment was listed and quantified in advance to create an onomatopoeia dictionary, and the impression was quantified through the onomatopoeia dictionary during preprocessing Fig. 4. Learning model The onomatopoeia for dances collected in the subject experiments are quantified using Sakamoto System, which quantifies the fine impressions of each onomatopoeia using a scale of 43-dimensional adjective pair scale, and the data set is prepared as one onomatopoeia sequence per one dance, which became input dataset with dance motions [36]. the output is the dance motion conditioned on the onomatopoeia. and generation at the time of evaluation. The onomatopoeic dictionary is then used for preprocessing and generation at the time of evaluation to create onomatopoeic series vectors. Learning and evaluation We used FACT, a model used in AI Choreographer, which generates dance motions from music. The input layer, which encodes music in 35 dimensions, was changed to encode onomatopoeia in 43 dimensions, and the collected data was trained with FACT. The input of the model contains a seed motion sequence with 120 frames (2 seconds) and a music sequence with 240 frames (4 seconds), where the two sequences are aligned on the first frame. The output of the model is the future motion sequence with N = 20 frames supervised by L2 loss. During inference FACT continually generate future motions in a autoregressive manner at 60 FPS, where only the first predicted motion is kept in every step. For the learning and evaluation phase, we employed the crossmodal learning of sound symbols (onomatopoeia) and dance motions using FACT [36]. In our learning model, a dance motion is generated by sequentially producing 20 frames of dance with 120 frames of dance motion and 240 frames of onomatopoeic sequences. The sound symbolic word features and motion features are initially embedded in an 800-dimensional hidden representation within a linear layer. A learnable position encoding is then added before being input into the transformer layer. This method allows for the generation of dance motions that correspond to the input onomatopoeic sequences, providing a unique and expressive way to create choreography. Despite potential limitations due to the size of the dataset or the AI Choreographer framework's age, our implementation demonstrates the viability of using onomatopoeia as a medium for generating dance motions. Table 2. Comparison of quantitative scores for each training model. The larger the data set, the better the score when compared in terms of the size of the data set, and the music score was better when compared in terms of whether the feature was music or onomatopoeia. RESULTS AND EVALUATIONS Learned model Quantitative evaluation FID scores, which quantify the difference from the distribution of grand-truth motions, and motion diversity scores, which calculate the average Euclidean distance in the feature space across 40 generated motions on the AIST++ test set to measure the diversity were calculated for the three models: AI Choreographer, a previous study that generates dances from music; AI Choreographer Mini, which has the same training data as this model; and this model. Both the FID score and the diversity score were calculated for AI Choreographer and this model scored worse than AI Choreographer Mini. In other words, there is room for improvement in both the amount of data set and the way feature vectors are created. Validity of the Generated Dance To verify the validity of the generated dance, we compare the input onomatopoeia's audio waveform with the generated dance. The audio waveform represents the rhythm of the onomatopoeia, and since the rhythm is closely related to the onomatopoeia meaning, we can investigate the semantic plausibility by looking at how the onomatopoeia meaning and waveform are related to the dance movement (see Figure 5). The first row (Figure 5a) represents the swaying motion in Japanese onomatopoeia, and from the waveform, we can see that there are four phonemes and a rhythm. Looking at the generated dance, we can see that the hands are swaying up and down with a slow, steady rhythm. The second row (Figure 5b) represents the sound of a cat stretching and meowing in Japanese onomatopoeia. From the waveform, we can see that the sound is being stretched. In the generated dance, the hands are stretched out and bent horizontally, reflecting the stretching image. The third row (Figure 5c) represents a Japanese onomatopoeia depicting the sound of heavy machinery with multiple parts hitting the floor or objects. From the waveform, we can see the bouncing sound. Looking at the generated motion, we can see the sharp lifting, lowering, and spreading of the hands, reflecting the bouncing sound. In this way, we could confirm that our method can generate reasonably plausible movements to a certain extent. Results in Various Languages To generate a dance intuitively even without knowledge of onomatopoeia, it is required that plausible dances can be generated even when onomatopoeia that comes to mind are used as is for input. The Sakamoto system we employed focuses on the phonemes and phonology of the input word, so there are virtually no restrictions on the onomatopoeia that can be used as input, and even created onomatopoeia can be used as input. To demonstrate this experimentally, we additionally generated a variety of onomatopoeia and attempted to generate dances. Onomatopoeia Generation in Various Languages. We used the large language model, ChatGPT (GPT-4) [40], to generate onomatopoeia. ChatGPT can generate plausible text responses when given text-based instructions. We instructed it for the 26 languages it can handle to "Please generate onomatopoeia in XXX (language). However, for languages without onomatopoeia, please generate words that sound like onomatopoeia. " We then had it generate four random onomatopoeia for each language. Results. Examples of the generated dances that exhibited good movements are shown in Figure 6. For other examples, please refer to the supplementary materials. We confirmed that our model can generate plausible motions as dances when given unknown onomatopoeia as input. Additionally, we found that the model can generate similar motions repeatedly for repeated onomatopoeia inputs, indicating a consistency in the generated motions corresponding to the onomatopoeia. However, further verification is needed to determine if the appearance of the motions truly matches human impression. DISCUSSION 6.1 Limitations and Future Work There are several limitations to this study that warrant further investigation. First, more updated model than AI Choreographer is available now which could generate more optimal dance motions. Future work should consider implementing more recent state-ofthe-art methods to improve the quality of the generated motions. Additionally, the dataset used in this study is relatively small, which may have limited the performance of our model. In future studies, increasing the size of the dataset and fine-tuning the model can be considered to enhance the quality of the generated motion. The proposed model has substantial potential applications in various domains. In the entertainment industry, this model can be used to create unique and engaging dance performances based on onomatopoeic input, adding a new dimension to the creative process. Furthermore, the model can be integrated into interactive applications, such as virtual reality or video games, to generate dynamic and immersive dance experiences for users. In the field of education, this model can serve as a valuable tool for teaching dance and movement. By providing onomatopoeic input, students can better understand and visualize the relationship between rhythm, meaning, and movement in the context of dance. Moreover, this model can also be used to study the impact of different onomatopoeic words on dance motion generation and explore the underlying principles that govern this relationship. Uniqueness of onomatopoeia Approach We would like to discuss the following points: 1. The difference between generating onomatopoeia and random generation, 2. How the generation method differs from time-series data based on phonetic symbols, 3. The significance of using onomatopoeia as an intermediate representation between meaning, words, and sound. Firstly, the generation of onomatopoeia is different from random generation in that it carries both meaning and a sense of rhythm. Unlike random generation, onomatopoeia can convey specific information while still maintaining an abstract representation that allows for creative interpretation. Secondly, the proposed method differs from timeseries data generation based on phonetic symbols, as it does not solely rely on the intensity of sound. Instead, the model generates dance motions that consider both the rhythm and meaning of the input onomatopoeia. This makes the method more sophisticated than simply following the intensity of sound in a time-series manner. Moreover, onomatopoeia serves as an intermediate representation between meaning, words, and sound. It is not strictly tied to the time-series data of sound intensity, allowing for more flexibility and expressiveness in the generated dance motions. This unique characteristic of onomatopoeia makes it a suitable input for generating dance movements that convey both rhythm and meaning. Although the dataset presented in this study is still relatively primitive, we believe that it holds significant potential as an intermediate representation between meaning and physical information. By further refining the dataset and improving the model, it is possible to create more expressive and meaningful dance motions that fully utilize the unique properties of onomatopoeia. Moreover the proposed method of using onomatopoeia for generating dance motions offers a novel and promising approach to computer graphics and animation research. By considering both rhythm and meaning, the model can generate more expressive and engaging dance performances that are not limited by the constraints of traditional time-series data or phonetic symbols. As the dataset and model continue to be developed, this research is expected to contribute to the advancement of the field and inspire new applications in various domains. Another area for future work is the development of an interactive application that generates dance movements based on onomatopoeic input. Such an application would allow users to create personalized dance experiences and further explore the potential of onomatopoeia in the context of dance motion generation. Qualitative evaluation will be conducted in the user study. It is conceivable that the onomatopoeic instructions to be annotated could be squeezed or annotated with voice. CONCLUSION In conclusion, this study has presented a novel approach to dance motion generation by leveraging the unique properties of onomatopoeia. The proposed machine learning model expands upon the existing AI Choreographer framework, introducing onomatopoeic input as a means to convey both rhythm and meaning in dance movement generation. The application of onomatopoeia as an input source is an innovative and linguistically interesting method, addressing the shortcomings of traditional dance generation inputs such as music, speech, and motion capture. Despite its limitations, this study has demonstrated the potential of onomatopoeia as a medium for dance motion generation and has introduced a new dataset for further exploration. Future work could involve fine-tuning the model, increasing the dataset size, and developing an interactive application that generates dance movements in response to onomatopoeic input. This would enable users to easily create unique and expressive dance sequences based on their linguistic creativity. ACKNOWLEDGMENTS This work was supported by READYFOR. Fig. 2 . 2Fig. 2. A questionnaire that asked respondents if they use onomatopoeia during dance practice in the pre-survey. Many respondents answered that they use onomatopoeia "often" or "often. Because one person did not respond, the result is the result of 15 respondents. Fig. 5 . 5The dances generated with the onomatopoeia included in the validation data, and the waveform of the sound when that onomatopoeia occurred. The section where the onomatopoeia is described indicates the moment when the input embedding was given, and the dashed line represents the section where a zero vector was given. The frames where the dance and the characters of the onomatopoeia are written are time-aligned, but the waveform only shows the waveform of a single occurrence of the onomatopoeia. The meaning of the sound symbolism of each onomatopoeia in ChatGPT is: a. The sound of some liquid boiling b. The sound of a cat meowing c. The sound of something hitting or being hit by a hard object. Fig. 6 . 6Generated motion results. The original language and the meaning of the sound symbolism of each onomatopoeia in ChatGPT is: a. German, Pukupukku and bubbling sound b. 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[ "STABJGL: A STABILITY APPROACH TO SPARSITY AND SIMILARITY SELECTION IN MULTIPLE NETWORK RECONSTRUCTION", "STABJGL: A STABILITY APPROACH TO SPARSITY AND SIMILARITY SELECTION IN MULTIPLE NETWORK RECONSTRUCTION" ]
[ "Camilla Lingjaerde [email protected] \nMRC Biostatistics Unit\nMRC Biostatistics Unit University of Cambridge\nUniversity of Cambridge\n\n", "Sylvia Richardson [email protected] \nMRC Biostatistics Unit\nMRC Biostatistics Unit University of Cambridge\nUniversity of Cambridge\n\n" ]
[ "MRC Biostatistics Unit\nMRC Biostatistics Unit University of Cambridge\nUniversity of Cambridge\n", "MRC Biostatistics Unit\nMRC Biostatistics Unit University of Cambridge\nUniversity of Cambridge\n" ]
[]
In recent years, network models have gained prominence for their ability to capture complex associations. In statistical omics, networks can be used to model and study the functional relationships between genes, proteins, and other types of omics data. If a Gaussian graphical model is assumed, a gene association network can be determined from the non-zero entries of the inverse covariance matrix of the data. Due to the high-dimensional nature of such problems, integrative methods that leverage similarities between multiple graphical structures have become increasingly popular. The joint graphical lasso is a powerful tool for this purpose, however, the current AIC-based selection criterion used to tune the network sparsities and similarities leads to poor performance in highdimensional settings. We propose stabJGL, which equips the joint graphical lasso with a stable and accurate penalty parameter selection approach that combines the notion of model stability with likelihood-based similarity selection. The resulting method makes the powerful joint graphical lasso available for use in omics settings, and outperforms the standard joint graphical lasso, as well as state-of-the-art joint methods, in terms of all performance measures we consider. Applying stab-JGL to proteomic data from a pan-cancer study, we demonstrate the potential for novel discoveries the method brings. A user-friendly R package for stabJGL with tutorials is available on Github: https://github.com/Camiling/stabJGL.If a Gaussian graphical model is assumed, a conditional (in)dependence network can be estimated by determining the non-zero entries of the inverse covariance (precision) matrix of the data. With its good performance in numerical studies, the graphical lasso of Friedman et al. [2008] is a state-of-the-art method for precision matrix estimation in the setting of Gaussian graphical models. The method combines L 1 regularization with maximum likelihood estimation. Other notable methods include the neighborhood selection approach of Meinshausen and Bühlmann [2006] and the graphical SCAD [Fan et al., 2009]. Notable Bayesian methods include the Bayesian graphical lasso [Wang et al., 2012], arXiv:2306.03212v1 [stat.ME] 5 Jun 2023StabJGL: stable sparsity and similarity selection for multiple networks Bayesian spike-and-slab approaches [Wang, 2015] and the graphical horseshoe [Li et al., 2019a]. If multiple related data sets are available, there are several ways to leverage common network structures. If focusing on one data type's network structure, data from other types can enhance inference via weighted graphical lasso methods Jackson, 2015, Lingjaerde et al., 2021]. However, to compare network structures across data sets, such as patient subgroups, a joint approach that leverages common information while preserving the differences can increase statistical power and provide interpretable insight.In the area of multiple Gaussian graphical models, existing methods include the group extension of the graphical lasso to multiple networks of Guo et al. [2011], the Bayesian spike-and-slab joint graphical lasso [Li et al., 2019b] and the Markov random field approach of Peterson et al. [2015]. The widely used joint graphical lasso (JGL) of Danaher et al. [2014] extends the graphical lasso to a multiple network setting and provides a powerful tool for inferring graphs with common traits. It employs two different penalty functions -group (GGL) and fused (FGL) -with the latter recommended for most applications. From this point forward, any mention of the joint graphical lasso will imply the fused version, unless otherwise specified. The method needs tuning of two regularization parameters for controlling (i) the number of non-zero effects, and (ii) the similarity between networks, respectively. However, the default parameter selection routine based on the AIC [Akaike et al., 1973] often results in severe over-selection in high-dimensional data, potentially impacting performance negatively [Liu et al., 2010, Foygel and Drton, 2010].We propose a stable and accurate penalty parameter selection method for the joint graphical lasso, combining the model stability principle of Liu et al. [2010] with likelihood-based selection for high-dimensional data [Foygel and Drton, 2010]. The resulting method inherits the powerful traits of the joint graphical lasso while mitigating the risk of severe under-or over-selection of edges in high-dimensional settings. We provide an R package, stabJGL (stable sparsity and similarity selection for the joint graphical lasso), which implements the method.The paper is organized as follows. In Section 2, we first describe the Gaussian graphical model framework and the penalized log-likelihood problem we aim to solve. We then describe our proposed algorithm. In Section 3, we demonstrate the performance of our proposed method on simulated data and apply it proteomic data from a pan-cancer study of hormonally responsive cancers. Finally, we highlight possible extensions in Section 4.
null
[ "https://export.arxiv.org/pdf/2306.03212v1.pdf" ]
259,088,931
2306.03212
47680c3d66d9f9e86bb87203dcae7219a4dc7522
STABJGL: A STABILITY APPROACH TO SPARSITY AND SIMILARITY SELECTION IN MULTIPLE NETWORK RECONSTRUCTION June 5, 2023 Camilla Lingjaerde [email protected] MRC Biostatistics Unit MRC Biostatistics Unit University of Cambridge University of Cambridge Sylvia Richardson [email protected] MRC Biostatistics Unit MRC Biostatistics Unit University of Cambridge University of Cambridge STABJGL: A STABILITY APPROACH TO SPARSITY AND SIMILARITY SELECTION IN MULTIPLE NETWORK RECONSTRUCTION June 5, 2023High-dimensional inference · Network models · Joint graphical model · Joint graphical lasso · Gaussian graphical model · Genomics · Gene networks · Protein-protein interaction networks · Integrative analysis In recent years, network models have gained prominence for their ability to capture complex associations. In statistical omics, networks can be used to model and study the functional relationships between genes, proteins, and other types of omics data. If a Gaussian graphical model is assumed, a gene association network can be determined from the non-zero entries of the inverse covariance matrix of the data. Due to the high-dimensional nature of such problems, integrative methods that leverage similarities between multiple graphical structures have become increasingly popular. The joint graphical lasso is a powerful tool for this purpose, however, the current AIC-based selection criterion used to tune the network sparsities and similarities leads to poor performance in highdimensional settings. We propose stabJGL, which equips the joint graphical lasso with a stable and accurate penalty parameter selection approach that combines the notion of model stability with likelihood-based similarity selection. The resulting method makes the powerful joint graphical lasso available for use in omics settings, and outperforms the standard joint graphical lasso, as well as state-of-the-art joint methods, in terms of all performance measures we consider. Applying stab-JGL to proteomic data from a pan-cancer study, we demonstrate the potential for novel discoveries the method brings. A user-friendly R package for stabJGL with tutorials is available on Github: https://github.com/Camiling/stabJGL.If a Gaussian graphical model is assumed, a conditional (in)dependence network can be estimated by determining the non-zero entries of the inverse covariance (precision) matrix of the data. With its good performance in numerical studies, the graphical lasso of Friedman et al. [2008] is a state-of-the-art method for precision matrix estimation in the setting of Gaussian graphical models. The method combines L 1 regularization with maximum likelihood estimation. Other notable methods include the neighborhood selection approach of Meinshausen and Bühlmann [2006] and the graphical SCAD [Fan et al., 2009]. Notable Bayesian methods include the Bayesian graphical lasso [Wang et al., 2012], arXiv:2306.03212v1 [stat.ME] 5 Jun 2023StabJGL: stable sparsity and similarity selection for multiple networks Bayesian spike-and-slab approaches [Wang, 2015] and the graphical horseshoe [Li et al., 2019a]. If multiple related data sets are available, there are several ways to leverage common network structures. If focusing on one data type's network structure, data from other types can enhance inference via weighted graphical lasso methods Jackson, 2015, Lingjaerde et al., 2021]. However, to compare network structures across data sets, such as patient subgroups, a joint approach that leverages common information while preserving the differences can increase statistical power and provide interpretable insight.In the area of multiple Gaussian graphical models, existing methods include the group extension of the graphical lasso to multiple networks of Guo et al. [2011], the Bayesian spike-and-slab joint graphical lasso [Li et al., 2019b] and the Markov random field approach of Peterson et al. [2015]. The widely used joint graphical lasso (JGL) of Danaher et al. [2014] extends the graphical lasso to a multiple network setting and provides a powerful tool for inferring graphs with common traits. It employs two different penalty functions -group (GGL) and fused (FGL) -with the latter recommended for most applications. From this point forward, any mention of the joint graphical lasso will imply the fused version, unless otherwise specified. The method needs tuning of two regularization parameters for controlling (i) the number of non-zero effects, and (ii) the similarity between networks, respectively. However, the default parameter selection routine based on the AIC [Akaike et al., 1973] often results in severe over-selection in high-dimensional data, potentially impacting performance negatively [Liu et al., 2010, Foygel and Drton, 2010].We propose a stable and accurate penalty parameter selection method for the joint graphical lasso, combining the model stability principle of Liu et al. [2010] with likelihood-based selection for high-dimensional data [Foygel and Drton, 2010]. The resulting method inherits the powerful traits of the joint graphical lasso while mitigating the risk of severe under-or over-selection of edges in high-dimensional settings. We provide an R package, stabJGL (stable sparsity and similarity selection for the joint graphical lasso), which implements the method.The paper is organized as follows. In Section 2, we first describe the Gaussian graphical model framework and the penalized log-likelihood problem we aim to solve. We then describe our proposed algorithm. In Section 3, we demonstrate the performance of our proposed method on simulated data and apply it proteomic data from a pan-cancer study of hormonally responsive cancers. Finally, we highlight possible extensions in Section 4. Introduction Network models have in recent years gained great popularity in many areas. In statistical omics, networks can be used to decode aspects of unknown structures, and hence study the relationships between genes, proteins, and other types of omics data. In health data sciences, rich data sets are more and more frequently encountered, enabling the development of models integrating a variety of biological resources. In the high-dimensional setting commonly found in omics, sharing information between data sources with shared structures -which could be different tissues, conditions, patient subgroups, or different omics types -can give a valuable increase in statistical power while elucidating shared biological function. A key question is how to combine the different data sources into a single model. Penalized log-likelihood problem Assume a network inference problem with K groups. We let {Θ} = (Θ (1) , . . . , Θ (K) ) be the set of their (unknown) precision matrices, and assume that the set of K k=1 n k observations are independent. We aim to solve the penalized log-likelihood problem [Danaher et al., 2014] { Θ} = arg max {Θ≻0} K k=1 n k [log(det(Θ (k) )) − tr(S (k) Θ (k) )] (1) − P({Θ}) where S (k) is the sample covariance matrix of group k and P(·) is a penalty function. In (1), det(·) denotes the determinant and tr(·) denotes the trace. The joint graphical lasso employs the fused penalty function P({Θ}) = λ 1 K k=1 i̸ =j abs(θ (k) ij ) + λ 2 k<k ′ ∥Θ (k) − Θ (k ′ ) ∥ 1(2) where λ 1 and λ 2 are positive penalty parameters, abs(·) denotes the absolute value function and ∥ · ∥ 1 denotes the L 1 penalty. This penalty applies L 1 penalties to each off-diagonal element of the K precision matrices as well as to the differences between corresponding elements of each pair of precision matrices. As for the graphical lasso, the parameter λ 1 controls the sparsity. The similarity parameter λ 2 controls the degree to which the K precision matrices are forced towards each other, encouraging not only similar network structures but also similar precision matrix entries. The current penalty parameter selection approach for λ 1 and λ 2 is based on the AIC [Danaher et al., 2014]. While suitable for determining network similarities, likelihood-based selection criteria can lead to severe under-or over-selection and thus poor performance in high-dimensional settings [Liu et al., 2010]. The stabJGL algorithm To improve the performance of the joint graphical lasso with the fused penalty for omics applications and other high-dimensional problems, we propose the stabJGL algorithm for stable sparsity and similarity selection in multiple network reconstruction. Below we outline the algorithm, where we first select the sparsity parameter λ 1 in the fused penalty (2) based on the notion of model stability, and then the similarity parameter λ 2 based on model likelihood. StabJGL jointly estimates multiple networks by leveraging their common information, and gives a basis for deeper exploration of their differences, as shown in Figure 1. Selecting λ 1 We select λ 1 by extending the framework introduced by Liu et al. [2010] in their Stability Approach to regularization Criterion (StARS) to a multiple network setting. The aim is to select the least amount of penalization that makes graphs sparse as well as reproducible under random subsampling. This is done by drawing many random subsamples from each of the K data types and using them to construct joint graphical lasso graphs over a range of λ 1 values. The smallest parameter value for which a given graph estimation variability measure does not surpass a specified threshold is then selected. We use a measure of edge assignment instability across subsamples to quantify the variability. Specifically, we consider a grid of λ 1 values in a suitable interval, i.e., (0, 1] and keep the similarity parameter λ 2 fixed to some small value such as 0.01 in the first instance. For η = 1, . . . , N sample , we draw a random subsample from each group k's set of n k observations without replacement, each of size b k < n k . Liu et al. [2010] show that in a single network setting, b k = ⌊10 √ n k ⌋ maintains theoretical properties for containing the true graph with high probability as well as high empirical performance, and this is the value we use. For each value of λ 1 to consider, we next construct the corresponding set of joint graphical lasso graphs {G η (k) (λ 1 )} K k=1 from these K sets of subsamples, using the fused penalty (2). The following is then done for each value of λ 1 we consider. For each group k = 1, . . . , K and all possible node pairs (i, j) we estimate the probability of an edge between the nodes over the N sample inferred sets of graphs ψ (k) ij (λ 1 ) = 1 N sample Nsample η=1 1 (i, j) ∈ G η (k) (λ 1 ) ,(3) where 1 [·] is the indicator function. Using this estimated probability, we find ξ (k) ij (λ 1 ) = 2 ψ (k) ij (λ 1 )(1 − ψ (k) ij (λ 1 )),(4) which is an estimate of two times the variance of the Bernoulli indicator of the edge (i, j) in group k. It lies in [0, 0.5] and can be regarded as an estimate of the fraction of times two inferred graphs for group k found with the joint graphical lasso with the given λ 1 value will disagree on the presence of the edge (i, j). Due to the L 1 penalty in (2), the number of inferred edges will decrease as λ 1 is increased. For a given λ 1 , ξ ij (λ 1 ) can be regarded as a measure of the variability of the edge (i, j) in group k across subsamples, and the total variability of graph k can be measured by averaging over all edges, yielding the estimate D (k) (λ 1 ) = 1 p 2 i<j ξ (k) ij (λ 1 ).(5) For each value of λ 1 , the total variability of the whole set of graphs found by the joint graphical lasso is then found by averaging the variability over all K networks D(λ 1 ) = 1 K K k=1 D (k) (λ 1 ).(6) For sufficiently large λ 1 , all edges are excluded from the model and so the variability D(λ 1 ) will be 0. The variability will in general increase as the penalty λ 1 decreases, however, for small enough λ 1 the graphs will become so dense that the variability starts to decrease again. As sparse network inference is the aim, we therefore monotonize the variability function by lettingD(λ 1 ) = sup 0≤t≤λ1 D(t). Finally, for a given variability threshold β 1 , the optimal penalty is chosen to be λ 1 = sup{λ 1 :D(λ 1 ) ≤ β 1 }. As opposed to λ 1 , β 1 is an interpretable quantity and we propose a default threshold of β 1 = 0.1 as suggested by Liu et al. for the original StARS algorithm, which reflects an acceptance of 10% variability in the edge assignments. Selecting λ 2 After λ 1 has been selected, we select λ 2 with a multiple-network version of the extended BIC (eBIC or BIC γ ) of Foygel and Drton [2010]. The eBIC is an extension of the Bayesian Information Criterion of Schwarz [1978], where the prior is reformulated to account for high-dimensional graphical settings. We propose an adaptation the eBIC to a multiple-network setting, BIC γ (λ 1 , λ 2 ) = K k=1 n k tr(S (k) Θ (k) λ1,λ2 ) − n k log(det( Θ (k) λ1,λ2 )) + |E k | log n k + 4|E k |γ log p ,(7) Algorithm 1 The stabJGL algorithm Require: n k × p data matrix X (k) for k = 1, . . . , K N sample ← 20 5: γ ← 0 6: b k ← ⌊10 √ n k ⌋ for k = 1, . . . , K 7: S (k) ← 1 n k −1 X (k) T X (k) for k = 1, . . . , K 8: for λ 1 in Λ 1 do 9: for η = 1 to N sample do 10: for k = 1 to K do 11: Sample b k indices I k ⊂ {1, . . . , n k } 12: X (k) sample ← X (k) [I k , ] 13: end for 14: {G η (k) (λ 1 )} K k=1 ← JGL {X (k) sample } K k=1 | λ 1 , λ(init)for i = 1 to j − 1 do 19: ψ (k) ij (λ 1 ) ← 1 Nsample Nsample η=1 1 (i, j) ∈ G η (k) (λ 1 ) 20: ξ (k) ij (λ 1 ) ← 2 ψ (k) ij (λ 1 )(1 − ψ (k) ij (λ 1 ))D (k) (λ 1 ) ← 1 ( p 2 ) i<j ξ (k) ij (λ 1 ) 24: end for 25: D(λ 1 ) ← 1 K K k=1 D (k) (λ 1 ) 26:D(λ 1 ) ← sup 0≤t≤λ1 D(t) 27: end for 28: λ 1 ← sup{λ 1 ∈ Λ 1 :D(λ 1 ) ≤ β 1 } 29: for λ 2 in Λ 2 do 30: { Θ (k) λ1λ2 , E k } K k=1 ← JGL {X (k) } K k=1 | λ 1 , λ 2 31: BIC γ ( λ 1 , λ 2 ) ← K k=1 n k tr(S (k) Θ (k) λ1,λ2 ) − n k log(det( Θ (k) λ1,λ2 )) + |E k | log n k + 4|E k |γ log p 32: end for 33: λ 2 ← arg min λ2∈Λ2 BIC γ ( λ 1 , λ 2 ) 34: { Θ (k) stabJGL } K k=1 ← JGL {X (k) } K k=1 | λ 1 , λ 2 where Θ (k) λ1,λ2 is the estimated precision matrix of network k obtained with the penalty parameters λ 1 and λ 2 , and |E k | is the size of the corresponding edge set. A grid of λ 2 values is considered, with λ 1 fixed to the value selected in the previous step. The value of λ 2 that minimizes (7) is selected. Like for the standard eBIC, the additional edge penalty parameter γ ∈ [0, 1] must be chosen. However, since we are using the eBIC for similarity selection rather than sparsity selection, the choice of γ is not as important because we are comparing graphs with the same value of λ 1 and hence similar levels of sparsity. We typically use γ = 0, which corresponds to the ordinary BIC, for most applications. Our implementation includes the eBIC generalization to give the user the option of additional penalization in extremely high-dimensional cases. Algorithm The full stabJGL algorithm is given in Algorithm 1. JGL(·) indicates that the joint graphical lasso function with the fused penalty is applied. The output of the JGL function can either be a set of graphs, a set of precision matrices or an edge set, depending on what is required Algorithm 1. Implementation details StabJGL is implemented in R, and available as an R package at https://github.com/Camiling/stabJGL. The subsampling routine is implemented so it can be done in parallel. The joint graphical lasso fittings are done as in Danaher et al. [2014], using an ADMM (Alternating Direction Method of Multipliers) algorithm [Boyd et al., 2011] for general penalty functions to solve the penalized log-likelihood problem (1), By default, 20 subsamples are used and we evaluate 20 values each of λ 1 ∈ [0.01, 1] and λ 2 ∈ [0, 0.1]. As in StARS, we use a subsample size of ⌊10 √ n k ⌋ for group k = 1, . . . , K [Liu et al., 2010]. The additional penalty parameter γ in the eBIC for similarity selection is set to 0 by default, corresponding to the standard BIC. We found this value to be suitable in most applications but leave the option to increase the penalization. We employ a default variability threshold of β 1 = 0.1. Results Simulated data We first assess the performance of stabJGL on simulated data. We compare the network reconstruction ability of stabJGL to that of state-of-the-art methods, including the joint graphical lasso with the fused penalty (FGL) and group penalty (GGL) with penalty parameters selected with the default AIC-based criterion [Danaher et al., 2014]. To assess the performance of another selection criterion specifically designed for high-dimensional graph selection, we also consider FGL with penalty parameters tuned by the extended BIC for multiple graphs (7) The latter estimates each network separately. We generate data that closely resembles our omics application of interest, featuring partial correlations between 0.1 and 0.2 in absolute value, while also exhibiting the scale-free property -a typical assumption for omics data where the degree distribution (i.e., the distribution of the number of edges that are connected to the nodes) adheres to a power-law distribution [Chen and Sharp, 2004]. In the main simulation scenario, we simulate K = 3 networks with p = 100 nodes, manipulating the degree of similarity in their "true" graphical structures to assess the performance of the method over a wide range of scenarios. We maintain a sparsity of 0.02 across all networks and generate data sets from the corresponding multivariate Gaussian distributions with n 1 = 150, n 2 = 200 and n 3 = 300 observations. We then apply different network reconstruction techniques to determine the networks from the data. For FGL and GGL, the two penalty parameters are chosen in a sequential fashion with the default AIC-based criterion proposed by Danaher et al. [2014], with 20 values of λ 1 ∈ [0.01, 1] and λ 2 ∈ [0, 0.1] respectively being evaluated. We consider the eBIC criterion on the same grid of values for FGL. We consider the same set of λ 1 and λ 2 values in the stabJGL algorithm and let γ = 0 in the eBIC criterion for similarity selection. For stabJGL and the graphical lasso tuned by StARS, we use a variability threshold of 0.1 and use 20 subsamples. For the Bayesian spike-and-slab joint graphical lasso all parameter specifications are as suggested by Li et al. [2019b]. In addition to the above setup, we consider additional settings with K ∈ {2, 4} graphs and p = 100 nodes. We only show a summarizing plot of these additional results, but the full tables for these simulations, as well as from additional scenarios with other values of K and p, are given in the Supplement. We also investigate the effect of the variability threshold β 1 in stabJGL on the results in a setting with p = 100 nodes and K = 2 networks. Finally, to compare the scalability of the respective methods we consider the time needed to infer networks for various p and K. Further details and code for the simulation study can be found at https://github.com/Camiling/stabJGL_simulations. Estimation accuracy is assessed with the precision (positive predictive value), and the recall (sensitivity). The precision gives the fraction of predicted edges that were correct, while the recall is the fraction of edges in the true graph that were identified by the inference. Because the sparsity of estimated networks will vary between methods, the precision-recall trade-off should be taken into consideration. In general, the recall will increase with the number of selected edges while the precision will decrease. Since sparsity selection is a main feature of our proposed method, we do not consider threshold-free comparison metrics such as the AUC. We therefore put emphasis on the following characteristics in our comparative simulation study; (i) suitable sparsity level selection, (ii) utilization of common information at any level of network similarity, i.e., inference improves with increased network similarity, and (iii) a suitable precision-recall trade-off that overly favours either measure. Simulation results The results are summarized in Table 1. First, we observe that the fused and group joint graphical lasso with the default AIC-based penalty parameter selection strongly over-select edges in all cases. This leads to high recall, but very low precision. Second, they do not appear to sufficiently utilize network similarities; the performance of the two methods, particularly GGL, differs little between completely unrelated and identical networks. Notably, in all cases the selected value of λ 2 is smaller for FGL and GGL tuned by AIC than it is for stabJGL. Consequently, similarity is not sufficiently encouraged even in settings where the networks are identical. The AIC criterion does not seem to provide sufficient penalization to encourage suitable sparsity and similarity. On the other hand, we observe that the alternative eBIC criterion gives extremely sparse FGL estimates, resulting in high precision but very low recall. In half of the cases, it selects an empty graph, i.e., no edges. Although the extended BIC is developed specifically for graphical model selection, likelihood-based criteria for sparsity selection tend to perform poorly in high-dimensional settings and risk both severe under-and over-selection [Foygel and Drton, 2010]. This issue is avoided in the stabJGL algorithm as the eBIC only is used to select similarity and not sparsity. The Bayesian spike-and-slab joint graphical lasso tends to select very few edges, leading to high precision but low recall. Its performance deteriorates drastically as the network differences increase, leading to extremely low recall. This implies a lack of flexibility to adapt to varying network similarity levels, as has previously been observed [Lingjaerde et al., 2022]. Out of all the joint methods, stabJGL gives the most accurate sparsity estimate. This ensures that we neither get very low precision like FGL and GGL tuned by AIC, nor very low recall like SSJGL and FGL tuned by eBIC. StabJGL also appears to adapt well to the similarity between networks, with the prediction accuracy increasing with the number of shared edges. As a result, the method either outperforms the graphical lasso tuned by StARS for highly similar networks or performs comparably to it for unrelated networks. The similar performance for unrelated networks can be explained by the fact that the sparsity controlling penalty parameter of both methods are tuned with a stability-based approach. The results suggest that stabJGL can be used agnostically in settings where there is no prior knowledge about the level of network similarity and does not run any risk of decreased accuracy should the networks have nothing in common. The results for K = 2 and K = 4 networks are summarized in Figure 2. The results for FGL tuned with eBIC are not shown as it did not select any edges in any of the settings. The findings from the K = 3 case are echoed here, with FGL and GGL having high recall but very low precision and particularly GGL exhibiting a lack of adaption to increased network similarity. On the contrary, SSJGL selects very few edges and thus has high precision but very low recall, with its performance quickly deteriorating for less similar networks. StabJGL achieves a balanced precision-recall trade-off and adapts well to the level of network similarity. Consequently, stabJGL performs comparably or better than the graphical lasso depending on the degree of similarity between the networks. A key question is whether stabJGL can achieve as high precision as the methods that give sparser networks (i.e., SSJGL) by using a lower variability threshold. Similarly, we want to see if stabJGL can achieve as high recall as the methods that infer more edges (i.e., FGL and GGL). To investigate this, we consider the same setting as in Figure 2 with K = 2 networks, focusing specifically on the case where the two networks have 20% edge agreement. Table 2 compares the performance of stabJGL for different values of the variability threshold β 1 to the other methods. For β 1 = 0.01, stabJGL gives very sparse estimates and obtains comparable precision and recall to SSJGL. For the higher threshold β 1 = 0.2, stabJGL selects a large number and obtains comparable recall to FGL and GGL while retaining a higher precision level. A complete comparison for all levels of edge agreement is given in the Supplement ( Figure S3), where we similarly find that by varying the variability threshold β 1 we can obtain at least as high precision and/or recall as the other methods at any level of similarity. The fact that stabJGL allows the user to obtain higher or lower sparsity by changing the variability threshold means that the method can be adapted to reflect the priorities of the user (i.e., concern for false positives versus false negatives). For most applications, a middle-ground value such as 0.1 yields a good balance between false positives and false negatives as demonstrated in the simulations. Figure 3 shows the CPU time used to jointly infer networks for K ∈ {2, 3, 4} networks and various numbers of nodes p, with n ∈ {100, 150} observations, for the joint graphical lasso with the fused penalty (FGL) with penalty parameters tuned with the AIC and stabJGL with the same parameter specifications as in the previously described simulations. Due to an efficient parallelized implementation, stabJGL has an almost identical run time to FGL when the same number of λ 1 and λ 2 values are considered. Thus, the increased estimation accuracy of stabJGL does not come at a computational cost. It is important to note that due to the generalized fused lasso problem having a closed-form solution in the special case of K = 2 [Danaher et al., 2014], stabJGL is substantially faster for only two networks than for K > 2. As stabJGL uses the fused penalty this comparison is the most relevant, but a run time comparison of all methods considered in our simulation study can be found in the Supplement ( Figure S1). In the Supplement, we also demonstrate that stabJGL can be applied to problems with p > 1, 000 nodes and K > 2 networks within reasonable time ( Figure S2). Table 1: Performance of the different graph reconstruction methods in simulations, reconstructing graphs with p = 100 nodes from K = 3 networks with various similarity of the true graph structures. The methods included are Glasso, FGL and GGL tuned by AIC, FGL tuned by eBIC, SSJGL and stabJGL. The similarity (percentage of edges that are in common) of the graphs is shown. The results are averaged over N = 100 simulations and shows the sparsity, precision, and recall of each of the K = 3 estimated graphs. The corresponding standard deviations are shown in parentheses. The graphs are reconstructed from n 1 = 150, n 2 = 200 and n 3 = 300 observations. All graphs have sparsity 0.02. The average selected values of the penalty parameters λ 1 and λ 2 for the relevant methods is shown as well. Runtime profiling Pan-cancer data We perform a proteomic network analysis of Reverse Phase Protein Array (RPPA) data from The Cancer Genome Atlas (TCGA) across different pan-Cancer tumor types [Cancer Genome Atlas Network and others, 2012]. In a large proteomic pan-Cancer study of 11 TCGA tumor types, Akbani et al. [2014] identified a major tumor super cluster consisting of hormonally responsive "women's cancers". (Luminal breast cancer, ovarian cystadenocarcinoma, and uterine corpus endometrial carcinoma). Our objective is to map the proteomic network structure of the respective tumor types, so that we can get a better grasp of the common mechanisms at play in the hormonally responsive tumors. We are also interested in highlighting the differences. We consider mature RPPA data from Luminal breast cancer (BRCA, n = 273), high-grade serous ovarian cystadenocarcinoma (OVCA, n = 412), and uterine corpus endometrial carcinoma (UCEC, n = 404). All data is downloaded from the UCSC Xena Browser [Goldman et al., 2020]. The data is measured with p = 131 high-quality antibodies that target (phospho)-proteins. To alleviate batch effects, the RPPA data is normalized with replicate-base normalization [Akbani et al., 2014]. We use stabJGL to jointly estimate the proteomic networks of the respective tumor types and interpret the results and their implications. We compare the output with that obtained with the fused joint graphical lasso (FGL) of Danaher et al. [2014] with the default penalty parameter tuning with AIC as described in Subsection 3.1. Further details and code for the analysis is given at https://github.com/Camiling/stabJGL_analysis. Pan-cancer analysis results Estimated proteomic networks The resulting stabJGL proteomic networks of the three tumor types are shown in Figure 4, where we observe plenty of common edges as well as network-specific ones. The sparsity as well as the selected penalty parameter values in the resulting stabJGL and FGL networks is shown in Table 3. The tendency as observed in the simulations of FGL tuned by the AIC to over-select edges appears to be consistent with the findings in this context. With more than two thirds of all potential edges being determined as present by FGL, the results are challenging to interpret and derive meaningful conclusions from. From a biological standpoint, we would not expect a proteomic network to be this saturated in terms of associations due to the expected scale-free property of the degree distribution [Barabasi and Oltvai, 2004]. While the degree distributions of the sparse stabJGL networks all follow a power-law with many low-degree nodes and fewer high-degree ones (hubs), an expected trait for omics data [Chen and Sharp, 2004], the degree distributions of the FGL networks do not. The full degree distributions are shown in the Supplement ( Figure S4). The methods are used to estimate graphs with p = 100 nodes from K = 2 networks, both of sparsity 0.02, of which 20% of their edges are in common. The performance of stabJGL is compared to that of Glasso, FGL, GGL and SSJGL. The results are averaged over N = 100 simulations and shows the sparsity, precision, and recall of each of the K = 2 estimated graphs. The corresponding standard deviations are shown in parentheses. The graphs are reconstructed from n 1 = 100 and n 2 = 150 observations. The average selected values of the penalty parameters λ 1 and λ 2 for the relevant methods is shown as well. In terms of penalty parameters, we see that just like for the simulated data the AIC selects very small penalty parameters for FGL, resulting in little sparsity and similarity encouragement. Given the findings of Akbani et al. [2014] about the presence of a super cluster consisting of the three hormonally responsive cancer types, it is not unreasonable to expect at least some proteomic network similarity to be encouraged by a joint method. This is achieved by stabJGL, which selects a large enough value of λ 2 to encourage similarity. A comparison of the pairwise similarities of the proteomic networks is given in Figure 5, where similarity is measured by Matthew's Correlation Coefficient (MCC), a discretized Pearson correlation coefficient that can be used to quantify pairwise network similarities (Matthews [1975]). StabJGL finds the networks of the three tumor types to be more similar than FGL, in accordance with the findings of Akbani et al. [2014]. Edge validation in STRING To compare the level of evidence supporting the edges detected by stabJGL and FGL tuned by the AIC in the literature, we conduct edge validation using the STRING database of known and predicted protein-protein interactions [Szklarczyk et al., 2019]. To ensure the reliability of the validation process, we only consider the experimentally validated interactions in STRING as evidence, with default confidence score threshold ≥ 0.4. The fraction of edges with supporting evidence in the STRING database is computed for the respective stabJGL and FGL networks and shown in Table 4. The analysis reveals that for all three tumor types investigated, a higher proportion of the edges detected by stabJGL had supporting evidence in the STRING database compared to those identified by FGL. Findings consistent with literature StabJGL successfully identifies protein-protein interactions known from literature. To highlight the findings of the proposed methodology, we only discuss edges and central proteins identified by stabJGL but not FGL. One example is the edge between activated (S345-phosphorylated) Checkpoint kinase 1 (Chk1) and DNA repair protein RAD51 homolog 1 (Rad51) in ovarian and breast cancer. The complex between the tumor suppressor BRCA2, which manifests predominantly in ovarian and breast cancer, and Rad51, is mediated by the DNA damage checkpoint Chk1 through Rad51 phosphorylation [Nair et al., 2020, Bahassi et al., 2008. It is also reassuring that stabJGL identifies many relevant tumor type-specific proteins as hubs in the relevant tumor type only, such as mammalian target of rapamycin (mTOR), Tuberous Sclerosis Complex 2 (Tuberin) and Ribosomal protein S6 in BRCA, all of which are involved or up/downstream of the PI3K/AKT/mTOR pathway known to frequently be deregulated in Luminal breast cancer [Miricescu et al., 2020]. Lists of the top hubs in the respective stabJGL and FGL networks of the different tumor types, and their node degree, are given in the Supplement (Tables S5 and S6). StabJGL also captures edges that we expect to be present in all three tumor types, such as the known interaction between the transcription factor Forkhead box O3 (FOXO3a) and 14-3-3-epsilon which facilitates cancer cell proliferation [Tzivion et al., 2011, Nielsen et al., 2008. This common interaction is documented in the STRING database. Figure 6 shows the network structure identified by stabJGL that is common to all three tumor types. Central proteins in this common network structure include Oncoprotein 18 (Stathmin), which is known to be relevant in all three hormonally responsive cancers due to its role in the regulation of cell growth and motility [Bieche et al., 1998, Trovik et al., 2011, Belletti and Baldassarre, 2011. Potential candidate hubs The recovery of documented links in the protein networks estimated by stabJGL highlights its capability to detect numerous relevant proteins and interactions. The potential for new discoveries is however an important aspect of stabJGL, as suggested by its good performance on simulated data. For example, stabJGL identifies phosphorylated epidermal growth factor receptor (EGFR) as a central hub protein in all three tumor types. While known to be relevant in ovarian cancer Zhang et al. [2016], Yang et al. [2013], the role of activated EGFR in uterine corpus endometrial carcinoma and Luminal breast cancer and is not yet clarified. Our findings suggest it could be relevant in all three hormonally responsive tumor types. Further, Platelet endothelial cell adhesion molecule (CD31) is found to be a central protein in UCEC only. The protein is important for angiogenesis and has been implicated in other tumor types such as haemangioma [Bergom et al., 2005]. Its prominence in the proteomic UCEC network suggests it may play a crucial role in this tumor type as well. Overall, these results showcase how stabJGL can aid in generating hypotheses by identifying central proteins and associations. Discussion Suitable sparsity and similarity selection is key for capturing and studying multiple related biological networks. We have proposed the stabJGL algorithm, which determines the penalty parameters in the fused graphical lasso for multiple networks based on the principle of model stability. StabJGL demonstrably avoids the under-or over-selection of edges observed in state-of-the-art selection methods based on information criteria, and succeeds at leveraging network similarities to a suitable degree. Consequently, the method can be employed in situations where the actual degree of similarity is uncertain, resulting in marked benefits with minimal risks associated with its use. StabJGL offers a fast parallelized implementation, particularly for K = 2 networks as a closed-form solution exists. We successfully apply the method to problems with p > 1, 000 nodes and K > 2 networks. With our novel approach, we can identify both common and distinct mechanisms in the proteomic networks of different types of hormonally responsive women's cancers. The results obtained with stabJGL are in line with known biology and compliment those of Akbani et al. [2014] by offering additional understanding of the underlying mechanisms in action. By recognizing various proteins as highly critical in the proteomic networks, stabJGL suggests their possible involvement in driving the diseases. The method both identifies proteins that are central in all three hormonally responsive cancers (e.g., phosphorylated EGFR) and proteins of tumor-specific relevance (e.g., CD31 in UCEC). Future extensions of the method can include alternative measures of variability, such as the entropy (see, e.g., Lartigue et al. [2020]). Further, while the method is formulated specifically for the joint graphical lasso with the fused penalty, it can in principle be used for any joint network approach requiring the tuning of sparsity-and similarity controlling parameters. One potential method of application is Jewel [Angelini et al., 2021], which is based on a group lasso penalty and currently employs a BIC-based selection routine. To conclude, stabJGL provides a reliable approach to joint network inference of omics data. The output can provide a better understanding of both common and data type-specific mechanisms, which can be used for hypothesis generation regarding potential therapeutic targets. A Appendix B Runtime profiling Figure B.1 compares the time used to infer K ∈ {2, 4} networks with various numbers of nodes p, for the different network reconstruction methods. We only consider the AIC selection for FGL as the eBIC considers the same grid of values and hence has identical running time. All methods are run with the same parameter specifications as in the main simulation study. The simulated networks are set to have 50% of their edges in common, generated with the same approach as in the main simulation study. As discussed by Danaher et al. [2014], the group joint graphical lasso is faster that its fused counterpart. The Bayesian spike-and-slab joint graphical lasso is substantially slower than the other methods, taking around ten times longer than the fused joint graphical lasso and stabJGL. Figure B.2 shows the time used by stabJGL to infer K ∈ {2, 3} networks with various numbers of nodes p and 50% of their edges in common. We see that for K = 2 networks, inference for p = 1, 400 nodes is feasible within half an hour, while for K = 3 inference for p = 1, 000 nodes is feasible within about eight hours. As discussed by Danaher et al. [2014], there is an explicit solution to the fused joint graphical lasso problem for K = 2 and hence inference is much faster for stabJGL as well in that case. C Additional simulation scenarios Additional simulation studies are conducted to assess a wider range of scenarios and compare the performance of the graphical lasso (Glasso), the fused joint graphical lasso (FGL) and the group joint graphical lasso (GGL) tuned by AIC, the fused joint graphical lasso tuned by eBIC, the Bayesian spike-and-slab joint graphical lasso (SSJGL) and stabJGL. Table C.1 shows the network reconstruction performance of the methods in a K = 3 network setting with p = 200 nodes and various similarity of the true graph structures, averaged over N = 100 simulations. Similarly, Table C.2 shows the results for a K = 4 setting with p = 100 nodes. In Table C.3 , the results are shown for a K = 2 network setting with p = 100 nodes. Finally, the results from a K = 2 network setting with p = 300 nodes are shown in Table C.4 . In the latter case, due to the longer run time of SSJGL as demonstrated in Section B, this method is omitted to make the simulation study feasible within reasonable time (< 48 hours). The results from the additional simulation are in line with those from the main simulation study; stabJGL succeeds at capturing both the sparsity level and similarity between the networks to a better degree than FGL and GGl, while either outperforming the standard graphical lasso for highly similar networks or getting comparable results for unrelated networks. FGL with the alternative eBIC selection mostly selects empty graphs. Finally, SSJGL select very few edges, leading to high precision but very low recall in all cases. Figure D.3 compares the performance of stabJGL for different values of the variability threshold β 1 to the the graphical lasso (Glasso), the fused joint graphical lasso (FGL), the group joint graphical lasso (GGL) and the Bayesian spike-andslab joint graphical lasso (SSJGL). The results for FGL tuned with eBIC are not shown as it selected an empty graph in all settings. The settings considered have K = 2 networks with p = 100 nodes of various similarity. As in the setting considered in the main manuscript, we find that by varying the variability threshold β 1 we can obtain at least as high precision and/or recall as the other methods at any level of similarity. Table E.4 shows the degree distribution of the proteomic networks identified by stabJGL and FGL. While the stabJGL networks all have degree distributions that follow clear power-law distributions, in line with biological expectations, the FGL networks have degree distributions that strongly contradict a power law with most nodes having node degree > 60. Table E.5 shows the node degree of the proteins with degree larger than the 90 th percentile in the respective stabJGL networks of the different tumor types. The same table for the FGL networks is shown in Table E.5 . Table C.3 : Performance of the different graph reconstruction methods in simulations, reconstructing graphs with p = 100 nodes from K = 2 classes with various similarity of the true graph structures. The methods included are the graphical lasso (Glasso), the fused joint graphical lasso tuned by the AIC (FGL) and by the extended BIC (eBIC), the group joint graphical lasso (GGL), the Bayesian spike-and-slab joint graphical lasso (SSJGL) and stabJGL. The similarity (percentage of edges that are in common) of the graphs is shown. The results are averaged over N = 100 simulations and shows the sparsity, precision, and recall of each of the K = 2 estimated graphs. The corresponding standard deviations are shown as well. The graphs are reconstructed from n 1 = 100 and n 2 = 150 observations. All graphs have sparsity 0.02. The average selected values of the penalty parameters λ 1 and λ 2 for the relevant methods is shown as well. Table C.4 : Performance of the different graph reconstruction methods in simulations, reconstructing graphs with p = 300 nodes from K = 2 classes with various similarity of the true graph structures. The methods included are the graphical lasso (Glasso), the fused joint graphical lasso tuned by the AIC (FGL) and by the extended BIC (eBIC), the group joint graphical lasso (GGL) and stabJGL. The similarity (percentage of edges that are in common) of the graphs is shown. The results are averaged over N = 100 simulations and shows the sparsity, precision, and recall of each of the K = 2 estimated graphs. The corresponding standard deviations are shown as well. The graphs are reconstructed from n 1 = 150 and n 2 = 200 observations. All graphs have sparsity 0.007. The average selected values of the penalty parameters λ 1 and λ 2 for the relevant methods is shown as well. on simulated data, compared to other graph reconstruction methods. The methods are used to estimate graphs with p = 100 nodes from K = 2 networks, both of sparsity 0.02, with various similarity of the true graph structures. The performance of stabJGL is compared to that of the graphical lasso (Glasso), the fused joint graphical lasso tuned by the AIC (FGL), the group joint graphical lasso (GGL) and the Bayesian spike-and-slab joint graphical lasso (SSJGL). The similarity (percentage of edges that are in common) of the graphs is shown. The results are averaged over N = 100 simulations and shows the precision and recall of each of the K = 2 estimated graphs. Standard deviation bars are shown for all methods. The graphs are reconstructed from n 1 = 100 and n 2 = 150 observations. D Choice of variability threshold E Additional Pan-Cancer analysis results E.1 Degree distributions E.2 Top hubs Figure 1 : 1The workflow of stabJGL, where the network structures of different data types or conditions are jointly estimated and can then be compared. Figure 2 : 2Performance of the Glasso, FGL and GGL tuned by AIC, SSJGL and stabJGL, reconstructing K ∈ {2, 4} graphs with p = 100 nodes and various similarity of the true graph structures. The similarity between the graphs is shown as the percentage of edges they have in common. The results are averaged over N = 100 replicates and show the precision and recall for the first estimated graph in each setting, reconstructed from n ∈ {100, 150} observations and n ∈ {150, 200, 250, 300} observations for K = 2 and K = 4 respectively. Standard deviation bars are shown for all methods. All graphs have true sparsity 0.02. Figure 3 : 3CPU time in seconds on a logarithmic scale used to jointly infer networks for K ∈ {2, 3, 4} networks and various numbers of nodes p, with n ∈ {100, 150} observations, for FGL tuned with AIC and stabJGL. The computations were performed on a 16-core Intel Xeon CPU, 2.60 GHz. Figure 4 : 4Proteomic network structure identified by stabJGL for the breast cancer (BRCA), ovarian cystadenocarcinoma (OVCA) and uterine corpus endometrial carcinoma (UCEC) tumors. The blue nodes represent proteins, and edges common to all three networks are marked in red, otherwise they are grey. Figure 5 : 5Pairwise Matthew's Correlation Coefficient between the proteomic network structures of the breast cancer (BRCA), ovarian cystadenocarcinoma (OVCA) and uterine corpus endometrial carcinoma (UCEC) tumors, identified by FGL tuned by the AIC and stabJGL respectively. Figure 6 : 6The proteomic network structure identified by stabJGL common to all three tumor types (BRCA, UCEC and OCVA). The node size indicates the degree in the common network structure, with proteins with more edges being represented by larger nodes. Figure B. 1 : 1CPU time in seconds on a logarithmic scale used to jointly infer networks for K ∈ {2, 4} networks and various numbers of nodes p, with n ∈ {100, 150} observations, for the fused and group joint graphical lasso with AIC penalty parameter selection (FGL and GGL), the Bayesian spike-and-slab joint graphical lasso (SSJGL) and stabJGL. The computations were performed on a 16-core Intel Xeon CPU, 2.60 GHz. Figure B. 2 : 2CPU time in seconds on a logarithmic scale used by stabJGL to jointly infer (a) K = 2 networks with various numbers of nodes p from n 1 = n 2 = 500 observations and (b) K = 3 networks with various numbers of nodes p from n 1 = n 2 = n 3 = 500 observations. The computations were performed on a 16-core Intel Xeon CPU, 2.60 GHz. C. 1 : 1Performance of the different graph reconstruction methods in simulations, reconstructing graphs with p = 200 nodes from K = 3 classes with various similarity of the true graph structures. The methods included are the graphical lasso (Glasso), the fused joint graphical lasso tuned by the AIC (FGL) and by the extended BIC (eBIC), the group joint graphical lasso (GGL), the Bayesian spike-and-slab joint graphical lasso (SSJGL) and stabJGL. The similarity (percentage of edges that are in common) of the graphs is shown. The results are averaged over N = 100 simulations and shows the sparsity, precision, and recall of each of the K = 3 estimated graphs. The corresponding standard deviations are shown as well. The graphs are reconstructed from n . All graphs have sparsity 0.01. The average selected values of the penalty parameters λ 1 and λ 2 for the relevant methods is shown as well. .2 : Performance of the different graph reconstruction methods in simulations, reconstructing graphs with p = 100 nodes from K = 4 classes with various similarity of the true graph structures. The methods included are the graphical lasso (Glasso), the fused joint graphical lasso tuned by the AIC (FGL) and by the extended BIC (eBIC), the group joint graphical lasso (GGL), the Bayesian spike-and-slab joint graphical lasso (SSJGL) and stabJGL. The similarity (percentage of edges that are in common) of the graphs is shown. The results are averaged over N = 100simulations and shows the sparsity, precision, and recall of each of the K = 4 estimated graphs. The corresponding standard deviations are shown as well. The graphs are reconstructed from n 300 observations. All graphs have sparsity 0.02. The average selected values of the penalty parameters λ 1 and λ 2 for the relevant methods is shown as well. .002 (0.000) 0.79 (0.11) 0.09 (0.02) 0.002 (0.000) 0.90 (0.08) 0.10 (0.02) stabJGL 0.218 0.088 0.010 (0.002) 0.58 (0.07) 0.29 (0.05) 0.007 (0.001) 0.76 (0.08) 0.26 (0.04) Figure D. 3 : 3Performance of stabJGL for different values of the variability threshold β 1 Figure E. 4 : 4Histogram of the node degrees of the proteomic network of each tumor type, for the (a) stabJGL and (b) FGL networks. Table 2 : 2Performance of stabJGL for different values of the variability threshold β 1 on simulated data, compared to other graph reconstruction methods. Table 3 : 3Network analysis results for stabJGL and FGL tuned by the AIC, applied to data from breast cancer (BRCA), ovarian cystadenocarcinoma (OVCA) and uterine corpus endometrial carcinoma (UCEC) tumors.Sparsity λ 1 λ 2 BRCA UCEC OVCA FGL 0.010 0.000 0.689 0.709 0.679 stabJGL 0.323 0.008 0.049 0.036 0.039 Table 4 : 4Comparison of evidence for edges in the respective FGL tuned by AIC and stabJGL proteomic networks of breast cancer (BRCA), ovarian cystadenocarcinoma (OVCA) and uterine corpus endometrial carcinoma (UCEC) tumors, considering experimentally determined protein-protein interactions documented in the STRING database. The highest percentage of edges with evidence is in bold.Edge evidence % Data set FGL stabJGL BRCA 5.4% 12.3% UVEC 5.6% 10.0% OVCA 5.7% 12.4% Table Table E . E5 : The genes with node degree larger than the 90 th percentile in the respective stabJGL networks of the different tumor types. 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[ "https://github.com/Camiling/stabJGL.If", "https://github.com/Camiling/stabJGL.", "https://github.com/Camiling/stabJGL_simulations.", "https://github.com/Camiling/stabJGL_analysis." ]
[ "Convergent Bregman Plug-and-Play Image Restoration for Poisson Inverse Problems", "Convergent Bregman Plug-and-Play Image Restoration for Poisson Inverse Problems" ]
[ "Samuel Hurault [email protected] \nUniv. Bordeaux\nBordeaux INP\nCNRS\nWashington University\nSt. LouisIMB\n", "Ulugbek Kamilov [email protected] \nUniv. Bordeaux\nBordeaux INP\n", "Arthur Leclaire [email protected] \nCNRS\nUniv. Bordeaux\nBordeaux INPIMB\n", "Nicolas Papadakis [email protected] \nCNRS\nIMB\n" ]
[ "Univ. Bordeaux\nBordeaux INP\nCNRS\nWashington University\nSt. LouisIMB", "Univ. Bordeaux\nBordeaux INP", "CNRS\nUniv. Bordeaux\nBordeaux INPIMB", "CNRS\nIMB" ]
[]
Plug-and-Play (PnP) methods are efficient iterative algorithms for solving ill-posed image inverse problems. PnP methods are obtained by using deep Gaussian denoisers instead of the proximal operator or the gradient-descent step within proximal algorithms. Current PnP schemes rely on data-fidelity terms that have either Lipschitz gradients or closed-form proximal operators, which is not applicable to Poisson inverse problems. Based on the observation that the Gaussian noise is not the adequate noise model in this setting, we propose to generalize PnP using the Bregman Proximal Gradient (BPG) method. BPG replaces the Euclidean distance with a Bregman divergence that can better capture the smoothness properties of the problem. We introduce the Bregman Score Denoiser specifically parametrized and trained for the new Bregman geometry and prove that it corresponds to the proximal operator of a nonconvex potential. We propose two PnP algorithms based on the Bregman Score Denoiser for solving Poisson inverse problems. Extending the convergence results of BPG in the nonconvex settings, we show that the proposed methods converge, targeting stationary points of an explicit global functional. Experimental evaluations conducted on various Poisson inverse problems validate the convergence results and showcase effective restoration performance.Preprint. Under review.
null
[ "https://export.arxiv.org/pdf/2306.03466v1.pdf" ]
259,089,010
2306.03466
56f514024462ae498bf69a9d419f90fa90902b52
Convergent Bregman Plug-and-Play Image Restoration for Poisson Inverse Problems Samuel Hurault [email protected] Univ. Bordeaux Bordeaux INP CNRS Washington University St. LouisIMB Ulugbek Kamilov [email protected] Univ. Bordeaux Bordeaux INP Arthur Leclaire [email protected] CNRS Univ. Bordeaux Bordeaux INPIMB Nicolas Papadakis [email protected] CNRS IMB Convergent Bregman Plug-and-Play Image Restoration for Poisson Inverse Problems Plug-and-Play (PnP) methods are efficient iterative algorithms for solving ill-posed image inverse problems. PnP methods are obtained by using deep Gaussian denoisers instead of the proximal operator or the gradient-descent step within proximal algorithms. Current PnP schemes rely on data-fidelity terms that have either Lipschitz gradients or closed-form proximal operators, which is not applicable to Poisson inverse problems. Based on the observation that the Gaussian noise is not the adequate noise model in this setting, we propose to generalize PnP using the Bregman Proximal Gradient (BPG) method. BPG replaces the Euclidean distance with a Bregman divergence that can better capture the smoothness properties of the problem. We introduce the Bregman Score Denoiser specifically parametrized and trained for the new Bregman geometry and prove that it corresponds to the proximal operator of a nonconvex potential. We propose two PnP algorithms based on the Bregman Score Denoiser for solving Poisson inverse problems. Extending the convergence results of BPG in the nonconvex settings, we show that the proposed methods converge, targeting stationary points of an explicit global functional. Experimental evaluations conducted on various Poisson inverse problems validate the convergence results and showcase effective restoration performance.Preprint. Under review. Introduction Ill-posed image inverse problems are classically formulated with a minimization problem of the form arg min x∈R n λf (x) + g(x)(1) where f is a data-fidelity term, g a regularization term, and λ > 0 a regularization parameter. The data-fidelity term is generally written as the negative log-likelihood f (x) = − log p(y|x) of the probabilistic observation model chosen to describe the physics of an acquisition y ∈ R m from linear measurements Ax of an image x ∈ R n . In applications such as Positron Emission Tomography (PET) or astronomical CCD cameras [Bertero et al., 2009], where images are obtained by counting particles (photons or electrons), it is common to use the Poisson noise model y ∼ P(αAx) with parameter α > 0. The corresponding negative log-likelihood corresponds to the Kullback-Leibler divergence f (x) = m i=1 y i log y i α(Ax) i + α(Ax) i − y i .(2) The minimization of (1) can be addressed with proximal splitting algorithms [Combettes and Pesquet, 2011]. Depending on the properties of the functions f and g, they consist in alternatively evaluating the proximal operator and/or performing a gradient-descent step on f and g. Plug-and-Play (PnP) [Venkatakrishnan et al., 2013] and Regularization-by-Denoising (RED) [Romano et al., 2017] methods build on proximal splitting algorithms by replacing the proximal or gradient descent updates with off-the-shelf Gaussian denoising operators, typically deep neural networks trained to remove Gaussian noise. The intuition behind PnP/RED is that the proximal (resp. gradientdescent) mapping of g writes as the maximum-a-posteriori (MAP) (resp. posterior mean) estimation from a Gaussian noise observation, under prior p = exp(−g). A remarkable property of these priors is that they are decoupled from the degradation model represented by f in the sense that one learned prior p can serve as a regularizer for many inverse problems. When using deep denoisers corresponding to exact gradient-step [Hurault et al., 2021, Cohen et al., 2021 or proximal [Hurault et al., 2022] maps, RED and PnP methods become real optimization algorithms with known convergence guarantees. However, for Poisson noise the data-fidelity term (2) is neither smooth with a Lipschitz gradient, nor proximable for A ̸ = Id (meaning that its proximal operator cannot be computed in a closed form), which limits the use of standard splitting procedures. Bauschke et al. [2017] addressed this issue by introducing a Proximal Gradient Descent (PGD) algorithm in the Bregman divergence paradigm, called Bregman Proximal Gradient (BPG). The benefit of BPG is that the smoothness condition on f for sufficient decrease of PGD is replaced by the "NoLip" condition "Lh − f is convex" for a convex potential h. For instance, Bauschke et al. [2017] show that the data-fidelity term (2) satisfies the NoLip condition for the Burg's entropy h(x) = − n i=1 log(x i ). Our primary goal is to exploit the BPG algorithm for minimizing (1) using PnP and RED priors. This requires an interpretation of the plug-in denoiser as a Bregman proximal operator. Prox h g (y) = arg min x g(x) + D h (x, y),(3) where D h (x, y) is the Bregman divergence associated with h. In Section 3, we show that by selecting a suitable noise model, other than the traditional Gaussian one, the MAP denoiser can be expressed as a Bregman proximal operator. Remarkably, the corresponding noise distribution belongs to an exponential family, which allows for a closed-form posterior mean (MMSE) denoiser generalizing the Tweedie's formula. Using this interpretation, we derive a RED prior tailored to the Bregman geometry. By presenting a prior compatible with the noise model, we highlight the limitation of the decoupling between prior and data-fidelity suggested in the existing PnP literature. In order to safely use our MAP and MMSE denoisers in the BPG method, we introduce the Bregman Score denoiser, which generalizes the Gradient Step denoiser from [Hurault et al., 2021, Cohen et al., 2021. Our denoiser provides an approximation of the log prior of the noisy distribution of images. Moreover, based on the characterization of the Bregman Proximal operator from Gribonval and Nikolova [2020], we show under mild conditions that our denoiser can be expressed as the Bregman proximal operator of an explicit nonconvex potential. In Section 4, we use the Bregman Score Denoiser within RED and PnP methods and propose the B-RED and B-PnP algorithms. Elaborating on the results from Bolte et al. [2018] for the BPG algorithm in the nonconvex setting, we demonstrate that RED-BPG and PnP-BPG are guaranteed to converge towards stationary points of an explicit functional. We finally show in Section 5 the relevance of the proposed framework in the context of Poisson inverse problems. Related Works Poisson Inverse Problems A variety of methods have been proposed to solve Poisson image restoration from the Bayesian variational formulation (1) with Poisson data-fidelity term (2). Although f is convex, this is a challenging optimization problem as f is non-smooth and does not have a closedform proximal operator for A ̸ = Id, thus precluding the direct application of splitting algorithms such as Proximal Gradient Descent or ADMM. Moreover, we wish to regularize (1) with an explicit denoising prior, which is generally nonconvex. PIDAL [Figueiredo andBioucas-Dias, 2009, 2010] and related methods [Ng et al., 2010, Setzer et al., 2010 solve (2) using modified versions of the alternating direction method of multipliers (ADMM). As f is proximable when A = Id, the idea is to add a supplementary constraint in the minimization problem and to adopt an augmented Lagrangian framework. Figueiredo and Bioucas-Dias [2010] prove the convergence of their algorithm with convex regularization. However, no convergence is established for nonconvex regularization. Boulanger et al. [2018] adopt the primal-dual PDHG algorithm which also splits A from the f update. Once again, there is not convergence guarantee for the primal-dual algorithm with nonconvex regularization. Plug-and-Play (PnP) PnP methods were successfully used for solving a variety of IR tasks by including deep denoisers into different optimization algorithms, including Gradient Descent [Romano et al., 2017], Half-Quadratic-Splitting [Zhang et al., 2017, 2021, ADMM [Ryu et al., 2019, Sun et al., 2021, and PGD [Kamilov et al., 2017, Terris et al., 2020. Variations of PnP have been proposed to solve Poisson inverse problems. Rond et al. [2016] use PnP-ADMM and approximates, at each iteration, the non-tractable proximal operator of the data-fidelity term (2) with an inner minimization procedure. Sanghvi et al. [2022] propose a PnP version of the PIDAL algorithm which is also unrolled for image deconvolution. Theoretical convergence of PnP algorithms with deep denoisers has recently been addressed by a variety of studies [Ryu et al., 2019, Sun et al., 2021, Terris et al., 2020 (see also a review in Kamilov et al. [2023]). Most of these works require non-realistic or sub-optimal constraints on the deep denoiser, such as nonexpansiveness. More recently, convergence was addressed by making PnP genuine optimization algorithms again. This is done by building deep denoisers as exact gradient descent operators (Gradient-Step denoiser) [Cohen et al., 2021, Hurault et al., 2021 or exact proximal operators [Hurault et al., 2022[Hurault et al., , 2023. These PnP algorithms thus minimize an explicit functional (2) with an explicit (nonconvex) deep regularization. Bregman optimization Bauschke et al. [2017] replace the smoothness condition of PGD by the NoLip assumption (4) with a Bregman generalization of PGD called Bregman Proximal Gradient (BPG). In the nonconvex setting, Bolte et al. [2018] prove global convergence of the algorithm to a critical point of (1). The analysis however requires assumptions that are not verified by the Poisson data-fidelity term and Burg's entropy Bregman potential. Al-Shabili et al. [2022] considered unrolled Bregman PnP and RED algorithms, but without any theoretical convergence analysis. Moreover, the interaction between the data fidelity, the Bregman potential, and the denoiser was not explored. Bregman denoising prior The overall objective of this work is to efficiently solve ill-posed image restoration (IR) problems involving a data-fidelity term f verifying the NoLip assumption for some convex potential h NoLip There is L > 0 such that Lh − f is convex on int dom h.(4) PnP provides an elegant framework for solving ill-posed inverse problems with a denoising prior. However, the intuition and efficiency of PnP methods inherit from the fact that Gaussian noise is well suited for the Euclidean L 2 distance, the latter naturally arising in the MAP formulation of the Gaussian denoising problem. When the Euclidean distance is replaced by a more general Bregman divergence, the noise model needs to be adapted accordingly for the prior. In Section 3.1, we first discuss the choice of the noise model associated to a Bregman divergence, leading to Bregman formulations of the MAP and MMSE estimators. Then we introduce in Section 3.2 the Bregman Score Denoiser that will be used to regularize the inverse problem (1). Notations For convenience, we assume throughout our analysis that the convex potential h : C h ⊆ R n → R ∪ {+∞} is C 2 and of Legendre type (definition in Appendix A.1). Its convex conjugate h * is then also C 2 of Legendre type. D h (x, y) denotes its associated Bregman divergence D h : R n × int dom h → [0, +∞] : (x, y) → h(x) − h(y) − ⟨∇h(y), x − y⟩ if x ∈ dom(h) +∞ otherwise.(5) Bregman noise model We consider the following observation noise model, referred to as Bregman noise 1 , for x, y ∈ dom(h) × int dom(h) p(y|x) := exp (−γD h (x, y) + ρ(x)) .(6) We assume that there is γ > 0 and a normalizing function ρ : dom(h) → R such that the expression (6) defines a probability measure. For instance, for h(x) = 1 2 ||x|| 2 , γ = 1 σ 2 and ρ = 0, we retrieve the Gaussian noise model with variance σ 2 . As shown in Section 5, for h given by Burg's entropy, p(y|x) corresponds to a multivariate Inverse Gamma (IG) distribution. Given a noisy observation y ∈ int dom(h), i.e. a realization of a random variable Y with conditional probability p(y|x), we now consider two optimal estimators of x, the MAP and the posterior mean. Maximum-A-Posteriori (MAP) estimator The MAP denoiser selects the mode of the a-posteriori probability distribution p(x|y). Given the prior p X , it writeŝ x M AP (y) = arg min x −log p(x|y) = arg min x −log p X (x) − log p(y|x) = Prox h − 1 γ (ρ+log p X ) (y). (7) Under the Bregman noise model (6), the MAP denoiser writes as the Bregman proximal operator (see relation (3)) of − 1 γ (log p X + ρ). This acknowledges for the fact that the introduced Bregman noise is the adequate noise model for generalizing PnP methods within the Bregman framework. Posterior mean (MMSE) estimator The MMSE denoiser is the expected value of the posterior probability distribution and the optimal Bayes estimator for the L 2 score. Note that our Bregman noise conditional probability (6) belongs to the regular exponential family of distributions p(y|x) = p 0 (y) exp (⟨x, T (y)⟩ − ψ(x))(8) with T (y) = γ∇h(y), ψ(x) = γh(x) − ρ(x) and p 0 (y) = exp (γh(y) − γ⟨∇h(y), y⟩). It is shown in [Efron, 2011] (for T = Id and generalized in [Kim and Ye, 2021] for T ̸ = Id) that the corresponding posterior mean estimator verifies a generalized Tweedie formula ∇T (y).x M M SE (y) = −∇ log p 0 (y) + ∇ log p Y (y), which translates to (see Appendix B for details) x M M SE (y) = E[x|y] = y − 1 γ (∇ 2 h(y)) −1 .∇(− log p Y )(y).(9) Note that for the Gaussian noise model, we have h(x) = 1 2 ||x|| 2 , γ = 1/σ 2 and (9) falls back to the more classical Tweedie formula of the Gaussian posterior mean denoiserx = y − σ 2 ∇(− log p Y )(y). Therefore, given an off-the-shelf "Bregman denoiser" B γ specially devised to remove Bregman noise (6) of level γ, if the denoiser approximates the posterior mean B γ (y) ≈x M M SE (y), then it provides an approximation of the score −∇ log p Y (y) ≈ γ∇ 2 h(y). (y − B γ (y)). Bregman Score Denoiser Based on previous observations, we propose to define a denoiser following the form of the MMSE (9) B γ (y) = y − (∇ 2 h(y)) −1 .∇g γ (y),(10) with g γ : R n → R a nonconvex potential parametrized by a neural network. When B γ is trained as a denoiser for the associated Bregman noise (6) with L 2 loss, it approximates the optimal estimator for the L 2 score, precisely the MMSE (9). Comparing (10) and (9), we get ∇g γ ≈ − 1 γ ∇ log p Y , i.e. the score is properly approximated with an explicit conservative vector field. We refer to this denoiser as the Bregman Score denoiser. Such a denoiser corresponds to the Bregman generalization of the Gaussian Noise Gradient-Step denoiser proposed in [Hurault et al., 2021, Cohen et al., 2021. Is the Bregman Score Denoiser a Bregman proximal operator? We showed in relation (7) that the optimal Bregman MAP denoiserx M AP is a Bregman proximal operator. We want to generalize this property to our Bregman denoiser (10). When trained with L 2 loss, the denoiser should approximate the MMSE rather than the MAP. For Gaussian noise, Gribonval [2011] re-conciliates the two views by showing that the Gaussian MMSE denoiser actually writes as an Euclidean proximal operator. Similarly, extending the characterization from [Gribonval and Nikolova, 2020] of Bregman proximal operators, we now prove that, under some convexity conditions, the proposed Bregman Score Denoiser (10) explicitly writes as the Bregman proximal operator of a nonconvex potential. Proposition 1 (Proof in Appendix C). Let h be C 2 and of Legendre type. Let g γ : R n → R ∪ {+∞} proper and differentiable and B γ (y) : int dom(h) → R n defined in (10). Assume Im(B γ ) ⊂ int dom(h). With ψ γ : R n → R ∪ {+∞} defined by ψ γ (y) = −h(y) + ⟨∇h(y), y⟩ − g γ (y) if y ∈ int dom(h) +∞ otherwise.(11) suppose that ψ γ • ∇h * is convex on int dom(h * ). Then for ϕ γ : R n → R ∪ {+∞} defined by ϕ γ (x) := g γ (y) − D h (x, y) for y ∈ B −1 γ (x) if x ∈ Im(B γ ) +∞ otherwise(12) we have that for each y ∈ int dom(h) B γ (y) ∈ arg min x∈R n {D h (x, y) + ϕ γ (x)}(13) Remark 1. Note that the order in the Bregman divergence is important in order to fit the definition of the Bregman proximal operator (3). In this order, [Gribonval and Nikolova, 2020, Theorem 3] does not directly apply. We propose instead in Appendix C a new version of their main theorem. This proposition generalizes the result of [Hurault et al., 2022, Prop. 1] to any Bregman geometry, which proves that the Gradient-Step Gaussian denoiser writes as an Euclidean proximal operator when ψ γ • ∇h * (x) = 1 2 ||x|| 2 − g γ (x) is convex. More generally, as exhibited for Poisson inverse problems in Section 5, such a convexity condition translates to a constraint on the deep potential g γ . To conclude, the Bregman Score Denoiser provides, via exact gradient or proximal mapping, two distinct explicit nonconvex priors g γ and ϕ γ that can be used for subsequent PnP image restoration. Plug-and-Play (PnP) image restoration with Bregman Score Denoiser We now regularize inverse problems with the explicit prior provided by the Bregman Score Denoiser (10). Properties of the Bregman Proximal Gradient (BPG) algorithm are recalled in Section 4.1. We show the convergence of our two B-RED and B-PnP algorithms in Sections 4.2 and 4.3. Bregman Proximal Gradient (BPG) algorithm Let F and R be two proper and lower semi-continuous functions with F of class C 1 on int dom(h). Bauschke et al. [2017] propose to minimize Ψ = F + R using the following BPG algorithm: x k+1 ∈ arg min x∈R n {R(x) + ⟨x − x k , ∇F (x k )⟩ + 1 τ D h (x, x k )}.(14) Recalling the general expression of proximal operators defined in relation (3), when ∇h(x k ) − τ ∇F (x k ) ∈ dom(h * ), the previous iteration can be written as (see Appendix D) x k+1 ∈ Prox h τ R •∇h * (∇h − τ ∇F )(x k ).(15) With formulation (15), the BPG algorithm generalizes the Proximal Gradient Descent (PGD) algorithm in a different geometry defined by h. Convergence of BPG If F verifies the NoLip condition (4) for some L F > 0 and if τ < 1 L F , one can prove that the objective function Ψ decreases along the iterates (15). Global convergence of the iterates for nonconvex F and R is also shown in [Bolte et al., 2018]. However, Bolte et al. [2018] take assumptions on F and h that are not satisfied in the context of Poisson inverse problems. For instance, h is assumed strongly convex on the full domain R n which is not satisfied by Burg's entropy. Additionally, F is assumed to have Lipschitz-gradient on bounded subset of R n , which is not true for F the Poisson data-fidelity term (2). Following the same structure of their proof, we extend in Appendix D.1 (Proposition 3 and Theorem 3) the convergence theory from [Bolte et al., 2018] with the more general Assumption 1 below, which is verified for Poisson inverse problems (Appendix E.3). Application to the IR problem In what follows, we consider two variants of BPG for minimizing (1) with gradient updates on the data-fidelity term f . These algorithms respectively correspond to Bregman generalizations of the RED Gradient-Descent (RED-GD) [Romano et al., 2017] and the Plug-and-Play PGD algorithms. For the rest of this section, we consider the following assumptions Assumption 1. (i) h : C h → R ∪ {+∞} is of class C 2 and of Legendre-type. (ii) f : R n → R ∪ {+∞} is proper, coercive, of class C 1 on int dom(h), with dom(h) ⊂ dom(f ), and is semi-algebraic. (iii) NoLip : L f h − f is convex on int dom(h). (iv) h is assumed strongly convex on any bounded convex subset of its domain and for all α > 0, ∇h and ∇f are Lipschitz continuous on {x ∈ dom(h), Ψ(x) ≤ α}. (v) g γ given by the Bregman Score Denoiser (10) and its associated ϕ γ obtained from Proposition 1 are lower-bounded and semi-algebraic. Even though the Poisson data-fidelity term (2) is convex, our convergence results also hold for more general nonconvex data-fidelity terms. Assumption (iv) generalizes [Bolte et al., 2018, Assumption D] and allows to prove global convergence of the iterates. The semi-algebraicity assumption is a sufficient conditions for the Kurdyka-Lojasiewicz (KL) property [Bolte et al., 2007] to be verified. The latter, defined in Appendix A.3 can be interpreted as the fact that, up to a reparameterization, the function is sharp. The KL property is widely used in nonconvex optimization [Attouch et al., 2013, Ochs et al., 2014. Semi-algebraicity is verified in practice by a very large class of functions and the sum of semi-algebraic functions is semi-algebraic. Assumption 1 thus ensures that λf + g γ and λf + ϕ γ verify the KL property. Bregman Regularization-by-Denoising (B-RED) We first generalize the RED Gradient-Descent (RED-GD) algorithm [Romano et al., 2017] in the Bregman framework. Classically, RED-GD is a simple gradient-descent algorithm applied to the functional λf + g γ where the gradient ∇g γ is assumed to be implicitly given by an image denoiser B γ (parametrized by γ) via ∇g γ = Id −B γ . Instead, our Bregman Score Denoiser (10) provides an explicit regularizing potential g γ whose gradient approximates the score via the Tweedie formula (9). We propose to minimize F λ,γ = λf + g γ on dom(h) using the Bregman Gradient Descent algorithm x k+1 = ∇h * (∇h − τ ∇F λ,γ )(x k )(16) which writes in a more general version as the BPG algorithm (14) with R = 0 x k+1 = arg min x∈R n {⟨x − x k , λ∇f (x k ) + ∇g γ (x k )⟩ + 1 τ D h (x, x k )}.(17) As detailed in Appendix E.3, in the context of Poisson inverse problems, for h being Burg's entropy, F λ,γ = λf + g γ verifies the NoLip condition only on bounded convex subsets of dom(h). Thus we select C a non-empty closed bounded convex subset of dom(h). For the algorithm (17) to be well-posed and to verify a sufficient decrease of (F λ,γ (x k )), the iterates need to verify x k ∈ C. We propose to modify (17) as the Bregman version of Projected Gradient Descent, which corresponds to the BPG algorithm (14) with R = i C , the characteristic function of the set C: (B-RED) x k+1 ∈ T τ (x k ) = arg min x∈R n {i C (x) + ⟨x − x k , ∇F λ,γ (x k )⟩ + 1 τ D h (x, x k )}. (18) For general convergence of B-RED, we need the following assumptions Assumption 2. (i) R = i C , with C a non-empty closed, bounded, convex and semi-algebraic subset of dom(h) such that C ∩ int dom(h) ̸ = ∅. (ii) g γ has Lipschitz continuous gradient and there is L γ > 0 such that L γ h − g γ is convex on C ∩ int dom(h). In [Bolte et al., 2018, Bauschke et al., 2017, the NoLip constant L needs to be known to set the stepsize of the BPG algorithm as τ L < 1. In practice, the NoLip constant which depends on f , g γ and C is either unknown or over-estimated. In order to avoid small stepsize, we adapt the backtracking strategy of [Beck, 2017, Chapter 10] to automatically adjust the stepsize while keeping convergence guarantees. Given γ ∈ (0, 1), η ∈ [0, 1) and an initial stepsize τ 0 > 0, the following backtracking update rule on τ is applied at each iteration k: while F λ,γ (x k ) − F λ,γ (T τ (x k )) < γ τ D h (T τ (x k ), x k ), τ ←− ητ.(19) Using the general nonconvex convergence analysis of BPG realized in Appendix D.1, we can show sufficient decrease of the objective and convergence of the iterates of B-RED. Theorem 1 (Proof in Appendix D.2). Under Assumption 1 and Assumption 2 , the iterates x k given by the B-RED algorithm (18) with the backtracking procedure (19) decrease F λ,γ and converge to a critical point of Ψ = i C + F λ,γ with rate min 0≤k≤K D h (x k+1 , x k ) = O(1/K). Bregman Plug-and-Play (B-PnP) We now consider the equivalent of PnP Proximal Gradient Descent algorithm in the Bregman framework. Given a denoiser B γ with Im(B γ ) ⊂ dom(h) and λ > 0 such that Im(∇h − λ∇f ) ⊆ dom(∇h * ), it writes (B-PnP) x k+1 = B γ • ∇h * (∇h − λ∇f )(x k ).(20) We use again as B γ the Bregman Score Denoiser (10). With ψ γ defined from g γ as in (11) and assuming that ψ γ • ∇h * is convex on int dom(h * ), Proposition 1 states that the Bregman Score denoiser B γ is the Bregman proximal operator of some nonconvex potential ϕ γ verifying (12). The algorithm B-PnP (20) then becomes x k+1 ∈ Prox h ϕγ •∇h * (∇h − λ∇f )(x k ), which writes as a Bregman Proximal Gradient algorithm, with stepsize τ = 1, x k+1 ∈ arg min x∈R n {ϕ γ (x) + ⟨x − x k , λ∇f (x k )⟩ + D h (x, x k )}.(21) With Proposition 1, we have B γ (y) ∈ Prox h ϕγ (y) i.e. a proximal step on ϕ γ with stepsize 1. We are thus forced to keep a fixed stepsize τ = 1 in the BPG algorithm (21) and no backtracking is possible. Using Appendix D.1, we can show that B-PnP converges towards a stationary point of λf + ϕ γ . Theorem 2 (Proof in Appendix D.3). Assume Assumption 1 and ψ γ • ∇h * convex on int dom(h * ). Then for Im(∇h − λ∇f ) ⊆ dom(∇h * ), Im(B γ ) ⊆ dom(h) and λL f < 1 (with L f specified in Assumption 1), the iterates x k given by the B-PnP algorithm (20) decrease λf + ϕ γ and converge to a critical point of λf + ϕ γ with rate min 0≤k≤K D h (x k+1 , x k ) = O(1/K). Remark 2. The condition Im(B γ ) ⊆ dom(h) and the required convexity of ψ γ • ∇h * come from Proposition 1 while the condition Im(∇h − λ∇f ) ⊆ dom(∇h * ) allows the algorithm B-PnP (20) to be well-posed. These assumptions will be discussed with more details in the context of Poisson image restoration in Section 5. Application to Poisson inverse problems We consider ill-posed inverse problems involving the Poisson data-fidelity term f introduced in (2). The Euclidean geometry (i.e. h(x) = 1 2 ||x|| 2 ) does not suit for such f , as it does not have a Lipschitz gradient. In [Bauschke et al., 2017, Lemma 7], it is shown that an adequate Bregman potential h (in the sense that there exists L f such that L f h − f is convex) for (2) is the Burg's entropy h(x) = − n i=1 log(x i ),(22) for which dom(h) = R n ++ and L f h−f is convex on int dom(h) = R n ++ for L f ≥ ||y|| 1 . For further computation, note that the Burg's entropy (22) satisfies ∇h(x) = ∇h * (x) = − 1 x and ∇ 2 h(x) = 1 x 2 . The Bregman score denoiser associated to the Burg's entropy is presented in Section 5.1. The corresponding Bregman RED and PnP algorithms are applied to Poisson Image deblurring in Section 5.2. Bregman Score Denoiser with Burg's entropy We now specify the study of Section 3 to the case of the Burg's entropy (22). In this case, the Bregman noise model (6) writes (see Appendix E.1 for detailed calculus) p(y|x) = exp(ρ(x) + nγ) n i=1 x i y i γ exp −γ x i y i .(23) For γ > 1, this is a product of Inverse Gamma (IG(α, β)) distributions with parameter β i = γx i and α i = γ − 1. This noise model has mean (for γ > 2) γ γ−2 x and variance (for γ > 3) γ 2 (γ−2) 2 (γ−3) x 2 . In particular, for large γ, the noise becomes centered on x with signal-dependent variance x 2 /γ. Furthermore, using Burg's entropy (22), the optimal posterior mean (9) and the Bregman Score Denoiser (10) respectively write, for y ∈ R n ++ , x M M SE (y) = y − 1 γ y 2 ∇(− log p Y )(y)(24) B γ (y) = y − y 2 ∇g γ (y). Bregman Proximal Operator Considering the Burg's entropy (22) in Proposition 1 we get that, if x → ψ γ • ∇h * (x) := ψ γ (− 1 x ) is convex on dom(h * ) = R −− , the Bregman Score Denoiser (25) satisfies B γ (y) = Prox h ϕγ , for the nonconvex potential ψ γ (y) = n i=1 log(y i ) − g γ (y) − 1. The convexity of ψ γ • ∇h * can be verified using the following characterization, which translates to a condition on the deep potential g γ (see Appendix E.2 for details) ∀x ∈ dom(h * ), ∀d ∈ R n , ⟨∇ 2 η γ (x)d, d⟩ ≥ 0 (26) ⇐⇒ ∀y ∈ R n ++ , ∀d ∈ R n , ⟨∇ 2 g γ (y)d, d⟩ ≤ n i=1 1 − 2y∇g γ (y) y 2 i d 2 i .(27) It is difficult to constrain g γ to satisfy the above condition. As discussed in Appendix E.2, we empirically observe on images that the trained deep potential g γ satisfies the above inequality. This suggests that the convexity condition for Proposition 1 is true at least locally, on the image manifold. Denoising in practice As Hurault et al. [2021], we chose to parametrize the deep potential g γ as g γ (y) = 1 2 ||x − N γ (x)|| 2 ,(28) where N σ is the deep convolutional neural network architecture DRUNet [Zhang et al., 2021] that contains Softplus activations and takes the noise level γ as input. ∇g γ is computed with automatic differentiation. We train B γ to denoise images corrupted with random Inverse Gamma noise of level γ, sampled from clean images via p (y|x) = n i=1 IG(α i , β i )(x i ). To sample y i ∼ IG(α i , β i ), we sample z i ∼ G(α i , β i ) and take y i = 1/z i . Denoting as p the distribution of a database of clean images, training is performed with the L 2 loss L(γ) = E x∼p,y∼IG γ (x) ||B γ (y) − x|| 2 .(29) Denoising performance We evaluate the performance of the proposed Bregman Score DRUNet (B-DRUNet) denoiser (28). It is trained with the loss (29), with 1/γ uniformly sampled in (0, 0.1). More details on the architecture and the training can be found in Appendix E.2. We compare the performance of B-DRUNet (28) with the same network DRUNet directly trained to denoise inverse Gamma noise with L 2 loss. Qualitative and quantitative results presented in Figure 1 and Table 1 show that the Bregman Score Denoiser (B-DRUNet), although constrained to be written as (25) with a conservative vector field ∇g γ , performs on par with the unconstrained denoiser (DRUNet). Bregman Plug-and-Play for Poisson Image Deblurring We now derive the explicit B-RED and B-PnP algorithms in the context of Poisson image restoration. Choosing C = [0, R] n for some R > 0, the B-RED and B-PnP algorithms (18) and (20) write (B-RED) x k+1 i = arg min{x∇F λ,γ (x k ) i + 1 τ x x k i − log x x k i : x ∈ [0, R]} 1 ≤ i ≤ n (30) = x k i 1+τ x k i ∇F λ,γ (x k )i if 0 ≤ x k i 1+τ x k i ∇F (x k )i ≤ R R else 1 ≤ i ≤ n (31) (B-PnP) x k+1 = B γ x k 1 + τ x k ∇F (x k ) .(32) Verification of the assumptions of Theorems 1 and 2 for the convergence of both algorithms is discussed in Appendix E.3. Poisson deblurring Equipped with the Bregman Score Denoiser, we now investigate the practical performance of the plug-and-play B-RED and B-PnP algorithms for image deblurring with Poisson noise. In this context, the degradation operator A is a convolution with a blur kernel. We verify the efficiency of both algorithms over a variety of blur kernels (real-world camera shake, uniform and Gaussian). The hyper-parameters γ, λ are optimized for each algorithm and for each noise level α by grid search and are given in Appendix E.4. In practice, we first initialize the algorithm with 100 steps with large τ and γ so as to quickly initialize the algorithm closer to the right stationary point. We show in Figures 2 and 3 that both B-PnP and B-RED algorithms provide good visual reconstruction. Moreover, we observe that, in practice, both algorithms satisfy the sufficient decrease property of the objective function as well as the convergence of the iterates. Additional quantitative performance and comparisons with other Poisson deblurring methods are given in Appendix E.3. (a) Clean (b) Observed (14.91dB) (c) B-RED (23.01dB) (d) B-PnP (22.96dB) (e) (λf + gγ)(x k ) B-RED (f) (λf + ϕγ)(x k ) B-PnP (g) ||xi+1 − xi|| 2 Conclusion In this paper, we derive a complete extension of the plug-and-play framework in the general Bregman paradigm for non-smooth image inverse problems. Given a convex potential h adapted to the geometry of the problem, we propose a new deep denoiser, parametrized by h, which provably writes as the Bregman proximal operator of a nonconvex potential. We argue that this denoiser should be trained on a particular noise model, called Bregman noise, that also depends on h. By plugging this denoiser in the BPG algorithm, we propose two new plug-and-play algorithms, called B-PnP and B-RED, and show that both algorithms converge to stationary points of explicit nonconvex functionals. We apply this framework to Poisson image inverse problem. Experiments on image deblurring illustrate numerically the convergence and the efficiency of the approach. (a) Clean (ii) of Legendre type if h is essentially smooth and strictly convex on int dom(h). (b) Observed (16.21dB) (c) B-RED (24.50dB) (d) B-PnP (24.51dB) (e) (λf + gγ)(x k ) B-RED (f) (λf + ϕγ)(x k ) B-PnP (g) ||xi+1 − xi|| 2 We also recall the following properties of Legendre functions that are used without justification in our analysis (see [Rockafellar, 1997, Section 26] for more details): • h is of Legendre type if and only if its convex conjugate h * is of Legendre type. • For h of Legendre type, dom(∇h) = int dom(h), ∇h is a bijection from int dom(h) to int dom(h * ) and (∇h) −1 = ∇h * . Examples We are interested in the two following Legendre functions • L 2 potential : h(x) = 1 2 ||x|| 2 , C = dom(h) = dom(h * ) = R n , D h (x, y) = 1 2 ||x − y|| 2 • Burg's entropy : h(x) = − n i=1 log(x i ), C = dom(h) = R n ++ , dom(h * ) = R n −− and D h (x, y) = n i=1 xi yi − log xi yi − 1. A.2 Nonconvex subdifferential Following Attouch et al. [2013], we use as notion of subdifferential of a proper, nonconvex function Φ the limiting subdifferential ∂Φ(x) = ω ∈ R n , ∃x k → x, f (x k ) → f (x), ω k → ω, ω k ∈∂Φ(x k )(33) with∂Φ the Fréchet subdifferential of Φ. We have • A subset S of R n is a real semi-algebraic set if there exists a finite number of real polynomial functions P i,j , Q i,j : R n → R such that {ω ∈ R n , ∃x k → x, f (x k ) → f (x), ω k → ω, ω k ∈ ∂Φ(x k )} ⊆ ∂Φ(x).(34)−→ R + such that ψ(0) = 0, ψ is C 1 on (0, η), ψ ′ > 0 on (0, η) and ∀x ∈ U ∩ [f (x * ) < f < f (x * ) + η], the Kurdyka-Lojasiewicz inequality holds: ψ ′ (f (x) − f (x * ))dist(0, ∂f (x)) ≥ 1.(35)S = ∪ p j=1 ∩ q i=1 {x ∈ R n , P i,j = 0, Q i,j < 0}(36) • A function f : R n → R m is called semi-algebraic if its graph {(x, y) ∈ R n × R m , y = f (x)} is a semi-algebraic subset of R n × R m . We refer to [Attouch et al., 2013, Coste, 2000 for more details on semi-algebraic functions and to [Bolte et al., 2007] for the proof that semi-algebraic functions are KL. B More details on the Bregman Denoising Prior We detail here the calculations realized in Section 3. We first verify that the Bregman noise conditional probability belongs to the regular exponential family of distributions. Indeed for T (y) = γ∇h(y), ψ(x) = γh(x) − ρ(x) and p 0 (y) = exp (γh(y) − γ⟨∇h(y), y⟩) p(y|x) = p 0 (y) exp (⟨x, T (y)⟩ − ψ(x)) = exp (γh(y) − γ⟨∇h(y), y⟩) exp (γ⟨x, ∇h(y)⟩ − γh(x) + ρ(x)) = exp (−γ(h(x) − h(y) − ⟨∇h(y), x − y⟩) + ρ(x)) = exp (−γD h (x, y) + ρ(x))(37) which corresponds to the Bregman noise conditional probability introduced in Equation (6). The Tweedie formula of the posterior mean is then [Kim and Ye, 2021] ∇T (y).x M M SE (y) = −∇ log p 0 (y) + ∇ log p Y (y). Using ∇ log p 0 (y) = γ∇h(y) − γ∇h(y) − γ∇ 2 h(y).y = −γ∇ 2 h(y).y, we get γ∇ 2 h(y).x M M SE (y) = γ∇ 2 h(y).y + ∇ log p Y (y).(40) As h is strictly convex, ∇ 2 h(y) is invertible and x M M SE (y) = y + 1 γ (∇ 2 h(y)) −1 .∇ log p Y (y).(41) C Proof of Proposition 1 A new characterization of Bregman proximal operators We first derive in Proposition 2 an extension of [Gribonval and Nikolova, 2020, Corollary 5 a)] when h is strictly convex and thus ∇h is invertible. We will then see that Proposition 1 is a direct application of this result. Proposition 2. Let h of Legendre type on R n . Let ζ : int dom(h) → R n . Suppose Im(ζ) ⊂ dom(h). The following properties are equivalent: (i) There is ϕ : R n → R ∪ {+∞} such that Im(ζ) ⊂ dom(ϕ) and for each y ∈ int dom(h) ζ(y) ∈ arg min x∈R n {D h (x, y) + ϕ(x)}.(42) (ii) There is a l.s.c ψ : R n → R ∪ {+∞} proper convex on int dom(h * ) such that ζ(∇h * (z)) ∈ ∂ψ(z) for each z ∈ int dom(h * ). When (i) holds, ψ can be chosen given ϕ with ψ(z) := ⟨ζ(∇h * (z)), z⟩ − h(ζ(∇h * (z))) − ϕ(ζ(∇h * (z))) if z ∈ int dom(h * ) +∞ otherwise.(43) When (ii) holds, ϕ can be chosen given ψ with ϕ(x) := ⟨ζ(y), ∇h(y)⟩ − h(ζ(y)) − ψ(∇h(y)) for y ∈ ζ −1 (x) if x ∈ Im(ζ) +∞ otherwise. Remark 3. [Gribonval and Nikolova, 2020, Corollary 5 a)] with Y = int dom(h) states that (i) is equivalent to (iii) There is a l.s.c g : R n → R ∪ {+∞} convex such that ∇h(ζ −1 (x)) ∈ ∂g(x) for each x ∈ Im(ζ). However (ii) and (iii) are not equivalent. We show here that (ii) implies (iii) but the converse is not true. Let ψ convex defined from (ii). Thanks to Legendre-Fenchel identity, we have that ∀z ∈ int dom(h * ), ζ(∇h * (z)) ∈ ∂ψ(z) (45) ⇔ ∀z ∈ int dom(h * ), z ∈ ∂ψ * (ζ(∇h * (z))) (46) ⇔ ∀y ∈ int dom(h), ∇h(y) ∈ ∂ψ * (ζ(y)) (47) ⇒ ∀x ∈ Im(ζ), ∇h(ζ −1 (x)) ∈ ∂ψ * (x).(48) Therefore (ii) implies (iii) with g = ψ * . However, the last lign is just an implication. Proof. We follow the same order of arguments from [Gribonval and Nikolova, 2020]. We first prove a general result reminiscent to [Gribonval and Nikolova, 2020, Theorem 3] for a general form of divergence function and then apply this result to Bregman divergences. Lemma 1. Let a : Y ⊆ R n → R ∪ {+∞}, b : R n → R ∪ {+∞}, A : Y → Z bijection from Y to Z (with Z ⊂ R n ). Consider ζ : Y → R n . Let D(x, y) := a(y) − ⟨x, A(y)⟩ + b(x). Suppose Im(ζ) ⊂ dom(b). The following properties are equivalent: (i) There is ϕ : R n → R ∪ {+∞} such that Im(ζ) ⊂ dom(ϕ) and for each y ∈ Y ζ(y) ∈ arg min x∈R n {D(x, y) + ϕ(x)}.(49) (ii) There is a l.s.c ψ : R n → R ∪ {+∞} proper convex such that ζ(A −1 (z)) ∈ ∂ψ(z) for each z ∈ Z. When (i) holds, ψ can be chosen given ϕ with ψ(z) := ⟨ζ(A −1 (z)), z⟩ − b(ζ(A −1 (z))) − ϕ(ζ(A −1 (z))) if z ∈ Z +∞ otherwise.(50) When (ii) holds, ϕ can be chosen given ψ with ϕ(x) := ⟨ζ(y), A(y)⟩ − b(ζ(y)) − ψ(A(y)) for y ∈ ζ −1 (x) if x ∈ Im(ζ) +∞ otherwise.(51) Before proving the Lemma, we can directly see that Proposition 2 is the specialization of Lemma 1 with Bregman divergences. Given h of Legendre-type, the divergence D(x, y) = a(y) − ⟨x, A(y)⟩ + b(x) becomes the Bregman divergence D h (x, y) defined in (5) for Y = int dom(h), Z = int dom(h * ), A(y) := ∇h(y), a(y) := ⟨∇h(y), y⟩ − h(y) and b(x) := h(x) if x ∈ dom(h) and + ∞ otherwise. For h of Legendre-type, ∇h is a bijection from int dom(h) to int dom(h * ) and (∇h) −1 = ∇h * . Proof. We now prove Lemma 1. We follow the same arguments as the proof of [Gribonval and Nikolova, 2020, Theorem 3(c)]. (i) ⇒ (ii) : Define ρ(z) = ⟨ζ(A −1 (z)), z⟩ − b(ζ(A −1 (z))) − ϕ(ζ(A −1 (z))) if z ∈ Z +∞ else.(52) As we assume Im(ζ) ⊂ dom(b) and Im(ζ) ⊂ dom(ψ), we have ρ : R n → R ∪ {+∞} and dom(ρ) = Z. Let z ∈ Z and y = A −1 (z). From (i), ζ(y) is the global minimizer of x → D(x, y) + ϕ(x) or of x → −⟨x, A(y)⟩ + b(x) + ϕ(x) and ∀z ′ ∈ Z, y ′ = A −1 (z ′ ), ρ(z ′ ) − ρ(z) = ⟨ζ(A −1 (z ′ )), z ′ ⟩ − b(ζ(A −1 (z ′ ))) − ϕ(ζ(A −1 (z ′ ))) − ⟨ζ(A −1 (z)), z⟩ + b(ζ(A −1 (z))) + ϕ(ζ(A −1 (z))) = ⟨ζ(y ′ ), A(y ′ )⟩ − b(ζ(y ′ )) − ϕ(ζ(y ′ )) − ⟨ζ(y), A(y)⟩ + b(ζ(y)) + ϕ(ζ(y)) = ⟨ζ(y), A(y ′ ) − A(y)⟩ + ⟨ζ(y ′ ), A(y ′ )⟩ − b(ζ(y ′ )) − ϕ(ζ(y ′ )) − ⟨ζ(y), A(y ′ )⟩ + b(ζ(y)) + ϕ(ζ(y)) ≥ ⟨ζ(y), A(y ′ ) − A(y)⟩ = ⟨ζ(A −1 (z)), z ′ − z⟩.(53) By definition of the subdifferential, this shows that ζ(A −1 (z)) ∈ ∂ρ(z).(54) Letρ the lower convex envelope of ρ (pointwise supremum of all the convex l.s.c functions below ρ). ρ is proper convex l.s.c. and ∀z ∈ Z, ∂ρ(z) ̸ = ∅. By [Gribonval and Nikolova, 2020, Proposition 3], ∀z ∈ Z, ρ(z) =ρ(z) and ∂ρ(z) = ∂ρ(z). Thus for ψ =ρ, we get (ii). (ii) ⇒ (i) : Define η : Y → R by η(y) := ⟨ζ(y), A(y)⟩ − ψ(A(y)). (55) The previous definition is valid because by (ii), Im(A) = Z ⊂ dom(∂ψ) ⊂ dom(ψ) and therefore ψ(A(y)) < +∞. By (ii), ∀z, z ′ ∈ Z, ψ(z) − ψ(z ′ ) ≥ ⟨ζ(A −1 (z ′ )), z − z ′ ⟩ (56) which gives ∀y, y ′ ∈ Y, ψ(A(y)) − ψ(A(y ′ )) ≥ ⟨ζ(y ′ ), A(y) − A(y ′ )⟩.(57) This yields η(y ′ ) − η(y) = ⟨ζ(y ′ ), A(y ′ )⟩ − ψ(A(y ′ )) − ⟨ζ(y), A(y)⟩ + ψ(A(y)) ≥ ⟨ζ(y ′ ) − ζ(y), A(y)⟩.(58) We define θ : R n → R ∪ {+∞} obeying dom(θ) = Im(ζ) with θ(x) := η(y) for y ∈ ζ −1 (x) if x ∈ Im(ζ) +∞ otherwise.(59) For y, y ′ ∈ ζ −1 (x), as ζ(y ′ ) = ζ(y), we have by (58) η(y ′ ) − η(y) ≥ 0 and η(y) − η(y ′ ) ≥ 0 and thus η(y ′ ) = η(y). The definition of θ is thus independent of the choice of y ∈ ζ −1 (x). For x ′ ∈ Im(ζ), x ′ = ζ(y ′ ). Using the previous inequality with η, we get θ(x ′ ) − θ(ζ(y)) = θ(ζ(y ′ )) − θ(ζ(y)) = η(y ′ ) − η(y) ≥ ⟨ζ(y ′ ) − ζ(y), A(y)⟩ = ⟨x ′ − ζ(y), A(y)⟩,(60) that is to say, ∀x ′ ∈ Im(ζ) θ(x ′ ) − ⟨x ′ , A(y)⟩ ≥ θ(ζ(y)) − ⟨ζ(y), A(y)⟩.(61) Given the definition of θ, this is also true for x ′ / ∈ Im(ζ). We set ϕ = θ − b. As Im(ζ) ⊂ dom(b), b(ζ(y)) < +∞ and ϕ : R n → R ∪ {+∞}. Adding a(y) on both sides, we get ∀x ′ ∈ R n , b(x ′ ) + ϕ(x ′ ) − ⟨x ′ , A(y)⟩ + a(y) ≥ b(ζ(y)) + ϕ(ζ(y)) − ⟨ζ(y), A(y)⟩ + a(y). ⇔ ∀x ′ ∈ R n , ϕ(x ′ ) + D(x ′ , y) ≥ ϕ(ζ(y)) + D(ζ(y), y) ⇔ ζ(y) ∈ arg min x ϕ(x) + D(x, y) As this is true for all y ∈ Y, we get the desired result. Eventually, Proposition 1 is a direct application of Proposition 2 with ζ = B γ : R n → R n defined on int dom(h) by B γ (y) = ∇(ψ γ • ∇h * ) • ∇h(y). (63) The function B γ verifies ∀z ∈ int dom(h * ), B γ (∇h * (z)) = ∇(ψ γ • ∇h * )(z)(64) and ψ = ψ γ • ∇h * is assumed convex on int dom(h * ). From Proposition 2, we get that there is ϕ γ : R n → R ∪ {+∞} such that for each y ∈ int dom(h) B γ (y) ∈ arg min{D h (x, y) + ϕ γ (x)}.(65) Moreover, for y ∈ int dom(h), B γ (y) = ∇(ψ γ • ∇h * ) • ∇h(y) (66) = ∇ 2 h * (∇h(y)).∇ψ γ • ∇h * • ∇h(y)(67) = ∇ 2 h * (∇h(y)).∇ψ γ (y). As h is assumed strictly convex on int dom(h), for y ∈ int dom(h), the Hessian of h, denoted as ∇ 2 h(y) is invertible. By differentiating ∇h * (∇h(y)) = y (69) we get ∇ 2 h * (∇h(y)) = (∇ 2 h(y)) −1 , (70) so that B γ (y) = (∇ 2 h(y)) −1 .∇ψ γ (y). (71) With the definition (11), we directly get B γ (y) = (∇ 2 h(y)) −1 .∇ψ γ (y) = y − (∇ 2 h(y)) −1 .∇g γ (y).(72) Finally, Proposition 2 also indicates that ϕ γ can be chosen given ψ γ with ϕ γ (x) := ⟨B γ (y), ∇h(y)⟩ − h(B γ (y)) − ψ γ • ∇h * (∇h(y)) for y ∈ B −1 γ (x) if x ∈ Im(B γ ) +∞ otherwise. (73) This gives ∀y ∈ int dom(h), ϕ γ (B γ (y)) = ⟨B γ (y), ∇h(y)⟩ − h(B γ (y)) − ψ γ • ∇h * (∇h(y)) = ⟨B γ (y) − y, ∇h(y)⟩ − h(B γ (y)) + h(y) + ⟨y, ∇h(y)⟩ − h(y) − ψ γ (y) = −D h (B γ (y), y) + ⟨y, ∇h(y)⟩ − h(y) − ψ γ (y) = −D h (B γ (y), y) + g γ (y). (74) D The Bregman Proximal Gradient (BPG) algorithm D.1 Convergence analysis of the nonconvex BPG algorithm We study in this section the convergence of the BPG algorithm x k+1 ∈ T τ (x k ) = arg min x∈R n {R(x) + ⟨x − x k , ∇F (x k )⟩ + 1 τ D h (x, x k )}.(75) for minimizing Ψ = F + R with nonconvex functions F and/or R. For the rest of the section, we take the following general assumptions. Assumption 3. (i) h : R n → R is of Legendre-type. (ii) F : R n → R is proper, C 1 on int dom(h), with dom(h) ⊂ dom(F ). (iii) R : R n → R is proper, lower semi-continuous with dom R ∩ int dom(h) ̸ = ∅. (iv) Ψ = F + R is lower-bounded, coercive and verifies the Kurdyka-Lojasiewicz (KL) property (defined in Appendix A.3). (v) For x ∈ int dom(h), T τ (x) is nonempty and included in int dom(h). Note that, since R is nonconvex, the mapping T τ is not in general single-valued. Assumption (v) is required for the algorithm to be well-posed. As shown in [Bauschke et al., 2017, Bolte et al., 2018, one sufficient condition for T τ (x) ̸ = ∅ is the supercoercivity of function h + λR for all λ > 0, that is lim ||x||→+∞ h(x)+λR(x) ||x|| = +∞. As T τ (x) ⊂ dom(h), T τ (x) ⊂ int dom(h) is true when dom(h) is open (which is the case for Burg's entropy for example). The convergence of the BPG algorithm in the nonconvex setting is studied by the authors of [Bolte et al., 2018]. Under the main assumption that Lh − F is convex on int dom(h), they show first the sufficient decrease property (and thus convergence) of the function values, and second, global convergence of the iterates. However, as we will develop in Appendix D.3, Lh − F is not convex on the full domain of int dom(h) but only on the compact subset dom(R). One can verify that all the iterates (75) belong to the convex set T τ (x) ⊂ Conv(dom R) ∩ int dom(h),(76) where Conv(E) stands for the convex envelope of E. We argue that it is enough to assume Lh − F convex on this convex subset with the following assumption. Assumption 4. There is L > 0 such that, Lh − F is convex on Conv(dom R) ∩ int dom(h). We can now prove a result similar than [Bolte et al., 2018, Proposition 4.1]. Proposition 3. Under Assumptions 3 and 4, let (x k ) k∈N be a sequence generated by (18) with 0 < τ L < 1. Then the following properties hold (i) (Ψ(x k )) k∈N is non-increasing and converges. (ii) k D h (x k+1 , x k ) < ∞ and min 0≤k≤K D h (x k+1 , x k ) = O(1/K). Proof. We adapt here the proof from [Bolte et al., 2018] to the case where Lh − F is not globally convex but only convex on the convex subset Conv(dom R) ∩ int dom(h). Sufficient decrease property We first show that the sufficient decrease property of Ψ(x k ) holds. This is true because the following characterisation of C 1 convex functions holds on a convex subset. We recall this classical proof for the sake of completeness. Lemma 2. Let f : X → R n be of class C 1 , then f is locally convex on C a convex subset of X = dom(f ) if and only if ∀x, y ∈ C, D f (x, y) = f (x) − f (y) − ⟨∇f (y), x − y⟩ ≥ 0. Proof. ⇒ Let x, y ∈ C, for all t ∈ (0, 1), x + t(y − x) ∈ C and by convexity of f f (y + t(x − y)) ≤ f (y) + t(f (x) − f (y)),(77)i.e. f (y + t(x − y)) − f (y) t ≤ f (x) − f (y)(78) and ⟨∇f (y), x − y⟩ = lim t→0 + f (y + t(x − y)) − f (y) t ≤ f (x) − f (y).(79) ⇐ Let x, y ∈ C, t ∈ (0, 1) and z = y + t(x − y) ∈ C. We have f (y) ≥ f (z) + ⟨∇f (z), y − z⟩ (80) f (x) ≥ f (z) + ⟨∇f (z), x − z⟩. (81) Combining both equations gives tf (x) + (1 − t)f (y) ≤ t(f (z) + ⟨∇f (z), x − z) + (1 − t)(f (z) + ⟨∇f (z), y − z⟩) = f (z) + ⟨∇f (z), tx + (1 − t)y − z⟩ = f (tx + (1 − t)y).(82) Therefore, we have Lh − F convex on C, if and only if, ∀x, y ∈ C, D Lh−F (x, y) ≥ 0, i.e. D F (x, y) ≤ LD h (x, y). The rest of the proof is identical to the one of [Bolte et al., 2018] and we recall it here. Given the optimality conditions in (75), all the iterates x k ∈ C satisfy R(x k+1 ) + ⟨x k+1 − x k , ∇F (x k )⟩ + 1 τ D h (x k+1 , x k ) ≤ R(x k )(83)using D F (x, y) ≤ LD h (x, y), we get (R(x k+1 ) + F (x k+1 )) − (R(x k ) + F (x k )) ≤ − 1 τ D h (x k+1 , x k ) + LD h (x k+1 , x k ) = (L − 1 τ )D h (x k+1 , x k ) ≤ 0(84) which, together with the fact that Ψ is lower bounded, proves (i). Summing the previous inequality from k = 0 to K − 1 gives 0 ≤ K−1 k=0 D h (x k+1 , x k ) ≤ τ 1 − τ L (Ψ(x 0 )−Ψ(x K )) ≤ τ 1 − τ L (Ψ(x 0 )− inf x∈C Ψ(x)) < +∞. (85) Thus (D h (x k+1 , x k )) k is summable and converges to 0 when k → +∞. Finally min 0≤k≤K D h (x k+1 , x k ) ≤ 1 K + 1 K k=0 D h (x k+1 , x k ) ≤ 1 K + 1 τ 1 − τ L (Ψ(x 0 ) − inf x∈C Ψ(x)). (86) To prove global convergence of the iterates upon the Kurdyka-Lojasiewicz (KL) property, [Bolte et al., 2018, Theorem 4.1] is based on the hypotheses (a) dom(h) = R n and h is strongly convex on R n and (b) ∇h and ∇F are Lipschitz continuous on any bounded subset of R n . These assumptions are clearly not verified for h being the Burg's entropy (22) or F the Poisson data-fidelity term (2). Indeed, in that case, dom(h) = R n ++ , and h is strongly convex only on bounded sets. Moreover F and h are not Lipschitz continuous near 0. However, thanks to the proven decrease of the iterates and as Ψ is assumed coercive, the iterates remain bounded. We can adopt the following weaker assumptions to ensure that the iterates do not tend to +∞ or 0. Assumption 5. (i) h is strongly convex on any bounded convex subset of its domain. (ii) For all α > 0, ∇h and ∇F are Lipschitz continuous on {Ψ(x) ≤ α}. Under these assumptions, we prove the equivalent of [Bolte et al., 2018, Theorem 4.1]. Theorem 3. Under Assumption 3, 4 and 5, the sequence (x k ) k∈N generated by (18) with 0 < τ L < 1 converges to a critical point of Ψ. We follow the proof given in [Bolte et al., 2018] with some updates to adapt to our weaker Assumption 5. In particular, we show in Lemma 3 that the sequence (x k ) k≥1 is still "gradient-like" i.e. it verifies the assumptions H1, H2 and H3 from [Attouch et al., 2013]. Once these conditions are verified, the result directly follows from [Bolte et al., 2018, Theorem 6.2]. Let us first note that by coercivity of Ψ and decrease of the iterates Ψ(x k ) (see Proposition 3), the iterates remain in the set ∀k ≥ 1, x k ∈ C(x 0 ) = {x ∈ dom(h), Ψ(x) < Ψ(x 0 )}.(87) Lemma 3. The sequence (x k ) k≥1 satisfies the following three conditions: H1 (Sufficient decrease condition) Ψ(x k ) − Ψ(x k+1 ) ≥ a||x k+1 − x k || 2 ,(88) H2 (Relative error condition) ∀k ≥ 1, there exists ω k+1 ∈ ∂Ψ(x k+1 ) such that ||ω k+1 || ≤ b||x k+1 − x k ||,(89) H3 (Continuity condition) Any subsequence (x ki ) converging towards x * verifies Ψ(x ki ) → Ψ(x * ).(90) Proof. H1 : Sufficient decrease condition. From (84), we get Ψ(x k ) − Ψ(x k+1 ) ≥ 1 τ − L D h (x k+1 , x k ).(91) Besides, h is assumed to be strongly convex on any bounded convex subset of its domain. Furthermore, notice that Conv(C(x 0 )) ∩ dom(h) is a convex subset of dom(h) as intersection of convex sets. Therefore, there is σ h > 0 such that ∀x, y ∈ Conv(C(x 0 )) ∩ dom(h), D h (x, y) ≥ σ h ||x − y|| 2 . (92) With the convention D h (x, y) = +∞ if x / ∈ dom(h) or y / ∈ int dom(h), ∀x, y ∈ Conv(C(x 0 )), D h (x, y) ≥ σ h ||x − y|| 2 . (93) As ∀k ≥ 1, x k ∈ C(x 0 ), we get Ψ(x k ) − Ψ(x k+1 ) ≥ σ h 1 τ − L ||x k+1 − x k || 2 ,(94) which proves (H1). H2 : Relative error condition. Given (75), the optimality condition for the update of x k+1 is 0 ∈ ∂R(x k+1 ) + ∇F (x k ) + 1 τ (∇h(x k+1 ) − ∇h(x k )).(95) For ω k+1 = ∇F (x k+1 ) − ∇F (x k ) + 1 τ (∇h(x k ) − ∇h(x k+1 ))(96) we have ω k+1 ∈ ∂Ψ(x k+1 ) = ∂R(x k+1 ) + ∇F (x k+1 ) (97) and ||ω k+1 || ≤ ||∇F (x k+1 ) − ∇F (x k )|| + 1 τ ||∇h(x k ) − ∇h(x k+1 )||.(98) By assumption, ∇F and ∇h are Lipschitz continuous on C(x 0 ) = {Ψ(x) < Ψ(x 0 )}. As seen before, ∀k ≥ 1, x k ∈ C(x 0 ). Thus, there is b > 0 such that ||ω k+1 || ≤ ||∇F (x k+1 ) − ∇F (x k )|| + 1 τ ||∇h(x k ) − ∇h(x k+1 )|| ≤ b||x k+1 − x k ||.(99) H3 : Continuity condition. Let a subsequence (x ki ) converging towards x * . Using the optimality in the update of x k we have R(x k ) + ⟨x k − x k−1 , ∇F (x k−1 )⟩ + 1 τ D h (x k , x k−1 ) (100) ≤ R(x * ) + ⟨x * − x k−1 , ∇F (x k−1 )⟩ + 1 τ D h (x * , x k−1 ) (101) ⇔ R(x k ) ≤ R(x * ) + ⟨x * − x k−1 , ∇F (x k−1 )⟩ + 1 τ D h (x * , x k−1 ) − 1 τ D h (x k , x k−1 ). (102) From (94) and the fact that (Ψ(x k )) k converges, we have that ||x k − x k−1 || → 0. Thus (x ki−1 ) i also converges towards x * . In addition, since h is continuously differentiable, D h (x * , x k−1 ) = h(x * ) − h(x k−1 ) − ⟨∇h(x k−1 ), x * − x k−1 ⟩ → 0. Passing to the limit in (102), we get lim sup i→+∞ R(x ki ) ≤ R(x * ).(103) By lower semicontinuity of R and continuity of F , we get the desired result: R(x ki ) + F (x ki ) → R(x * ) + F (x * ).(104) Backtracking The convergence actually requires to control the NoLip constant. In order to avoid small stepsizes, we adapt the backtracking strategy of [Beck, 2017, Chapter 10] to the BPG algorithm. Given γ ∈ (0, 1), η ∈ [0, 1) and an initial stepsize τ 0 > 0, the following backtracking update rule on τ is applied at each iteration k: while Ψ(x k ) − Ψ(T τ (x k )) < γ τ D h (T τ (x k ), x k ), τ ←− ητ.(105) Proposition 4. At each iteration of the algorithm, the backtracking procedure (105) is finite and with backtracking, the convergence results of Proposition 3 and Theorem 3 still hold. Proof. For a given stepsize τ , we showed in equation (84) that Φ(x k ) − Φ(T τ (x k )) ≥ 1 τ − L D h (T τ (x k ), x k ).(106) Taking τ < 1−γ L , we get 1 τ − L > γ τ so that Φ(x k ) − Φ(T τ (x k )) > γ τ D h (T τ (x k ), x k ).(107) Hence, when τ < 1−γ L , the sufficient decrease condition is satisfied and the backtracking procedure (τ ←− ητ ) must end. Replacing the former sufficient decrease condition (106) with (107), the rest of the proofs from Proposition 3 and Theorem 3 are identical. D.2 Proof of Theorem 1 We recall the B-RED algorithm (B-RED) x k+1 ∈ T τ (x k ) = arg min x∈R n {i C (x) + ⟨x − x k , ∇F λ,γ (x k )⟩ + 1 τ D h (x, x k )}. (108) It corresponds to the BPG algorithm (14) with F = F λ,γ = λf + g γ and R = i C . Theorem 1 is a direct application of Proposition 4 that is to say of the convergence results of Proposition 3 and Theorem 3 with backtracking. Given Assumptions 1 and 2, we verify that Assumptions 3, 4 and 5 are verified: Assumption 3. R = i C verifies dom(R) ∩ int dom(h) = C ∩ int dom(h) ̸ = ∅. Moreover R is semi-algebraic as the indicator function of a closed semi-algebraic set. g γ and f being assumed semi-algebraic, Ψ = F λ,γ + i C is semi-algebraic, and thus KL. Ψ is also lower-bounded and coercive as F is lower-bounded and coercive and g γ is lower-bounded. Finally, for x ∈ int dom(h), T τ (x) is non-empty as h + λi C is supercoercive. Assumption 4. By summing convex functions, using Assumption 1(iii) and Assumption 2(ii), L = λL f + L γ verifies Lh − (λf + g γ ) convex on Conv(dom R) ∩ int dom(h) = C ∩ int dom(h). Assumption 5. As g γ is assumed to have globally Lipschitz continuous gradient, this follows directly from Assumption 1(iv). D.3 Proof of Theorem 2 (B-PnP) x k+1 = B γ • ∇h * (∇h − λ∇f )(x k ).(109) It corresponds to the BPG algorithm (14) with F = λf and R = ϕ γ . Theorem 2 is a direct application of the convergence results of Proposition 3 and Theorem 3. We now denote Ψ = λf + ψ γ . Given Assumptions 1, we verify that Assumptions 3, 4 and 5 are verified. Assumption 3. For R = ϕ γ , we have Im(B γ ) ⊂ dom(ϕ γ ) and as for y ∈ int dom(h), Im(B γ ) ⊂ int dom(h) (by equation 13), we get dom(R) ∩ int dom(h) ̸ = ∅. We assumed in Assumption 1 that ϕ γ is semi-algebraic and thus KL. Ψ is coercive because f is coercive and ϕ γ is lower-bounded. Finally, here the well-posedness of T τ (x) is ensured via Proposition 1. Assumption 4. This is the L-smad property of f given by Assumption 1. Assumption 5. This is directly given by Assumption 1(iv). E Application to Poisson Inverse Problems E.1 Burg's entropy Bregman noise model For h being the Burg's entropy (22), the Bregman noise model (6) writes for x, y ∈ R n ++ as p(y|x) = exp (−γD h (x, y) + ρ(x)) = exp(ρ(x)) exp − γ(h(x) − h(y) − ⟨∇h(y), x − y⟩) = exp(ρ(x)) exp −γ n i=1 − log(x i ) + log(y i ) − 1 + x i y i = exp(ρ(x) + nγ) n i=1 x i y i γ exp −γ x i y i .(110) E.2 Inverse Gamma Bregman denoiser Derivation of (26) We first derive here the condition for convexity of η γ := ψ γ • ∇h * introduced in Section 5.1. We have ∇η γ (x) = ∇ 2 h * (x).∇ψ γ (∇h * (x)) (111) and ∇ 2 η γ (x) = (∇ 2 h * (x)) 2 .∇ 2 ψ γ (∇h * (x)) + ∇ 3 h * (x).∇ψ γ (∇h * (x)) (112) which gives ∇ 2 η γ (∇h(y)) = (∇ 2 h * (∇h(y))) 2 .∇ 2 ψ γ (y) + ∇ 3 h * (∇h(y)).∇ψ γ (y). For Burg's entropy and y = ∇h * (x) = − 1 x with x < 0, this writes ∇ 2 η γ (x) = y 4 ∇ 2 ψ γ (y) + 2y 3 Diag(∇ψ γ (y)). Using ψ γ (y) = −h(y) − g γ (y) − 1,(115) ∇ψ γ (y) = −∇h(y) − ∇g γ (y) = 1 y − ∇g γ (y) and ∇ 2 ψ γ (y) = −∇ 2 h(y) − ∇ 2 g γ (y) = − 1 y 2 − ∇ 2 g γ (y),(117) we get ∇ 2 η γ (x) = y 2 y 2 ∇ 2 ψ γ (y) + 2yDiag(∇ψ γ (y)) = y 2 −1 − y 2 ∇ 2 g γ (y) + 2 − 2y∇g γ (y) = y 2 1 − y 2 ∇ 2 g γ (y) − 2y∇g γ (y) 1 y 4 ∇ 2 η γ (x) = Diag 1 − 2y∇g γ (y) y 2 − ∇ 2 g γ (y).(118) For η γ to be convex, a necessary condition is that ∀y ∈ R n ++ and d ∈ R n ⟨∇ 2 η γ (y)d, d⟩ ≥ 0 i.e. ⟨∇ 2 g γ (y)d, d⟩ ≤ n i=1 1 − 2y∇g γ (y) y 2 i d 2 i .(120) Training details For N γ , we use the DRUNet architecture from [Zhang et al., 2021] with 2 residual blocks at each scale and softplus activation functions. We condition the network N γ on γ similarly to what is done for DRUNet. We stack to 3 color channels of the input image an additional channel containing an image with constant pixel value equal to 1/γ. We use the same training dataset as [Zhang et al., 2021]. Training is performed with ADAM during 1200 epochs. The learning rate is initialized with learning rate 10 −4 and is divided by 2 at epochs 300, 600 and 900. E.3 Convergence of B-RED and B-PnP algorithm for Poisson inverse problems In this section we verify that the Burg's entropy h in (22) and the Poisson data likelihood f defined in (2) verify the assumptions required for convergence of B-RED and B-PnP. We remind the expression of f and h. h(x) = − n i=1 log(x i ),(121)f (x) = m i=1 y i log y i α(Ax) i + α(Ax) i − y i ,(122) for A ∈ R m×n . Note that as done in Bauschke et al. [2017], denoting (a i ) 1≤i≤n the columns of A, we assume that a i ̸ = 0 m and ∀1 ≤ j ≤ m, n i=1 a i,j > 0 such that Ax ∈ R m ++ if x ∈ R n ++ . This is verified for A representing the blur with circular boundary conditions with the (normalized) kernels used in Section 5.2. We first check Assumptions 1. Verifying (i) and (ii) is straightforward. We now discuss the three other assumptions. L f h − f is convex on R n ++ , for L f ≥ ||y|| 1 . y stands for the Poisson degraded observation appearing in the definition of f . (iv) First, h is strongly convex everywhere on its domain except in +∞. For C a bounded subset of R n ++ , as ∇ 2 h(x) = 1 x 2 Id, we have ∀x ∈ C, ∀d ∈ R n , ⟨∇ 2 h(x)d, d⟩ > 1 sup x∈C x 2 ||d|| 2 indicating that h is strongly convex on bounded subsets of R n ++ . Second, h and f are Lipschitz continuous everywhere on R n ++ except close to 0. For both B-RED and B-PnP Ψ(x) → +∞ when x → 0 and {Ψ(x) ≤ α} avoids the case x → 0. (v) We remind the parametrizations B γ = Id −∇g γ and g γ (y) = 1 2 ||x − N γ (x)|| 2 with a U-Net N γ (with softplus activations). Using the fact that the composition and sum of semi-algebraic mappings are semi-algebraic mappings (Attouch et al. [2013], Coste [2000]), we easily verify that g γ and B γ are semi-algebraic. We also assumed that ϕ γ is semi-algebraic. We give more details now. From (12), we have that ∀x ∈ Im(B γ ) = dom(ϕ γ ), ϕ γ (x) = g γ (y) − D h (x, y), y ∈ B −1 γ (x)(123) As shown in [Coste, 2000, Corollary 2.9], the inverse image of a semi-algebraic set by a semi-algebraic mapping is a semi-algebraic set. The graph of ϕ γ is then a semi-algebraic subset of R n × R and ϕ γ is semi-algebraic. Eventually, g γ is positive and thus lower-bounded. Moreover we now prove that we have ∀y ∈ R n , ϕ γ (y) ≥ g γ (y). If y / ∈ int dom(h), as Im(B γ ) ⊂ int dom(h), ϕ γ (y) = +∞, this is verified. If y ∈ dom(h), as D h (y, y) = 0, we have ϕ γ (y) = ϕ γ (y) + D h (y, y) ≥ ϕ γ (B γ (y)) + D h (B γ (y), y) = g γ (y),(124) where the inequality comes from (13) and the last equality from (12). Second, we verify Assumption 2 required for the convergence of B-RED. (i) [0, R] n is a non-empty closed, bounded, convex and semi-algebraic subset of R n ++ . (ii) With the parametrization g γ (y) = 1 2 ||x − N γ (x)|| 2 with a neural network N γ . g γ can be shown to have Lipschitz gradient (see Hurault et al. [2021, Appendix B] for a proof). We can not show a global NoLip property for g γ . However, as ∇g γ is Lip(g γ )-Lipschitz, we have ∀x ∈ (0, R] n , ∀d ∈ R n , ⟨∇ 2 g γ (x)d, d⟩ ≤ Lip(g γ )||d|| 2 ≤ Lip(g γ )R 2 n i=1 d 2 i x 2 i = Lip(g γ )R 2 ⟨∇ 2 h(x)d, d⟩,(125) which proves that, for L γ = Lip(g γ )R 2 , L γ h − g γ is convex on (0, R] n . B-PnP additional assumptions We finally discuss the additional assumptions required for the convergence of B-PnP in Theorem 2. • Im(B γ ) ⊆ dom(h) = R n ++ . We train the denoiser B γ to restore images in [ϵ, 1] n (with ϵ = 10 −3 ), the denoiser is thus softly enforced to have its image in this range. In practice, we empirically verify during the iterations that we always get x k > 0. • ψ γ • ∇h * convex on int dom(h * ). As shown in Appendix E.2, a necessary condition for this convexity to hold is ⟨∇ 2 g γ (y)d, d⟩ ≤ n i=1 1 − 2y∇g γ (y) y 2 i d 2 i .(126) After training g γ , we empirically verifies that the above condition holds for random d ∼ U[0, 1] n and y sampled from the validation dataset with random noise (Inverse Gamma or Gaussian noise) of different intensity. We can then assume that the convexity condition is verified locally, around the image manifold. • Im(∇h − λ∇f ) ⊆ dom(∇h * ). We now show that this condition is true if λ||y|| 1 < 1. For x > 0 ∇h(x) − λ∇f (x) = − 1 x − λ∇f (x) = − x + xλ∇f (x) x .(127) Thus we need to verify that ∀1 ≤ i ≤ n, 1 + λx i ∇f (x) i > 0. For f Poisson data-fidelity term, using (Ax) j = n k=1 a j,k x k , we have ∀1 ≤ i ≤ n, ∇f (x) i = m j=1 −y j a j,i n k=1 a j,k x k + αa j,i(128) and 1 + λx i ∇f (x) i = 1 + αλ m j=1 a j,i x i − λ m j=1 y j a j,i x i n k=1 , a j,k x k .(129) We assumed that A has positive entries and m j=1 a j,i = r i > 0. Therefore, using 0 ≤ aj,ixi n k=1 ,a j,k x k < 1, we get 1 + λx i ∇f (x) i ≥ 1 − λ||y|| 1 (130) which is positive if λ||y|| 1 < 1. • The stepsize condition λL f < 1. Using the NoLip constant proposed in [Bauschke et al., 2017, Lemma 7] L f = ||y|| 1 , the condition boils down to λ||y|| 1 < 1. The condition λ||y|| 1 < 1 is too restrictive in practice, as ||y|| 1 could get very big, especially for large images. This is due to the fact that the NoLip constant L f ≥ ||y|| 1 can be largely overestimated. Indeed, in the proof of [Bauschke et al., 2017, Lemma 7] as well as in the proof of the previous point, the upper bound a j,i x i n k=1 , a j,k x k < 1 can be very loose in practice. For B-RED this is not a problem, as we use automatic stepsize backtracking. However, for B-PnP backtracking is not possible as the stepsize is fixed. In practice, we employ the following empirical procedure to adjust λ, reminiscent of stepsize backtracking. We first run B-PnP with λ > 0 without restriction. We then empirically check that a sufficient decrease condition of the objective function is verified. If not, we decrease λ until verification. E.4 Experiments We give here more details and results on the evaluation of B-PnP and B-RED algorithms for Poisson image deblurring. We present in Figure 4 the four blur kernels used for evaluation. Initialization is done with x 0 = A T y. The algorithm terminates when the relative difference between consecutive values of the objective function is less than 10 −8 or the number of iterations exceeds K = 500. Choice of hyperparameters For B-RED, stepsize backtracking is performed with γ = 0.8 and η = 0.5. When performing plug-and-play image deblurring with our Bregman Score Denoiser trained with Inverse Gamma noise, for the right choice of hyperparameters λ and γ, we may observe the following behavior. The algorithm first converges towards a meaningful solution. After hundreds of iterations, it can converge towards a different stationary point that does not correspond to a visually good reconstruction. We illustrate this behavior Figure 5 where we plot the evolution of the PSNR and of the function values f (x k ) and g γ (x k ) along the algorithm. (a) PSNR(x k ) (b) f (x k ) (c) gγ(x k ) Figure 5: Evolution of the PSNR, f (x k ) and g γ (x k ) when deblurring with B-RED with the initialization parameters from Table 2 and without hyper-parameter update after 100 iterations. We observe a first phase of fast decrease of both the data-fidelity term and regularization term values, resulting in a fast PSNR increase. After approximately 100 iterations, the regularization continue decreasing and the iterates converge towards a different stationary point with low PSNR. This phenomenon can be mitigated by using small stepsize, large γ and small λ values at the expense of slowing down significantly the algorithm. To circumvent this issue, we propose to first initialize the algorithm with 100 steps with initial τ , γ and λ values and then to change this parameters for the actual algorithm. Note that it is possible for B-RED to change the stepsize τ but not for B-PnP which has fixed stepsize τ 1 =. For B-PnP, as done in Hurault et al. [2022] in the Euclidean setting, we propose to multiply g θ by a parameter 0 < α < 1 such that the Bregman Score denoiser becomes B α γ (y) = y−α(∇ 2 h(y)) −1 .∇g γ (y). The convergence of B-PnP with this denoiser follows identically. The overall hyperparameters λ, γ, τ and α for B-PnP and B-RED algorithms for initialization and for the actual algorithm are given in Table 2. α 20 40 60 B-RED Initialization τ = 1 γ = 50 λ = 1.5 λ = 2. λ = 2.5 Algorithm τ = 0.05 γ = 500 λ = 0.5 λ = 0.5 λ = 0.5 B-PnP Initialization α = 1 γ = 50 λ = 1.5 λ = 2. λ = 2.5 Algorithm α = 0.05 γ = 500 λ = 0.025 λ = 0.025 λ = 0.025 Table 2: B-RED and B-PnP hyperparameters Additional experimental results We provide in Table 3 a quantitative comparison between our 2 algorithms B-RED and B-PnP and 3 other methods. (a) PnP-PGD corresponds to the plug-and-play proximal gradient descent algorithm x k+1 = D σ • (Id −τ ∇f ) with D σ the DRUNet denoiser (same architecture than B-RED and B-PnP) trained to denoiser Gaussian noise. (b) PnP-BPG corresponds to the B-PnP algorithm x k+1 = D σ •∇h * (∇h−τ ∇f )(x k ) with again the DRUNet denoiser D σ trained for Gaussian noise. For both (a) and (b) the parameters σ and τ are optimized for each noise level α. (c) ALM Unfolded [Sanghvi et al., 2022] uses the Augmented Lagrangian Method for deriving a 3-operator splitting algorithm that is then trained specifically in an unfolded fashion for image deblurring with a variety of blurs and noise levels α. The publicly available model being trained on grayscale images, for restoring our color images, we treat each color channel independently. Note that contrary to the proposed B-PnP and B-RED algorithms, the 3 compared methods do not have any convergence guarantees. We observe that our algorithms performs the best when the Poisson noise is not too intense (α = 40 and α = 60) but that PSNR performance decreases for intense noise (α = 20). We assume that this is due to the fact that the denoising prior trained on Inverse Gamma noise is not powerful enough for such a strong noise. A visual example for α = 20 is given Figure 6. As a future direction, we plan on investigating how to increase the regularization capacity of the deep inverse gamma noise denoiser to better handle intense noise. Table 3: PSNR (dB) of Poisson deblurring methods on the CBSD68 dataset. PSNR averaged over the 4 blur kernels represented Figure 4 for each noise levels α. Figure 1 : 1Denoising of a 256 × 256 image corrupted with Inverse Gamma noise of level γ = 25. Figure 2 : 2Deblurring from the indicated motion kernel and Poisson noise with α = 40. Figure 3 : 3Deblurring from the indicated Gaussian blur kernel and Poisson noise with α = 60. A. 3 3Kurdyka-Lojasiewicz (KL) property and semi-algebraicity Definition 2 (Kurdyka-Lojasiewicz (KL) [Attouch et al., 2013]). A function f : R n −→ R ∪ +∞ is said to have the Kurdyka-Lojasiewicz property at x * ∈ dom(f ) if there exists η ∈ (0, +∞), a neighborhood U of x * and a continuous concave function ψ : [0, η) ( iii) It is shown in[Bauschke et al., 2017, Lemma 7] that f verifies the NoLip assumption, i.e. Figure 4 : 4The 4 blur kernels used for deblurring evaluation. (a) and (b) are real-world camera shake kernels from Levin et al. [2009]. (c) is a 9 × 9 uniform kernel. (d) is a 25 × 25 Gaussian kernel with standard deviation 1.6. Figure 6 : 6Observed (14.91dB) (c) B-RED (22.44dB) (d) B-PnP (22.40dB) Deblurring from the indicated motion kernel and Poisson noise with α = 20. DRUNet 28.38 30.88 32.76 34.74 36.71 Table 1: Average denoising PSNR performance of Inverse Gamma noise denoisers B-DRUNet and DRUNet on 256 × 256 center-cropped images from the CBSD68 dataset, for various noise levels γ.γ 10 25 50 100 200 DRUNet 28.42 30.91 32.80 34.76 36.79 B- 7 Acknowledgements 7This work was funded by the French ministry of research through a CDSN grant of ENS Paris-Saclay. It has also been carried out with financial support from the French Research Agency through the PostProdLEAP and Mistic projects (ANR-19-CE23-0027-01 and ANR-19-CE40-005). It has also been supported by the NSF CAREER award under grant CCF-2043134.Samuel Hurault, Arthur Leclaire, and Nicolas Papadakis. Gradient step denoiser for convergent plug-and-play. arXiv preprint arXiv:2110.03220, 2021.Samuel Hurault, Arthur Leclaire, and Nicolas Papadakis. Proximal denoiser for convergent plugand-play optimization with nonconvex regularization. In International Conference on Machine Learning, pages 9483-9505. PMLR, 2022. Yash Sanghvi, Abhiram Gnanasambandam, and Stanley H Chan. Photon limited non-blind deblurring using algorithm unrolling. IEEE Transactions on Computational Imaging, 8:851-864, 2022. Simon Setzer, Gabriele Steidl, and Tanja Teuber. Deblurring poissonian images by split bregman techniques. Journal of Visual Communication and Image Representation, 21(3):193-199, 2010. Yu Sun, Zihui Wu, Xiaojian Xu, Brendt Wohlberg, and Ulugbek S Kamilov. Scalable plug-and-play admm with convergence guarantees. Yawei Li, Wangmeng Zuo, Lei Zhang, Luc Van Gool, and Radu Timofte. Plug-and-play image restoration with deep denoiser prior. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2021. Definition 1 (Legendre function, Rockafellar [1997]). Let h : C ⊆ R n → R ∪ {+∞} be a proper lower semi-continuous convex function. It is called: (i) essentially smooth, if h is differentiable on int dom(h), with moreover ||∇h(x k )|| → ∞ for every sequence {x k } k∈N of int dom(h) converging towards a boundary point of dom(h).References Abdullah H Al-Shabili, Xiaojian Xu, Ivan Selesnick, and Ulugbek S Kamilov. Bregman plug-and-play priors. In 2022 IEEE International Conference on Image Processing (ICIP), pages 241-245. IEEE, 2022. Hédy Attouch, Jérôme Bolte, Patrick Redont, and Antoine Soubeyran. Proximal alternating mini- mization and projection methods for nonconvex problems: An approach based on the kurdyka- łojasiewicz inequality. Mathematics of operations research, 35(2):438-457, 2010. Hedy Attouch, Jérôme Bolte, and Benar Fux Svaiter. Convergence of descent methods for semi- algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized gauss-seidel methods. Mathematical Programming, 137(1-2):91-129, 2013. 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Ernest Ryu, Jialin Liu, Sicheng Wang, Xiaohan Chen, Zhangyang Wang, and Wotao Yin. Plug-and- play methods provably converge with properly trained denoisers. In International Conference on Machine Learning, pages 5546-5557. PMLR, 2019. IEEE Transactions on Computational Imaging, 7:849-863, 2021. Matthieu Terris, Audrey Repetti, Jean-Christophe Pesquet, and Yves Wiaux. Building firmly nonex- pansive convolutional neural networks. In ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 8658-8662. IEEE, 2020. Singanallur V Venkatakrishnan, Charles A Bouman, and Brendt Wohlberg. Plug-and-play priors for model based reconstruction. In 2013 IEEE Global Conference on Signal and Information Processing, pages 945-948. IEEE, 2013. Kai Zhang, Wangmeng Zuo, Shuhang Gu, and Lei Zhang. Learning deep cnn denoiser prior for image restoration. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3929-3938, 2017. Kai Zhang, A Definitions A.1 Legendre functions, Bregman divergence Proper lower semicontinuous functions which satisfy the Kurdyka-Lojasiewicz inequality at each point of dom(∂f ) are called KL functions.This condition can be interpreted as the fact that, up to a reparameterization, the function is sharp i.e. we can bound its subgradients away from 0. For more details and interpretations, we refer to [Attouch et al., 2010] and [Bolte et al., 2010]. A large class of functions that have the KL-property is given by semi-algebraic functions. Definition 3 (Semi-algebraicity [Attouch et al., 2013]). ALM Unfolded [Sanghvi et al., 2022] 23.39 23.91 24.22 B-RED 23.58 24.54 24.90 B-PnP 23.29 24.54 24.80α 20 40 60 PnP-PGD 23.81 24.41 24.45 PnP-BPG 23.85 24.26 24.71 The Bregman divergence being non-symmetric, the order of the variables (x, y) in D h is important. Distributions of the form (6) with reverse order in D h have been characterized in[Banerjee et al., 2005] but this analysis does not apply here.
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[ "Characterization of stellar companion from high-contrast long-slit spectroscopy data The EXtraction Of SPEctrum of COmpanion (Exospeco) algorithm", "Characterization of stellar companion from high-contrast long-slit spectroscopy data The EXtraction Of SPEctrum of COmpanion (Exospeco) algorithm" ]
[ "Samuel Thé \nCentre de Recherche Astrophysique de Lyon\nUMR 5574\nUniversité de Lyon\nUniversité Lyon1\nENS de Lyon\nCNRS\nF-69230Saint-Genis-LavalFrance\n", "Éric Thiébaut \nCentre de Recherche Astrophysique de Lyon\nUMR 5574\nUniversité de Lyon\nUniversité Lyon1\nENS de Lyon\nCNRS\nF-69230Saint-Genis-LavalFrance\n", "Loïc Denis \nInstitut d Optique Graduate School\nLaboratoire Hubert Curien UMR 5516\nUniversité Jean Monnet Saint-Etienne\nCNRS\nF-42023SAINT-ETIENNEFrance\n", "Thibault Wanner \nCentre de Recherche Astrophysique de Lyon\nUMR 5574\nUniversité de Lyon\nUniversité Lyon1\nENS de Lyon\nCNRS\nF-69230Saint-Genis-LavalFrance\n", "Rémi Thiébaut \nCentre de Recherche Astrophysique de Lyon\nUMR 5574\nUniversité de Lyon\nUniversité Lyon1\nENS de Lyon\nCNRS\nF-69230Saint-Genis-LavalFrance\n", "Maud Langlois \nCentre de Recherche Astrophysique de Lyon\nUMR 5574\nUniversité de Lyon\nUniversité Lyon1\nENS de Lyon\nCNRS\nF-69230Saint-Genis-LavalFrance\n", "Ferréol Soulez \nCentre de Recherche Astrophysique de Lyon\nUMR 5574\nUniversité de Lyon\nUniversité Lyon1\nENS de Lyon\nCNRS\nF-69230Saint-Genis-LavalFrance\n" ]
[ "Centre de Recherche Astrophysique de Lyon\nUMR 5574\nUniversité de Lyon\nUniversité Lyon1\nENS de Lyon\nCNRS\nF-69230Saint-Genis-LavalFrance", "Centre de Recherche Astrophysique de Lyon\nUMR 5574\nUniversité de Lyon\nUniversité Lyon1\nENS de Lyon\nCNRS\nF-69230Saint-Genis-LavalFrance", "Institut d Optique Graduate School\nLaboratoire Hubert Curien UMR 5516\nUniversité Jean Monnet Saint-Etienne\nCNRS\nF-42023SAINT-ETIENNEFrance", "Centre de Recherche Astrophysique de Lyon\nUMR 5574\nUniversité de Lyon\nUniversité Lyon1\nENS de Lyon\nCNRS\nF-69230Saint-Genis-LavalFrance", "Centre de Recherche Astrophysique de Lyon\nUMR 5574\nUniversité de Lyon\nUniversité Lyon1\nENS de Lyon\nCNRS\nF-69230Saint-Genis-LavalFrance", "Centre de Recherche Astrophysique de Lyon\nUMR 5574\nUniversité de Lyon\nUniversité Lyon1\nENS de Lyon\nCNRS\nF-69230Saint-Genis-LavalFrance", "Centre de Recherche Astrophysique de Lyon\nUMR 5574\nUniversité de Lyon\nUniversité Lyon1\nENS de Lyon\nCNRS\nF-69230Saint-Genis-LavalFrance" ]
[]
Aims. High-contrast long-slit spectrographs can be used to characterize exoplanets. High-contrast long-slit spectroscopic data are however corrupted by stellar leakages which largely dominate other signals and make the process of extracting the companion spectrum very challenging. This paper presents a complete method to calibrate the spectrograph and extract the signal of interest. Methods. The proposed method is based on a flexible direct model of the high-contrast long-slit spectroscopic data. This model explicitly accounts for the instrumental response and for the contributions of both the star and the companion. The contributions of these two components and the calibration parameters are jointly estimated by solving a regularized inverse problem. This problem having no closed-form solution, we propose an alternating minimization strategy to effectively find the solution. Results. We have tested our method on empirical long-slit spectroscopic data and by injecting synthetic companion signals in these data. The proposed initialization and the alternating strategy effectively avoid the self-subtraction bias, even for companions observed very close to the coronagraphic mask. Careful modeling and calibration of the angular and spectral dispersion laws of the instrument clearly reduce the contamination by the stellar leakages. In practice, the outputs of the method are mostly driven by a single hyperparameter which tunes the level of regularization of the companion SED.
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[ "https://export.arxiv.org/pdf/2306.03467v1.pdf" ]
259,089,189
2306.03467
1a7811376d592e53a4f2324d889bac250cc3a54a
Characterization of stellar companion from high-contrast long-slit spectroscopy data The EXtraction Of SPEctrum of COmpanion (Exospeco) algorithm June 5, 2023 Samuel Thé Centre de Recherche Astrophysique de Lyon UMR 5574 Université de Lyon Université Lyon1 ENS de Lyon CNRS F-69230Saint-Genis-LavalFrance Éric Thiébaut Centre de Recherche Astrophysique de Lyon UMR 5574 Université de Lyon Université Lyon1 ENS de Lyon CNRS F-69230Saint-Genis-LavalFrance Loïc Denis Institut d Optique Graduate School Laboratoire Hubert Curien UMR 5516 Université Jean Monnet Saint-Etienne CNRS F-42023SAINT-ETIENNEFrance Thibault Wanner Centre de Recherche Astrophysique de Lyon UMR 5574 Université de Lyon Université Lyon1 ENS de Lyon CNRS F-69230Saint-Genis-LavalFrance Rémi Thiébaut Centre de Recherche Astrophysique de Lyon UMR 5574 Université de Lyon Université Lyon1 ENS de Lyon CNRS F-69230Saint-Genis-LavalFrance Maud Langlois Centre de Recherche Astrophysique de Lyon UMR 5574 Université de Lyon Université Lyon1 ENS de Lyon CNRS F-69230Saint-Genis-LavalFrance Ferréol Soulez Centre de Recherche Astrophysique de Lyon UMR 5574 Université de Lyon Université Lyon1 ENS de Lyon CNRS F-69230Saint-Genis-LavalFrance Characterization of stellar companion from high-contrast long-slit spectroscopy data The EXtraction Of SPEctrum of COmpanion (Exospeco) algorithm June 5, 2023Pre-print version, article under review (DOI: will be inserted by hand later) Astronomy & AstrophysicsInfrared: planetary systems -Methods: data analysis -Techniques: imaging spectroscopy -Instrumentation: spectro- graphs -Instrumentation: adaptive optics Aims. High-contrast long-slit spectrographs can be used to characterize exoplanets. High-contrast long-slit spectroscopic data are however corrupted by stellar leakages which largely dominate other signals and make the process of extracting the companion spectrum very challenging. This paper presents a complete method to calibrate the spectrograph and extract the signal of interest. Methods. The proposed method is based on a flexible direct model of the high-contrast long-slit spectroscopic data. This model explicitly accounts for the instrumental response and for the contributions of both the star and the companion. The contributions of these two components and the calibration parameters are jointly estimated by solving a regularized inverse problem. This problem having no closed-form solution, we propose an alternating minimization strategy to effectively find the solution. Results. We have tested our method on empirical long-slit spectroscopic data and by injecting synthetic companion signals in these data. The proposed initialization and the alternating strategy effectively avoid the self-subtraction bias, even for companions observed very close to the coronagraphic mask. Careful modeling and calibration of the angular and spectral dispersion laws of the instrument clearly reduce the contamination by the stellar leakages. In practice, the outputs of the method are mostly driven by a single hyperparameter which tunes the level of regularization of the companion SED. Introduction High-contrast extreme adaptive optics (AO) systems such as SPHERE (Spectro-Polarimetry High-contrast Exoplanet REsearch, Beuzit et al. 2019), GPI (Gemini Planet Imager, Macintosh et al. 2006Macintosh et al. , 2014, or SCExAO (Jovanovic et al. 2015) have been developed to directly observe the close environment of stars in the visible and the near infrared. The study of exoplanets and their formation is one of the main scientific objective of these instruments. One of the advantages of high-contrast extreme AO systems is that they can provide direct access to the light from the exoplanet which is crucial to perform spectral characterizations. Substantial contamination by the light from the host star occurs, though: in the visible and the near infrared, in spite of the real-time correction by the AO system and of the masking of the host star by a coronagraph, the residual stellar light diffracted by the instrument is much brighter than that received from most exoplanets of interest. For this reason, dedicated post-processing methods have been developed to track evidences of exoplanet presence in data corrupted by strong stellar leakages. The number of published detection algorithms, Loci (Lafreniere et al. 2007), Tloci (Marois et al. 2013), Klip (Soummer et al. 2012), Moods (Smith et al. 2009), Andromeda (Mugnier et al. 2009), PeX (Devaney & Thiébaut 2017), and Paco (Flasseur et al. 2018(Flasseur et al. , 2020b to name a few, reflects the scientific interest but also the intrinsic difficulty of trustfully detecting an exoplanet from sequences of high-contrast images. The most successful of these methods are the ones that take into account the statistics of the stellar leakages (notably their correlations) whether they consist in sequences of images (Smith et al. 2009;Flasseur et al. 2018Flasseur et al. , 2020b, in sequences of multi-spectral images from Integral field spectrographs (IFS) (Flasseur et al. 2020a), or even in multi-epoch sequences of images (Dallant et al. 2022). After its detection, the direct characterization of an exoplanet is possible with high-contrast extreme AO systems equipped with a spectrograph. Both SPHERE and GPI are equipped with low resolution IFS. In addition SPHERE/IRDIS is equipped with a medium (MRS) resolution long-slit spectrograph (LSS) in J, H, and K bands, the latter being also available at low (LRS) resolution 1 . With SPHERE/IRDIS/LSS, the spectrum of a detected companion can then be measured by aligning the slit of the spectrograph with the host star and the companion while the host star is occulted by an opaque mask combined with the slit. In an LSS image, the stellar leakages take the form of speckles spectrally dispersed along oblique lines generally brighter than the companion spectrum (see Fig. 1 for an example). In order to get rid of these stellar leakages, Vigan et al. (2008Vigan et al. ( , 2012 have developed a spectral deconvolution (SD) method following the work of Sparks & Ford (2002). The SD method consists in a filtering of the LSS image after a geometrical transform to align the speckles along a given direction. In practice, the SD method is quite sensitive to the alignment of the instrument, requires to fix defective pixels, and suffers from a self-subtraction bias. The latter is due to an overestimation of the stellar leakages caused by the presence of the companion. To improve on the SD method and reduce the self-subtraction bias, Mesa, D. et al. (2016) have adapted the strategy implemented in Tloci (Marois et al. 2013) to the case of LSS data. In spite of these improvements, existing extraction methods suffer from a number of defaults, most of them steming from the requirement to geometrically transform the LSS data to align the dispersed speckles. In particular, they provide, at best, a least squares estimation of the stellar leakages which is sub-optimal as the noise is not independent and identically distributed (i.i.d.) in the geometrically transformed images (Thiébaut et al. 2016). To overcome the drawbacks of existing methods, we propose to formulate the extraction of the spectrum of a companion as an inverse problem. The inverse problem corresponds to the joint estimation of the contributions of the star and of its companion from the LSS data. Not only does this approach require no transform of the LSS data (thus avoiding the introduction of correlations) but it also yields statistically optimal estimators. To cope both with possible instrumental misalignment and the lack of a closed-form solution for the inverse problem, we implemented an alternating optimization strategy with optional self-calibration stages to solve the problem. The outline of the paper is as follows. In Section 2, we present a model of the distribution of the light on the detector of a LSS instrument. This model is used to illustrate how, after a geometrical transform, stellar leakages can be partially removed by a truncated singular value decomposition (TSVD) before extracting the companion signal. Such an approach is representative of the performance that can be reached, at best, by standard methods. We then present in Section 3 our approach to jointly estimate the stellar leakages and the contribution of the companion without transforming the data. For the processing of LSS data, knowing the spectro-angular coordinates of each detector pixel is mandatory and we describe in Section 4 a numerical method to estimate the spatio-spectral dispersion laws from given calibration data. In Section 5, we validate the proposed method on both real data from SPHERE/IRDIS/MRS and on injections of synthetic companions in real data. We show the importance of the calibration (described in Appendix A) and compare our method with more standard approaches. State of the art processing In spite of the coronagraphic mask in high-contrast data, the stellar leakages, in the form of quasi-static dispersed speckles, largely dominate the signal of interest, i.e., the spectrum of the companion. These speckles, whose contribution cannot be precisely determined by using other stars (i.e. by reference differential imaging, Xie et al. 2022) or by rotating the slit to hide the planet (Vigan 2016) are a major source of nuisance for extracting the companion spectrum. This section introduces a modeling of the LSS data that is used in Section 3 to design our spectrum extraction method. This model is also useful to explain how previous approaches perform the suppression of the stellar leakages (Vigan et al. 2008;Mesa, D. et al. 2016). λ ρ Fig. 1. Long-slit medium resolution spectroscopy data of HR 3549 taken by IRDIS, with horizontally the spectral axis λ and vertically the angular separation axis ρ. The blue arrows indicate the position of the companion. Figure 1 shows a single exposure captured by the LSS of SPHERE/IRDIS. The vector d ∈ R N , with N the number of pixels, can be modeled by: Image formation d n = m(ρ n , λ n ) + ε n(1) with m(ρ, λ) the distribution of light in the detector plane at angular coordinate ρ along the slit and wavelength λ, ρ n and λ n the angular and spectral coordinates at the n-th pixel, and ε n the contribution of the noise. Our notations are summarized in Table 1. The light distribution in the detector plane is the sum of the contributions by the star and by the companion: m(ρ, λ) = f (λ) h (ρ, λ) + f ⊕ (λ) h ⊕ (ρ, λ)(2) with f and f ⊕ the spectral energy distributions (SEDs) of the star and of the companion as seen by the detector 2 , h and h ⊕ the point spread functions (PSFs) for a source at the respective angular positions of the star and of the companion, the so-called on-axis and off-axis PSFs. The on-axis PSF, h (ρ, λ) explains the oblique bright lines due to stellar leakages in Fig. 1 while the stellar SED f (λ) explains the variations of intensity along these lines. As can be seen in Fig. 1, the companion signal, that is f ⊕ (λ) h ⊕ (ρ, λ), is barely distinguishable in the LSS data and it is mandatory to get rid of the stellar leakages f (λ) h (ρ, λ). Low rank approximation of the stellar leakages Devaney & Thiébaut (2017) have shown that, except in the vicinity of the coronagraphic mask, the chromatic PSF can be written in the form of a series expansion. Applying their model to the star and taking into account that our data have one angular dimension instead of two yields: h (ρ, λ) = k≥1 γ(λ) k h ,k γ(λ) (ρ − ρ )(3) with γ(λ) = λ ref /λ a chromatic magnification factor relative to some arbitrary reference wavelength λ ref , ρ the angular position of the star along the slit to account for a possible pointing error of the instrument, and {h ,k } k=1,... a family of spatial PSF modes at the reference wavelength. Since the stellar leakages dominate the signal in the LSS image d, Eq. (3) suggests to apply specific image warping so as to form a 2-D image d warp whose, say, first dimension varies along Table 1. Notations. Lowercase letters are for continuous functions and scalars (e.g. f ), boldface lowercase letters for vectors (e.g. x), and boldface uppercase letters for linear mappings, a.k.a. matrices, (e.g. F ). Vectors with a hat (e.g.x) are estimators. The main unknowns of the problem are x, the sampled star SED, y, the sampled on-axis PSF, and z, the sampled companion SED. Notation Description subscript Parameters of the stellar model subscript ⊕ Parameters of the companion model d ∈ R N Science data m ∈ R N Sampled model of d w ∈ R N Diagonal of the precision matrix of d λ ∈ R N Pixel-wise wavelengths ρ ∈ R N Pixel-wise angular positions f Continuous star SED x ∈ R N x Sampled star SED f λ grd ∈ R N x Sampling wavelengths for x F ∈ R N×N x Interpolation operator: x to pixel-wise f h Continuous on-axis PSF y ∈ R N y Sampled on-axis PSF h at λ ref ρ grd ∈ R N y Sampling angles for y ρ Angular position of the star ν ∈ Ω Calibration parameters of h Ω Feasible set h parameters H ∈ R N×N y Interpolation operator: y to pixel-wise h at λ ref f ⊕ Continuous companion SED z ∈ R N z Sampled companion SED f ⊕ λ grd ⊕ ∈ R N x Sampling wavelengths for z F ⊕ ∈ R N×N z Interpolation operator: z to pixel-wise f ⊕ h ⊕ Continuous off-axis PSF h ⊕ ∈ R N Sampled off-axis PSF h ⊕ at λ ref ν ⊕ ∈ Ω ⊕ Calibration parameters of h ⊕ Ω ⊕ Feasible set of h ⊕ parameters ρ ⊕ ∈ R Angular position of the companion µ = (µ x , µ y , µ z ) Hyper-parameters γ ∈ R N Pixel-wise chromatic scaling factors ν = (ν , ν ⊕ ) ∈ Ω Calibration parameters Ω = Ω × Ω ⊕ Feasible set of calibration parameters Λ Spectral dispersion law a Parameters of the spectral dispersion law Angular dispersion law s Parameters of the angular dispersion law ∆ρ Width of the coronagraphic mask the wavelength while its second dimension varies along coordinate s = γ(λ) (ρ − ρ ) (see Fig. 2). According to Eqs. (1) and (3), the warped image is modeled and then approximated by: d warp i, j = m ρ + s warp j /γ λ warp i , λ warp i + ε warp i, j (4) ≈ k≥1 γ λ warp i k f λ warp i h ,k s warp j(5) where ε warp in Eq. (4) denotes the contribution of the noise in the warped image while the ≈ symbol in Eq. (5) is to account for the contributions of the potential companion and for the noise which have been neglected. In words, the stellar leakages appear to have a simple separable decomposition in the warped image. Fig. 1, corresponding to the side where lies the companion, warped so as to align the dispersed speckles of the stellar leakages. The warping is defined by the coordinates ρ and λ of the pixels given by the complex calibration model of the spectral and angular dispersion laws described in Appendix A. The companion signal can be seen as a faint curved track indicated by the blue arrows. The Singular Value Decomposition (SVD) of the warped image 3 writes: d warp = min(N 1 ,N 2 ) k=1 u k σ k v k(6) where N 1 and N 2 are the dimensions of the warped image, u k ∈ R N 1 is the k-th left singular vector of the decomposition, σ k ≥ 0 is the k-th singular value, and v k ∈ R N 2 is the k-th right singular vector. Comparing Eq. (5) and Eq. (6), the SVD of d warp readily provides a decomposition similar to the contribution of the stellar leakages with, for each index k, the left singular vector u k sampling γ(λ) k f (λ) as a function of λ and the right singular vector v k sampling h ,k (s) as a function of s (both up to a normalization factor that depends on k). The truncated singular value decomposition (TSVD) of the warped image is obtained by limiting the sum in the right-hand side of Eq. (6) to the k max ≤ min(N 1 , N 2 ) first terms and, by Eckart-Young-Mirsky (Eckart & Young 1936;Mirsky 1960) theorem, it is the best possible approximation of d warp of rank k max in the least squares sense. Hence, it may be assumed that, for a suitable choice of k max , the TSVD of d warp provides a good approximation of the stellar leakages without being too much affected by the companion signal (if the companion is not too bright) and by the noise. A residual image that mostly depends on the companion can then be formed by subtracting the un-warped TSVD of the warped image d warp from the LSS image d: r ⊕ = d − U         k max k=1 u k σ k v k        (7) where U denotes the un-warping operation 4 . As illustrated by Fig. 3, the signal of interest, the companion SED, is then easier to extract from the residual image r ⊕ . This can be done by standard aperture photometry tools. As pointed by Devaney & Thiébaut (2017), there are a number of issues in using the TSVD to get rid of the stellar leakages in multi-wavelengths high-contrast data. First, to produce a Fig. 1, the companion signal appears more distinctly (indicated by the 2 arrows). rectangular warped image (that can be interpreted as a matrix to perform the SVD), quite substantial regions of the original data d have to be discarded (near the coronagraphic mask and the edges of the formed image). This limits the range of admissible angular positions for the companion and gets rid of data that might be valuable to improve the estimation of the stellar leakages. Second, the presence of a companion in the original data d yields a positive bias in the approximation of the stellar leakages by the TSVD. This results in a negative bias in the residual image and, hence, in the estimated companion SED. This artifact is known in the literature as "self-subtraction". Third, the least squares fit performed by the TSVD of d warp is sub-optimal regarding the distribution of the noise in the warped image. Indeed, least squares are only optimal for independent identically distributed (i.i.d.) noise which is certainly not the case for ε warp i, j : at least, the shot noise in the image d has a non-uniform distribution and a side effect of the transform of d to yield the warped image d warp is to introduce correlations. Moreover, defective pixels, which are quite numerous for the kind of detectors used by NIR instruments such as LSS, must be corrected, usually by averaging their neighbors values, before warping the image. This correction can only introduce additional correlations. In spite of these drawbacks, proposed processing methods (Vigan et al. 2008;Mesa, D. et al. 2016) are similar to the TSVD approximation of warped LSS images (optimal linear combination of a set of images). Some refinements have been proposed to limit the self-subtraction bias (Mesa, D. et al. 2016) but the other issues have been largely left unaddressed. In the remaining of this paper, we propose a new inverse problems approach to solve all aforementioned limitations. Inverse problems approach To avoid the issues resulting from warping the LSS image, we propose to solve an inverse problem which consists in jointly estimating the parameters of the direct model of the data given in Eqs. (1) and (2) without transforming the data themselves. For an optimal information extraction, we model the likelihood of the data to consider the uneven quality of the data and, therefore, account for defective pixels or missing data in a consistent way. Besides, our approach relies on a precise calibration of the spectro-spatial instrumental dispersion as seen by the detector. The proposed method includes auto-calibration stages to refine the calibration parameters and thus accounts for a possible misalignment of the science exposures. Assumed continuous model To simplify the on-axis PSF model in Eq. (3), we keep only the first and most significant of these modes and thus assume that: h (ρ, λ) = γ(λ) h γ(λ) (ρ − ρ )(8) with h (ρ) = h ,1 (ρ) the first spatial mode of the on-axis PSF. As shown in Section 5, this simple model of the stellar leakages already gives excellent results. Likewise, the chromatic off-axis PSF h ⊕ can also be written as: h ⊕ (ρ, λ) ≈ γ(λ) h ⊕ γ(λ) (ρ − ρ ⊕ )(9) with h ⊕ (ρ) = h ⊕ (ρ, λ ref ) the off-axis PSF at the reference wavelength λ ref and ρ ⊕ the angular position of the companion along the slit. Note that the γ(λ) factor ensures that the on-axis and offaxis PSFs be normalized at all wavelengths provided the PSF at the reference wavelength be also normalized, i.e. h(ρ, λ) dρ = 1 (∀λ). These approximations for the on-axis and off-axis PSFs yield the following simplified model that we consider in the rest of the paper: m(ρ, λ) = γ(λ) f (λ) h γ(λ) (ρ − ρ ) + f ⊕ (λ) h ⊕ γ(λ) (ρ − ρ ⊕ ) .(10) Discretized distribution In order to fit the data, the model m(ρ, λ) in Eq. (10) has to be estimated at each angular and spectral coordinates (ρ n , λ n ) of the N pixels of the detector. These pixel coordinates can be identified by fitting angular and spectral dispersion laws to calibration data, as explained in Appendix A. Because of these angular and spectral dispersion laws, continuous functions are needed to model the SEDs of the star and the companion ( f and f ⊕ ) and their respective PSFs (h and h ⊕ ) on the sensor pixel grid. We explain next how we parameterize these functions. Our models of the star SED f (λ), of the on-axis PSF h (ρ), and of the companion SED f ⊕ (λ) are given by the following linear interpolations: f (λ) = N x j=1 ϕ λ − λ grd , j f λ grd , j x j ,(11a)h (ρ) = N y j=1 ψ ρ − ρ grd , j h ρ grd , j y j , (11b) f ⊕ (λ) = N z j=1 ϕ ⊕ λ − λ grd ⊕, j f ⊕ λ grd ⊕, j z j ,(11c) with ϕ : R → R, ψ : R → R, and ϕ ⊕ : R → R chosen interpolation functions, and where λ grd ∈ R N x is an evenly spaced grid of wavelengths to sample the star SED f , ρ grd ∈ R N y is an evenly spaced grid of angles to sample the on-axis PSF h , and λ grd ⊕ ∈ R N z is an evenly spaced grid of wavelengths to sample the companion SED f ⊕ . At the coordinates (ρ n , λ n ) of any pixel Fig. 4. Illustration of the direct model for high-contrast long-slit spectroscopy given in Eq.(10): the data are modeled as the sum of two components: a stellar component and a companion component. Extracting the SED of the companion also requires the estimation of the on-axis PSF and the SED of the host star. The labels "available via calibration" denote components that may be self-calibrated by Exospeco directly from the science data (see Section 3.5 for details). n ∈ 1, N of the detector, our linear interpolation yields: f ,n = f (λ n ) = N x j=1 ϕ λ n − λ grd , j F ,n, j x j = (F x) n ,(12a)h ,n = h γ n (ρ n − ρ ) = N y j=1 ψ γ n ρ n − ρ − ρ grd , j H ,n, j y j = (H y) n ,(12b)f ⊕,n = f ⊕ (λ n ) = N z i=1 ϕ ⊕ λ n − λ grd ⊕, j F ⊕,n,i z i = (F ⊕ z) n ,(12c) with γ n = γ(λ n ), and the matrices F ∈ R N×N x , H ∈ R N×N y , and F ⊕ ∈ R N×N z defined in Eqs. (12a)-(12c) represent interpolation operators 5 . These operators are applied to the vectors 6 x ∈ R N x , y ∈ R N y , and z ∈ R N z defined in Eqs. (11a)-(11c). They form the unknown parameters of our models of the star SED f , of the on-axis PSF h , and of the companion SED f ⊕ . The interpolation functions (ϕ , ψ , and ϕ ⊕ ) and the sampling lists (λ grd , ρ grd , and λ grd ⊕ ) may be chosen differently for each component of the model. If the spectral sampling lists and spectral interpolation functions are the same (as we chose for our experiments), then the two spectral interpolation operators F ⊕ and F are the same. In our implementation of the method, we selected the Catmull & Rom (1974) cardinal cubic spline ϕ as the interpolation function: ϕ (λ) = ϕ(λ/∆λ grd ), ψ (ρ) = ϕ(ρ/∆ρ grd ), and ϕ ⊕ (λ) = ϕ(λ/∆λ grd ⊕ ) with ∆λ grd , ∆ρ grd , and ∆λ grd ⊕ the sampling steps of λ grd , ρ grd , and λ grd ⊕ . For the off-axis PSF h ⊕ (ρ) at the reference wavelength, we consider a simple parametric model. Since the principal lobe of the off-axis PSF represents most of the energy received from the companion, we assume a Gaussian approximation: h ⊕ (ρ) = 1 √ 2πσ ⊕ exp − ρ 2 2σ 2 ⊕ .(13) Hence h ⊕ , the sampled off-axis PSF at the reference wavelength for the companion, depends on the angular position of the companion ρ ⊕ and on σ ⊕ the standard deviation of the PSF at the reference wavelength. Other parametric models of the off-axis PSF could be considered with a simple adaptation of the algorithm proposed in Section 3.4. Finally, we introduce the N-vectors m ∈ R N , γ ∈ R N and h ⊕ ∈ R N defined by: m n = m(ρ n , λ n ),(14a)γ n = γ(λ n ) = λ ref /λ n , (14b) h ⊕,n = h ⊕ γ n (ρ n − ρ ⊕ ) ,(14c) for n ∈ 1, N . The discretized model of the light distribution in Eq. (10) then writes: m(x, y, z, ν) = γ (H (ν) y) (F x) + γ h ⊕ (ν) (F ⊕ z) (15) with the Hadamard product (entry-wise multiplication) and ν = (ν , ν ⊕ ) the calibration parameters of the model which are the other unknown parameters than x, y, or z. With our Gaussian approximation of the off-axis PSF at the reference wavelength, the calibration parameters for the companion are ν ⊕ = (ρ ⊕ , σ ⊕ ). To account for a possible misalignment between the coronagraphic mask and the star, the calibration parameters for the star are just ν = (ρ ), with ρ the angular position of the star along the slit. The signal-processing problem then amounts to estimating the companion's SED z as well as the other nuisance parameters of the model, x, y, and ν. A method to perform this task is proposed in the next section. Objective function and regularization After proper calibration of the detector, raw images are preprocessed to compensate for bias and gain non-uniformity and to identify defective pixels (i.e. pixels with a non-linear response). This pre-processing produces the long-slit spectroscopy data d ∈ R N considered here and modeled by m(x, y, z, ν) in Eq. (15). Due to photon and detector noises as well as modeling inaccuracies, some discrepancies are expected between the data d and our model m(x, y, z, ν). Due to the observed flux level, there are enough photons detected per pixel for the data d to approximately follow a Gaussian distribution of mean the model m(x, y, z, ν) and of precision matrix 7 W. Since we directly consider the data without any pixel interpolation (i.e. no image warping to align the dispersed speckles and no attempt to fix defective pixels), no correlations are introduced in the data and the pixels can be considered as mutually independent. The precision 7 the precision matrix is the inverse of the covariance matrix matrix is thus diagonal, W = diag(w) where w ∈ R N collects the diagonal entries of W and is given by: w n = 0 if n-th pixel is invalid, 1/Var(d n ) otherwise.(16) where Var(d n ) can be estimated by different pre-processing methods (Mugnier et al. 2004;Berdeu et al. 2020). We consider as invalid all pixels for which the model is incorrect, this includes defective pixels, pixels too much impacted by the coronagraphic mask, and pixels located outside of the field of view (see Fig. 6). We assume that the estimation of the variances and the identification of defective pixels are part of the pre-processing stage. The definition of the precision matrix in Eq. (16) amounts to assuming that the variance of invalid pixels is infinite. In other words, this expresses that the values of invalid pixels should not be considered at all. Given the large number of unknowns, the estimation of the stellar and companion components x, y, and z cannot be performed solely by fitting the data: regularity constraints are necessary to prevent noise amplification and cope with missing data (Titterington 1985). We consider regularized estimators obtained by minimizing the following criterion: C(x, y, z, ν, µ) = d − m(x, y, z, ν) 2 W + R x y z (x, y, z, µ) ,(17) where the first term is a statistical distance between the model and the data (the co-log-likelihood) while R x y z (x, y, z, µ) is a regularization term parameterized by the vector µ of so-called hyper-parameters. In the above equation, u 2 W = u W u denotes the squared Mahalanobis (1936) norm. Our estimatorsx, y,ẑ, andν of the parameters of interest are the ones that jointly minimize the criterion in Eq. (17): (x(µ),ŷ(µ),ẑ(µ),ν(µ)) = arg min x≥0, y≥0, z≥0, ν∈Ω C(x, y, z, ν, µ).(18) These estimators depend on the hyper-parameters µ, as made explicit by the notation. As the parameters x, y, and z represent nonnegative quantities, their estimators are improved by enforcing nonnegativity as indicated by the inequality constraints in Eq. (18) such as x ≥ 0 which hold element-wise. The calibration parameters ν = (ν , ν ⊕ ) are constrained to belong to a set Ω = Ω ×Ω ⊕ where Ω and Ω ⊕ are the respective feasible sets for the stellar and companion calibration parameters defined based on physical considerations. The SEDs and the on-axis PSF at the reference wavelength being mutually independent, the regularization function can be decomposed as: R x y z (x, y, z, µ) = µ x R x (x) + µ y R y (y) + µ z R z (z) .(19) The complete set of hyper-parameters is then: µ = (µ x , µ y , µ z )(20) where µ x > 0, µ y > 0, and µ z > 0 tune the weights of the different regularization terms. There are many regularizations that are suitable for our problem. Regularization terms should enforce some kind of continuity or smoothness of the sought uni-dimensional distributions. In the following and for the sake of simplicity, we consider simple smoothness regularizations imposed by the quadratic penalty (Tikhonov & Arsenin 1977): R(u) = N u −1 j=1 u j+1 − u j (D u) j 2 = D u 2 2 ,(21) with N u the size of u = x, y, or z, and D ∈ R (N u −1)×N u a finite difference operator. Alternating minimization strategy The joint minimization of the criterion defined in Eq. (17) requires to cope with a highly non-linear function whose conditioning may be very bad and depends on the scaling of the parameters. We propose to solve the problem by an alternated minimization strategy, that is estimating each set of parameters given the others. Such a strategy consists in sequentially solving the following sub-problems: x(y, r , ν , µ x ) = arg min x≥0 C(x, y, z, ν, µ) = arg min x≥0 A x − r 2 W + µ x R x (x) ,(22a)y(x, r , ν , µ y ) = arg min y≥0 C(x, y, z, ν, µ) = arg min y≥0 B y − r 2 W + µ y R y (y) , (22b) ν (x, y, r ) = arg min ν ∈Ω C(x, y, z, ν, µ) = arg min ν ∈Ω m (x, y, ν ) − r 2 W ,(22c)z(r ⊕ , ν ⊕ , µ z ) = arg min z≥0 C(x, y, z, ν, µ) = arg min z≥0 A ⊕ z − r ⊕ 2 W + µ z R z (z) , (22d) ν ⊕ (z, r ⊕ ) = arg min ν ⊕ ∈Ω ⊕ C(x, y, z, ν, µ) = arg min ν ⊕ ∈Ω ⊕ m ⊕ (z, ν ⊕ ) − r ⊕ 2 W ,(22e) with: ∀x, A x = γ H (ν) y (F x), (23a) ∀y, B y = γ F x (H (ν)y), (23b) ∀z, A ⊕ z = γ h ⊕ (ν) (F ⊕ z), (23c) r = d − m ⊕ (z, ν ⊕ ), (23d) r ⊕ = d − m (x, y, ν ), (23e) m (x, y, ν ) = γ (H (ν) y) (F x) = A x = B y, (23f) m ⊕ (z, ν ⊕ ) = γ h ⊕ (ν) (F ⊕ z) = A ⊕ z.(23g) We enforce positivity constraints for the variables x, y, and z, while Ω and Ω ⊕ respectively denote the feasible set of parameters ν and ν ⊕ . Note that m (x, y, ν ) and m ⊕ (z, ν ⊕ ) defined in Eqs. (23f) and (23g) are the respective contributions of the star and companion. When the convex regularization defined in (21) is chosen and A WA , B WB , and A ⊕ WA ⊕ are invertible, each of the Problems (22a), (22b), and (22d) is strictly convex and thus has a unique solution which can be found by using existing algorithms 8 . This is another advantage of the alternated strategy. Since the original minimization problem (18) is not jointly convex with respect to all unknowns, only a local minimum is reached by the alternating minimization scheme, though. We solve for the two stellar components x and y following the alternated method proposed by Thé et al. (2020) to exploit the scaling indetermination of this problem (see Appendix C for details). This method is implemented by Algorithm 1 and takes Algorithm 1: FitStar -fit stellar parameters. Input: r ∈ R N , W ∈ R N×N , x [0] ∈ R N x , ν [0] , µ x > 0, and α 0 > 0. Output:x,ŷ, andν a local minimum of C in x, y, and ν . k = 0 µ y = 1 while not converged do while true do Update on-axis PSF y [k+1] =ŷ x [k] , r , ν [k] , α −2 k µ y Eq. (22b) α k+1/2 =α x [k] , y [k+1] , µ x , µ y Eq. (C.4) if k ≥ 1 or α k+1/2 ≈ α k then break α k = α k+1/2 Update star SED x [k+1] =x y [k+1] , r , ν [k] , α 2 k+1/2 µ x Eq. (22a) α k+1 =α x [k+1] , y [k+1] , µ x , µ y Eq. (C.4) Auto-calibration (optional) ν [k+1] ←ν x [k+1] , y [k+1] , r Eq. (22c) k ← k + 1 x ← α k x [k] y ← y [k] /α k ν ← ν [k] as inputs the residuals r = d − m ⊕ (z, ν ⊕ ) (i.e. the data without the contribution of the companion), the precision matrix W, initial calibration parameters ν [0] , the hyper-parameters µ x > 0 (hyper-parameter µ y is set to the arbitrary value 1 in Algorithm 1), initial estimates x [0] of the stellar SED, and initial estimate α 0 > 0 of the scaling parameter. Algorithm 1 deserves some remarks: 1. The initial stellar SED x [0] must be such that R x (x [0] ) > 0 to be able to apply formula (C.4) to compute the optimal scaling factor (i.e., a non-flat SED). The initial stellar SED can be provided by calibration data (see Appendix B); otherwise it can be computed from the science data d by the following weighted mean: ∀ j ∈ 1, N x : x [0] , j = n∈X j w n d n n∈X j w n(24) with w n = W n,n the n-th diagonal term of the precision matrix and: X j = n ∈ 1, N λ grd , j − λ n = min j ∈ 1,N x λ grd , j − λ n(25) the set of pixels whose nearest wavelength in the model grid is the j-th one. Since Algorithm 1 scales the final components x [k] and y [k] by the corresponding optimal scaling factor, α 0 = 1 is a natural choice for the initial scaling factor in subsequent calls to Algorithm 1 (to refine the solution or after having improved the other parameters). 2. The inner loop of Algorithm 1 avoids sensitivity to the initial scaling of the parameters (Thé et al. 2020). 3. The convergence criterion of Algorithm 1 is left unspecified. In our implementation, we chose to stop the algorithm when the relative change, in norm, between two consecutive iterates is smaller than 10 −3 . Algorithm 2: FitCompanion -fit companion parameters. Input: residuals r ⊕ ∈ R N , precision matrix W, initial off-axis PSF parameters ν [0] ⊕ , µ z > 0. Output:ẑ andν ⊕ , a local minimum of C in z and ν ⊕ given m (x, y, ν ) the model of the stellar contribution. k = 0 while not converged do Update companion SED z [k+1] =ẑ r ⊕ , ν [k] ⊕ , µ z Eq. (22d) Update off-axis PSF (optional) ν [k+1] ⊕ =ν ⊕ z [k+1] , r ⊕ Eq. (22e) k ← k + 1 z ← z [k] ν ⊕ ← ν [k]∀ j ∈ 1, N y : y [0] , j = n∈Y j w n d n n∈Y j w n(26) with: Y j = n ∈ 1, N ρ grd , j − ρ n = min j ∈ 1,N y ρ grd , j − ρ n(27) the set of pixels whose nearest angular position in the model grid is the j-th one. 5. When the SED z of the companion is not yet known, it is sufficient to call Algorithm 1 with the weights of the pixels the most impacted by the companion set to zero (we write the corresponding precision matrix W ) to estimate the components x and y of the stellar leakages without introducing a significant bias due to the contribution of the companion. Algorithm 2 (FitCompanion) implements an alternated strategy to estimate the parameters z and ν of the companion SED and its off-axis PSF at the reference wavelength. It takes as inputs the residuals r ⊕ = d − m (x, y, ν ) (i.e. the data without the contribution of the star) and their respective weights W, the hyper-parameter µ z > 0 and an initial estimate ν [0] ⊕ ∈ Ω ⊕ of the parameters of the off-axis PSF at the reference wavelength. These latter parameters can be given by the calibration described in Appendix B. Algorithm 2 also deserves some remarks: 1. The outputs of the algorithm only depend on the residual data r ⊕ defined in Eq. (23e) that need to be computed only once (on entry of the algorithm and not at each iterations). 2. Like for Algorithm 1, various stopping criteria may be implemented to break the loop. 3. Like for Algorithm 1, we can use the VMLM-B algorithm (Thiébaut 2002) to solve Problem (22d) to estimate z under a non-negativity constraint. ⊕ , and hyper-parameters µ x > 0 and µ z > 0. Output:x,ŷ,ẑ,ν , andν ⊕ a local minimum of C. µ y = 1 z [0] = 0 α = 1 k = 0 while not converged do if k = 0 then Hide companion W = W Eq. (28) else Account for companion W = W Update star leakage model r [k] = d − m ⊕ z [k] , ν [k] ⊕ Eq. (23d) x [k+1] , y [k+1] , ν [k+1] = FitStar r [k] , W , x [k] , ν [k] , µ x ,α Update companion model r [k+1] ⊕ = d − m x [k+1] , y [k+1] , ν [k+1] Eq. (23e) z [k+1] , ν [k+1] ⊕ = FitCompanion r [k+1] ⊕ , W, ν [k] ⊕ , µ z k ← k + 1 (x,ŷ,ẑ,ν ,ν ⊕ ) ← x [k] , y [k] , z [k] , ν [k] , ν [k] ⊕ In both algorithms, there are optional self-calibration steps performed by solving Problem (22c) in Algorithm 1 (FitStar) and Problem (22e) in Algorithm 2 (FitCompanion) to estimate the parameters of the on-axis and off-axis PSFs. These minimizations can be carried out by a derivative-free minimization algorithm. When there is a single calibration parameter, we use Brent (2013) Fmin algorithm; if there are several parameters, we use one of Powell's derivative-free methods Newuoa or Bobyqa (Powell 2006(Powell , 2009) depending on the constraints defined by Ω. The Exospeco algorithm FitStar (Algorithm 1) and FitCompanion (Algorithm 2) are the building blocks of the Exospeco method given in Algorithm 3 for estimating all unknowns. Only a few additional remarks are worth being made: 1. For the first estimation of the stellar leakage parameters, it is beneficial to define a masked version W of the precision matrix of the data to avoid a significant bias of the first estimates due to the signal from the companion, which would slow down the convergence of Algorithm 3. The masked precision matrix is simply given by: W = diag(w ),(28) where the weights w are those of the precision matrix W of the data except that they are set to zero for the pixels that are the most impacted by the companion: ∀n ∈ 1, N : w ,n = 0 if γ n |ρ n − ρ ⊕ | ≤ τ w n otherwise(29) with τ > 0 the angular half-width at the reference wavelength of the impacted region. In practice τ is taken to be 2-3 times σ [0] ⊕ the initial angular standard deviation of the off-axis PSF at the reference wavelength. 2. The model of the stellar leakage only depends on either µ x or µ y , the other being arbitrarily chosen. For this reason Algorithm 3 takes as inputs only 2 hyper-parameters µ x and µ z , the remaining hyper-parameter being set to µ y = 1. 3. After extracting the companion's spectrum by Exospeco Algorithm 3, it is possible to express it as a contrast relative to the host star which can be multiplied by a reference spectrum of the star to get rid of the atmospheric absorption (see Appendix B). 4. The auto-calibration steps in FitStar (Algorithm 1) and Fit-Companion (Algorithm 2) are optional and consist in the resolution of Problems (22c) and (22e). As these problems are non-convex, activating the auto-calibration at the beginning of the method can lead to a local minimum. To avoid such a behavior, it is possible to start the self-calibration of ν and ν ⊕ only after a few iterations of Exospeco (Algorithm 3). 5. Controlling the number of inner iterations to solve each subproblem could be done by changing the value of the stopping parameter (cf. remark 3 in Section 3.4): the smaller the more inner iterations are needed and conversely. But this is expected to also impact the number of outer iterations. Owing to the modest amount of time (2-3 min.) taken by our implementation of Exospeco to solve the entire problem, we did not investigate whether the algorithm can be effectively accelerated by changing and keep the value = 10 −3 suggested before. Calibration The direct model in Eq. (15) assumes known the physical coordinates (ρ n , λ n ) of each pixel n of the detector. We describe in this section a consistent approach to derive the spectro-angular dispersion laws of the instrument from calibration data. Calibration data Calibration data takes the form of an image such as the one shown in Fig. 5 which is obtained by illuminating the spectrograph slit with N λ laser sources 9 . This produces N λ monochromatic lines on the detector, each being interrupted by the coronagraphic mask. The calibration image d cal is of size I × J and, for the calibration procedure, we denote by n ∼ (i, j) the one to one mapping between the pixel number n and its indices i ∈ 1, I and j ∈ 1, J along the first and second dimensions of the detector. The calibration image shall have been pre-processed to compensate for bias and non-uniform response of the detector. Furthermore, we assume known a mask of valid pixels: w msk,i, j = 1 if the pixel (i, j) is valid, 0 else.(30) We consider a pixel as being invalid if its value cannot follow the assumed direct model given in Eq. (15). Invalid pixels include pixels outside the field of view, pixels under the coronagraphic mask or close to this mask, and defective pixels whose level does not linearly depend on the illumination. Figure 6 shows the mask of valid pixels for the HR 3549 data: the field of view and the coronagraphic mask are outlined by the two green trapezes while the defective pixels are marked by green dots. 9 N λ = 6 at wavelengths 0.9877 µm, 1.1237 µm, 1.3094 µm, 1.5451 µm, 1.73 µm, and 2.015 µm for SPHERE/IRDIS Dispersion laws There are two dispersion laws to calibrate: Λ(i, j) for the wavelength and (i, j) for the separation angle along the slit. To determine the best approximation of these spectro-angular dispersion laws, we compared three models: -The standard model which assumes that the dispersion laws are uni-dimensional polynomials with spectral and angular directions aligned with the detector axes: Λ sta (i, j) = P λ p=0 a p j p ,(31a)sta (i, j) = P ρ p=0 s p i p ,(31b) with P λ and P ρ the degrees of the polynomials and {a p } p∈ 0,P λ and {s p } p∈ 0,P ρ their coefficients. For P ρ = 1 and P λ = 3 − 5, the standard model reproduces what is done in the software by Vigan et al. (2008) usually used to process SPHERE/LSS data. -A model of medium complexity, also assuming 1D polynomials for the dispersion laws but accounting for misalignment angles φ λ and φ ρ ≈ φ λ +90 • respectively between the spectral and angular directions and the detector axes: Λ med (i, j) = P λ p=0 a p (i sin φ λ + j cos φ λ ) p ,(32a)med (i, j) = P ρ p=0 s p (i sin φ ρ + j cos φ ρ ) p .(32b) Note that taking φ λ = 0 • and φ = 90 • yields the standard model. -A more complex model which assumes 2D polynomials for the dispersion laws and, depending on the degree of these polynomials, can account for more complex image distortions than a simple rotation: Λ Exospeco (i, j) = P λ p 1 =0 P λ −p 1 p 2 =0 a p 1 ,p 2 i p 1 j p 2 ,(33a)Exospeco (i, j) = P ρ p 1 =0 P ρ −p 1 p 2 =0 s p 1 ,p 2 i p 1 j p 2 .(33b) To summarize, the considered dispersion laws are polynomials of respective degree P λ and P ρ . Their calibration amounts to fitting their coefficients a and s given the calibration image d cal as explained in the next sub-sections. Calibration of the spectral dispersion law Λ To calibrate the spectral dispersion law Λ, we extract from the calibration image d cal (see Fig. 5) N λ lists of pixel coordinates following the path of each spectral line on the detector and estimate the coefficients a by a least squares fit: a = arg min a N λ =1 (i, j)∈C (φ λ ) λ − Λ(i, j) 2(34) where C (φ λ ) denotes the list of, possibly fractional, pixel coordinates (i, j) along the -th spectral line on the detector. Since Λ(i, j) linearly depends on the coefficients a, the solutionâ of the above problem has a closed form expression (Lawson & Hanson 1974) that is easy to compute. To extract the paths C (φ λ ) of the spectral lines, we first compute a transverse projection q ⊥ (φ λ ) of the calibration image d cal tuning the projection angle φ λ so as to maximize the peak values in of the resulting projection. This transverse projection is plotted in red in the top panel of Fig. 5 and corresponds to φ λ ≈ 0 • for the considered calibration data. Equations (A.1) in Appendix A.1 formally define how we carefully compute the projection avoiding invalid pixels. We then use the procedure described in Appendix A.2 to locate the position of the N λ most significant peaks in the transverse projection q ⊥ (φ λ ) which can be seen as a mean cross section of the spectral lines. Finally, we use the method de-scribed in Appendix A.3 to extract the coordinates of the points defining the N λ paths C (φ λ ). These coordinates are given by the centers of gravity (again accounting for invalid pixels thanks to the mask) of the calibration data in small sliding rectangular windows along each spectral lines (see Appendix A.3 for details). Calibration of the angular dispersion law To calibrate the angular dispersion law , we extract from the calibration image d cal (see Fig. 5) the positions of the edges of the coronagraphic mask for each of the N λ spectral lines and estimate the coefficients s of the polynomial and the width ∆ρ of the mask by a least squares fit: (φ ρ ,∆ρ,ŝ) = arg min φ ρ ,∆ρ,s N λ =1 i down , j down + ∆ρ/2 2 + N λ =1 i up , j up − ∆ρ/2 2(35) where i down , j down and i up , j up denote the coordinates of the edges of the coronagraphic mask respectively on the downhill and uphill sides along the profile of the -th spectral line. To solve this problem, we exploit that the criterion is quadratic in the unknowns s and ∆ρ which thus have a closed-form solution (Lawson & Hanson 1974) which depends on φ ρ . Replacing this closed-form solution in the criterion yields an uni-variate objective function that only depends on φ ρ and which we minimize by Brent (2013) Fmin method starting at φ ρ = φ λ + 90 • . As explained in Appendix A.4, the coordinates of the edges of the coronagraphic mask for the -th spectral line are obtained from the longitudinal profile q // (φ λ ) of the line which is the weighted projection, in a direction perpendicular to the transversal projection q ⊥ (φ λ ), of the calibration data in a window encompassing the line. The longitudinal profile q // (φ λ ) for the second ( = 2) line is plotted in green in the left panel of Fig. 5 and the corresponding window is outlined in green in the central panel of Fig. 5. Comparison of the calibration models To compare the calibration models considered in Section 4.2, we apply Exospeco (Algorithm 3) on a scientific dataset of the star HR 3549 observed in MRS mode of SPHERE/IRDIS on 2015/12/28. Figure 7 shows the residuals r = d − m, that is the difference between the data and their model, computed for the different spectro-angular dispersion laws. This figure shows that the Root Mean Square (RMS) of the residuals are significantly reduced when using more flexible calibration models than the standard one. The improvement brought by the medium complexity model compared to the standard model proves that accounting for a slight angular misalignment between the spatial and spectral directions and the detector axes is important. Compared to the medium model, the complex model is able to account other image distortions than a simple rotation and thus achieves a better suppression of the stellar leakages. These results motivate the choice of the complex dispersion model in Exospeco to reduce the RMS level of the residual stellar leakages by a factor of ∼ 2 compared to the standard model and should therefore result in a better extraction of the companion contribution. Remember that the simple standard calibration model is similar to what is usually done by others for these data. Validation and tuning of the method To fully validate the method, we propose in this section a study of the method on both real data and data where a synthetic companion was injected. This study allows us to both evaluate the modeling of the stellar leakages, and so its subtraction in the residuals, and the extraction of the companion, by comparing with a ground truth spectrum. Reduction of the self-subtraction One of the feature of Exospeco is that it jointly estimates the contributions of the star and of the companion whose parameters are iteratively refined until convergence. Figure 8 shows the residuals close to the companion (in the region outlined by the black rectangle in the bottom of Fig. 7) with the complex model of the spatial and spectral dispersion laws in two cases: after the first outer iteration of Exospeco and after convergence of the algorithm. In the first outer iteration of Exospeco, the model of the stellar leakages is estimated by masking the region most impacted by the companion, see Eq. (28), which is similar to what is done by conventional methods. In all other outer iterations of Exospeco, the contribution of the other component is taken into account when fitting a given component (star or companion). As shown by the bottom panel of Fig. 8, there is a noticeable bias in the estimated companion's SED after the first outer iteration. This so-called self-subtraction bias is mostly avoided by the proposed alternating strategy. figure), medium (center), and complex (bottom) calibration models. The RMS values of the residuals are given for each model. The iso-levels of ρ/λ which are approximately followed by the dispersed stellar speckles are plotted as green dashed lines. The position of the companion at the different wavelengths (i.e., at ρ = ρ ⊕ ) is plotted as an orange dashed line. Fig. 7) and for the complex model of the spatial and spectral dispersion laws after one iteration (top) and after convergence (middle) of Expospec. The RMS of the residuals in this region are significantly reduced after convergence. The orange dashed line indicates the position of the companion at the different wavelengths. The SEDs of the companion (including atmospheric absorption) extracted from these residuals are plotted in the bottom-most panel. Tuning of the regularization parameters of this component by a single hyper-parameter, the other hyperparameter being held fixed. Thanks to this, the solution found by Exospeco only depends on 2 hyper-parameters, one for the star, say, µ x (while µ y = 1 is imposed) and one for the companion, µ z . In this section, we highlight the incidence on the companion SED extracted by Exospeco of these remaining hyper-parameters using the same scientific data set as in Section 4.5. Figure 9 shows the SEDs of the companion estimated by Exospeco for different values of the star regularization hyperparameters (µ y = 1 and µ x = 10 −3 , 10, and 10 5 ). For such a large range of values, the differences between the extracted companion SEDs are smaller than 1%. The stellar regularization hyperparameters have thus a limited impact on the resulting companion SED. The tuning of µ x can thus reasonably be done by visual inspection. On the contrary, as Fig. 10 shows, the hyper-parameter µ z has a strong impact on the resulting companion SED. This is expected as µ z directly tunes the strength of the smoothness constraint for the companion SED z. This hyper-parameter has thus to be carefully chosen to find the best compromise between a solution that is too smooth (e.g. for µ z = 10 7 in Fig. 10) or too noisy (e.g. for µ z = 10 2 in Fig. 10). It is worth noticing that the correct value of µ z strongly depends on the considered data, so µ z = 10 5 , which seems to be a good choice for the HR 3549 data (see Fig. 10), should not be considered as a universal value. Many methods have been proposed to automatically tune the hyper-parameter(s) of an inverse problem: the Generalized Cross Validation (CGV, Golub et al. 1979), Stein's Unbiased Risk Fig. 10. Top: Profiles of the companion SED z for different levels of the companion regularization (µ z = 10 2 in dashed orange, µ z = 10 5 in blue, and µ z = 10 7 in dotted green). Bottom: differences between the profile for µ z = 10 5 and the profiles for µ z = 10 2 (dashed orange) and µ z = 10 7 (dotted green). For all these results, the stellar hyperparameters are µ x = 10 and µ y = 1. Estimate (SURE, Stein 1981), the hierachical Bayesian method (Molina 1994), or the L-curve (Hansen & O'Leary 1993) to mention a few that could be used with our extraction algorithm. Implementing and testing these methods for Exospeco is out of the scope of this paper. However, since the companion SED found by Exospeco does not strongly depend on the tuning of the stellar regularization, our method is mostly driven by a single hyperparameter, µ z , the level of the regularization for the companion SED. This greatly reduces the complexity of tuning the Exospeco algorithm. Extraction of simulated spectrum in real data To validate the Exospeco method, we injected the contribution of a synthetic companion in existing SPHERE/IRDIS MRS data d of the star HIP 65426 observed on 2019-05-20. Although HIP 65426 star hosts a planet (Chauvin et al. 2017;Carter et al. 2022), the frame was selected for the derotation angles hiding the planet outside the slit. The off-axis PSF h ⊕ of the synthetic companion follows the model in Eqs. (13) and (14c) with σ ⊕ set to match the diffraction limit of the telescope at the reference wave- The ground truth SED of the synthetic companion is z gt = χ x flux where χ > 0 is the mean contrast of the companion relative to the star (without a coronagraph) and x flux is the SED of the star HIP 65426 calibrated as explained in Appendix B. We use a constant contrast for all wavelengths (i.e., the SED of the star and of the companion are the same, up to the contrast χ). Figure 11 shows examples of generated data with a synthetic companion whose contrast with respect to the star is χ = 2 · 10 −4 and which is injected at different angular separations ρ ⊕ − ρ . To assess the quality of the extracted companion's SEDẑ, we compute the following relative error: q = N z j=1 z gt, j −ẑ j N z j=1 z gt, j .(36) In the following tests, the value of µ z , the regularization level of the companion's SED, has been tuned so as to minimize the relative error q. Figure 12 plots the relative error q for synthetic companions injected at angular separations ρ ⊕ − ρ ranging from 200 mas to 1850 mas and with contrasts χ = 3 · 10 −5 , 2 · 10 −4 , and 2 · 10 −3 . Clearly, the quality of the recovered SEDs degrades as the companion gets closer to the mask. This is expected because, when getting closer to the mask, not only are the stellar leakages brighter (hence causing more photon noise in the residuals) but the approximation by the assumed off-axis PSF model also worsens. For angular separations larger than ∼ 600 mas and for all considered contrasts, the quality of the recovered SEDs improves as the separation increases until a plateau is reached at ρ ⊕ − ρ ∼ 1400 mas where the dominant source of nuisance is the readout noise. Figure 13 shows examples of recovered companion SEDsẑ at angular separations ρ ⊕ − ρ = 273 mas (A), 890 mas (B), and 1353 mas (C) for the same contrasts χ as in Fig. 12. Figure 13 confirms that the relative error q does reflect the ability of our method to reliably recover the companion SED. When q ≤ 0.1 (the green curves for angular separations B and C and the orange curve for angular separation C), all the features of the SED are correctly recovered. For q ∼ 0.2 (the green curve for case A, the orange curve for case B, and the blue curve for case C), the global shape of the SED is restored but with small spectral features smoothed out and some photometric biases. These cases prove that it is possible to extract a coarse but still exploitable SED for bright companions quite close to the mask, typically χ ≥ 10 −3 for ρ ⊕ − ρ ∼ 250 mas, from a single MRS exposure. The angular separation must be larger for fainter companions; for example, ρ ⊕ −ρ ≥ 1200 mas for χ ∼ 2 · 10 −5 . The photometric biases in the most difficult cases (the green curve in case A, the orange one in case B, and the blue one in case C) clearly indicates that the removal of the modeled stellar contribution leaves non-negligible residuals compared to the companion. A possible improvement could be to use a more complex model of the on-axis PSF and consider more than one mode in the series expansion of Eq. (3). To summarize the performances of the current version of Exospeco for a single data frame of the HIP 65426 observations, Fig. 14 plots the minimal contrast needed to achieve a given relative error q as a function of the angular separation. The figure shows that by tolerating a relative error as high as q = 0.3, a companion with a contrast up to χ ∼ 2 · 10 −5 can be characterized. In our conclusions, we explain how to extend Exospeco to jointly process several data frames in order to increase the sensitivity of the algorithm. Fig. 13. Examples of recovered companion SEDsẑ in the same conditions as in Fig. 11 and Fig. 12 for synthetic companions injected at angular separations ρ ⊕ −ρ = 273 mas (A), 890 mas (B), and 1353 mas (C) with contrasts χ = 3 · 10 −5 (blue curves), 2 · 10 −4 (orange curves), and 2 · 10 −2 (green curves). The ground truth SED z gt is plotted in dashed lines. The normalized residuals are plotted below each panel. 200 400 600 800 1000 1200 1400 1600 1800 angular separation [mas] 10 5 10 4 10 3 q = 10 1 q = 2.10 1 q = 3.10 1 Fig. 14. Minimal contrast χ required to achieve a given relative error q as a function of the angular separation. The conditions are the same as in Fig. 11 and Fig. 12. Comparison with TSVD extraction We compared Exospeco to a standard approach based on the TSVD method described in Section 2.2 to remove the stellar leakages. Figure 15 shows the companion SED extracted from the HR 3549 data by Exospeco and by the TSVD approach. For the latter method, the SED of the companion was extracted by local averaging in a 7 pixel height sliding window along the companion signal in the residual image given by Eq. 7 and shown in Fig. 3. In both cases, the same complex calibration model of the spectral and angular dispersion laws described in Appendix A has been used. In spite of this identical calibration, the two extracted SEDs are notably different. Thanks to the optimal extraction in the maximum likelihood sense and to the spectral regularization, the SED extracted by Exospeco is smoother and less noisy. At a coarser resolution, the two SEDs display quite different spectral features. However, without a known ground truth, the two SEDs cannot be ranked. For this reason, we also compared the results given by the two methods on a synthetic injection done as described in Section 5.3. Figure 16 clearly demonstrates that not only does Exospeco produce less noisy results, but that they also better reflect reality. Conclusion In this paper we presented a novel algorithm, Exospeco, to extract the spectrum of a companion from high-contrast long-slit spectroscopic data. The most challenging part of such a process- Fig. 16. Comparison of the SEDs extracted from semi-synthetic data by Exospeco and by a standard TSVD method. The angular separation and the contrast of the injected companion are respectively ρ ⊕ − ρ = 785 mas and χ = 2×10 −4 . The green dashed curve represents the ground truth injected spectrum which is that of HIP 65426 multiplied by χ. ing is to disentangle the signal of interest from the stellar leakages which are much brighter. Compared to existing methods, our algorithm avoids any transform of the data, whether it is to align the speckles of the stellar leakages at all wavelengths or to fix defective pixels. Exospeco has also the advantage of jointly extracting the parameters describing the stellar leakages (the star spectrum and the on-axis PSF), the companion spectrum, the offaxis PSF, and, optionally, some calibration parameters. By using non-uniform statistical weights for the data pixels, our approach is optimal in the maximum likelihood sense, it takes into account all available measurements and consistently treats defective pixels as missing data. The joint optimization problem having no closed-form solution, we proposed an alternating minimization strategy which has proven to be effective. In spite of the numerous parameters coming into play in the algorithm, the outputs of the method are, in practice, mostly driven by a single hyperparameter that tunes the level of regularization of the companion SED. Although it is not directly part of the spectrum extraction algorithm, we have shown that careful calibration of the instrument is critical to get rid of the contamination by the stellar leakages. For that purpose, we described a refined calibration method of the spectral and spatial dispersion laws from available calibration data. In particular, SPHERE/LSS data present a misalignment of the principal directions of dispersion with the detector axes as well as a geometrical shear. If not accounted for, we show that these distortions have a detrimental impact on the result of the processing, whether it is by Exospeco or by the current state-of-the-art method. A few remaining calibration parameters that may depend on the observing conditions, such as the off-axis PSF size and the precise locations of the star and of the companion along the slit, can be optionally adjusted by a selfcalibration procedure built into Exospeco. Thanks to this calibration step, our method significantly reduces the self-subtracting bias by better disentangling the stellar leakages component from the companion component. A Julia (Bezanson et al. 2017) implementation of Exospeco is freely available at https://github.com/SJJThe/ Exospeco, while an implementation of the calibration method described in Section 4 and Appendix A is at https://github. com/SJJThe/ExospecoCalibration. Based on tests carried on empirical long-slit spectroscopic data and on injections of a synthetic companion signal in these data, we demonstrated that the proposed approach effectively avoids the self-subtraction bias, even very close to the coronagraphic mask. We provided curves to predict the minimal contrast required to achieve a given quality of extraction of the com-panion SED. Reliable extraction of a companion SED can be achieved from a single data frame at contrasts as low as a few 10 −5 . The proposed method could boost the characterization of known (faint) exoplanets at a spectral resolution substantially higher than currently possible with SPHERE IFS (R ∼ 35 − 50) and for contrasts much better than achievable with IRDIS MRS using state of the art methods. By capturing more efficiently the stellar contamination, the method we propose does not require independent and thus imperfect calibration of the speckles by rotating the slit to hide the planet signal. This will typically gain at least 50% telescope time while reaching, and even surpassing, the same contrast limit. This new method also paves the way to combining polarimetry and spectroscopic measurements with IRDIS LSS mode (R. Holstein private communication). Being based on an inverse problems framework, Exospeco is very flexible and can be adapted to various kinds of data (such as data sequences or data from other instruments). An example of such an extension of Exospeco is the joint processing of multiple frames that can be done as follows. Assuming T LSS exposures d 1 to d T of the same object are collected during a night, they can be combined into a single criterion that extends Eq. (17): C(x, y 1 , y 2 , . . . , y T , z, ν, µ) = T t=1 d t − m(x, y t , z, ν) 2 W t + µ x R x (x t ) + µ y T t=1 R y (y t ) + µ z R z (z),(37) where the statistical independence of noise between frames is considered (a natural assumption). This criterion can be minimized in x, y 1 , . . . , y T , z, and ν following the same alternating method as described in Section 3.4, only with more steps in order to estimate the on-axis PSF at each of the T frames. Such a joint processing has the potential to improve the estimation of companion SEDs and push further the achievable contrast limit. Finally, to better disentangle the stellar leakages from the companion spectrum, the model of the on-axis PSF could be improved by taking into account more spatial modes of the series expansion in Eq. (3). Indeed, as demonstrated in Devaney & Thiébaut (2017), accounting for more such modes significantly improves the modeling of the stellar leakages, especially near the coronagraphic mask. Such an improvement would not call into question the founding principles of Exospeco, but would require to adapt the optimization strategy. iso-angular distance (green) curves, on a zoom in of the HR3549 dataset. This figure highlights how well our proposed models for the dispersion laws are following the speckles, compared to the standard model. A strong shear effect due to the dispersive elements is visible and taken into account by our model. For all these models, we took polynomials of degrees P λ = 5 and P ρ = 1 for the spectral and spatial dispersion laws. Appendix C: Exploiting the scaling indetermination The estimated componentsx,ŷ,ẑ, andν of the direct model defined in Eq. (15) depend on hyper-parameters which include the regularization weights µ x , µ y , and µ z . In this Appendix, we show how to adapt the approach of Thé et al. (2020) to reduce the effective number of regularization parameters and also accelerate the minimization. We first note that the regularizations considered for the stellar components are homogeneous functions of degree 2, i.e. the following property holds: R u (α u) = α 2 R u (u), (C.1) whatever the component u = x or y considered and α ≥ 0. The contribution of the star, the first right-hand side term in Eq. (15), is a bilinear function of the parameters x and y. As a consequence: m(α x, y/α, z, ν) = m(x, y, z, ν), (C.2) holds for any scaling factor α > 0. Combining the properties in Eqs. (C.1) and (C.2) with the definition of the objective function in Eq. (17) and that of the regularization in Eq. (19), it can be seen that for any α > 0: C(α x, y/α, z, ν, µ x , µ y , µ z ) = C(x, y, z, ν, α 2 µ x , α −2 µ y , µ z ). (C.3) In other words, scaling the unknowns x and y without changing the model is equivalent to scaling their regularization weights. Exploiting this, it is possible to compute an optimal scaling factor: α(x, y, µ x , µ y ) = arg min α>0 C(α x, y/α, z, ν, µ x , µ y , µ z ) = arg min α>0 C(x, y, z, ν, α 2 µ x , α −2 µ y , µ z ) = arg min α>0 α 2 µ x R x (x) + α −2 µ y R y (y) = µ y R y (y) µ x R x (x) 1 4 . (C.4) Plugging this expression in the definition of the criterion yields: C + (x, y, z, ν, µ x , µ y , µ z ) = min α>0 C(α x, y/α, z, ν, µ x , µ y , µ z ) = d − m(x, y, z, ν) 2 W + µ x,y R x,y (x, y) + µ z R z (z) (C.5) where: R x,y (x, y) = R x (x) R y (y), (C.6) and : µ x,y = 2 √ µ x µ y . (C.7) This shows that the regularizations of the stellar components x and y are entangled and that the effect of these regularizations on the shape of these components is effectively controlled by a single hyper-parameter, here µ x,y . As shown by Eq. (C.4), the ratio µ x /µ y of the hyper-parameters controls the scaling, not the shape, of x and y, one of the two can be fixed. This saves us the hassle of adjusting both parameters at the same time. Fig. 2 . 2Warped HR 3549 image. This figure shows the bottom half of the data shown in Fig. 3 . 3Bottom half of the residual image r ⊕ for the HR 3549 data with stellar leakages estimated by the TSVD method as defined in Eq. 7 and with the warped image shown inFig. 2. Compared to the original data shown in ⊕ 4 . 4Although they represent very different physical quantities, the problem is quite symmetric in variables x and y. Thus a variant of Algorithm 1 can be easily implemented to start with an initial estimate y [0] of the stellar on-axis PSF at the reference wavelength instead of an initial estimate x [0] of the stellar SED. For the very first run, this variant of Algorithm 1 is started with the weighted average of the on-axis PSF defined by: Fig. 5 .Fig. 6 . 56Calibration data for the SPHERE/IRDIS instrument and for the observations of HR 3549 on 2015/12/28. Central panel: calibration image. Left and top panels: projections of the calibration data along the 2nd spectral line (in green) and across all spectral lines (in red). Valid pixel mask for the HR 3549 data observed on 2015-12-28 in MRS mode. Fig. 7 . 7Residuals between the HR 3549 scientific data and the model of the stellar leakages, assuming a standard (top Fig. 8 . 8As described in Appendix C, the fact that the model of the star leakages is bi-linear makes it possible to tune the regularization Residuals between the HR 3549 scientific data and the model of the stellar leakages near the companion (defined by the black rectangle in the bottom of Fig. 9 . 9Top: Profiles of the companion SED z, for different levels of the stellar hyper-parameter µ x and with µ y = 1 and µ z = 10 5 . Bottom: differences between the profile for µ x = 10 and the profiles for µ x = 10 −3 (dashed orange) and µ x = 10 5 (dotted green) Fig. 11 . 11Scientific data of HIP 65426 with a synthetic companion whose contrast is χ = 2 · 10 −4 relative to the star and injected at angular separations ρ ⊕ −ρ = 273 mas (A), 890 mas (B), and 1353 mas (C) indicated by the arrows. length λ ref and with different angular positions ρ ⊕ on the side of the coronagraphic mask where no companion was detected. Fig. 12 . 12Relative error q defined in Eq. (36) for synthetic companions injected in the scientific data of HIP 65426 (with the same spectra as the host star) as a function of the angular separation ρ ⊕ −ρ and for contrasts χ = 3 · 10 −5 (blue), 2 · 10 −4 (orange), and 2 · 10 −2 (green). The grayed area represents the region invalidated by the coronagraphic mask. The angular separations of the three cases presented inFig. 11are highlighted by the dashed lines labeled A, B, and C. Fig. 15 . 15Comparison of the SEDs extracted from the HR 3549 data by Exospeco and by a standard TSVD method. See text for details. Fig. A. 1 . 1Iso-wavelength curves at the wavelengths of the calibration sources (blue lines) and iso-angular distance of the center of the coronagraphic mask of (green lines) presented on top of the calibration data d cal . The upper panel presents the results for the simple model, the central panel shows the results for the medium model, while the bottom panel shows the results of using the complex model. Fig. A. 2 . 2Magnified images of the two regions outlined by the yellow and purple rectangles for the three models described in Section 4.2. To best see the differences between models, the magnifications are different in the two dimensions. Fig . A.3. Iso-wavelength and iso-angular distance superposed to the HR 3549 data observed on 2015-12-28 with IRDIS in MRS mode. Fig . B.1. Calibration of the SED of the star HIP 65426. Top: Calibration image d flux observed on 2019-05-20 with the MRS mode of SPHERE/IRDIS. Bottom: SED of the star extracted by FitStar (Algorithm 2) and corrected from the density filter. λ/∆λ = 35 for the LRS mode or λ/∆λ = 400 for the MRS mode Article number, page 1 of 19 arXiv:2306.03467v1 [astro-ph.IM] 6 Jun 2023 these SEDs are affected by the chromatic transmission of the atmosphere and of the instrument considered as a simple matrix 4 e.g., a simple separable 2-D interpolation Article number, page 3 of 19Pre-print version, article under review In practice, the interpolation operators are very sparse and only their non-zero entries need to be stored or computed on the fly.6 We use boldface lowercase letters to denotes vectors, that is quantities that depend on a single index, and boldface uppercase letters to denotes linear operators, that is quantities that depend on two indices. Article number, page 5 of 19Pre-print version, article under review For example, in the unconstrained case and with quadratic regularizations, the solution of one of these sub-problem has a closed-form expression. Otherwise, each of these sub-problems can be solved by optimization algorithms such as quasi-Newton methods with bound constraints (e.g.,Thiébaut 2002). Article number, page 7 of 19Pre-print version, article under review Article number, page 10 of 19 Samuel Thé et al.: Characterization of stellar companion from high-contrast long-slit spectroscopy data Article number, page 14 of 19 in the sense that it contains at least one valid pixel Article number, page 17 of 19 Pre-print version, article under review Article number, page 18 of 19 Appendix A: Calibration of the spectro-angular dispersion lawsThis appendix provides some details about the methods used for the calibration of the spectro-angular laws described in Section 4 and some figures to support the results discussed in Section 4.5. The considered calibration data d cal is the image in the central panel ofFig. 5.Appendix A.1: Transverse projectionTo locate the positions of the spectral lines, we compute a weighted transverse projection of the calibration image d cal :with weights given by:and for a projection angle φ λ chosen so as to maximize the peak values of the resulting projection (plotted in red in the top panel ofFig. 5for φ λ ≈ 0 • ). In practice, we take ϕ proj (t) = max(1 − |t|, 0), the linear B-spline, as the interpolating function for the projection. Note that, thanks to the weighting by the mask of valid pixels w msk defined in Eq. (30), invalid pixels have no incidence on the computed projection.Appendix A.2: Detection of the spectral peaksWe use Algorithm 4 with tolerance parameter δ ⊥ = 10 pixels to find the N λ most significant peaks in the transverse projection q ⊥ (φ λ ) ∈ R N q computed according to Eq. (A.1).Algorithm 4: Find the most significant peaks Input: N λ , q ⊥ (φ λ ), and δ ⊥ . Output: P(φ λ ). P ← ∅ start with an empty list z = q ⊥ (φ λ ) copy profile in workspace array: Extraction of the paths of the spectral linesGiven the projection angle φ λ and the list P(φ λ ) of the N λ most significant peaks in the transverse projection q ⊥ (φ λ ), we build the -th spectral path C as a list of points along the -th spectral line. The coordinates i path ,m , j path ,m of the m-th such point are given by computing the center of gravity of the calibration data in a small rectangular window W ,m (φ λ ) sliding along the considered spectral lines:computed for all non-empty 10 sliding window W ,m of size ∼ 1 pixel along the spectral line and 2 δ ⊥ + 1 pixels in the perpendicular direction:where k is the -th index in the list P(φ λ ) of the N λ most significant peaks in the transverse projection q ⊥ (φ λ ). Again note that, thanks to the weighting by the mask of valid pixels, invalid pixels have no incidence on the computed coordinates. In practice, we use the same value for the half-width of the sliding windows and for the minimal separation between peaks in the transverse projection, that is δ ⊥ = 10 pixels for the considered calibration data.Appendix A.4: Detection of the edges of the spectral bandsGiven the projection angle φ λ and the list P(φ λ ) of the N λ most significant peaks in the transverse projection q ⊥ (φ λ ), we compute the longitudinal profile of each spectral line as a the following weighted projection:with weights given by:and where k is the index along the projection and D (φ λ ) is a narrow rectangular window (in green in the central panel ofFig. 5) to isolate the pixels of the calibration image d cal impacted by the considered spectral line:with, as before, δ ⊥ ≈ 10 pixels the half-width of the region. Detecting the edges of the central hole due to the coronagraphic mask in the resulting profile (plotted in green in the left panel ofFig. 5)can be done by a quite simple procedure. Each value at index k of the profile is compared with the next one up to a threshold value τ. If q // ,k τ q // ,k+1 and q // ,k < q // ,k+1 , then an ascending edge is detected. If q // ,k τ q // ,k+1 and q // ,k > q // ,k+1 , it is a descending edge. From the four edges detected in the -th spectral line (indicated by the crosses in the left panel ofFig. 5), the second and third ones correspond to the coronagraphic mask. Retrieving these edges in C (φ λ ) yields the coordinates i down , j down and i up , j up required in Section 4.4 for the calibration of the angular dispersion law.Appendix A.5: Results on the calibration dataThe results of the proposed calibration models are displayed inFig. A.1, as blue lines for the spectral law and green lines for the spatial law. In practice, degrees P λ = 5 and P ρ = 1 (limited by the small number of points for ρ) were chosen. The three models described in Section 4.2 are tested, that is a standard model, one of a medium complexity and one more complex.As can be seen on the different zooms shown inFig. A.2, 2D polynomials are needed to explain local distortions. Choosing this model, we plot onFig. A.3 some iso-wavelength (blue) andAppendix B: Calibration of the contrastIn order to express the SED of the companion in terms of contrast with respect to the star, we use specific calibration data d flux (shown inFig. B.1 top)for which the star is placed in the spectrograph slit but shifted away from the coronagraphic mask and with a neutral density inserted in the optical path to avoid detector saturation. Applying FitCompanion (Algorithm 2) to d flux and dividing the resulting SED by the spectral transmission of the neutral filter yields the SED of the star x flux (shown inFig. B.1 bottom). 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[ "https://github.com/SJJThe/" ]
[ "Under-Counted Tensor Completion with Neural Incorporation of Attributes", "Under-Counted Tensor Completion with Neural Incorporation of Attributes" ]
[ "Shahana Ibrahim ", "Xiao Fu ", "Rebecca Hutchinson ", "Eugene Seo " ]
[]
[]
Systematic under-counting effects are observed in data collected across many disciplines, e.g., epidemiology and ecology. Under-counted tensor completion (UC-TC) is well-motivated for many data analytics tasks, e.g., inferring the case numbers of infectious diseases at unobserved locations from under-counted case numbers in neighboring regions. However, existing methods for similar problems often lack supports in theory, making it hard to understand the underlying principles and conditions beyond empirical successes. In this work, a low-rank Poisson tensor model with an expressive unknown nonlinear side information extractor is proposed for undercounted multi-aspect data. A joint low-rank tensor completion and neural network learning algorithm is designed to recover the model. Moreover, the UC-TC formulation is supported by theoretical analysis showing that the fully counted entries of the tensor and each entry's undercounting probability can be provably recovered from partial observations-under reasonable conditions. To our best knowledge, the result is the first to offer theoretical supports for undercounted multi-aspect data completion. Simulations and real-data experiments corroborate the theoretical claims.
null
[ "https://export.arxiv.org/pdf/2306.03273v1.pdf" ]
259,089,226
2306.03273
36f8bb2da9a0e1b0bdc5b629a54783f57257a33e
Under-Counted Tensor Completion with Neural Incorporation of Attributes 5 Jun 2023 Shahana Ibrahim Xiao Fu Rebecca Hutchinson Eugene Seo Under-Counted Tensor Completion with Neural Incorporation of Attributes 5 Jun 2023 Systematic under-counting effects are observed in data collected across many disciplines, e.g., epidemiology and ecology. Under-counted tensor completion (UC-TC) is well-motivated for many data analytics tasks, e.g., inferring the case numbers of infectious diseases at unobserved locations from under-counted case numbers in neighboring regions. However, existing methods for similar problems often lack supports in theory, making it hard to understand the underlying principles and conditions beyond empirical successes. In this work, a low-rank Poisson tensor model with an expressive unknown nonlinear side information extractor is proposed for undercounted multi-aspect data. A joint low-rank tensor completion and neural network learning algorithm is designed to recover the model. Moreover, the UC-TC formulation is supported by theoretical analysis showing that the fully counted entries of the tensor and each entry's undercounting probability can be provably recovered from partial observations-under reasonable conditions. To our best knowledge, the result is the first to offer theoretical supports for undercounted multi-aspect data completion. Simulations and real-data experiments corroborate the theoretical claims. Introduction Tensor completion (TC) has been a workhorse for a large variety of data analytics tasks, e.g., image/video inpainting (Liu et al., 2013), hyperspectral denoising (Wang et al., 2018), sampling and recovery in magnetic resonance imaging (MRI) (Kanatsoulis et al., 2020), and multimodal data mining (Papalexakis et al., 2017)-just to However, less attention has been paid to integer data TC problems where the observed entries suffer from systematical under-counting-but (strong) under-counting effects are encountered in many disciplines, e.g., epidemiology and ecology. In epidemiology, the cases of an infectious disease (e.g., COVID-19) may be under-counted due to the existence of symptom-free patients and the lack of testing (Schneider, 2020). In ecology, when an observer sees a species at a certain location, the observer likely just observes a small portion of this species' population (MacKenzie et al., 2006;Tylianakis et al., 2010). Hence, taking under-counting effects into consideration for TC is meaningful in these domains. For example, recovering the "true" (fully counted) number of COVID-19 cases at a certain location and time helps estimate/understand the infection situation. The task of under-counted tensor completion (UC-TC) is naturally more challenging than the conventional (integer data) TC, as it implicitly involves an extra task of compensating the unknown under-counting effect. Prior Work and Challenges In the literature, there exist some limited efforts towards dealing with under-counted data completion/prediction. For example, the line of work for user exposure matrix completion considered link prediction under undetected links and unobserved links simultaneously (Liang et al., 2016). The work on "zero-inflated" matrix/tensor factorization and related models (Billio et al., 2022;Simchowitz, 2013;Durif et al., 2019) considered a similar scenario, where the datasets' observed entries are zeros with a certain nonzero probability. The recent work by (Fu et al., 2021) proposed an approach to tackle under-counted matrix completion (UC-MC) using a Poisson-Binomial matrix factorization model -which was inspired by the species' abundance modeling in ecology (Joseph et al., 2009;Royle, 2004;Hutchinson et al., 2011). Challenges. The existing works by (Liang et al., 2016;Billio et al., 2022;Simchowitz, 2013;Durif et al., 2019;Fu et al., 2021) were shown effective on many UC-MC tasks. With proper modifications, these computational frameworks may be generalized to handle tensor data. However, some notable challenges remain. First, how to effectively model the under-counting effect has been an open question. For example, the works by (Liang et al., 2016;Billio et al., 2022;Simchowitz, 2013;Durif et al., 2019) all considered zero-inflated MC models, which dealt with a special case of under-counting, i.e., the observed matrix entries are either the true counts or zeros. For more general under-counted data (i.e., the observed counts are smaller than or equal to the true counts), the recent work by (Fu et al., 2021) proposed to model the probability of missing a count (e.g., an interaction of a pollinator and a plant) as a linear function of side attributes, e.g., the location, temperature, and humidity-which is reminiscent of the classic species distribution models (SDMs) (Hutchinson et al., 2011). However, using simple linear functions is a compromise between model expressiveness and computational convenience. Second, the existing works demonstrated their effectiveness only empirically. It has been unclear under what conditions the underlying true counts and some key model parameters (e.g., the detection probabilities of the counts) could be provably recovered. Contributions In this work, we propose a UC-TC framework based on the Poisson-Binomial model (Joseph et al., 2009;Royle, 2004;Hutchinson et al., 2011) as in the MC work by (Fu et al., 2021). Unlike (Fu et al., 2021) that used a simple side information model and was largely empirical, we address a number of key challenges in modeling, computation, and theory. Our contribution is twofold: UC-TC with Nonlinear Incorporation of Side Information. To model and learn the under-counting effect in tensor data, we propose a UC-TC framework with neural networkbased incorporation of attributes. Specifically, we extend the UC-MC idea in (Fu et al., 2021) to the higher-order tensor regime and model the underlying fully counted data as a Poisson tensor that admits a low-rank canonical polyadic decomposition (CPD) (Harshman, 1970); the Poisson tensor is then under-counted via a Binomial detection proce-dure. Different from the existing work that uses a linear function to model the relationship between the attributes and the detection probability of a count, we represent the relationship as an unknown nonlinear function, which substantially generalizes the modeling capacity. We recast the UC-TC problem as a maximum likelihood estimation (MLE) criterion, where the unknown nonlinear function is represented by a neural network (NN) for side information incorporation. We also design a block coordinate descent (BCD) algorithm (Razaviyayn et al., 2013) to tackle the formulated MLE effectively. Recoverability Analysis of UC-TC. We present theoretical characterizations of the proposed UC-TC criterion under the Poisson-Binomial framework. We show that, under reasonable conditions, the fully counted tensor is recoverable, and the associated detection probabilities of all the entries are identifiable-both are up to a global scaling/counter-scaling ambiguity. Our analysis reveals interesting performance-deciding factors such as the similarity of the under-counting effects across some entries. The result also shows that the nonlinear incorporation of side information imposes an intuitive trade-off between model complexity and recovery accuracy, given a fixed amount of samples-which also justifies our proposal of using neural nonlinear models. To our best knowledge, this is the first provable learning criterion for under-counted factor analysis models. We conduct extensive evaluation of the proposed approach on synthetic data to validate our theoretical findings. We also test our approach on real-world prediction tasks in epidemiology (Zhang et al., 2020b) and ecology (Seo et al., 2022;Daly et al., 2019). Background UC-TC: Problem Statement A Kth-order tensor Y ∈ R I1×...×IK is a multi-way array whose entries are indexed by K coordinates. An entry of such a tensor can be expressed as follows: y i = [Y ] i , i = (i 1 , . . . , i K ) ∈ [I 1 ] × . . . × [I K ]. (1) Tensors are widely used to represent multi-aspect data. For example, in a four-aspect epidemic data, y i can represent the number of recorded infected cases in city i 1 , during week i 2 , among age group i 3 and ethnicity group i 4 . TC and MC tasks naturally arise in many applications, where multi-aspect data are only observed partially. A classic use case is recommender systems (Hu et al., 2008;Karatzoglou et al., 2010), where the ratings that users provide to products are used to predict the unseen ratings-which can be done via completing the rating matrix/tensor. TC problems were primarily considered for continuous data (Gandy et al., 2011;Montanari & Sun, 2016;Yuan & Zhang, 2016;Zhang & Aeron, 2017;Sørensen & De Lathauwer, 2019;Zhang et al., 2020a), but TC for other data formats (e.g., binary and integer data) were also studied in the literature (Ghadermarzy et al., 2018;Chi & Kolda, 2012; Lee & Wang, 2020)-as the latter found many applications in real data analytics problems, e.g., 5-star rating-based recommender systems (Karatzoglou et al., 2010), adjacency network analysis (Acar et al., 2009) and traffic flow prediction (Tan et al., 2016). In this work, our interest lies in integer TC problems where the partially observed tensor data suffer from systematical under-counting. To be specific, we consider integer tensor data as in (Chi & Kolda, 2012;Lee & Wang, 2020). Like in conventional integer TC problems, the observed tensor Y is highly incomplete; i.e., Ω ⊆ [I 1 ] × . . . × [I K ] are the indices of the observed entries, where |Ω| ≪ K k=1 I k . The key difference from conventional integer TC is that in our setting, the observed entries are smaller than or equal to their actual values, i.e., y i ≤ n i , i ∈ Ω, where n i = [N ] i , N ∈ Z I1×...×IK + represents the (unobserved) true-count tensor, and y i ∈ Z + is the observed and under-counted version of n i . We assume that for each entry of Y , there is an associated attribute/feature vector z i ∈ R D , capturing side information regarding the observation. The attributes reflect the detection probability of each count in n i . We note that such entry attributes are often available in real-world applications. Two examples are as follows: • Epidemiology Data: Consider tensor data in epidemiology that records infected case counts. The count tensor could be indexed by 'city'×'age group'×'profession'. Then, z i may contain observation-affecting features such as the testing capacity of city i 1 , population of age group i 2 , and the level of exposure to the virus for profession i 3 . • Ecology Data: In ecology, the counts of interactions between pollinators and plants at different times could be represented by a tensor (i.e., a 'pollinator'×'plant'×'time' tensor). There, z i could contain the properties of the pollinator i 1 and plant i 2 , the temperature, humidity and other observation-affecting factors of site i 3 . In this work, we assume that the entry features z i are available for a subset of entries indexed by Ξ ⊆ [I 1 ]×. . .×[I K ]. Note that Ξ and Ω need not be identical. Existing Work: The Poisson-Binomial MC Model The work in (Fu et al., 2021) proposed an MC framework with the following generative model for the under-counted entries of a pollinator-plant interaction count matrix: n i,j ∼ Poisson(λ i,j ), λ i,j = u ⊤ i v j , u i , v j ≥ 0, (2a) y i,j ∼ Binomial(n i,j , p i,j ), p i,j = z ⊤ i,j θ,(2b) where u i ∈ R F , v j ∈ R F and θ ∈ R D are the model parameters to estimate. To be more specific, u i and v j are the F -dimensional latent representations of pollinator i and plant j. The parameter λ i,j stands for the average number of interactions between pollinator i and plant j, and the ground-truth number of interactions n i,j is the realization of a Poisson sampling process with parameter λ i,j . The observed data y i,j is under-counted through a Binomial detection process, with the detection probability p i,j . The detection probability p i,j was modeled as a linear function of the entry features z i,j . Under this model, Λ = U V ⊤ where [Λ] i,j = λ i,j is a low-rank matrix model if F is relatively small. Hence, estimating u i 's, v j 's, and θ can be regarded as a variant of Poisson matrix completion problem. Challenges To extend the Poisson-Binomial model to tensor cases and more general settings, there are a couple of key challenges to be addressed. First, the linear model in (2b), i.e., p i,j = z ⊤ i,j θ, may not be expressive enough, as the relation between the features and the detection probability is highly likely to be nonlinear. The linear model assumption was used for computational convenience, but is a performancelimiting factor in the model of (Fu et al., 2021) (as will be seen in the experiments). Second, perhaps more importantly, there were no theoretical characterizations of the Poisson-Binomial model, despite of its popularity in ecological data analysis (Fu et al., 2021;Hutchinson et al., 2011;Royle, 2004;Dennis et al., 2015). In the context of UC-MC, it is unclear if the underlying Poisson parameters λ i,j for all (i, j) could be recovered from a subset of observations y i,j , (i, j) ∈ Ω. The recoverability analysis can not be covered by existing TC theory, as the "extra" task of estimating p i,j makes UC-TC a much harder learning problem compared to the traditional TC problems. Problem Formulation In this work, we generalize the Poisson-Binomial model to cover tensor data, with nonilinear attribute incorporationand more importantly-with recoverability support. Generative Model We generalize the model (2) by considering under-counted tensors with a size of I 1 × · · · × I K generated as follows: λ i = F f =1 K k=1 U k (i k , f ), U k ≥ 0, ∀k ∈ [K],(3a)n i ∼ Poisson(λ i ), (3b) p i = g (z i ) , 0 ≤ p i ≤ 1, (3c) y i ∼ Binomial(n i , p i ),(3d) where i = (i 1 , . . . , i K ) is a shorthand notation as defined before, U k (i k , :)∈ R F denotes the F -dimensional latent representation of entity i k of aspect k, y i denotes the observed count indexed by i, n i is the corresponding true count, λ i stands for the Poisson parameter (which is the expectation of the true count), and p i represents the detection probability as in the model of (2). The vector z i ∈ R D collects the features of entry i. The unknown nonlinear function g(·) : R D → [0, 1] is used to model the dependence between the detection probability p i and z i . In (3), we have imposed nonnegativity constraints on U k 's in order to make sure that the Poisson parameters are all nonnegative. Given y i for i ∈ Ω ⊂ [I 1 ] × . . . × [I K ] and z i for i ∈ Ξ ⊂ [I 1 ] × . . . × [I K ] , our goal is to estimate λ i and p i for all i. Maximum Likelihood Estimation-Based UC-TC Assuming that y i 's are independently sampled from the generative model in (3), the log-likelihood of the observations can be expressed as follows: log i∈Ω Pr(y i ; λ i , p i ) = log i∈Ω ∞ n=y i Pr(N i = n; λ i )Pr(y i |N i = n; p i ) = i∈Ω log ∞ n=y i λ n i e −λ i n! n!p y i i (1 − p i ) n−y i y i !(n − y i )! = i∈Ω [y i log λ i + y i log p i − λ i p i − log y i !] ;(4) see the detailed derivation in the supplementary material in Sec. B. Under the generative model in (3), we have p i = g(z i ). Since g(·) is unknown and nonlinear, we represent it using a neural network denoted as g θ : R D → [0, 1]: g θ (z) = σ(w ⊤ L σ(W L−1 σ(. . . σ(W 1 z)))),(5) where W ℓ ∈ R d ℓ ×d ℓ−1 denotes the weight matrix at ℓth layer (d 0 = D), w L ∈ R dL−1 denotes the last layer's weights, θ denotes the parameters of the NN ({W ℓ } L−1 ℓ=1 , w L ), and σ(·) = [σ(·), . . . , σ(·)] ⊤ denotes the activation function with a proper dimension. Hence, combining (4) with (3a) and (3c), we formulate the MLE criterion as follows: minimize {U k },θ,{p i } L(U , θ, P ) := i∈Ω F f =1 K k=1 U k (i k , f ) p i −y i log F f =1 K k=1 U k (i k , f ) − y i log p i , (6a) subject to U k ∈ U k , ∀k ∈ [K], (6b) p i = g θ (z i ), ∀i ∈ Ξ, g θ ∈ G,(6c) where U k is a constraint set that encodes the prior knowledge of U k (e.g., nonnegativity), and G represents the function class where g θ is taken from. Given the formulated MLE, there are a couple of key aspects that are of interest. First, if one solves the formulated learning criterion in (6), can the optimal solution recover the latent parameters λ i and p i over all i? This question will be answered in Sec. 4. Second, the criterion (6) presents a challenging optimization problem that involves tensor decomposition and neural network learning-both are nontrivial problems. A BCD algorithm to effectively tackle this problem will be proposed in Sec. 5. Recoverability Analysis Our goal is to show that the optimal solution given by (6) provably recovers λ i 's and p i 's under the generative model (3) (up to certain reasonable ambiguities). Analysis Setup To better present the analysis, we use " ♮ " to denote the ground-truth parameters in the model (3)-e.g., λ ♮ i and p ♮ i denote the ground-truth Poisson parameter and groundtruth detection probability of entry i, respectively. We have the corresponding tensor notations [Λ ♮ ] i = λ ♮ i and [P ♮ ] i = p ♮ i . The "hat" notation is used to denote the terms constructed from the optimal solution given by (6). For instance, { U k } K k=1 , g θ and p i , ∀i denote the estimates given by the solution obtained from solving (6). We consider the following assumptions: Assumption 4.1 (Bounded Parameters). There exist scalars p min , p max , β u , α u > 0 such that p ♮ i = g ♮ (z i ), 0 < p min ≤ p ♮ i ≤ p max , ∀i and 0 < β u ≤ U ♮ k (i k , f ) ≤ α u , ∀i k ∈ [I k ], f ∈ [F ], k ∈ [K]. In addition, the constraints in (6), namely, U k and G satisfy U k = {U ∈ R I k ×F | β u ≤ U (i k , f ) ≤ α u , ∀i k , f } and G = {g θ : R D → [p min , p max ]}, respectively. Assumption 4.2 (Approximation Error). Assume that there exists g θ ∈ G such that | g θ (z) − g ♮ (z)| ≤ ν for all z ∈ R D , where 0 ≤ ν < ∞. In addition, assume that the class G has a complexity measure R G . Assumption 4.3 (Similar Attribute Subset). There exists an index set Θ whose elements are sampled uniformly at random (without replacement) from [I 1 ] × . . . × [I K ] and a scalar ζ > 0 such that max i,j∈Θ z i − z j 2 ≤ ζ. Assumption 4.4 (Lipschitz Continuity). Assume that the ground-truth function g ♮ and any function g θ ∈ G are Lipschitz continuous, i.e., for certain L g , L θ > 0 and for any pair of z i , z j , |g ♮ (z i ) − g ♮ (z j )| ≤ L g z i − z j 2 , and |g θ (z i ) − g θ (z j )| ≤ L θ z i − z j 2 hold. Assumption 4.2 requires that the neural network class G contains a function g θ that is closer to the ground-truth function g ♮ . In practice, if deeper/wider neural networks are employed, the value of ν is smaller. For the complexity measure R G used in Assumption 4.2, we use the sensitive complexity parameter introduced in (Lin & Zhang, 2019). For deeper and wider neural network function classes, the parameter R G gets larger-also see supplementary material Sec. M for more details. Assumptions 4.3-4.4 together imply that there exists a set of p i 's that are similar. For example, in an epidemic dataset (e.g., the COVID-19 dataset (Zhang et al., 2020b)) that records the number of detected infectious disease cases, when the testing capacity and the population of a number of locations are similar with each other, it is reasonable to assume that the corresponding detection probabilities p i 's are similar. The Lipschitz continuity assumption on g θ given by Assumption 4.4 requires that the learned g θ is smooth enough, which can be achieved via bounding (or penalizing) the norm of the neural network parameter θ during the learning process. Main Result Our main result is as follows: Theorem 4.5. Suppose that the Assumptions 4.1-4.4 hold true. Assume that the entries of index sets Ω and Ξ are sampled from [I 1 ] × . . . × [I K ] uniformly at random with replacement and satisfies Ω ⊆ Ξ. In addition, let z i , . . . , z iS be the set of observed features. Assume that each z i is drawn from a distribution D. Then, for δ > 0, the following hold with probability of at least 1 − 5δ − 3e −α(e 2 −3) : Λ ♮ − ξ Λ 2 F k I k ≤ O 1 p 2 min ̺ 1 ,(7a)E z i ∼D (p ♮ i − 1 / ξ g θ (z i )) 2 ≤ O p max βp 2 min ̺ 1 + ̺ 2 , (7b) where ̺ 1 = F α K u η + ζ 2 (L g + L θ ) 2 + log(1/δ) |Θ| , ̺ 2 = log(1/δ) S + ( Z F R G ) 1/4 √ S , η 2 = K k I k + Z F R G √ T + c log(1/δ) T + αν, c = α max{| log β|, log α}, α = F α K u p max , β = F β K u p min , Z = [z i1 , . . . , z iS ] ∈ R D×S , T = |Ω|, and ξ denote the global scaling ambiguity. The proof of Theorem 4.5 is relegated to the supplementary material in Sec. D. The result reveals several interesting insights. First, the results in (7) show that the proposed MLE criterion can correctly recover the average true counts λ ♮ i 's and the detection probabilities p ♮ i 's, up to a certain global scaling ambiguity between the entries. Second, the bounds indicate that the estimation accuracy is better when there are more observations (i.e., larger |Ω|) and more similar features (i.e., smaller ζ and larger |Θ|). Third, the result suggests that an appropriate choice of the neural network class plays a role in ensuring good estimation accuracy. This is because if one uses deeper/wider neural networks, the approximation error ν becomes smaller, but the complexity paramater R G is larger. Hence, there is a trade-off to strike while choosing the network architecture of g θ . Algorithm Design The proposed MLE in (6) is a combination of Poisson tensor decomposition and neural network learning-both are hard optimization problems. To tackle this problem, we design a BCD-based (Razaviyayn et al., 2013;Bertsekas, 1999) algorithm. We first re-formulate (6) using a regularized form by considering nonnegativity constraints for U k 's: minimize {U k },θ,{p i } 1 |Ω| L(U , θ, P ) + µ |Ξ| i∈Ξ ℓ(g θ (z i ), p i ),(8a)subject to U k ≥ 0, ∀k ∈ [K], (8b) 0 ≤ p i ≤ 1, ∀i ∈ Ω ∪ Ξ, g θ ∈ G,(8c) where µ > 0 is a regularization parameter and ℓ(·, ·) denotes a certain distance/divergence measure, e.g., the least squares function ℓ(x, y) = (x − y) 2 . Note that in the re-formulated problem (8), we did not explicitly use the bounds on U k and G in Assumption 4.1. The reason is twofold: First, the bounds serve for analytical purposes, yet not easy to know exactly in practice. Second, ignoring the bounds is inconsequential in terms of performance (as will be seen in experiments), but substantially simplifies the algorithm design. Nonetheless, as one will see, the update of U k naturally results in positive and bounded solutions, under reasonable conditions (cf. the comments after Eq. 10). In addition, as g θ can be constructed to have a sigmoid output layer, the value of g θ (z i ) is naturally bounded away from 0 and 1. Also note that we adopt this regularized optimization design since it gives flexibility to handle different cases such as Ω ⊆ Ξ and Ω ⊃ Ξ-as we will see in the following section. Our BCD updates are designed as follows: The U k -Subproblem. When θ, p i , U j , j = k are fixed, the subproblem w.r.t. U k is given by minimize U k ≥0 i∈Ω   F f =1 U k (i k , f ) j =k U j (i j , f ) p i − y i log F f =1 U k (i k , f ) j =k U j (i j , f )   .(9) From (9), the update for U k is as follows: U k (i k , f ) ← i∈Ω y i α (f ) i i∈Ω j =k U j (i j , f )p i , ∀i, f,(10) where α (f ) i = U k (i k ,f ) j =k Uj (ij ,f ) F f =1 U k (i k ,f ) j =k Uj (ij ,f ) and U k denotes U k from the previous iteration. Note that the above solution is always strictly positive if U k > 0 for all k and p i > 0 for all i. Our update rule design for U k follows the ideas in (Chi & Kolda, 2012;Fu et al., 2021), which is essentially a majorization minimization step w.r.t. U k ; see Sec. C in the supplementary material. The p-Subproblem. To update p i , we fix U k 's and θ and consider different cases. Case 1. When both y i and z i are available, i.e., i ∈ Ω ∩ Ξ, p i is updated via optimizing the following subproblem: min p i ∈[0,1] 1 |Ω| (λ i p i − y i log(p i )) + µ |Ξ| ℓ(g θ (z i ), p i ),(11)where λ i = F f =1 K k=1 U k (i k , f ) is obtained using the current estimates of U k 's. Case 2. Suppose that the observation y i is not available, but z i is available, i.e., i ∈ Ξ − Ω ∩ Ξ. Then, we consider the following subproblem to update such p i 's: min p i ∈[0,1] ℓ(g θ (z i ), p i ).(12) If we choose a convex ℓ(·) in (11)-(12), the problems can be optimally solved using any standard techniques such as projected gradient decent. Moreover, if we set ℓ(·) to be one of the popular choices such Euclidean distance and KL divergence, closed-form solutions for the problems in (11)-(12) exist-see Table 6 in the supplementary material. Case 3. 1 When y i is observed, but z i is not accessible, i.e., i ∈ Ω − Ω ∩ Ξ, we update p i via solving the following subproblem: min p i ∈(0,1] λ i p i − y i log(p i ),(13) which gives the update rule for p i as p i ← [ y i/λ i ] [0,1] . The θ-Subproblem. By fixing the U k 's and all the p i 's, the subproblem for updating θ is as follows: min θ i∈Ξ ℓ(g θ (z i ), p i ). This is an unconstrained neural network learning problem. Many off-the-shelf neural network learning algorithms that use gradient, e.g., Adam (Kingma & Ba, 2015) and Adagrad (Duchi et al., 2011), can be used to handle this subproblem. More design details and the description of the algorithm are provided in supplementary material in Sec. C. The proposed algorithm is referred to as the Under-Counted Data Prediction Via Learner-Aided Tensor Completion (UncleTC) algorithm. A remark is that our algorithm uses a batch (instead of stochastic) update rule for the U k 's. This can also be replaced by stochastic tensor decomposition algorithms (see, e.g., (Fu et al., 2020a;Pu et al., 2022) and the references in (Fu et al., 2020b)). Using stochastic algorithms for the U k -subproblems may further improve efficiency. Experiments In this section, we evaluate our proposed method through a series of synthetic and real data experiments. The source code is available at https://github.com/ shahanaibrahimosu/undercounted-tensorcompletion. Baselines. We employ a number of lowrank tensor completion-based baselines, . Among them, both NTF-CPD-KL and BPTF consider Poisson modeling, but they do not have a Binomial detection stage in their models to accommodate the under-counting effect. We also consider two recent 1 Our recoverability analysis is built upon the assumption Ω ⊆ Ξ; see Theorem 4.5. However, in practice, there are cases where y i is observed, but z i is not available (i.e., Ω − Ω ∩ Ξ = ∅). Our algorithm can still operate under such cases. In order to incorporate the side features, the CostCo method is slightly modified from its original implementation by concatenating side features with the indices of the observed entries as the input to their networks. In addition, we also include a tensor version of (Fu et al., 2021), where a linear function g θ (z i ) = θ ⊤ 1 z i + θ 2 is used to learn g function. This baseline is referred to as UncleTC(Linear). More details on the implementation are provided in the supplementary material in Sec. N. Synthetic-Data Simulations Data Generation. The data generating process follows (3). We first generate U k ∈ R I k ×F , ∀k by randomly sampling its entries from a uniform distribution between 0 and κ. We fix K = 3, F = 3 and I k = 20, ∀k, unless specified otherwise. We also fix κ = 15 so that λ i 's are not unreasonably small. The feature vectors z i ∈ R D with D = 10 are generated by randomly sampling its entries from the standard normal distribution. We use the ground-truth nonlinear function (unknown to our algorithm) g(z) = sigmoid(ν ⊤ (0.5z 3 + 0.2z)), where the vector ν ∈ R D is generated by randomly sampling its entries from the uniform distribution between 0 and 1. We observe only a subset of y i 's such that each i belongs to Ω with probability γ Ω ∈ (0, 1]. Similarly, only a portion of z i 's are observed such that each i belongs to Ξ with probability γ Ξ ∈ (0, 1]. We also make a subset of z i 's similar such that z i = ϕ + n i , i ∈ Θ, Θ ⊆ Ξ, with each i belongs to Θ with probability γ Θ ∈ (0, 1]. Here, ϕ ∈ R D is a random vector whose entries are drawn from the standard normal distribution and n i denotes additive noise whose entries are zero mean Gaussian with variance σ 2 . Under this setting, we define the signal-to-noise (SNR) ratio SNR = 10 log 10 ϕ 2 2/Dσ 2 dB. As the value of SNR increases, the feature vectors z i ∈ Θ are more similar. Metric. To characterize the performance of recovering p i 's and λ i 's, we employ mean absolute error (MAE) which is defined as MAE λ = 1 / k I k i λ i/ λ (avg) − λ ♮ i/λ ♮(avg) , where λ i is the estimated value corresponding to λ ♮ i , and the subscript (avg) denotes the average across all values, e.g., λ (avg) = 1 / k I k i λ i . Note that the metric MAE λ involves entry-wise normalization of the estimates and the ground truth by the average in order to remove the scaling ambiguity. The metric MAE p is defined in the same way, with all the λ-terms replaced by their p counterparts.. Implementation Settings. To learn the nonlinear function, we use a neural network g θ (·) with 3 hidden layers and 20 ReLU activation functions in each hidden layer. The regularization parameter µ is fixed to be 2000. More implementation details and experiments using different µ's are provided in the supplementary material in Sec. N. Results. Table. 1 shows the performance under various γ Ω 's of all the methods under test. Note that for the baselines that do not take detection probability into their models, only the MAE λ results are presented. The results show that all the non-Poisson baselines including the NN-based methods struggle to estimate the true counts λ i . This shows the importance of tailored modeling for integer data. Among the algorithms using Poisson models, the proposed UncleTC mothod estimates λ i 's with much higher accuracy relative to NTF-CPD-KL, as our method explicitly models the under-counting effect. One can also note that UncleTC significantly outperforms Table 4. Statistics of λ i/λ ♮ i and p i/p ♮ i for different trials of the proposed UncleTC with settings γΩ = 0.3, γΞ = 0.3, γΘ = 0.2, ζ = 40dB, and g(z) = sigmoid(ν ⊤ (0.1 log(z 2 ) + 0.1z 2 )) UncleTC(Linear), in terms of both MAE p and MAE λ . This is due to the ability of UncleTC to better handle the nonlinear relationships between the detection probabilities p i and the side information z i , by using its NN-based learning. In addition, we note that as γ Ω is higher, i.e., more observations are available, the performance becomes better, as expected and indicated in our theoretical claim. #Trial mean±std of λi /λ ♮ i mean±std of pi /p ♮ i mean±std of λi pi /λ ♮ i p ♮ i 1 1.16 ± 0 In Table 2 and Table 3, we show the performance of the algorithms under vairous SNR and γ Θ , respectively. In Table 2, as SNR increases from 0 to 40dB, i.e., ζ in Assumption 4.3 becomes smaller and the feature vectors z i 's become more similar, the MAEs decrease noticeably. The impact on MAE p is obviously spelled out, as a reduction of around 7 times is observed. Table 3 shows that as the size of Θ gets larger, the estimation accuracy of the proposed method is positively impacted. Both tables echo the results presented by Theorem 4.5. Table 4 validates a key result from Theorem 4.5. That is, there exists a global scaling factor and a counter-scaling factor for λ i and p i , respectively. In other words, assuming perfect estimation for λ i and p i by the proposed UncleTC, we have λ i = ξλ ♮ i , p i = 1 /ξp ♮ i , ∀i for a certain scalar ξ > 0. To see if the UncleTC algorithm attains good estimation of λ i and p i up to a global scaling/counter-scaling ambiguity, we report the mean and standard deviation of λ i/λ ♮ i and p i/p ♮ i taken over all the i's in each random trial. One can see that only small standard deviations exist in all the trials, suggesting that the scaling/counter-scaling factors are approximately identical over all the indices. Similar observations are made on p i/p ♮ i . One can also note that the scaling factors of λ i and p i are approximately reciprocal to each other (cf. the last column of Table 4). This observations corroborate the findings in Theorem 4.5. Real-Data Experiments COVID-19 Data. The dataset is obtained from the COVID-19 impact analysis platform hosted by the University of Maryland, College Park, United States (Zhang et al., The dataset also includes about 37 entry attributes corresponding to each observation, such as flower color of the plant, pollinator tongue length, weather conditions, and so on. For both datasets, since the features are in different numerical units and range, we normalize each feature value using min-max normalization (Jain et al., 2005), across all observations. More details on the datasets and tensor construction are given in the supplementary material in Sec. N. Quantitative Evaluation. Since the ground-truth λ ♮ i 's and p ♮ i 's are unknown for real data, we employ a count data prediction metric, i.e., relative root mean square (rRMSE) (Fu et al., 2021) for evaluation. Let {y i } i∈∆ be the set of actual holdout observations and { y i } i∈∆ be the corresponding predicted values. Then, the rRMSE is defined as rRMSE = 1 /y i∈∆ |y i − y i | 2 /|∆|, where y is the mean value of the actual observations {y i } i∈∆ . Note that, for the proposed method, the observations are predicted via y i = λ i p i . the COVID-19 and PPI datasets. One can see that the proposed UncleTC exhibits promising performance on both datasets. One can also see that the performance of the Poisson model-based method NTF-CPD-KL has a comparable rRMSE performance relative to our method. This is expected, as the Poisson-Binomial model with parameters λ and p can be considered as a Poisson model with the parameter λp (see Lemma B.1). Nontheless, our method still outperforms NTF-CPD-KL, as the incorporation of the entry attributes in our framework may be beneficial. Compared to the non-Poisson methods, our approach stands out with obviously lower rRMSEs, which is similar to what was observed in the simulations. It can also be noted that UncleTC outperforms its linear function-based counterpart UncleTC(Linear) in both cases, which is also consistent with our simulation results. Qualitative Evaluation. In Fig. 1, we analyze the detection probabilities output by the proposed UncleTC for COVID-19 dataset. Here, we present the top 10 counties having the highest and lowest detection probabilities for COVID-19 cases found by the proposed UncleTC (averaged over all days). Note that the estimated detection probabilities have a global scaling ambiguity-see Theorem 4.5-but their relative ranking order is not affected by this ambiguity. We also report the average number of COVID-19 tests done per 1000 people during the selected period (we removed this feature while training the model to avoid correlated results). One can see that the counties with more tests done are mostly aligned with higher detection probabilities estimated by our algorithm. Also, most counties reported with high detection probabilities are populous, urban counties, which enjoy better infrastructure. These observations suggest that the proposed algorithm outputs plausible results for the detection probabilities-also see the detection probability results for the PPI dataset in the supplementary material in Sec. N. Conclusion We proposed an under-counted tensor completion framework, which is motivated by the commonly seen undercounting effects in data acquisition across many domains, e.g., ecology and epidemiology. Our model uses a Poisson-Binomial data-generating perspective as often advocated in computational ecology, where the true integer data are under-counted by a Binomial detection process. Unlike existing approaches that treat the under-counting effects using relatively simple models, we used a neural network-based design to account for a wide range of unknown nonlinear relations between the side information (attributes) and the under-counting effects. More importantly, we showed that the proposed under-counted tensor completion criterion can provably recover the underlying Poisson tensor and its entries' under-counting characteristics, under reasonable conditions-which is the first theoretical result of its kind. We validated our theorem and algorithm using simulated and real data and observed intuitively plausible results from our real data results. Limitations. A key limitation lies in the model assumption that the under-counted probabilities are functions of observed side attributes. However, such attributes may not always be available. Provable UC-TC criteria that do not rely on such assumption are highly desirable, yet the current formulation and analysis cannot cover such settings without substantial changes and re-design. A. Notation The notations x, x, X and X represent a scalar, a vector, a matrix, and a tensor, respectively. x i denotes the ith element of the vector x. [X] i,j and X(i, j) both mean the (i, j)th entry of X. Similarly, [X] i belongs to the ith entry of the tensor X ∈ R I1×...×IK where i = (i 1 , . . . , i K ). Z + denote the set of all nonnegative integers. x ≥ 0, X ≥ 0, and X ≥ 0 imply that all the associated entries are greater than 0. x 2 , X F and X F all mean the Euclidean (Frobenius) norm of the augment. [X] Ω F indicates the Euclidean norm of the vector formed by concatenating the entries indexed by all unique i's belonging to Ω, i.e., [X] Ω 2 F = i∈Supp(Ω) x 2 i , where Supp(Ω) denotes the set of all unique entries in Ω. X ∞ denotes the max norm of X such that X ∞ = max i [X] i . [I] means an integer set {1, 2, . . . , I}. ⊤ and † denote transpose and pseudo-inverse, respectively. x • y denotes the outer product of two vectors x and y, i.e., x • y = xy ⊤ . ⊛ denotes the element-wise product (also known as Hadamard product). The function sigmoid(x) represents B. MLE Formulation Assuming that y i 's are independently sampled from the generative model in (3), the log-likelihood of the observations can be expressed as follows: log i∈Ω Pr(Y i = y i ; λ i , p i ) = log i∈Ω ∞ n=y i Pr(N i = n; λ i )Pr(y i |N i = n; p i ) = i∈Ω log ∞ n=y i λ n i e −λ i n! n!p y i i (1 − p i ) n−y i y i !(n − y i )! ,(14) where Y i denotes the random variable associated with the observation y i and N i denotes the random variable associated with the true count n i . To circumvent the infinite sum in the log-likelihood, we invoke the folowing lemma: Applying the above equality in (14), we obtain: log i∈Ω Pr(Y i = y i ; λ i , p i ) = i∈Ω log (p i λ i ) y i e −λ i p i y i ! = i∈Ω [y i log λ i + y i log p i − λ i p i − log y i !] .(16) Here, the MLE is formulated and derived under the same spirit of the MC case in (Fu et al., 2021). C. More Details on Algorithm Design In this section, we provide more details on the implementation of the proposed algorithm UncleTC. Our formulation is given below: minimize {U k } K k=1 ,θ,{p i } 1 |Ω| i∈Ω F f =1 K k=1 U k (i k , f ) p i − y i log F f =1 K k=1 U k (i k , f ) − y i log p i + µ |Ξ| i∈Ξ ℓ(g θ (z i ), p i ),(17a)subject to U k ≥ 0, ∀k ∈ [K], (17b) 0 ≤ p i ≤ 1, ∀i ∈ Ω ∪ Ξ,(17c) where µ > 0 is a regularization parameter and ℓ(·, ·) denotes a certain distance/divergence measure, e.g., the least squares function ℓ(x, y) = (x − y) 2 . C.1. The U k -Subproblem The subproblem for each U k is given below: minimize U k ≥0 w(U k ) := i∈Ω   F f =1 U k (i k , f ) j =k U j (i j , f ) p i − y i log F f =1 U k (i k , f ) j =k U j (i j , f )   .(18) To obtain the update rule for U k from (18), we have the following result: Lemma C.1. Let U k denote the current estimate of U k . Then, the objective function in (18) can be upper-bounded by the following function: s(U k ; U k ) = i∈Ω F f =1 U k (i k , f ) j =k U j (i j , f ) p i − y i F f =1 α (f ) i log U k (i k , f ) j =k U j (i j , f ) α (f ) i ,(19) where α (f ) i = U k (i k , f ) j =k U j (i j , f ) F f =1 U k (i k , f ) j =k U j (i j , f ) . The construction of the surrogate function s(U k ; U k ) follows the idea in (Chi & Kolda, 2012;Fu et al., 2021); The differences include that the model in (Chi & Kolda, 2012) did not have p i terms and that the work in (Fu et al., 2021) did not consider the tensor case. Here, we use the Jensen's inequality to construct a "tight" upper bound, i.e., w(U k ) ≤ s(U k ; U k ) and the equality can be attained at U k = U k ; see Sec. L for the proof of Lemma C.1. To be more precise, the update of U k is through solving the following subproblem: U k ← arg min U k ≥0 s(U k ; U k ).(20) By taking the derivative of s(·; U k ) w.r.t. U k and equating the derivative to zero, we obtain: U k (i k , f ) ← i∈Ω y i α (f ) i i∈Ω j =k U j (i j , f )p i .(21) For efficient implementation of the above updates in the presence of missing data, we adopt a strategy proposed in (Chi & Kolda, 2012) for handling sparse tensors. Table 6 summarizes the update rules for the p i -subproblem in Sec. 5 under different choices of the regularization function ℓ(·). The complete description of the proposed UncleTC algorithm is provided in Algorithm 1. C.2. The p-Subproblem ℓ(x, y) = (x − y) 2 ℓ(x, y) = x log x y − x + y Case 1: ∀i ∈ Ω ∩ Ξ p i ← (2μg θ (z i )−λ i )+ √ (2μg θ (z i )−λ i ) 2 +8μȳ i 4μ p i ←ȳ i +μg θ (z i ) λ i +μ Case 2: ∀i ∈ Ξ − Ω ∩ Ξ p i ← g θ (z i ) p i ← g θ (z i ) Case 3: ∀i ∈ Ω − Ω ∩ Ξ p i ← [ y i/λ i ] [0,1] p i ← [ y i/λ i ] [0,1] µ = µ /|Ξ|,λ i = λ i/|Ω|,ȳ i = y i/|Ω| Algorithm 1 UncleTC input :Observations y i , ∀i ∈ Ω, feature vectors z i , ∀i ∈ Ξ, Initializations θ (0) , p (0) i , ∀i ∈ Ξ, and U (0) k , ∀k. 1 t ← 0; 2 repeat 3 λ (t+1) i ← F f =1 K k=1 U (t) k (i k , f ), ∀i; 4p (t+1) i ← g θ (t) (z i ), i ∈ Ξ; 5 p (t+1) i ← arg min p i ∈[0,1] 1 |Ω| (λ (t+1) i p (t) i − y i log(p (t) i )) + µ |Ξ| ℓ(p (t+1) i , p (t) i ), ∀i ∈ Ω ∩ Ξ; 6 p (t+1) i ← arg min p i ∈[0,1] ℓ(p (t+1) i , p (t) i ), ∀i ∈ Ξ − Ω ∩ Ξ; 7 p (t+1) i ← [ y i/λ (t+1) i ] [0,1] , ∀i ∈ Ω − Ω ∩ Ξ; 8 θ (t+1) ← arg min θ i∈Ξ ℓ(g θ (z i ), p (t+1) i ); 9 for k = 1 to K do 10 r ← 0; 11 repeat 12 α (f ) i ← U (r) k (i k ,f ) j =k U (t) j (ij ,f ) F f =1 U (r) k (i k ,f ) j =k U (t) j (ij ,f ) , ∀i, f ; 13 U (r+1) k (i k , f ) ← i∈Ω y i α (f ) i i∈Ω j =k U (t) j (ij ,f )p (t+1) i , ∀i k ∈ [I k ], D. Proof of the Main Result (Theorem 4.5) Let us first construct the following constraint sets: L = {Λ | Λ = F f =1 U 1 (:, f ) • . . . • U K (:, f ), U k (:, f ) 2 ≤ u, U k ≥ 0, ∀k, f },(22a)M = M = Λ ⊛ P | β ≤ [M ] i ≤ α, ∀i, Λ ∈ L, [P ] i = g θ (z i ), g θ ∈ G, z i ∈ R D , ∀i ∈ Ξ .(22b) Under Assumption 4.1, we can choose the parameters u, α, and β as follows: u = √ F α u , α = F α K u p max , β = F β K u p min .(23) Using these notations, the MLE in (6) can be re-expressed as follows: M = arg min M ∈M i∈Ω f (m i ; y i , z i ),(24)where f (m i ; y i , z i ) = m i − y i log m i , where m i = λ i g θ (z i ). Estimating M ♮ − M F / k I k : Our goal is to characterize the estimation accuracies of λ ♮ i and p ♮ i . To achieve this goal, we first characterize the term M ♮ − M F . We have the following result: (3) and that z i1 , . . . , z iS are the set of observed features. Then,under (24), with probability at least 1 − 2δ − 3e −α(e 2 −3) , we have the following relation, for any 0 < δ < 1: M ♮ − M 2 F k I k ≤ C(α, β) 2R T (F ) + (5f max − f min ) 2 log(4/δ) T + F α K u 1 + p max (e 2 − 2) p min ν ,(25) where C(α, β) = 4αγ 1−e −γ , γ = 1 8β (α − β) 2 , f max ≤ α(1 + (e 2 − 2) max{| log β|, log α}), f min = β − α(e 2 − 2) log α, R T ≤   4 √ T + 12 √ T F k I k log 6KL f √ T (F α u ) K + F α K u L f √ T Z F R G   , L f = 1 + α(e 2 −2) β , Z = [z i1 , . . . , z iS ] ∈ R D×S , and R G denotes the complexity measure of the class G. The proof is relegated to Appendix E. Estimating [M ♮ − M ] Θ F /|Θ|: Next, we proceed to estimate [M ♮ − M ]Θ F /|Θ| using the result in Theorem D.1. Consider the following result: Lemma D.2. Assume that i ∈ Θ is drawn from [I 1 ] × . . . × [I K ] uniformly at random (without replacement). Consider that M 1 , M 2 ∈ {M | M ∈ R I1×···×IK + , M ∞ ≤ α} holds. Let us define τ (Θ; M 1 , M 2 ) = 1 |Θ| (M 1 − M 2 ) Θ F − 1 √ I K M 1 − M 2 F . Then, with probability greater than 1 − δ, we have τ (Θ; M 1 , M 2 ) ≤ α log 2 δ 1 − (|Θ| − 1) k I k 1 2|Θ| 1/4 . The proof is provided in Appendix J. Combining Lemma D.2 (here, α = F α K u p max using Assumption 4.1) with Theorem D.1 and denoting η = 1 √ k I k M ♮ − M F , we have the following with probability at least 1 − 3δ − 3e −α(e 2 −3) : M ♮ − M Θ F |Θ| ≤ η + τ (Θ; M ♮ , M ),(26) where η 2 is given by (25) from Theorem D.1. Recovering λ i 's: Next, we proceed to characterize the estimation accuracies of λ i 's using the result in (26). Under Assumptions 4.3 and 4.4, we have the following representation: p ♮ i = g ♮ (z i ) = ξ + s i , |s i | ≤ ζL g , ∀i ∈ Θ, p i = g θ (z i ) = ξ ′ + s ′ i , |s ′ i | ≤ ζL θ , ∀i ∈ Θ, where 0 ≤ ξ, ξ ′ ≤ 1 are certain scalars. Hence, we get the following relation, with probability at least 1 − 3δ − 3e −α(e 2 −3) : (η + τ (Θ; M ♮ , M )) |Θ| ≥ M ♮ − M Θ F = i∈Θ (m ♮ i − m i ) 2 = i∈Θ (λ ♮ i p ♮ i − λ i p i ) 2 = i∈Θ (λ ♮ i (ξ + s i ) − λ i (ξ ′ + s ′ i )) 2 = i∈Θ (ξλ ♮ i − ξ ′ λ i + s i λ ♮ i − s ′ i λ i ) 2 ≥ i∈Θ (ξλ ♮ i − ξ ′ λ i ) 2 − i∈Θ (s i λ ♮ i − s ′ i λ i ) 2 . Since |s i | ≤ ζL g , |s ′ i | ≤ ζL θ , and λ i ≤ F α K u , ∀Λ ∈ L by (23), the above relation can be further expressed as (η + τ (Θ; M ♮ , M )) |Θ| + ζ(L g + L θ )F α K u |Θ| ≥ ξ i∈Θ (λ ♮ i − ξ ′ /ξ λ i ) 2 = ξ Λ ♮ − ξ ′ /ξ Λ Θ F =⇒ Λ ♮ − ξ ′ /ξ Λ Θ F |Θ| ≤ (η + τ (Θ; M ♮ , M )) + ζ(L g + L θ )F α K u ξ ≤ (η + τ (Θ; M ♮ , M )) + ζ(L g + L θ )F α K u p min .(27) Let ξ ′ /ξ = ξ. Our next goal is to characterize Λ ♮ − ξ Λ F using the result in (27). To proceed, consider the term τ (Θ; Λ ♮ , ξ Λ) = Λ ♮ − ξ Λ Θ F |Θ| − Λ ♮ − ξ Λ F k I k .(28) We again invoke Lemma D.2 for characterizing the term τ (Θ; Λ ♮ , ξ Λ), and obtain that with probability of at least 1 − δ, τ (Θ; Λ ♮ , ξ Λ) ≤ F α K u log 2 δ 1 − (|Θ| − 1) k I k 1 2|Θ| 1/4 .(29) Combining the result in (28) and (29) with (27), we obtain the following result with probability at least 1−4δ −3e −α(e 2 −3) : Λ ♮ − ξ Λ F k I k ≤ τ (Θ; Λ ♮ , ξ Λ) + (η + τ (Θ; M ♮ , M )) + ζ(L g + L θ )F α K u p min η ′ . Under-Counted Tensor Completion with Neural Incorporation of Attributes Recovering p i 's: Next, we proceed to bound the estimation accuracies for p i 's. Consider the following chain of inequalities: M ♮ − M F = Λ ♮ ⊛ P ♮ − ξ Λ ⊛ 1 / ξ P F = Λ ♮ ⊛ P ♮ − Λ ♮ ⊛ 1 / ξ P + Λ ♮ ⊛ 1 / ξ P − ξ Λ ⊛ 1 / ξ P F ≥ Λ ♮ ⊛ (P ♮ − ⊛ 1 / ξ P ) F − (Λ ♮ − ξ Λ) ⊛ 1 / ξ P F ≥ λ min P ♮ − 1 / ξ P F − (Λ ♮ − ξ Λ) ⊛ 1 / ξ P F ≥ β p max P ♮ − 1 / ξ P F − (Λ ♮ − ξ Λ) ⊛ 1 / ξ P F , where λ min = min i λ i = min i m i max i p i = β pmax . The above inequalities imply that with probability at least 1 − 4δ − 3e −α(e 2 −3) P ♮ − 1 / ξ P F k I k ≤ p max β   Λ ♮ − ξ Λ F p min k I k + M ♮ − M F k I k   , = p max β η ′ p min + η ,(30) where we have used the result that (Λ ♮ − ξ Λ) ⊛ 1 / ξ P F ≤ 1 pmin Λ − ξ Λ F , since [ 1 / ξ P ] i = 1 / ξ p i = ξ /ξ ′ p i ≤ 1 /pmin. Next, we proceed to characterize the generalization error of the learned nonlinear function g θ . Towards this, we first bound the term [P ♮ − 1 / ξ P ]Ξ F √ S using the result in (30). Let us consider τ (Ξ; P ♮ , 1 / ξ P ) = P ♮ − 1 / ξ P Ξ F √ S − P ♮ − 1 / ξ P F k I k . By applying Lemma D.2 combined with (30), we get that with a probability greater than 1 − 5δ − 3e −α(e 2 −3) P ♮ − 1 / ξ P Ξ F √ S ≤ τ (Ξ; P ♮ , 1 / ξ P ) + p max β η ′ p min + η ,(31) where τ (Ξ; P ♮ , 1 / ξ P ) ≤ p max log 2 δ 1 − (S − 1) k I k 1 2S 1/4 . Using the result in (31), we proceed to bound the generalization performance of the learned nonlinear function g θ . Towards this goal, consider the following result: Lemma D.3. Under the Assumptions given by Theorem 4.5, the following holds with probability greater than 1 − δ: E z∼D (g ♮ (z) − 1 / ξ g θ (z)) 2 ≤ P ♮ − 1 / ξ P Ξ 2 F S + 32 (2 Z F R G ) 1 4 S 5/8 + 4 2 log(4/δ) S ,(32) where Z = [z i1 , . . . , z iS ] and R G denotes the complexity measure as given by Assumption 4.2. The proof is given in Appendix K. Combining Lemma D.3 with (31), we get the final bound for E z∼D (g ♮ (z) − 1 / ξ g θ (z)) 2 with probability greater than 1 − 6δ − 3e −α(e 2 −3) . E. Proof of Theorem D.1 For a set Ω = {i 1 , . . . , i T }, we assume that Ω ∼ Π where Π is uniform, i.e., each y i is observed independently at random with probability 1/( k I k ) 2 . Hence, we define the expected risk as follows: D Π (M ; Y ) := E Ω∼Π [f (m i ; y i , z i )] = i 1 k I k f (m i ; y i , z i ).(33) Accordingly, we define the empirical risk as follows: D Ω (M ; Y ) = 1 T i∈Ω f (m i ; y i , z i ).(34) Eq.(24) implies that D Ω ( M ; Y ) ≤ D Ω ( M ; Y )(35) where [ M ] i = λ ♮ i g θ (z i ) and g θ is given by Assumption 4.2. To proceed, consider the following chain of inequalities: E y [D Π ( M ; Y ) − D Π (M ♮ ; Y )] = E y [D Π ( M ; Y )] − D Ω ( M ; Y ) + D Ω (M ♮ ; Y ) − E y [D Π (M ♮ ; Y )] + D Ω ( M ; Y ) − D Ω ( M ; Y ) + D Ω ( M ; Y ) − D Ω (M ♮ ; Y ) ≤ E y [D Π ( M ; Y )] − D Ω ( M ; Y ) + D Ω (M ♮ ; Y ) − E y [D Π (M ♮ ; Y )] + |D Ω ( M ; Y ) − D Ω (M ♮ ; Y )| ≤ sup M ∈M |D Ω (M ; Y ) − E y [D Π (M ; Y )]| + |D Ω (M ♮ ; Y ) − E y [D Π (M ♮ ; Y )]| + |D Ω ( M ; Y ) − D Ω (M ♮ ; Y )|,(36) where expectation is taken w.r.t. y i 's and the first inequality is by (35). Let us consider the L.H.S. of (36): E y [D Π ( M ; Y ) − D Π (M ♮ ; Y )] = 1 k I k i E y ( m i − y i log m i ) − (m ♮ i − y i log m ♮ i ) = 1 k I k i ( m i − m ♮ i log m i ) − (m ♮ i − m ♮ i log m ♮ i ) = 1 k I k i m ♮ i log m ♮ i m i − (m ♮ i − m i ) := KL(M ♮ || M ),(37) where the second equality utilizes Lemma B.1 to get E[ y i ] = λ ♮ i p ♮ i = m ♮ i . Upper-bounding the first term on the R.H.S of (36). Next, we characterize the first term on the R.H.S. of (36). Towards this, we invoke the following theorem (Theorem 26.5 in (Shalev-Shwartz & Ben-David, 2014)) : Theorem 1. (Shalev-Shwartz & Ben-David, 2014, Theorem 26.5) Assume that for all y and for all x, we have |f (x; y)| ≤ f max . Then for any M ∈ M, the following holds with probability greater than 1 − δ: |D Ω (M ; Y ) − E y [D Π (M ; Y )]| ≤ 2R T (F ) + 4f max 2 log(4/δ) T ,(38) where F denotes the set F [f (m i1 ; y i1 , z i1 ), . . . , f (m iT ; y iT , z iT )] ⊤ | M ∈ M, i t ∈ Ω and R T (F ) denotes the empirical Rademacher complexity of the set F . Applying Theorem 1, with probability greater than 1 − δ, we have sup M ∈M |D Ω (M ; Y ) − E y [D Π (M ; Y )]| ≤ 2R T (F ) + 4f max 2 log(4/δ) T .(39) Let us characterize f max in (39). Lemma E.1. Let f (m i ; y i , z i ) = m i − y i log m i where m i = λ i p i = λ i g θ (z i ). Then, under Assumption 4.1, the maximum and minimum value taken by the function f (m i ; y i , z i ), denoted as f max and f min are given by f max = α + c max{| log β|, log α} f min = β − c log α, with probability greater than 1 − e −α(e 2 −3) where c = α(e 2 − 2). The proof of the lemma is given in Appendix G. Next, we aim to characterize the Rademacher complexity R T (F F = {f ∈ R T | f = [f (m i1 ; y i1 , z i1 ), . . . , f (m iT ; y iT , z iT )] ⊤ | m it = [M ] it , i t ∈ Ω, M ∈ M}. With probability greater than 1 − e −α(e 2 −3) , the empirical Rademacher complexity of F is bounded by R T (F ) ≤   4 √ T + 12 √ T F k I k log 6KL f √ T (F α u ) K + F α K u L f √ T Z F R G   , where L f = 1 + α(e 2 −2) β , Z = [z i1 , . . . , z iS ] ∈ R D×S and and R G denotes the complexity measure of the class G. The proof is provided in Appendix F. By combining Lemma E.1 and Lemma E.2, we completely characterize (39). Upper-bounding the second term on the R.H.S of (36). Next, we upper bound the second term on the R.H.S of (36). Let us consider the Hoeffding's inequality Lemma E.3. Let F 1 , . . . , F T be independent bounded random variables with F t ∈ [f min , f max ] for all t where −∞ < f min ≤ f max < ∞. Then for all t ≥ 0, Pr 1 T T t=1 (F t − E[F t ]) ≥ q ≤ exp − 2T q 2 (f max − f min ) 2 . To use Lemma E.3, let us define the random variable F t as follows: F t f (m it ; y it , z it ). Then, invoking Lemma E.3, one can obtain Pr D Ω (M ♮ ; Y ) − E y [D Π (M ♮ ; Y )] ≥ q ≤ exp − 2T q 2 (f max − f min ) 2 .(40) where f max and f min are given by Lemma E.1. Hence, by substituting q = (f max − f min ) log( 1 δ ) 2T , where δ ∈ (0, 1) in (40), we get that with probability greater than 1 − δ D Ω (M ♮ ; Y ) − E y [D Π (M ♮ ; Y )] ≤ (f max − f min ) log 1 δ 2T .(41) Upper-bounding the third term on the R.H.S of (36). Consider the following chain of equations: D Ω ( M ; Y ) − D Ω (M ♮ ; Y ) = 1 T i∈Ω f ( m i ; y i , z i ) − 1 T i∈Ω f (m ♮ i ; y i , z i ) = 1 T i∈Ω ( m i − y i log m i ) − (m ♮ i − y i log m ♮ i ) = 1 T i∈Ω (λ ♮ i ( g θ (z i ) − g ♮ θ (z i )) − y i log λ ♮ i g θ (z i ) − log λ ♮ i g ♮ θ (z i ) = 1 T i∈Ω (λ ♮ i ( g θ (z i ) − g ♮ θ (z i )) − y i log g θ (z i ) − log g ♮ θ (z i ) ≤ max i λ ♮ i | g θ (z i ) − g ♮ θ (z i )| + max i y i | g θ (z i ) − g ♮ θ (z i )| p min , ≤ (c λ + c/p min )ν, = F α K u (1 + p max (e 2 − 2)/p min )ν,(42) where the first inequality employs the triangle inequality and the Lipschitz continuity of the log function. The first inequality also employs the Assumptions 4.1 that the lowerbound of both g θ (z i ) and g ♮ θ (z i ) are p min . The second inequality is obtained by applying c λ = F α K u from Assumption 4.1, c = α + α(e 2 − 3) from Lemma H.1 with probability greater than 1 − e −α(e 2 −3) and α = F α K u p max . Putting Together. Combining (36), (37) with the upperbounds (39), (41), and (42), we get with probability greater than 1 − 2δ − 3e −α(e 2 −3) . KL(M ♮ || M ) ≤ 2R T (F ) + (5f max − f min ) 2 log(4/δ) T + F α K u 1 + p max (e 2 − 2) p min ν,(43) where f max = α + c max{| log β|, log α} and f min = β − c log α given by Lemma E.1 . By (Cao & Xie, 2015, Lemma 8), we have KL(M ♮ || M ) ≥ M ♮ − M 2 F C(α, β) k I k ,(44) where C(α, β) = 4αγ 1−e −γ and γ = 1 8β (α − β) 2 . Hence, combining (43) and (44), we have the following with probability greater than 1 − 2δ − 3e −α(e 2 −3) : M ♮ − M 2 F k I k ≤ C(α, β) 2R T (F ) + (5f max − f min ) 2 log(4/δ) T + F α K u 1 + p max (e 2 − 2) p min ν .(45) F. Proof of Lemma E.2 Let us consider the following vector f : f = [f (m i1 ; y i1 , z i1 ), . . . , f (m iT ; y iT , z iT )] ⊤ ∈ F , where i t ∈ Ω, ∀t and F = {f ∈ R T | f = [f (m i1 ; y i1 , z i1 ), . . . , f (m iT ; y iT , z iT )] ⊤ | m it = [M ] it , M ∈ M}.(46) The Rademacher complexity of set F , denoted as R T (F ), is defined as follows: Definition F.1. (Shalev-Shwartz & Ben-David, 2014) The empirical Rademacher complexity of a set of vectors F ⊂ R T is defined as follows: R T (F ) = 1 T E sup f ∈F T i=1 ǫ i f i ,(47) where expectation is w.r.t. ǫ i 's which are i.i.d. Rademacher random variables taking values from {−1, 1}. By Definition F.1, the empirical Rademacher complexity of the set F defined in (46) is given by R T (F ) = 1 T E σ sup f ∈F T i=1 σ i f (m i ; y i , z i ) .(48) To characterize the Rademacher complexity R T (F ), we proceed to characterize the covering number of F which is defined as follows: Definition F.2. (Vershynin, 2012) The ε-net of a set F represented by F ε is a finite set such that for any f ∈ F , there is a f ∈ F ε satisfying f − f 2 ≤ ε. The covering number of F is N(F , ε) = |F ε |. Consider the following: f − f 2 2 = i∈Ω (f (m i , y i , z i ) − f (m i ; y i , z i )) 2 ≤ i∈Ω L 2 f (m i − m i ) 2 ≤ L 2 f T [M − M ] Ω 2 F ≤ L 2 f T M − M Ξ 2 F .(49) where f ∈ F ε belongs to an ε-net for F , M ∈ M ε belongs to an ε-net of M, m i = [M ] i , and T = |Ω|. The last inequality is by applying the assumption that Ω ⊆ Ξ. To characterize the term L f in (49), we have the following result: Fact F.3. Assume that there exist two real numbers β > 0 and α < ∞ such that β ≤ m i ≤ α for all i ∈ Ω. Then, the following hold for all i ∈ Ω with probability greater than 1 − e −α(e 2 −3) : |f ′ i (m i ; y i , z i )| ≤ 1 + α(e 2 − 2) β . The proof is provided in Appendix H. Hence, applying Fact F.3, in (49), we have the following relation with probability greater than 1 − e −α(e 2 −3) , L f = 1 + (α(e 2 − 2))/β.(50) Next, we consider the term (49): [M − M ] Ξ 2 F in[M − M ] Ξ F = Λ ⊛ P − Λ ⊛ P Ξ F = Λ ⊛ P + Λ ⊛ P − Λ ⊛ P − Λ ⊛ P Ξ F ≤ [(Λ − Λ) ⊛ P ] Ξ F + [Λ ⊛ (P − P )] Ξ F ≤ max i∈Ξ p i [Λ − Λ] Ξ F + max i∈Ξ λ i [P − P ] Ξ F ≤ max i p i Λ − Λ F + max i λ i g θ ([Z] Ξ ) − g θ ([Z] Ξ ) F ≤ Λ − Λ F + c λ g θ ([Z] Ξ ) − g θ ([Z] Ξ ) F ,(51) where g θ ([Z] Ξ ) = [g θ (z i1 ), . . . , g θ (z iS )] ⊤ , g θ ∈ G, g θ ([Z] Ξ ) = [g θ (z i1 ), . . . , g θ (z iS )] ⊤ , g θ ∈ G ε belongs to the ε-net of G, z i1 , . . . , z iS are the set of observed features, and Λ ∈ L ε belongs to an ε-net of L. The first inequality uses the triangle inequality and the last inequality uses the facts that max i p i ≤ 1 and max i λ i ≤ c λ where we have c λ = F α K u under Assumption 4.1. Eq. (49) and (51) imply that that if there exists an ε /2c λ L f √ T -net covering for G • Ξ, where G • Ξ = {g θ ([Z] Ξ ) = [g(z i1 ), . . . , g(z iS )] ⊤ , g θ ∈ G},(52) and an ε /2L f √ T -net covering for L, then we can construct an ε-net to cover F . Since Λ which is a low-rank tensor, we invoke the following result to get the covering number of for L: (Fan et al., 2020) Lemma F.4.Let L = {X ∈ R I1×...×IK | X = F f =1 U 1 (:, f ) • . . . • U K (:, f ), U k (:, f ) 2 ≤ u}. Then, the covering number of L with respect to the Frobenius norm satisfies N(L, ε) ≤ 3K ε (F u 2 ) K/2 F k I k .(53) The proof of Lemma F.4 is also detailed in Sec. I. Invoking Lemma F.4, we have: N(L, ε /2L f √ T ) ≤ 6KL f √ T (F α u ) K ε F k I k ,(54) where we have applied U k (:, f ) 2 ≤ √ F α u . The covering number of the set G • Ξ is given by Lemma 14 of (Lin & Zhang, 2019) (cf. Lemma M.1 in Sec. M) as below: N(G • Ξ, ε) ≤ exp Z F R G ε 1/2 ,(55) where Z = [z i1 , . . . , z iS ] ∈ R D×S and R G denotes the complexity measure of the class G. Applying (55), we get N(G • Ξ, ε /2c λ L f √ T ) ≤ exp 2c λ L f √ T Z F R G ε 1/2 .(56) From (54) and (56), combined with (49) and (51), one can obtain that N(F , ε) ≤ N(L, ε /2L f √ T ) × N(G, ε /2c λ L f √ T ) ≤ 6KL f √ T (F α u ) K ε F k I k × exp 2c λ L f √ T Z F R G ε 1/2 .(57) To characterize the Rademacher complexity R T (F ) using the covering number N(F , ε), we invoke the following lemma: Lemma F.5. (Bartlett et al., 2017, Lemma A.5) The empirical Rademacher complexity of the set F ⊂ R T is upper bounded as follows: R T (F ) ≤ inf a>0 4a √ T + 12 T √ T a log N(F , µ)dµ .(58) Applying Lemma F.5, we have the following set of relations: R T (F ) ≤ inf a>0 4a √ T + 12 T √ T a log N(F , µ)dµ ≤ inf a>0 4a √ T + 12 √ T log N(F , a) ≤ inf a>0   4a √ T + 12 √ T F k I k log 6KL f √ T (F α u ) K a + c λ L f √ T Z F R G a   ≤   4 √ T + 12 √ T F k I k log 6KL f √ T (F α u ) K + F α K u L f √ T Z F R G   ,(59) where the second inequality is obtained since log N(F , µ) increases monotonically with the decrease of µ and hence √ T a log N(F , µ)dµ ≤ √ T log N(F , a), the third inequality is by applying (57), the last inequality is by setting a = 1, applying c λ = F α K u and L f is given by (50) with probability greater than 1 − e −α(e 2 −3) . G. Proof of Lemma E.1 We start by noting that β ≤ m i ≤ α (see (22) and (23)). By Lemma H.1, we have Pr (y i ≤ c) ≥ 1 − e −α(e 2 −3) , c = α(e 2 − 2).(60)Also, if β < 1, we have f i = m i − y i log m i ≤ α + c| log β|. Suppose β ≥ 1, we have f i = m i − y log m i ≤ m i + y i log m i ≤ α + c log α. Similarly, we get that f i ≥ β − c log α. Hence, we get f max ≤ α + c max{| log β|, log α} f min ≤ β − c log α, which hold with probability greater than 1 − e −α(e 2 −3) . H. Proof of Fact F.3 We have f i (m i ) = m i − y i log m i . Hence, we get f ′ i (m i ) = 1 − y i m i .(61) To proceed, we invoke the following lemma: Lemma H.1. (Cao & Xie, 2015), Lemma 9 (Tail Bound of Poisson) For y ∼ Poisson(m) with m ≤ α, we have Pr(y − m ≥ t) ≤ e −t(62) for all t ≥ t 0 where t 0 = α(e 2 − 3), where e is the Euler's number. Utilizing Lemma H.1, it can be seen that Pr(y i − α ≥ α(e 2 − 3)) ≤ e −α(e 2 −3) =⇒ Pr(y i ≤ α(e 2 − 3) + α) ≥ 1 − e −α(e 2 −3) . Combining this result with (61), we get that with probability greater than 1 − e −α(e 2 −3) , |f ′ i (m i )| ≤ 1 + y i m i ≤ 1 + α(e 2 − 2) β , where the last inequality uses the assumption that m i ≥ β. I. Proof of Lemma F.4 Let L ε denote the ε-net of L. Consider X ∈ L ε . In addition, let us define the following set: R k = {U ∈ R I k ×F | U (:, f ) 2 ≤ u}. Let U k belongs to ε-net of R k such that U k −U k F ≤ ε. Then, the cardinality of the ε-net of R k is given by (Wainwright, 2019) N(R k , ε) ≤ 3 √ F u 2 ε I k F .(63) In order to derive the covering number of the set L, we consider the following chain of relations: X − X F = F f =1 U 1 (:, f ) • . . . • U K (:, f ) F − F f =1 U 1 (:, f ) • . . . • U K (:, f ) F (64a) ≤ F f =1 U 1 (:, f ) − U 1 (:, f ) • U 2 (:, f ) . . . • U K (:, f ) F + F f =1 U 1 (:, f ) •   F f =1 U 2 (:, f ) . . . • U K (:, f ) − U 2 (:, f ) . . . • U K (:, f )   F (64b) ≤ ( √ F u 2 ) K−1 F f =1 U 1 (:, f ) − U 1 (:, f ) F + √ F u 2 F f =1 U 2 (:, f ) • . . . • U K (:, f ) F − F f =1 U 2 (:, f ) • . . . • U K (:, f ) F QU 1 (64c) ≤ ( √ F u 2 ) K−1 U 1 − U 1 F + √ F u 2 Q U1 .(64d) In a similar way as followed in (64), we can derive upper-bounds for every Q U k , k ∈ [K]. Then, we can finally establish the below relationship: X − X F ≤ ( √ F u 2 ) K−1 K k=1 U k − U k F .(65) The result in (65) implies that if there exists an ε /K( √ F u 2 ) K−1 -net to cover each R k , then we can construct an ε-net to cover L. Then, the cardinality of the ε-net of L is given by N(L, ε) ≤ K k=1 N (R k , ε /K( √ F u 2 ) K−1 ) ≤ 3K ε (F u 2 ) K/2 F k I k , where the inequality is by applying (63). Hence the proof. J. Proof of Lemma D.2 Let us define u = 1 |Θ| [M 1 − M 2 ] Θ 2 F ,(66a)u = 1 k I k M 1 − M 2 2 F .(66b) Next, we consider the following lemma which is the Serfling's sampling-without-replacement extension of the Hoeffding's inequality (Serfling, 1974): Lemma J.1. Let X 1 , X 2 , . . . , X M be a set of samples taken without replacement from {x 1 , x 2 , . . . , x N } of mean µ. Denote a = min i x i and b = max i x i . Then Pr 1 M M i=1 X i − µ ≥ t ≤ 2 exp − 2M t 2 (1 − (M − 1)/N )(b − a) 2 . One can see that u in (66b) denotes the mean of k I k terms of ( m i − m ♮ i ) 2 , u denotes the mean of |Θ| samples randomly drawn from {( m i − m ♮ ) 2 } without replacement. Hence, using the assumption that i ∈ Θ is drawn uniformly at random from [I 1 ] × . . . × [I K ] and invoking Lemma J.1, we have Pr [| u − u| ≥ t] ≤2 exp − 2|Θ|t 2 (1 − (|Θ| − 1)/ k I k )α 4 . where we also applied ( m i − m ♮ i ) 2 ≤ α 2 . Let δ = 2 exp − 2|Θ|t 2 (1−(|Θ|−1)/ k I k )α 4 , we have the following result with probability 1 − δ: | u − u| ≤α 2 log 2 δ 1 − (|Θ| − 1) k I k 1 2|Θ| .(67) Using | √ a − √ b| ≤ |a − b| for nonnegative a and b, we have √ u − √ u ≤ | u − u|, which implies that √ u − √ u ≤ α log 2 δ 1 − (|Θ| − 1) k I k 1 2|Θ| 1/4 ,(68) holds with probability greater than 1 − δ. K. Proof of Lemma D.3 First, let us define the following notations w.r.t the loss function ℓ(p 1 , p 2 ) = (p 1 − p 2 ) 2 , p 1 , p 2 ∈ [0, 1] and the observed features Z = [z i1 , . . . , z iS ] where each z is is sampled i.i.d. from the distribution D: L Z ( g θ ) 1 S S s=1 ℓ(g ♮ (z is ), 1 / ξ g θ (z is )) = 1 S S s=1 (g ♮ (z is ) − 1 / ξ g θ (z is )) 2 = P ♮ − 1 / ξ P Ξ 2 F S (69a) L D ( g θ ) E z∼D ℓ(g ♮ (z), 1 / ξ g θ (z)) = E z∼D (g ♮ (z) − 1 / ξ g θ (z)) 2 .(69b) We invoke Theorem 26.5 in (Shalev-Shwartz & Ben-David, 2014) and get that with probability greater than 1 − δ: L D ( g θ ) ≤ L Z ( g θ ) + 2R S (ℓ • G • Ξ) + 4c 2 log(4/δ) S =⇒ E z∼D (g ♮ (z) − 1 / ξ g θ (z)) 2 ≤ P ♮ − 1 / ξ P Ξ 2 F S + 2R S (G • Ξ) + 4 2 log(4/δ) S(70) where the last inequality utilizes the definitions in (69), the contraction lemma (Lemma 26.9 from (Shalev-Shwartz & Ben-David, 2014)) and also appliedc = 1 since |ℓ(f , y)| ≤ 1 in our case. The term R S (G • Ξ) denotes the empirical Rademacher complexity of the function class G • Ξ (see its definition in (52)) which is upperbounded via the sensitive complexity parameter R G as follows (Lin & Zhang, 2019): R S (G • Ξ) ≤ 16S −5/8 (2 Z F R G ) 1 4 . This completes the proof. L. Proof of Lemma C.1 Consider the following fact due to the cocavity of the log function and Jensen's inequality (Boyd & Vandenberghe, 2004): Fact L.1. Assume that α f 's are certain scalars such that α f > 0, f α f = 1. Then, for any u f > 0, log f α f u f ≥ f α f log u f . To proceed, we define the following scalars for our case: α (f ) i := U k (i k , f ) j =k U j (i j , f ) F f =1 U k (i k , f ) j =k U j (i j , f ) , ∀f One can see that F f =1 α (f ) i = 1, α (f ) i > 0, ∀f . Hence, we can use Fact L.1 to have the following inequality: log F f =1 U k (i k , f ) j =k U j (i j , f ) ≥ F f =1 α (f ) i log U k (i k , f ) j =k U j (i j , f ) α (f ) i . Applying the above relation, we immediately have w(U k ) ≤ s(U k ; U k ), ∀U k .(71)By substituting U k = U k in s(U k ; U k ), we have s(U k ; U k ) = i∈Ω F f =1 U k (i k , f ) j =k U j (i j , f )p i − y i F f =1 α (f ) i log F f =1 U k (i k , f ) j =k U j (i j , f ) = i∈Ω   F f =1 U k (i k , f ) j =k U j (i j , f ) p i − y i log F f =1 U k (i k , f ) j =k U j (i j , f )   = w(U k ),(72) where the second equality is due to f α (f ) i = 1. By taking the derivative of s(U k ; U k ) w.r.t. U k (i k , f ), we have ∇ U k (i k ,f ) s(U k ; U k ) = i −k ∈Ω −k j =k U j (i j , f )p i − y i α (f ) i α (f ) i U k (i k , f ) j =k U j (i j , f ) j =k U j (i j , f ) α (f ) i = i −k ∈Ω −k j =k U j (i j , f )p i − y i α (f ) i U k (i k , f ) = i −k ∈Ω −k j =k U j (i j , f )p i − y i U k (i k , f ) j =k U j (i j , f ) U k (i k , f )( F f =1 U k (i k , f ) j =k U j (i j , f )) ,(73) where i −k = (i 1 , . . . , i k−1 , i k+1 , . . . , i K ), Ω −k collects all i −k 's such that the corresponding y i 's are observed and the last equality (73) is obtained by substituting the definition of α (f ) i . Further, by taking the derivative of w(U k ) w.r.t. U k (i k , f ), we get ∇ U k (i k ,f ) w(U k ) = i −k ∈Ω −k j =k U j (i j , f )p i − y i j =k U j (i j , f ) ( F f =1 U k (i k , f ) j =k U j (i j , f ) .(74) From (73) and (74), one can see that ∇ U k (i k ,f ) w(U k ) = ∇ U k (i k ,f ) s(U k ; U k ).(75G = {g θ : R D → [0, 1] | g θ (z) = σ(w ⊤ L σ(W L−1 σ(. . . σ(W 1 z + b 1 ) + . . .) + b L )), ∀z ∈ R d }, where σ(·) = [σ(·) , . . . , σ(·)] ⊤ denote the activation function, L denotes the number of layers, W ℓ ∈ R d ℓ ×d ℓ−1 , d 0 = D, b ℓ ∈ R d ℓ , and θ represents the parameters such that θ = ({W ℓ , b ℓ } L−1 ℓ=1 , w L , b L ). Then, the covering number of G • Ξ with respect to the Frobenius norm satisfies N(G • Ξ, ε) ≤ exp Z F R G ε 1/2 ,(76) where G • Ξ = {g θ ([Z] Ξ ) = [g(z i1 ), . . . , g(z iS )] ⊤ , g θ ∈ G}, Z = [z i1 , . . . , z iS ] ∈ R D×S represents the training data and R G = 2 L ℓ=1 ρ ℓ s ℓ L ℓ=1 d 2 ℓ d 2 ℓ−1 a ℓ s ℓ L 2 , where W ℓ F ≤ a ℓ , W ℓ 2 ≤ s ℓ , the activation function σ ℓ is ρ ℓ -Lipschitz and σ ℓ (0) = 0. and Ω ⊃ Ξ,. K = 3, I k = 20, ∀k, F = 3, D = 10, SNR = 40dB, g(z) = sigmoid(ν ⊤ (3z 3 + 0.2z)), where the vector ν ∈ R D is generated by randomly sampling its entries a uniform distribution between 0 and 1. N. More Details on Experiments N.1. Synthetic Data Experiments -Implementation Settings. To learn g(·), we use a neural network g θ (·) with 3 hidden layers and 20 ReLU activation functions in each hidden layer. The optimizer for handling the subproblems of θ is Adam (Kingma & Ba, 2015) with an initial learning rate of 10 −3 . The optimizer uses a batch size of 1024. The subproblems are stopped when the relative change in the corresponding objective functions is smaller than 10 −6 . Also, the algorithm is stopped if the relative change in the overall objective function is less than 10 −6 or if 100 BCD iterations are completed. We fix the regularization term in the loss function to be ℓ(x, y) = (x − y) 2 . The regularization parameter µ is chosen to be a high value, i.e., µ = 2000 across all experiments. N.2. Additional Synthetic Data Experiments In Fig. 2, we compare the performance of the proposed approach and two baselines by plotting MAE p and MAE λ over iterations, under different conditions such as Ω ⊂ Ξ, Ω ⊃ Ξ, and Ω = Ξ. The baseline NTF-CPD-KL does not consider the detection probability in their model. Hence, we only plot its MAE λ . It can be observed that UncleTC outperforms the baselines after around 10 iterations under all the settings under test. Table 7 shows the results on a different synthetic dataset. Here, we consider another nonlinear function for the data generation, i.e., g(z) = sigmoid ν ⊤ (0.1 log(z 2 ) + 0.1z 2 ) , where the vector ν ∈ R D is generated by randomly sampling its entries from the uniform distribution between 0 and 1. We fix γ Ω = 0.2, γ Ξ = 0.3, γ Θ = 0.2, SNR = 40dB, and vary the rank F of the ground-truth tensor Λ ♮ . One can note that under these settings, the proposed algorithm still consistently gives better performance relative to the baselines. Table 8 shows the results where wrong tensor ranks F 's were used in our algorithms. In practice, the underlying data generation process is unknown and hence, the true rank F is hard to know or estimate. Hence, it is of interest to understand the algorithm's robustness to using a wrong F . One can see that the MAE values are worse when F < F but become better when F ≥ F . This makes sense since underestimating the rank means that less information of the true model is captured by the algorithms. Table 9 shows the performance under various µ's, i.e., the regularization parameter of ℓ in UncleTC. The settings follow those of Table 7 with rank F = 3. The results indicate that using reasonably large µ (µ ≥ 1000) may be preferable, as we hope that the regularization term can enforce equality. In practice, one may also use a validation set to choose µ. In order to extract a count-type tensor from the COVID-19 dataset, we adopt the following procedure: Each county is represented using two coordinates, i.e., the rounded value of latitude and longitude of its geographic center (we collected this information from a publicly available website 3 ). The rounded value of latitude points ranges between 20 and 69, out of which 43 values correspond to the latitudes of at least one county. On the other hand, the discretized longitude points ranges between -165 and -69, out of which 80 discrete points correspond to the longitudes of at least one county. The third coordinate of the tensor is the dates on which the COVID-19 cases were reported. Hence, each entry of the tensor represents the number of reported COVID-19 cases associated with a latitude point and a longitude point on a particular day. We use the data corresponding to 60 days from January 1, 2021 to March 1, 2021. We form a 43 × 80 × 60 tensor, which has 19.73% of observed entries, out of which 15.43% are nonzeros entries and 4.30% are zeros. subset of the dataset by selecting 50 plants and 50 pollinators that exhibit the highest number of interactions. This leads to a 50 × 50 × 37 integer tensor with 22.06% observed entries, out of which 4.01% are nonzero entries and 18.05% are zero entries. The zero entries indicate that the plant and pollinator were both observed by the researcher during the visit, but they were not observed to be interacting with one another. The dataset has 37 attributes: • 12 plant species static traits: the estimate of reward per flower, the basic ecological life form of a plant, floral structure, peduncle feebleness, the level of exclusion, exclusion by flower color, exclusion by pendant position, time of diel availability, size of pollen grain, the ability of the inflorescence to support the arrival of the flower visitor, the possible affinity of plant species to edaphic characteristics, and dispersal mechanism. • 11 pollinator species static traits: the length of a species body size, the width of a species body size, the depth of a species body size, individual biomass, size, energy requirement, a species behavior characterized by active time, the level of exclusion, an indicator showing whether the platform is required to approach flowers, tongue length, and tube size. • 1 plant temporal feature: flower abundance • 4 temporal features: month, day, wind, and cloud • 7 temperature features: mean, max, min, max time, min time, cumulative degree days with 5 degrees, and cumulative degree days with 10 degrees. • 2 precipitation features: daily precipitation, and antecedent with k = 0.9 N.4. Real Data Experiments -Implementation Settings. For the proposed UncleTC, we fix ReLU as the neural network activation function and Adam as the the optimizer. Other parameters such as rank F , the regularization parameter µ, batch size, learning rates, and neural network size are selected via the grid search hyper-parameter tuning strategy on the validation set where the rRMSE is used to measure the validation score. For the proposed UncleTC, we choose the rank F of the tensor from [5,10,15,20], the regularization parameter µ from [500,1000,2000,3000], the number of hidden units from [10,20,30], the number of hidden layers from [1, 3, 5], the learning rate from [0.001, 0.005, 0.01] and the batch size from [256,512,1024]. For all the low-rank tensor decompositionbased baselines, we select the rank from [5,10,15,20]. In the experiments, we perform 5-fold cross-validation to characterize the prediction performance with k chosen to be 5, where 3 splits serve for training, and the other two for validation and testing, respectively. The roles of the sets are switched in a cyclical way in different trials. N.5. Real Data Experiments -Additional Results. Fig. 3 displays the top 10 interactions with the highest and lowest average detection probabilities, as discovered by our proposed UncleTC method. Generally, interactions involving larger, more common species are more likely to have higher detection probabilities, while smaller, rarer species tend to have lower detection probabilities. The species with the highest average detection probabilities have a larger number of observed interactions compared to those with the lowest average detection probabilities. However, it is worth noting that not all species follow this trend. Other factors such as species rareness, environmental conditions, or seasonal variability may contribute to these differences (Dorado et al., 2011;Chacoff et al., 2012). In Fig. 4, we present the periodic variation of detection probabilities for specific species interactions over the years. Both figures illustrate temporal variation across visits and years for two different interactions-one with the highest average detection probability and the other with the lowest. Despite the variations, the overall range remains relatively consistent, with probabilities ranging from approximately 0.4 to 0.8 in the left figure and from 0.2 to 0.6 in the right figure, ultimately leading to the highest and lowest average detection probabilities. Considering the exceptions where detection probability does not align with species activity levels, we further examined interactions that show the most and least significant variations in detection probabilities over time. Fig. 5 plots the detection probabilities over visits for interactions with the highest (left) and lowest (right) standard deviation of detection probabilities across visits and annually. The left figure confirms that the detection probability of the interaction between two well-known common species (Gilia capitata and Apis mellifera) is significantly influenced by the time of year, resulting in a relatively lower average detection probability. The right figure provides another example of an interaction between another plant and pollinator species (Eriophyllum lanatum and Bombus mixtus), with relatively lower detection probabilities over time. Such variation may be a product of the phenology of the species (i.e., when flowers are blooming and when insects are flying). namely NTF-CPD-KL(Chi & Kolda, 2012), HaLRTC (Liu et al., 2013), NTF-CPD-LS (Shashua & Hazan, 2005), BPTF-CPD (Schein et al., 2015), and NTF-Tucker-LS (Kim & Choi, 2007) Figure 1 . 1The top-10 counties that have the highest UncleTCestimated average detection probabilities (left) and lowest average detection probabilities (right) for COVID-19. Inside the bar, we show the average numbers of COVID-19 tests done per 1000 people during the selected period. ) . The constant e denotes the Euler's number. [x] [0,1] denotes min{max{x, 0}, 1}. Lemma B. 1 . 1(Dennis et al., 2015) Assume that the observation y follows the below generative model:n ∼ Poisson(λ), (15a) y ∼ Binomial(n, p). (15b)Then, the model in(15)is equivalent to y ∼ Poisson(λp).Lemma B.1 states that the Poisson-Binomial model admits an equivalent Poisson model whose Poisson parameter is the multiplication of the original Poisson parameter and the detection probability. The derivation is based on an interesting equality:∞ n=y λ n e −λ n! n!p y (1 − p) n−y y!(n − y)! = (pλ) y e −λp y! ; see the derivation in(Fu et al., 2021). 2 Figure 2 . 22) Ω ⊂ Ξ, γΩ = 0.1, γΞ = 0.3, γΘ = 0) Ω = Ξ, γΩ = 0.3, γΞ = 0.3, γΘ = 0) Ω ⊃ Ξ, γΩ = 0.4, γΞ = 0.3, γΘ = 0.Average MAEp (top) and MAE λ (bottom) over 20 random trials plotted against iterations for different settings Ω ⊂ Ξ, Ω = Ξ, Figure 3 . 3Pollinator-Plant Interaction (PPI) Data. The Plant-Pollinator Interaction (PPI) dataset is a publicly available dataset, collected by the researchers at the H.J. Andrews (HJA) Long-term Ecological Research site in Oregon,USA (Seo et al., 2022; Daly et al., 2019). The dataset comprises plant-pollinator interactions observed at about 12 meadows over a period of seven years. Each meadow was visited 5-7 times per year by student researchers. The dataset contains interactions between 69 plant species and 215 pollinator species observed during 37 visits. Only 1.07% of the entries are recorded. The counts in this dataset are widely believed to be under-counted(Fu et al., 2021; Seo & Hutchinson, 2018). We consider aTable 9. Average MAEp and MAE λ for UncleTC over 20 random trials under different µ values. We fix K = 3, F = 3, D = 10, The top 10 plant-pollinator interactions having highest average detection probabilities (left) and lowest average average detection probabilities (right) as identified by the proposed UncleTC. Figure 4 . 4(left) The detection probabilities of the interaction pair Senecio integerrimus-Apis mellifera, which is reported by our algorithm to have the highest average detection probability. (right) The detection probabilities of the pair Aquilegia formosa-Bombus flavifrons, which is reported by our algorithm to have the lowest average detection probability. In addition, the sparsity parameter of the BPTF-CPD (Schein et al., 2015) is selected from [0.02, 0.05, 0.1, 0.2] and the regularization parameter of the HaLRTC (Liu et al., 2013) is selected from [10 −7 , 10 −6 , 10 −5 , 10 −4 ], based on the recommended range of values from the respective papers. The number of iterations to stop training all the iterative algorithms is also chosen using the validation sets. Figure 5 . 5(left) The detection probability of the interaction Gilia capitata-Apis mellifera over different years, which is reported by our algorithm to have the highest standard deviation of the detection probability across 37 visits. (right) The detection probability pattern for the interaction Eriophyllum lanatum-Bombus mixtus, which is reported by our method to have the lowest standard deviation of the detection probability across 37 visits. Table 1 . 1Average MAEp and MAE λ over 20 random trials under various values of γΩ; γΞ = 0.3, γΘ = 0.2, SNR = 40dB with Θ ∪ Ω ⊆ Ξ.Algorithm Metric γΩ = 0.05 γΩ = 0.1 γΩ = 0.3 UncleTC MAEp 0.156 ± 0.055 0.111 ± 0.038 0.063 ± 0.047 MAEλ 0.104 ± 0.063 0.063 ± 0.041 0.037 ± 0.031 UncleTC(Linear) MAEp 0.231 ± 0.072 0.189 ± 0.046 0.193 ± 0.180 MAEλ 0.135 ± 0.069 0.092 ± 0.039 0.090 ± 0.089 NTF-CPD-KL MAEλ 0.306 ± 0.119 0.258 ± 0.128 0.190 ± 0.124 BPTF MAEλ 0.547 ± 0.143 0.493± 0.161 0.345 ± 0.104 HaLRTC MAEλ 0.503 ± 0.382 0.495 ± 0.391 0.436 ± 0.282 NTF-LS MAEλ 0.742 ± 0.462 0.684 ± 0.421 0.403 ± 0.099 NTF-Tucker-LS MAEλ 0.639 ± 0.494 0.526 ± 0.299 0.341 ± 0.072 CostCo MAEλ 0.964 ± 0.235 0.853 ± 0.213 0.809 ± 0.147 POND MAEλ 0.841 ± 0.146 0.877 ± 0.098 0.885 ± 0.087 Table 2. Average MAEp and MAE λ over 20 random trials under various values of ζ; γΩ = 0.2, γΞ = 0.3, γΘ = 0.2 with Θ ∪ Ω ⊆ Ξ. Algorithm Metric SNR = 0dB SNR = 10dB SNR = 40dB UncleTC MAEp 0.154 ± 0.007 0.047 ± 0.018 0.035 ± 0.001 MAEλ 0.089 ± 0.010 0.026 ± 0.007 0.023 ± 0.001 UncleTC(Linear) MAEp 0.304 ± 0.012 0.109 ± 0.041 0.129 ± 0.049 MAEλ 0.193 ± 0.054 0.052 ± 0.016 0.065 ± 0.020 NTF-CPD-KL MAEλ 0.299 ± 0.055 0.111 ± 0.001 0.107 ± 0.001 BPTF MAEλ 0.420 ± 0.051 0.414 ± 0.047 0.427 ± 0.070 HaLRTC MAEλ 0.423 ± 0.068 0.441 ± 0.109 0.440 ± 0.126 NTF-LS MAEλ 0.479 ± 0.062 0.474 ± 0.057 0.472 ± 0.046 NTF-Tucker-LS MAEλ 0.366 ± 0.047 0.375 ± 0.050 0.378 ± 0.045 CostCo MAEλ 0.987 ± 0.256 0.876 ± 0.432 0.890 ± 0.357 POND MAEλ 0.824 ± 0.096 0.785 ± 0.131 0.837 ± 0.087 NN-based tensor completion methods, CostCo (Liu et al., 2019) and POND (Tillinghast et al., 2020), as baselines. Table 3 . 3Average MAEp and MAE λ over 20 random trials under various values of γΘ; γΩ = 0.2, γΞ = 0.7, SNR = 40dB with Θ ∪ Ω ⊆ Ξ.Algorithm Metric γΘ = 0 γΘ = 0.3 γΘ = 0.5 UncleTC MAEp 0.163 ± 0.007 0.126 ± 0.044 0.104 ± 0.076 MAEλ 0.062 ± 0.003 0.055 ± 0.028 0.054 ± 0.148 UncleTC(Linear) MAEp 0.231 ± 0.005 0.252 ± 0.029 0.201 ± 0.081 MAEλ 0.078 ± 0.005 0.104 ± 0.041 0.214 ± 0.174 NTF-CPD-KL MAEλ 0.185 ± 0.006 0.107 ± 0.002 0.156 ± 0.078 BPTF MAEλ 0.381 ± 0.024 0.445 ± 0.092 0.473 ± 0.174 HaLRTC MAEλ 0.436 ± 0.030 0.533 ± 0.212 0.561 ± 0.352 NTF-LS MAEλ 0.434 ± 0.030 0.472 ± 0.068 0.644 ± 0.412 NTF-Tucker-LS MAEλ 0.345 ± 0.039 0.411 ± 0.080 0.564 ± 0.337 CostCo MAEλ 0.876 ± 0.178 0.814 ± 0.301 0.987 ± 0.239 POND MAEλ 0.926 ± 0.158 0.851 ± 0.165 0.859 ± 0.112 Table 5 . 5The rRMSE of various methods on the COVID-19 and PPI datasets. population density, testing capacity and so on. In order to extract a count-type tensor from the COVID-19 dataset, we represent each county using two coordinates, i.e., the rounded value of latitude and longitude of its geographical center. This way, we obtain an integer tensor of size 43×80×60 (with 18.42% of entries observed), recording the COVID-19 case numbers of 642 counties over 60 days from January 1, 2021 to March 1, 2021.Pollinator-Plant Interaction (PPI) Data. The Plant-Pollinator Interaction (PPI) dataset is a publicly available dataset, collected by the researchers at the H.J. Andrews (HJA) Long-term Ecological Research site in Oregon,USA (Seo et al., 2022; Daly et al., 2019). We consider a subset of the dataset by selecting 50 plants and 50 pollinators that have the highest numbers of interactions. This leads to a 50 × 50 × 37 integer tensor with 22.06% observed entries.Method COVID-19 PPI UncleTC 3.52 ± 0.38 9.89 ± 1.67 UncleTC(Linear) 3.90 ± 0.33 10.21 ± 1.45 HaLRTC 6.14 ± 0.45 10.77 ± 1.25 BPTF 6.72 ± 0.42 11.02 ± 1.39 NTF-CPD-KL 3.61 ± 0.52 10.17 ± 1.12 NTF-CPD-LS 7.00 ± 0.38 11.25 ± 1.27 NTF-Tucker-LS 6.82 ± 0.48 11.22 ± 1.27 CostCo 7.12 ± 0.34 11.01 ± 1.14 POND 6.92 ± 0.51 10.95 ± 1.31 2020b). The dataset includes the number of reported COVID-19 cases of about 2270 US counties during 2020- 2021, with 25 different attributes such as social distancing index, Table 5 5presents the performance of various approaches on normalized margin bounds for neural networks. InAdvances in Neural Information Processing Systems, volume 30, 2017. Bertsekas, D. P. Nonlinear programming. Athena Scientific, 1999.Billio, M., Casarin, R., and Iacopini, M. Bayesian markovswitching tensor regression for time-varying networks.Chacoff, N. P., Vázquez, D. P., Lomáscolo, S. B., Stevani, E. 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Table 6 . 6Updates for p i subproblems for different loss functions f ; 20 until the stopping criterion is reached; output :estimates { U k } K k=1 , θ, and p i , ∀i ∈ Ω ∪ Ξ14 r ← r + 1; 15 until the stopping criterion is reached; 16 U (t+1) k ← U (r) k ; 17 U (0) k ← U (r) k ; 18 end 19 t ← t + 1; Theorem D.1. Suppose that the assumptions of Theorem 4.5 hold true. Let T = |Ω|. Assume that {y i } i∈Ω are i.i.d. samples under the generative model ) M . MCharacterization of the Complexity Measure R G The work in (Lin & Zhang, 2019) introduces a complexity measure, called the sensitive complexity and denoted as R G , to assess the expressive power of various families of neural network classes. For example, regrading the fully connected neural network function classes, the following result is presented: Lemma M.1. (Lin & Zhang, 2019, Lemma 14) Consider the following function class: Table 7 . 7Average MAEp and MAE λ over 20 random trials under different values of F . K = 3, I k = 20, ∀k, D = 10, γΩ = 0.2, γΞ = 0.3, γΘ = 0.2, SNR = 40dB, g(z) = sigmoid ν ⊤ (0.1 log(z 2 ) + 0.1z 2 ) .Table 8. Average MAEp and MAE λ over 20 random trials for different F with true rank F = 7. K = 3, I k = 20, ∀k, D = 10, γΩ = 0.2, γΞ = 0.3, γΘ = 0.2, SNR = 40dB, g(z) = sigmoid ν ⊤ (0.1 log(z 2 ) + 0.1z 2 ) .Algorithm Metric F = 5 F = 7 F = 10 UncleTC MAE p 0.059 ± 0.001 0.049 ± 0.029 0.030 ± 0.002 MAE λ 0.058 ± 0.007 0.052 ± 0.021 0.074 ± 0.001 UncleTC (Linear) MAE p 0.148 ± 0.018 0.187 ± 0.026 0.077 ± 0.005 MAE λ 0.102 ± 0.024 0.202 ± 0.024 0.131 ± 0.014 NTF-CPD-KL MAE λ 0.166 ± 0.050 0.214 ± 0.085 0.120 ± 0.001 Algorithm Metric F = 3 F = 5 F = 7 F = 10 F = 12 UncleTC MAE p 0.052 ± 0.015 0.042 ± 0.014 0.046 ± 0.024 0.051 ± 0.018 0.046 ± 0.019 MAE λ 0.125 ± 0.008 0.084 ± 0.011 0.060 ± 0.018 0.075 ± 0.019 0.076 ± 0.019 UncleTC (Linear) MAE p 0.110 ± 0.041 0.108 ± 0.057 0.130 ± 0.045 0.112 ± 0.041 0.116 ± 0.046 MAE λ 0.141 ± 0.017 0.120 ± 0.039 0.140 ± 0.046 0.155 ± 0.064 0.197 ± 0.143 NTF-CPD-KL MAE p 0.151 ± 0.029 0.134 ± 0.046 0.159 ± 0.071 0.157 ± 0.072 0.151 ± 0.069 N.3. Real Data Experiments -Dataset Details.COVID-19 Data. The dataset(Zhang et al., 2020b) includes the number of reported COVID-19 cases of about 2270 US counties during 462 days in 2020-2021. The dataset has 25 attributes, including social distancing index, percentage of people staying home, trips per person, percentage of out-of-county trips, percentage of out-of-state trips, transit mode share, miles per person, work trips per person, non-work trips per person, COVID exposure per 1000 people, percentage of people older than 60, median income, percentage of African Americans, percentage of Hispanic Americans, population density, number of hot spots per 1000 people, total population, percentage of people working from home, imported COVID cases, hospital beds per 1000 people, testing capacity gap, tests done per 1000 people, unemployment rate, cumulative inflation rate, and unemployment claims per 1000 people. Here, the set Ω can be a multiset, meaning it can contain multiple copies of an index since the the assumed sampling scheme is with replacement. https://public.opendatasoft.com/explore/dataset/us-county-boundaries Acknowledgement. This work is supported in part by the National Science Foundation (NSF) under Project NSF IIS-1910118. The PPI dataset is provided by the HJ Andrews Experimental Forest research program, funded by the National Science Foundation's Long-Term Ecological Research Program (DEB 2025755), US Forest Service Pacific Northwest Research Station, and Oregon State University. The authors thank Dr. Julia A. Jones for helpful guidance with the PPI data and associated features. Link prediction on evolving data using matrix and tensor factorizations. E Acar, D M Dunlavy, T G Kolda, IEEE International Conference on Data Mining Workshop. Acar, E., Dunlavy, D. M., and Kolda, T. G. Link prediction on evolving data using matrix and tensor factorizations. In IEEE International Conference on Data Mining Work- shop, pp. 262-269, 2009. Tensor factorization for knowledge graph completion. I Balazevic, C Allen, T Hospedales, Tucker, Proceedings of Empirical Methods in Natural Language Processing and International Joint Conference on Natural Language Processing. Empirical Methods in Natural Language Processing and International Joint Conference on Natural Language ProcessingBalazevic, I., Allen, C., and Hospedales, T. TuckER: Ten- sor factorization for knowledge graph completion. In Proceedings of Empirical Methods in Natural Language Processing and International Joint Conference on Natu- ral Language Processing, pp. 5185-5194, 2019. . P L Bartlett, D J Foster, M J Telgarsky, Spectrally, Bartlett, P. L., Foster, D. J., and Telgarsky, M. J. Spectrally- Hieracium cynoglossoides--Mordella atrata albosuturalis Rudbeckia occidentalis--Calliprobola pulchra Mimulus guttatus--Syrphus opinator Gilia capitata--Bombylius major. Hieracium cynoglossoides--Mordella atrata albosuturalis Rudbeckia occidentalis--Calliprobola pulchra Mimulus guttatus--Syrphus opinator Gilia capitata--Bombylius major Eriophyllum lanatum--Euphydryas colon Angelica arguta--Platycheirus stegnus. Eriophyllum lanatum--Euphydryas colon Angelica arguta--Platycheirus stegnus Collinsia parviflora--Halictus rubicundus. Collinsia parviflora--Halictus rubicundus Anaphalis margaritacea--Asemosyrphus polygrammus Senecio integerrimus--Apis mellifera. Anaphalis margaritacea--Asemosyrphus polygrammus Senecio integerrimus--Apis mellifera Perideridia gairdneri--Lepturopsis dolorosa Aquilegia formosa--Tapinoma sessile Potentilla gracilis--Plebejus acmon Mimulus tilingii--Anaspis rufa. Perideridia gairdneri--Lepturopsis dolorosa Aquilegia formosa--Tapinoma sessile Potentilla gracilis--Plebejus acmon Mimulus tilingii--Anaspis rufa Lomatium martindalei--Muscoid genus 1. Lomatium martindalei--Muscoid genus 1 Aquilegia formosa--Heringia sp 1. Aquilegia formosa--Heringia sp 1 Prunella vulgaris--Megachile perihirta Aquilegia formosa--Bombus flavifrons. Prunella vulgaris--Megachile perihirta Aquilegia formosa--Bombus flavifrons
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[ "Bootstrapped Training of Score-Conditioned Generator for Offline Design of Biological Sequences", "Bootstrapped Training of Score-Conditioned Generator for Offline Design of Biological Sequences" ]
[ "Minsu Kim \nKorea Advanced Institute of Science and Technology (KAIST)\n\n", "Federico Berto [email protected] \nKorea Advanced Institute of Science and Technology (KAIST)\n\n", "Sungsoo Ahn [email protected] \nPohang University of Science and Technology (POSTECH)\n\n", "Jinkyoo Park [email protected] \nKorea Advanced Institute of Science and Technology (KAIST)\n\n" ]
[ "Korea Advanced Institute of Science and Technology (KAIST)\n", "Korea Advanced Institute of Science and Technology (KAIST)\n", "Pohang University of Science and Technology (POSTECH)\n", "Korea Advanced Institute of Science and Technology (KAIST)\n" ]
[]
We study the problem of optimizing biological sequences, e.g., proteins, DNA, and RNA, to maximize a black-box score function that is only evaluated in an offline dataset. We propose a novel solution, bootstrapped training of scoreconditioned generator (BOOTGEN) algorithm. Our algorithm repeats a two-stage process. In the first stage, our algorithm trains the biological sequence generator with rank-based weights to enhance the accuracy of sequence generation based on high scores. The subsequent stage involves bootstrapping, which augments the training dataset with self-generated data labeled by a proxy score function. Our key idea is to align the score-based generation with a proxy score function, which distills the knowledge of the proxy score function to the generator. After training, we aggregate samples from multiple bootstrapped generators and proxies to produce a diverse design. Extensive experiments show that our method outperforms competitive baselines on biological sequential design tasks. We provide reproducible source code: https://github.com/kaist-silab/bootgen. Preprint. Under review.
null
[ "https://export.arxiv.org/pdf/2306.03111v1.pdf" ]
259,089,247
2306.03111
61f466f8b12df44009ab6a827bdb63f3f93a453d
Bootstrapped Training of Score-Conditioned Generator for Offline Design of Biological Sequences Minsu Kim Korea Advanced Institute of Science and Technology (KAIST) Federico Berto [email protected] Korea Advanced Institute of Science and Technology (KAIST) Sungsoo Ahn [email protected] Pohang University of Science and Technology (POSTECH) Jinkyoo Park [email protected] Korea Advanced Institute of Science and Technology (KAIST) Bootstrapped Training of Score-Conditioned Generator for Offline Design of Biological Sequences We study the problem of optimizing biological sequences, e.g., proteins, DNA, and RNA, to maximize a black-box score function that is only evaluated in an offline dataset. We propose a novel solution, bootstrapped training of scoreconditioned generator (BOOTGEN) algorithm. Our algorithm repeats a two-stage process. In the first stage, our algorithm trains the biological sequence generator with rank-based weights to enhance the accuracy of sequence generation based on high scores. The subsequent stage involves bootstrapping, which augments the training dataset with self-generated data labeled by a proxy score function. Our key idea is to align the score-based generation with a proxy score function, which distills the knowledge of the proxy score function to the generator. After training, we aggregate samples from multiple bootstrapped generators and proxies to produce a diverse design. Extensive experiments show that our method outperforms competitive baselines on biological sequential design tasks. We provide reproducible source code: https://github.com/kaist-silab/bootgen. Preprint. Under review. Introduction The automatic design of biological sequences, e.g., DNA, RNA, and proteins, with a specific property, e.g., high binding affinity, is a vital task within the field of biotechnology [6,48,37,30]. To solve this problem, researchers have developed algorithms to optimize a biological sequence to maximize a score function [36,10,11,3,26]. Here, the main challenge is the expensive evaluation of the score function that requires experiments in a laboratory setting or clinical trials. To resolve this issue, recent works have investigated offline model-based optimization [29,18,39,47,12,40,MBO]. Given an offline dataset of biological sequences paired with scores, offline MBO algorithms train a proxy for the score function, e.g., a deep neural network (DNN), and maximize the proxy function without querying the true score function. Therefore, such offline MBO algorithms bypass the expense of iteratively querying the true score function whenever a new solution is proposed. However, even optimizing such a proxy function is challenging due to the vast search space over the biological sequences. On one hand, several works [18,39,47,12] considered applying gradient-based maximization of the proxy function. However, when the proxy function is parameterized using a DNN, these methods often generate solutions where the true score is low despite the high proxy score. This is due to the fragility of DNNs against adversarial optimization of inputs [47,39,18]. Furthermore, the gradientbased methods additionally require reformulating biological sequence optimization as a continuous optimization, e.g., continuous relaxation [18,39,12] or mapping discrete designs to a continuous latent space [47]. On the other hand, one may consider training deep generative models to learn a distribution over high-scoring designs [29,26]. They learn to generate solutions from scratch, which amortizes optimization over the design space. To be specific, Kumar and Levine [29] suggests learning an inverse map from a score to a solution with a focus on generating high-scoring solutions. Next, Jain et al. [26] proposed training a generative flow network [8,GFN] as the generative distribution of high-scoring solutions. Contribution In this paper, we propose a novel algorithm, coined bootstrapped training of score conditioned generator (BOOTGEN), for the offline design of biological sequences. Our key idea is to enhance the score-conditioned ge nerator by suggesting a novel variation of the classical ensemble strategy of bootstrapping and aggregating. We train multiple generators using bootstrapped datasets from training and combine them with proxy models to create a reliable and diverse sampling solution. In the bootstrapped training, we aim to align a score-conditioned generator with a proxy function by bootstrapping the training dataset of the generator. To be specific, we repeat multiple stages of (1) training the conditional generator on the training dataset with a focus on high-scoring sequences and (2) augmenting the training dataset using sequences that are sampled from the generator and labeled using the proxy function. Intuitively, our framework improves the score-to-sequence mapping (generator) to be consistent with the sequence-to-score mapping (proxy function), which is typically more accurate. When training the score-conditioned generator, we assign high rank-based weights [41] to highscoring sequences. Sequences that are highly ranked among the training dataset are more frequently sampled to train the generator. This leads to shifting the training distribution towards an accurate generation of high-scoring samples. Compared with the value-based weighting scheme previously proposed by Kumar and Levine [29], the rank-based weighting scheme is more robust to the change of training dataset from bootstrapping. To further boost the performance of our algorithm, we propose two post-processing processes after the training: filtering and diversity aggregation (DA). The filtering process aims to filter samples from generators using the proxy function to gather samples with cross-agreement between the proxy and generator. On the other hand, DA collects sub-samples from multiple generators and combines them into complete samples. DA enables diverse decision-making with reduced variance in generating quality, as it collects samples from multiple bootstrapped generators. We perform extensive experiments on six offline biological design tasks: green fluorescent protein design [48,GFP], DNA optimization for expression level on an untranslated region [37,UTR], and transcription factor binding [6,TFBind8], and RNA optimization for binding to three types of transcription factors [30,. Our BOOTGEN demonstrates superior performance, surpassing the 100th percentile score and 50th percentile score of the design-bench baselines [40], a generative flow network (GFN)-based work [26] and bidirectional learning method (BDI) [12]. Furthermore, we additionally verify the superior performance of BOOTGEN in various design scenarios, particularly when given a few opportunities to propose solutions. Related Works Automatic design of biological sequences Researchers have investigated machine learning methods to automatically design biological sequences, e.g., Bayesian optimization [45,7,32,34,38], evolutionary methods [4,9,15,5,36], model-based reinforcement learning [2], and generative methods [10,29,17,26,16]. The aim of these methods is to optimize biological sequence (e.g., protein, RNA, and DNA) for maximizing the target objective of binding activity and folding, which has crucial application in drug discovery and health care [26]. These methods make different assumptions on the cost of evaluating the ground truth function, e.g., online optimization, sample-efficient optimization, and offline optimization. In this work, we focus on the offline optimization. Step A: Rank-based Weighted Training High Score † Figure 2.1: Illustration of the bootstrapped training process for learning score-conditioned generator. Offline Model-based Optimization Offline model-based optimization (MBO) aims to find the design x that maximizes a score function f (x), using only pre-collected offline data. The most common approach is to use gradient-based optimization on differentiable proxy models trained on an offline dataset [44,39,47,18,12]. However, they can lead to poor scores due to the non-smoothness of the proxy landscape. To address this issue, Trabucco et al. [39] proposed conservative objective models (COMs), which use adversarial training to create a smooth proxy. Yu et al. [47] suggests the direct imposition of a Gaussian prior on the proxy model to create a smooth landscape and model adaptation to perform robust estimation on the specific set of candidate design space. While these methods are effective for high-dimensional continuous design tasks, their performance on discrete spaces is often inferior to classical methods, e.g., gradient ascent [40]. Bootstrapping Bootstrapping means maximizing the utilization of existing resources. In a narrow sense, it refers to a statistical method where the original dataset is repeatedly sampled to create various datasets [22]. In a broader sense, it also encompasses concepts frequently used in machine learning to improve machine learning scenarios where the label is expensive, e.g., self-training and semi-supervised learning [33,1,19]. These methods utilize an iterative training scheme to augment the dataset with self-labeled samples with high confidence. The bootstrapping strategy at machine learning showed great success in various domains, e.g., fullylabeled classification [46], self-supervised learning [14] and offline reinforcement learning [43]. We introduce a novel bootstrapping strategy utilizing score-conditioned generators and apply it to offline biological sequence design, addressing the challenge of working with a limited amount of poor-quality offline datasets. Design by Conditional Generation Conditional generation is a promising method with several high-impact applications, e.g., classconditional image generation [31], language-to-image generation [35], reinforcement learning [13]. With the success of conditional generation, several studies proposed to use it for design tasks, e.g., molecule and biological sequence design. Hottung et al. [24] proposed an instance-conditioned variational auto-encoder [28] for routing problems, which can generate near-optimal routing paths conditioned on routing instances. Igashov et al. [25] suggested a conditional diffusion model to generate 3D molecules given their fragments. Specifically, the molecular fragments are injected into the latent space of the diffusion model, and the diffusion model generates links between fragments to make the 3D molecular compound. 3 Bootstrapped Training of Score-Conditioned Generator (BOOTGEN) Problem definition We are interested in optimizing a biological sequence x to maximize a given score function f (x). We consider an offline setting where, during optimization, we do not have access to the score function f (x). Instead, we optimize the biological sequences using a static dataset D = {(x n , y n )} N n=1 consisting of offline queries y n = f (x n ) to the score function. Finally, we consider evaluating a set of sequences {x m } M m=1 as an output of offline design algorithms. Algorithm 1 Bootrapped Training of Score-conditioned generators 1: Input: Offline dataset D = {x n , y n } N n=1 . 2: Update φ to minimize (x,y)∈D (f φ (x) − y) 2 . 3: for j = 1, . . . , N gen do Training multiple generaters 4: Initialize D tr ← D. 5: for i = 1, . . . , I do Rank-based weighted training 6: Update θ n to maximize (x,y)∈Dtr w(y, D tr ) log p θj (x|y). 7: Sample x * ∼ p θn (x|y † ) for = 1, . . . , L. Bootstrapping 8: Set y * ← f φ (x * ) for = 1, . . . , L. 9: Set D aug as top-K scoring samples in {x * , y * } L =1 . 10: Set D tr ← D tr ∪ D aug . 11: end for 12: end for 13: Output: trained score-conditioned generators p θ1 (x|y), ..., p θ N (x|y). Overview of BOOTGEN We first provide a high-level description of our bootstrapped training of score-conditioned generator, coined BOOTGEN. Our key idea is to align the score-conditioned generation with a proxy model via bootstrapped training (i.e., we train the generator on sequences labeled using the proxy model) and aggregate the decisions over multiple generators and proxies for reliable and diverse sampling of solutions. Before BOOTGEN training, we pre-train proxy score function f φ (x) ≈ f (x) only leveraging offline dataset D. After that, our BOOTGEN first initializes a training dataset D tr as the offline dataset D and then repeats the following steps: A. BootGen optimizes the score-conditioned generator p θ (x|y) using the training dataset D tr . During training, it assigns rank-based weights to each sequence for the generator to focus on high scores samples. B. BOOTGEN bootstraps the training dataset D tr using samples from the generator p θ (x|y † ) conditioned on the desired score y † . It uses a proxy f φ (x) of the score function to label the new samples. After BOOTGEN training for multiple score-conditioned generators p θ1 (x|y), ..., p θn (x|y), we aggregate samples from the generators with filtering of proxy score function f φ to generate diverse and reliable samples. We provide the pseudo-code of the overall procedure in Algorithm 1 for training, and Algorithm 2 for generating solutions from the trained model. Rank-based Weighted Training Here, we introduce our framework to train the score-conditioned generator. Our algorithm aims to train the score-conditioned generator with more focus on generating high-scoring designs. Such a goal is helpful for bootstrapping and evaluation of our framework, where we query the generator conditioned on a high score. Given a training dataset D tr , our BOOTGEN minimizes the following loss function: L(θ) := − (x,y)∈Dtr w(y, D tr ) log p θ (x|y), w(y, D tr ) = (k|D tr | + rank(y, D tr )) −1 (x,y)∈Dtr (k|D tr | + rank(y, D tr )) −1 . where w(y, D tr ) is the score-wise rank-based weight [41]. Here, k is a weight-shifting factor, and rank(y, D tr ) denotes the relative ranking of a score y with respect to the set of scores in the dataset D tr . We note that a small weight-shifting factor k assigns high weights to high-scoring samples. For mini-batch training of the score-conditioned generator, we approximate the loss function L(θ) via sampling with probability w(y, D tr ) for each sample (x, y). We note that Tripp et al. [41] proposed the rank-based weighting scheme for training unconditional generators to solve online design problems. At a high level, the weighting scheme guides the generator to focus more on generating high-scoring samples. Compared to weights that are proportional Algorithm 2 Aggregation Strategy for Sample Generation 1: Input: Trained score-conditioned generators p θ1 (x|y), ..., p θ Ngen (x|y), and trained proxy function f φ (x). 2: Initialize D samples ← ∅. 3: for i = 1, . . . , N gen do 4: Sample x * m ∼ p θi (x|y † ) for m ∈ [M ]. 5: Set y * m ← f φ (x * m ) for m ∈ [M ]. 6: Set D sub-samples as Top-K scoring samples in {x * m , y * m } M m=1 .Filtering 7: Set D samples ← D samples ∪ D sub-samples Diversity Aggregation 8: end for 9: Output: D samples . to scores [29], using the rank-based weights promotes the training to be more robust against outliers, e.g., samples with abnormally high weights. To be specific, the weighting factor w(y, D) is less affected by outliers due to its upper bound that is achieved when rank(y, D) = 1. Bootstrapping Next, we introduce our bootstrapping strategy to augment a training dataset with high-scoring samples that are collected from the score-conditioned generator and labeled using a proxy model. Our key idea is to enlarge the dataset so that the score-conditioned generation is consistent with predictions of the proxy model, in particular for the high-scoring samples. This enables self-training by utilizing the extrapolation capabilities of the generator and allows the proxy model to transfer its knowledge to the score-conditioned generation process. We first generate a set of samples x * 1 , . . . , x * L from the generator p θ (x|y † ) conditioned on the desired score y † 1 Then we compute the corresponding labels y * 1 , . . . , y * L using the proxy model, i.e., we set y = f φ (x ) for = 1, . . . , L. Finally, we augment the training dataset using the set of top-K samples D aug with respect to the proxy model, i.e., we set D tr ∪ D aug as the new training dataset D tr . Aggregation Strategy for Sample Generation Here, we introduce additional post-hoc aggregation strategies that can be used to further boost the quality of samples from our generator. See Algorithm 2 for a detailed process. Filtering We follow Kumar and Levine [29] to exploit the knowledge of the proxy function for filtering high-scoring samples from the generator. To be specific, when evaluating our model, we sample a set of candidate solutions and select the top samples with respect to the proxy function. Diverse aggregation To enhance the diversity of candidate samples while maintaining reliable generating performances with low variance, we gather cross-aggregated samples from multiple score-conditioned generators. These generators are independently trained using our proposed bootstrapped training approach. Since each bootstrapped training process introduces high randomness due to varying training datasets, combining the generative spaces of multiple generators yields a more diverse space compared to a single generator. Moreover, this process helps reduce the variance in generating quality. By creating ensemble candidate samples from multiple generators, we ensure stability and mitigate the risk of potential failure cases caused by adversarial samples. These samples may receive high scores from the proxy function but have low actual scores. This approach resembles the classical ensemble strategy known as "bagging," which aggregates noisy bootstrapped samples from decision trees to reduce variances. [40] 0.462 ± 0.080 0.437 ± 0.033 0.463 ± 0.043 0.936 ± 0.041 0.865 ± 0.003 0.685 ± 0.012 0.643 CMA-ES [21] 0.841 ± 0.058 0.822 ± 0.046 0.803 ± 0.039 0.904 ± 0.040 0.055 ± 0.003 0.737 ± 0.013 0.694 BO-qEI [45] 0.724 ± 0.055 0.729 ± 0.038 0.707 ± 0.034 0.798 ± 0.083 0.254 ± 0.352 0.684 ± 0.000 0.649 CbAS [10] 0.541 ± 0.042 0.647 ± 0.057 0.644 ± 0.071 0.913 ± 0.025 0.865 ± 0.004 0.692 ± 0.008 0.717 Auto. CbAS [17] 0.524 ± 0.055 0.562 ± 0.031 0.495 ± 0.048 0.890 ± 0.050 0.865 ± 0.003 0.693 ± 0.009 0.672 MIN [29] 0.376 ± 0.039 0.374 ± 0.041 0.404 ± 0.047 0.892 ± 0.060 0.865 ± 0.001 0.691 ± 0.011 0.600 Grad [40] 0.821 ± 0.048 0.720 ± 0.047 0.688 ± 0.035 0.965 ± 0.030 0.862 ± 0.003 0.682 ± 0.013 0.792 COMs [39] 0.403 ± 0.062 0.393 ± 0.076 0.494 ± 0.098 0.945 ± 0.033 0.861 ± 0.009 0.699 ± 0.011 0.633 GFN-AL [26] 0.630 ± 0.054 0.677 ± 0.079 0.623 ± 0.045 0.956 ± 0.018 0.059 ± 0.006 0.695 ± 0.021 0.607 BDI [12] 0.700 ± 0.000 0.560 ± 0.000 0.632 ± 0.000 0.973 ± 0.000 0.864 ± 0.000 0.667 ± 0.000 0.733 [21] 0.558 ± 0.012 0.531 ± 0.010 0.535 ± 0.012 0.526 ± 0.017 0.047 ± 0.000 0.497 ± 0.009 0.449 BO-qEI [45] 0.389 ± 0.009 0.397 ± 0.015 0.391 ± 0.012 0.439 ± 0.000 0.246 ± 0.341 0.571 ± 0.000 0.406 CbAS [10] 0.246 ± 0.008 0.267 ± 0.021 0.281 ± 0.015 0.467 ± 0.008 0.852 ± 0.004 0.566 ± 0.018 0.447 Auto. CbAS [17] 0.241 ± 0.022 0.237 ± 0.009 0.193 ± 0.007 0.413 ± 0.012 0.847 ± 0.003 0.563 ± 0.019 0.420 MIN [29] 0.146 ± 0.009 0.143 ± 0.007 0.174 ± 0.007 0.417 ± 0.012 0.830 ± 0.011 0.586 ± 0.000 0.383 Grad [40] 0.473 ± 0.025 0.462 ± 0.016 0.393 ± 0.017 0.513 ± 0.007 0.763 ± 0.181 0.611 ± 0.000 0.531 COMs [39] 0.172 ± 0.026 0.184 ± 0.039 0.228 ± 0.061 0.512 ± 0.051 0.737 ± 0.262 0.608 ± 0.000 0.407 GFN-AL [26] 0.312 ± 0.013 0.300 ± 0.012 0.324 ± 0.009 0.538 ± 0.045 0.051 ± 0.003 0.597 ± 0.021 0.354 BDI [12] 0.411 ± 0.000 0.308 ± 0.000 0.345 ± 0.000 0.595 ± 0.000 0.837 ± 0.010 0.527 ± 0.000 0.504 Experiments We present experimental results on six representative biological sequence design tasks to verify the effectiveness of the proposed method. We also conduct ablation studies to verify the effectiveness of each component in our method. For training, we use a single GPU of NVIDIA A100, where the training time of one generator spends approximately 10 minutes. Experimental Setting Tasks. We evaluate an offline design algorithm by (1) training it on an offline dataset and (2) using it to generate 128 samples for high scores. We measure the 50th percentile and 100th percentile scores of the generated samples. All the results are measured using eight independent random seeds. We consider six biological sequence design tasks: green fluorescent protein (GFP), DNA optimization for expression level on untranslated region (UTR), DNA optimization tasks for transcription factor binding (TFBind8), and three RNA optimization tasks for transcription factor binding (RNA-Binding-A, RNA-Binding-B, and RNA-Binding-C). The scores of the biological sequences range in [0, 1]. We report the statistics of the offline datasets used for each task in Implementation We parameterize the conditional distribution p θ (x t |x 1:t−1 , y) using a 2-layer long short-term memory [23,LSTM] network with 512 hidden dimensions. The condition y is injected into the LSTM using a linear projection layer. We parameterize the proxy model using a multi-layer perceptron (MLP) with 2048 hidden dimensions and a sigmoid activation function. Our parameterization is consistent across all the tasks. We provide a detailed description of the hyperparameters in Appendix A. We also note the importance of the desired score y † to condition during bootstrapping and evaluation. In this regard, we set it as the maximum score that is achievable for the given problem, i.e., we set y † = 1. We assume that such a value is known following [12]. Performance Evaluation In Table 4.1 and Table 4.2, we report the performance of our BOOTGEN along with other baselines. One can observe how our BOOTGEN consistently outperforms the considered baselines across all six tasks. In particular, one can observe how our BOOTGEN achieves large gains in 50th percentile metrics. This highlights how our algorithm is able to create a reliable set of candidates. For TFbind8, which has a relatively small search space (4 8 ), having high performances on the 100th percentile is relatively easy. Indeed, the classical method of CMA-ES and Grad. gave pretty good performances. However, for the 50th percentile score, which is a metric for measuring the method's reliability, BDI outperformed previous baselines by a large margin. Our method outperformed even BDI and achieved an overwhelming score. For higher dimensional tasks of UTR, even the 50th percentile score of BOOTGEN outperforms the 100th percentile score of other baselines by a large margin. We note that our bootstrapping strategies and aggregation strategy greatly contributed to improving performances on UTR. For additional tasks of RNA, we achieved the best score for both the 50th percentile and the 100th percentile. This result verifies that our method is task-expandable. Varying the evaluation budget In real-world scenarios, there may be situations where only a few samples can be evaluated due to the expensive score function. For example, in an extreme scenario, for the clinical trial of a new protein drug, there may be only one or two chances to be evaluated. As we measure the 50th percentile and 100th percentile score among 128 samples following the design-bench [40] at Tables 4.1 and 4.2, we also provide a 100th percentile score report at the fewer samples from 1 sample to the 128 samples to evaluate the model's robustness on the low-budget evaluation scenarios. To account for this, we also provide a budget-performance graph that compares the performance of our model to the baselines using different numbers of evaluations. This allows us to observe the trade-off between performance and the number of samples generated. Note that we select baselines as the Top 5 methods in terms of average percentile 100 scores reported at Table 4.1. As shown in Fig. 4.1, our method outperforms every baseline for almost every evaluation budget. For the UTR task, our performance on a single evaluation budget gives a better score than the other baselines' scores when they have a budget of 128 evaluations. For RNA tasks, our method with an approximate budget of 30 achieves superior performance compared to other methods with a budget of 128. These results show that our method is the most reliable as its performance is most robust when the evaluation budget is limited. Average Score with Diversity For biological sequence design, measuring the diversity and novelty of the generated sequence is also crucial [26]. Following the evaluation metric of [26] we compare the performance of models in terms of diversity and novelty. Here is measurement of diversity for sampled design dataset D = {x 1 , ..., x M } from generator which is average of Levenshtein distance [20], denoted by d (x i , x j ), between arbitrary two biological sequences x i , x j from the generated design candidates D: Diversity(D) := 1 |D|(|D| − 1) x∈D s∈D\{x} d (x, s) . Next, we measure the minimum distance from the offline dataset D offline which measures the novelty of generated design candidates D as: Novelty(D, D offline ) = 1 |D| x∈D min s∈Doffline d (x, s) . Our method surpasses all baselines, including GFN-AL [26], in the UTR task, as evidenced by the Pareto frontier depicted in Table 4.3 and Fig. 4.2. Given the highly dimensional nature of the UTR task and its expansive search space, the discovery of novel and diverse candidates appears to be directly related to their average score. This implies that extensive exploration of the highdimensional space is crucial for improving scores in the UTR task. It is worth noting that GFN-AL, which is specifically designed to generate diverse, high-quality samples through an explorative policy, secures a second place for diversity. Although GFN-AL occasionally exhibits better diversity than our method and achieves a second-place average score, it consistently delivers poor average scores in the GFP and RNA tasks Diverse aggregation strategy Our diverse aggregation (DA) strategy significantly enhances diversity, novelty, and score variance, as demonstrated in Table 4.3. This is especially beneficial for the UTR task, which necessitates extensive exploration of a vast solution space, posing a substantial risk to the bootstrapped training process. In this context, certain bootstrapped generators may yield exceedingly high scores, while others may produce low scores due to random exploration scenarios. By employing DA, we combine multiple generators to generate candidate samples, thereby greatly stabilizing the quality of the bootstrapped generator. Ablation study The effectiveness of our components, namely rank-based reweighting (RR), bootstrapping (B), and filtering (F), in improving performance is evident in Table 4.4. Across all tasks, these components consistently contribute to performance enhancements. The bootstrapping process is particularly more beneficial for high-dimensional tasks like UTR and GFP. This correlation is intuitive since high-dimensional tasks require a larger amount of data for effective exploration. The bootstrapped training dataset augmentation facilitates this search process by leveraging proxy knowledge. Additionally, the filtering technique proves to be powerful in improving scores. As we observed from the diverse aggregation and filtering, the ensemble strategy greatly enhances score-conditioned generators. Limitations While our bootstrapping method shows promise for offline bio-sequential design tasks, it has inherent technical limitations. The assumption that the generator produces superior data to the training dataset may backfire if the generator samples have poor quality designs and the proxy used is inaccurate. While the current aggregation strategy effectively manages this risk, we can address this limitation by utilizing conservative proxies [39] and robust model adaptation techniques [47] for further improvement. Conclusion In this study, we introduce a novel approach to stabilize and enhance score-conditioned generators for offline biological sequence design, incorporating the classical concepts of bootstrapping and aggregation. Our novel method, named BOOTGEN, consistently outperformed all baselines across six offline biological sequence design tasks, encompassing RNA, DNA, and protein optimization. Our strategy of bootstrapping and aggregation yielded remarkable improvements in terms of achieving high scores, generating diverse samples, and minimizing performance variance. A.1 Datasets A Additional Experimental Settings • GFP [37] is a task to optimize a protein sequence of length 237 consisting of one of 20 amino acids, i.e., the search space is 20 237 . Its objective is to find a protein with high fluorescence. Following Trabucco et al. [40], we prepare the offline dataset using 5000 samples with 50 to 60 percentile scores in the original data. • UTR [37] is a task to optimize a DNA sequence of length 50 consisting of one of four nucleobases: adenine (A), guanine (G), cytosine (C), thymine (T). Its objective is to maximize the expression level of the corresponding 5'UTR region. For the construction of the offline dataset D, following Trabucco et al. [40], we provide samples with scores under the 50th percentile data of 140, 000 examples. • TFBind8 [6] is a task that optimizes DNA similar to the UTR. The objective is to find a length 8 sequence to maximize the binding activity with human transcription factors. For the offline dataset D, we provide under 50th percentile data of 32,898 examples following Trabucco et al. [40]. • RNA-Binding [30] is a task that optimizes RNA, a sequence that contains four vocab words of nucleobases: adenine (A), uracil (U), cytosine (C), and guanine (G). The objective is to find a length 14 sequence to maximize the binding activity with the target transcription factor. We present three target transcriptions of RNA termed RNA-Binding-A (for L14 RNA1), RNA-Binding-B (for L14 RNA2), and RNA-Binding-C (for L14 RNA3). We provide under 0.12 scored data for the offline dataset D among randomly generated 5,000 sequences using opensource code 2 . A.2 Implementation of Baselines This section provides a detailed implementation of baselines of offline biological sequence design. Baselines from Design Bench [40]. Most baselines are from the offline model-based optimization (MBO) benchmark called design-bench [40]. The design bench contains biological sequence tasks of the GFP, UTR, and TFbind8, where it contains baselines of REINFORCE, CMA-ES [21], BO-qEI [45], CbAS [10], Auto. Cbas [17], MIN [29], gradient ascent (Grad.), and COMS [39]. We reproduce them by following the official source code 3 . For the RNA tasks, we follow hyperparameters of TFBind8 as the number of vocab are same as 4 and the sequence length is similar where the TFBind8 has length 8 and RNA tasks have length 14 as our method follows the same. BDI [12]. For BDI, we follow hyperparameter setting at the paper [12] and implementation at the opensource code 4 . For RNA tasks, we follow the hyperparameter for TFBind8 tasks, as our method follows the same. GFN-AL [26]. For GFN-AL we follow hyperparameters setting at the paper [26] and implementation on open-source code 5 . Because they only reported the TFbind8 and the GFP tasks, we use the hyperparameter of the GFP for the UTR tasks hyperparameter of the TFBind8 for RNA tasks, as our method follows the same. A.3 Hyperparameters Training. We give consistency hyperparameters for all tasks except the learning rate. We set the generator's learning rate to 10 −5 for short-length tasks (lengths 8 and 14) of TFBind and RNA tasks and 5 × 10 −5 for longer-length tasks (lengths 50 and 237) of UTR and GFP. We trained the generator with 12,500 steps before bootstrapping. Bootstrapping is applied with 2, 500 in additional steps. The batch size of training is 256. We set the weighting parameter k = 10 −2 . Note that we early stopped the generator iteration of GFP with the 3, 000 step based on monitoring the calibration model of Appendix B. For bootstrapping, the generator samples 2 candidates every 5 steps. For Top-K sampling at the bootstrapping, we sample with L = 1, 000 and select the Top 2 samples to augment the training dataset. Testing. For filtering, we generated M = 1, 280 candidate samples and collected the Top-K samples where K = 128 based on the proxy score. For diverse aggregation, we collect K = 16 samples from 8 generators, making total of 128 samples. Proxy model. For the proxy model, we applied a weight regularization of 10 −4 , set the learning rate to 10 −4 , and used a dropout rate of 0.1. We used early stopping with a tolerance of 5 and a train/validate ratio of 9:1 following Jain et al. [26]. We used the Adam optimizer [27] for the training generator, proxy, and calibration model. Tuning the hyperparameters of offline design algorithms is challenging due to the lack of access to the true score function. Therefore, existing works have proposed various strategies to circumvent this issue, e.g., choosing a hyperparameter that is transferrable between different tasks [40] or tuning the hyperparameter based on training statistics [47]. In this work, we leverage the calibration function. Inspired by Wang et al. [42], we train the calibration function on the offline dataset to approximate the true score function similar to the proxy function. Then we use the calibration function to select a score-conditioned model which achieves higher performance with respect to the calibration function. We also choose the number of training steps and early stopping points using the same criterion. As shown in Fig. B.1, the calibration model accurately predicts early stopping points as the GFP task is unstable and has a narrow high score region which gives a high chance to be overfitted into the low-scored region ( Table 4.1 shows that score of GFP is highly polarized). By using the calibration function, we can simply choose an early stopping point for GFP. Note we simply leverage the proxy model as a calibration model with an exact sample training scheme and hyperparameters. Building upon the § 4.4 findings of the UTR, we present further multi-objective experimental results for the remaining 5 tasks, comparing them closely with the GFN-AL [26] approach. The GFN-AL model aims to achieve extensive exploration by prioritizing sample diversity and novelty, leading to the generation of diverse, high-quality biological sequences. Nevertheless, the diversity measure occasionally introduces a trade-off between sample scores, particularly when certain tasks exhibit a narrow score landscape, resulting in only a limited number of samples with high scores. C Diversity and Novelty Comparison with GFN-AL [26] We conducted a detailed analysis to shed light on the relationship between score metrics (average, 100th percentile, 50th percentile) and diversity metrics (diversity and novelty). The results, presented in Table C.2, demonstrate that our proposed method, BOOTGEN, outperforms GFN-AL in terms of score performance. However, it is noteworthy that GFN-AL exhibits high diversity, particularly in the case of GFP. On the contrary, GFN-AL generates extremely low scores for GFP, almost comparable to those produced by a uniform random generator. This observation indicates that the GFP task possesses a narrow score landscape, making it relatively easy to generate diverse yet low-scoring samples. For the TFbind8 and RNA tasks, GFN-AL achieves high diversity but a low novelty. This suggests that GFN-AL struggles to discover samples beyond the scope of the offline training dataset, resulting in less novel but diverse samples with low scores. In contrast, BOOTGEN successfully identifies high-scoring and novel samples. Consequently, in this scenario, we consider high diversity coupled with low novelty and score to be somewhat meaningless, as such results can also be achieved by a random generator. To substantiate our claim regarding diversity, we present experimental results of an enhanced diversity version of BOOTGEN. In order to achieve increased diversity, BOOTGEN makes certain sacrifices in terms of score performance. One approach we employ is interpolation with a uniform random sequence generator. Specifically, we combine our generator with the uniform random gen-erator to generate random samples in a portion of the sequence (we make 3/4 samples from the random generator and 1/4 from BOOTGEN). Additionally, we can filter out low-diversity sequences without requiring score evaluation, thereby generating a more diverse set of samples by referring code of GFN-AL [26]. To this end, we introduce the diversity-improved version of our method, denoted as BOOTGEN † . It is important to note that BOOTGEN † sacrifices score performance, as diversity and score are inherent trade-offs, and it focuses primarily on diversity to provide a more direct comparison with GFN-AL by manually adjusting diversity. As shown in Table C.2, BOOTGEN † exhibits similar diversity levels in RNA-A, RNA-B, RNA-C, and TFBind8, while achieving higher score metrics and novelty. We attribute these results to GFN's underfitting issue, as it fails to adequately fit within the high-scoring region of the score landscape, particularly for the high-dimensional tasks of UTR and GFP. We verify the contribution of the rank-based weighting (RW) scheme compared to ours with no weighting (NW) and the existing value-based weighting (VW) proposed by Kumar and Levine [29]. To implement VW, we set the sample-wise weight proportional to exp(|y − y * |/T ), where y * is the maximum score in the training dataset and T ∈ {0.1, 0.3, 0.7, 1.0} is a hyperparameter. As shown in Fig. D.1, the results indicate that RW outperforms both NW and VW. This validates our design choice for BootGen. BOOTGEN 0 . 0902 ± 0.039 0.931 ± 0.055 0.831 ± 0.044 0.979 ± 0.001 0.865 ± 0.000 0.865 ± 0.000 0.895 BOOTGEN0.707 ± 0.005 0.717 ± 0.006 0.596 ± 0.006 0.833 ± 0.007 0.853 ± 0.017 0.701 ± 0.004 0.731 Figure 4 . 1 : 41Evaluation-performance graph to compare with representative offline biological design baselines. The number of evaluations K ∈ [1, 128] stands for the number of candidate designs to be evaluated by the Oracle score function. The average value and standard deviation error bar for 8 independent runs are reported. Our method outperforms other baselines at every task for almost all K. Figure 4 . 2 : 42Multi-objectivity comparison of diversity and novelty on the average score for the UTR task. Each datapoint for 8 independent runs is depicted. Figure B. 1 : 1Calibration model's tendency. Figure D. 1 : 1Comparison of rank-based weighting (RW) and value-based weighting (VW) methods. The NW represents the case where no weighting is applied to the training distribution. In the VW case, we explored different weighting temperatures, T ∈ {0.1, 0.3, 0.7, 1.0}. The 50th percentile scores of TFBind8 are reported, and the results include a bootstrapping procedure applied from iteration 12,500 to 15,000. Table 4 . 41: Experimental results on 100th percentile scores. The mean and standard deviation are reported for 8 independent solution generations. D(best) indicate the maximum score of the offline dataset. The best-scored value is marked in bold.Method RNA-A RNA-B RNA-C TFBind8 GFP UTR Avg. D (best) 0.120 0.122 0.125 0.439 0.789 0.593 0.365 REINFORCE Table 4 . 42: Experimental results on 50th percentile scores. The mean and standard deviation are reported for 8 independent solution generations. D(best) indicate the maximum score of the offline dataset. The best-scored value is marked in bold. 40] 0.159 ± 0.011 0.162 ± 0.007 0.177 ± 0.011 0.450 ± 0.017 0.845 ± 0.003 0.575 ± 0.018 0.395 CMA-ESMethod RNA-A RNA-B RNA-C TFBind8 GFP UTR Avg. D (best) 0.120 0.122 0.125 0.439 0.789 0.593 0.365 REINFORCE [ Table A.1. We also provide a detailed description of the tasks in Appendix A.1.Baselines We compare our BOOTGEN with the following baselines: gradient ascent with respect to a proxy score model [40, Grad.], REINFORCE [44], Bayesian optimization quasi-expected- improvement [45, BO-qEI], covariance matrix adaptation evolution strategy [21, CMA-ES], condi- tioning by adaptative sampling [10, CbAS], autofocused CbAS [17, Auto. CbAS], model inversion network [29, MIN], where these are in the official design bench [40]. We compare with additional baselines of conservative objective models [39, COMs], generative flow network for active learning [26, GFN-AL] and bidirectional learning [12, BDI]. Table 4 . 43: Experimental results on 100th percentile scores (100th Per.), 50th percentile scores (50th Per.), average score (Avg. Score), diversity, and novelty, among 128 samples of UTR task. The mean and standard deviation of 8 independent runs for producing 128 samples is reported. The best-scored value is marked in bold. The lowest standard deviation is marked as the underline. The DA stands for the diverse aggregation strategy.Methods 100th Per. 50th Per. Avg. Score Diversity Novelty MIN [29] 0.691 ± 0.011 0.587 ± 0.012 0.554 ± 0.010 28.53 ± 0.095 18.32 ± 0.091 CMA-ES [21] 0.746 ± 0.018 0.498 ± 0.012 0.520 ± 0.013 24.69 ± 0.150 19.95 ± 0.925 Grad. [40] 0.682 ± 0.013 0.513 ± 0.007 0.521 ± 0.006 25.63 ± 0.615 16.89 ± 0.426 GFN-AL [26] 0.700 ± 0.015 0.602 ± 0.014 0.580 ± 0.014 30.89 ± 1.220 20.25 ± 2.272 BOOTGEN w/o DA. 0.729 ± 0.074 0.672 ± 0.082 0.652 ± 0.081 17.83 ± 5.378 20.49 ± 1.904 BOOTGEN w/ DA. (ours) 0.858 ± 0.003 0.701 ± 0.004 0.698 ± 0.001 31.57 ± 0.073 21.40 ± 0.057 22 24 26 28 30 32 34 Diversity 0.50 0.55 0.60 0.65 0.70 Avg. Score BootGen GFN-AL Grad. CMA-ES CbAS Auto. CbAS MIN REINFORCE BDI 16 17 18 19 20 21 22 Novelty 0.50 0.55 0.60 0.65 0.70 Avg. Score BootGen GFN-AL Grad. CMA-ES CbAS Auto. CbAS MIN REINFORCE BDI Table 4 . 44: Ablation study for BOOTGEN. The average score among 128 samples is reported. We make 8 independent runs to produce 128 samples where the mean and the standard deviation are reported. For every method, an aggregation strategy is applied by default. The best-scored value is marked in bold. The lowest standard deviation is underlined. The RR stands for rank-based reweighting, the B stands for bootstrapping, and the F stands for filtering.Components RNA-A RNA-B RNA-C TFbind8 UTR GFP ∅ 0.388 ± 0.007 0.350 ± 0.008 0.394 ± 0.010 0.579 ± 0.010 0.549 ± 0.009 0.457 ± 0.044 {RR} 0.483 ± 0.006 0.468 ± 0.008 0.441 ± 0.010 0.662 ± 0.009 0.586 ± 0.008 0.281 ± 0.031 {RR, B} 0.408 ± 0.009 0.379 ± 0.009 0.417 ± 0.006 0.666 ± 0.009 0.689 ± 0.003 0.470 ± 0.034 {RR, F} 0.576 ± 0.005 0.586 ± 0.004 0.536 ± 0.007 0.833 ± 0.004 0.621 ± 0.003 0.783 ± 0.011 {RR, F, B} 0.607 ± 0.009 0.612 ± 0.005 0.554 ± 0.007 0.840 ± 0.004 0.698 ± 0.001 0.804 ± 0.002 Table 4 . 42. This drawback Table A . A1: Details of the offline datasets. We let |X | and |D| denote the sizes of the search space and the offline dataset, respectively. Seq. Length Vocab size |X | |D| GFP 20 237 20 237 5,000 UTR 50 4 50 4 140,000 TFBind8 8 8 4 8 32,898 RNA-Binding 14 4 4 14 5,000 Table C . C1: Experimental results on 100th percentile scores (100th Per.), 50th percentile scores (50th Per.), average score (Avg. Score), diversity, and novelty, among 128 samples of low dimensional tasks comparing with GFN-AL. The mean and standard deviation of 8 independent runs for producing 128 samples is reported. The best-scored value is marked in bold. BOOTGEN † 0.750 ± 0.041 0.382 ± 0.014 0.399 ± 0.008 8.917 ± 0.078 4.957 ± 0.052 BOOTGEN † 0.686 ± 0.052 0.355 ± 0.018 0.371 ± 0.017 8.929 ± 0.121 4.967 ± 0.132 BOOTGEN † 0.651 ± 0.056 0.370 ± 0.011 0.376 ± 0.013 8.913 ± 0.087 4.597 ± 0.073 BOOTGEN † 0.970 ± 0.018 0.613 ± 0.017 0.627 ± 0.014 5.048 ± 0.039 0.965 ± 0.033Table C.2: Experimental results on 100th percentile scores (100th Per.), 50th percentile scores (50th Per.), average score (Avg. Score), diversity, and novelty, among 128 samples of 6 six biological sequential tasks comparing with GFN-AL. The mean and standard deviation of 8 independent runs for producing 128 samples is reported. The best-scored value is marked in bold. The 'Random' stands for uniform random generator.Methods 100th Per. 50th Per. Avg. Score Diversity Novelty RNA-A GFN-AL 0.630 ± 0.054 0.312 ± 0.013 0.320 ± 0.010 8.858 ± 0.045 4.269 ± 0.130 BOOTGEN 0.898 ± 0.039 0.694 ± 0.009 0.699 ± 0.008 5.694 ± 0.008 7.509 ± 0.049 RNA-B GFN-AL 0.677 ± 0.080 0.300 ± 0.012 0.311 ± 0.011 8.846 ± 0.050 4.342 ± 0.128 BOOTGEN 0.886 ± 0.028 0.689 ± 0.007 0.693 ± 0.007 5.192 ± 0.073 7.981 ± 0.036 RNA-C GFN-AL 0.623 ± 0.045 0.324 ± 0.010 0.333 ± 0.010 8.831 ± 0.046 4.151 ± 0.088 BOOTGEN 0.837 ± 0.045 0.598 ± 0.006 0.606 ± 0.006 4.451 ± 0.071 7.243 ± 0.051 TFBind8 GFN-AL 0.951 ± 0.026 0.537 ± 0.055 0.575 ± 0.037 5.001 ± 0.178 0.778 ± 0.143 BOOTGEN 0.977 ± 0.004 0.848 ± 0.010 0.839 ± 0.009 3.118 ± 0.045 1.802 ± 0.025 Methods 100th Per. 50th Per. Avg. 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[ "https://github.com/kaist-silab/bootgen.", "https://github.com/samsinai/FLEXS", "https://github.com/brandontrabucco/design-baselines", "https://github.com/GGchen1997/BDI", "https://github.com/MJ10/BioSeq-GFN-AL" ]
[ "Beyond Tree Level with Solar Neutrinos: Towards Measuring the Flavor Composition and CP Violation", "Beyond Tree Level with Solar Neutrinos: Towards Measuring the Flavor Composition and CP Violation" ]
[ "Vedran Brdar \nTheoretical Physics Department\nCERN\n1 Esplanade des Particules1211Geneva 23Switzerland\n", "Xun-Jie Xu \nInstitute of High Energy Physics\nChinese Academy of Sciences\n100049BeijingChina\n" ]
[ "Theoretical Physics Department\nCERN\n1 Esplanade des Particules1211Geneva 23Switzerland", "Institute of High Energy Physics\nChinese Academy of Sciences\n100049BeijingChina" ]
[]
After being produced as electron neutrinos (νe), solar neutrinos partially change their flavor to νµ and ντ en route to Earth. Although the flavor ratio of the νe flux to the total flux has been well measured, the νµ : ντ composition has not yet been experimentally probed. In this work we show that the νµ : ντ flavor ratio could be measured by utilizing flavor-dependent radiative corrections in the cross sections for νµ and ντ scattering. Moreover, since the transition probabilities of νe to νµ and ντ depend on the leptonic CP phase, we also demonstrate that the method proposed in this work will allow next-generation neutrino experiments to probe leptonic CP violation through the observation of solar neutrinos.
null
[ "https://export.arxiv.org/pdf/2306.03160v1.pdf" ]
259,089,319
2306.03160
805e80fee7bccd1381a4a5a29a30f1766cf227db
Beyond Tree Level with Solar Neutrinos: Towards Measuring the Flavor Composition and CP Violation Vedran Brdar Theoretical Physics Department CERN 1 Esplanade des Particules1211Geneva 23Switzerland Xun-Jie Xu Institute of High Energy Physics Chinese Academy of Sciences 100049BeijingChina Beyond Tree Level with Solar Neutrinos: Towards Measuring the Flavor Composition and CP Violation After being produced as electron neutrinos (νe), solar neutrinos partially change their flavor to νµ and ντ en route to Earth. Although the flavor ratio of the νe flux to the total flux has been well measured, the νµ : ντ composition has not yet been experimentally probed. In this work we show that the νµ : ντ flavor ratio could be measured by utilizing flavor-dependent radiative corrections in the cross sections for νµ and ντ scattering. Moreover, since the transition probabilities of νe to νµ and ντ depend on the leptonic CP phase, we also demonstrate that the method proposed in this work will allow next-generation neutrino experiments to probe leptonic CP violation through the observation of solar neutrinos. Introduction. The first observation of solar neutrinos at the Homestake experiment [1] was not consistent with the theoretical predictions from Bahcall [2] and this turned out to be the first experimental hint for neutrino oscillation [3][4][5][6][7]. This phenomenon implies that neutrinos produced in the Sun change flavor en route to Earth, which has by now been confirmed with a number of experiments including SNO [8,9], Super-Kamiokande [10,11], and Borexino [12,13]. For recent reviews on solar neutrino physics, see [14,15]. The modern-day solar neutrino observations have established that only about a third of neutrinos produced in the Sun arrive to Earth as electron neutrinos (ν e ), while the remaining fraction is composed of muon (ν µ ) and tau neutrinos (ν τ ). The ratio between ν µ and ν τ fluxes is theoretically known but it has never been measured. A full measurement of the flavor composition (ν e : ν µ : ν τ ) would be valuable, as it would not only allow us to gain a better understanding of our nearest star, but would also make it possible to probe the transition probabilities P νe→νµ and P νe→ντ which depend on the level of the CP violation in the lepton sector (parameterized by the phase δ CP ). Given such dependence, solar neutrinos could serve as a complementary probe of CP violation to the near-future acceleration-based neutrino program led by DUNE [16] and Hyper-Kamiokande (HK) [17]. In this letter we propose a viable method to differentiate between solar ν µ and ν τ . Since solar neutrino energies do not exceed ∼ 20 MeV, ν µ and ν τ can only be detected via elastic scattering and, at the leading order, cross sections for this process are identical for both ν µ and ν τ . At the next-to-leading order (NLO), however, differences arise from flavor-dependent radiative corrections [18][19][20]; see e.g. Fig. 1. In this work we will be focused on solar neutrinos and the phenomenological consequences that radiative corrections in the cross section for neutrino-electron elastic scattering (eES) [21,22] can induce. One of the advantages for considering ν µ,τ + e − → ν µ,τ + e − is rather small theoretical uncertainty in the cross section which is at sub-percent level [23,24]. With sufficiently high The diagram on the left shows the leading order contribution whereas the one on the right illustrates the flavor-dependent NLO contribution. statistics, the difference between ν µ and ν τ cross sections could manifest itself in the data, namely in the total number of eES events. We investigate that by considering large next-generation neutrino detectors such as HK [17], DUNE [16], JUNO [25] and THEIA [26]. Radiative corrections for eES cross section. As already introduced, in this work we will mainly consider eES, the experimental signature of which is the detection of electron recoil kinetic energy, denoted as T . The differential cross section without the inclusion of any radiative corrections reads [22] dσ dT = 2G 2 F m e π s 2 W ± 1 2 2 + s 4 W 1 − T E ν 2 − s 2 W ± 1 2 s 2 W m e T E 2 ν ,(1) where the + (−) sign applies for ν e (ν µ or ν τ ) scattering on electron. In Eq. (1), G F is the Fermi constant, s W stands for the sine of the weak mixing angle and m e is the electron mass. The difference between ν µ and ν τ scattering cross sections comes about at NLO [21,22,27]. We are in particular interested in flavor-dependent corrections, which arise from higher-order diagrams involving µ and τ leptons in the loops, such as the right diagram in 3.0 Figure 2. Comparison between the differential cross sections, normalized to the leading order one (see Eq. (1)). In particular, we would like to draw reader's attention to the cyan line that demonstrates O(1%) difference between νµ and ντ scattering cross sections. This is the effect our phenomenological study is based on. [19,27] in Eq. (1); here ∆ accounts for a subset of 1-loop corrections for eES and we are mostly interested in the flavor-dependent contribu- T [MeV] 100⨯ ( σ A dT - σ B dT )/ σ LO dT E ν =11 MeV A = μ , B = τ A = LO , B= μ A = L O , B = τ2 W → s 2 W (1 − ∆)tion ∆ l ≡ α(6πs 2 W ) −1 Log(m 2 W /m 2 l ) where m W and m l are W boson and charged lepton masses, respectively. ∆ l is evaluated assuming a vanishing momentum transfer which is a reasonable approximation given the magnitude of the considered E ν . It turns out that ∆ µ − ∆ τ ≈ 0.01, and this propagates to O(1%) difference in the cross section; the ν µ cross section is larger than the ν τ one, see the cyan line in Fig. 2. In the figure, we also compare each of these cross sections at 1-loop level with the respective tree-level expression (see Eq. (1)); these O(α) effects are shown by purple and green lines, respectively. Let us stress that in making Fig. 2 as well as for our analysis presented in the next sections, we utilize results from [23,24]. There, both electroweak and QED corrections as well as the emission of soft photons is taken into account for eES. Note that, as a cross check, we also explicitly implemented the expressions from [21] and found consistent results; for instance, the difference between ν µ and ν τ cross sections was found, for any value of T , to deviate from cyan line in Fig. 2 by no more than 0.1%. The above discussion on the cross sections is important for the detection of neutrinos. However, one may also wonder about the impact of radiative corrections in neutrino propagation; after all, for the computation of the Mikheyev-Smirnov-Wolfenstein (MSW) matter potential [5][6][7] the same diagrams as those presented in Fig. 1 should be evaluated at a zero momentum transfer. The flavor-dependent NLO effects in the propagation were studied in [28] where it was found that when summing the relevant contributions, including diagrams with neutrino scattering on both electrons and quarks (nucleons), there is a cancellation at O(α) level for neutrinos traveling through a neutral unpolarized medium. In turn, the flavor-dependent effects in neutrino propagation arise only at O(α(m 2 l /m 2 W )Log(m 2 W /m 2 l )) ≈ 10 −6 which is rather small. This led the authors of [29] to conclude that such smallness of flavor-dependent terms leads to virtually no sensitivity to δ CP when studying solar neutrinos. In this paper we will oppose such a claim and demonstrate the sensitivity to CP violation by utilizing O(α) differences in eES for different neutrino flavors. Measuring the Flavor Composition. Now, let us utilize the flavor-dependent cross sections to scrutinize the potential for measuring the solar neutrino flavor composition (ϕ νe : ϕ νµ : ϕ ντ ) where ϕ να denotes the flux of ν α . For convenience, we define R α ≡ ϕ να ϕ total , ϕ total ≡ ϕ νe + ϕ νµ + ϕ ντ .(2) Since R e + R µ + R τ = 1, only two of the ratios are independent. We also define R µ ≡ ϕ νµ /(ϕ νµ + ϕ ντ ) which can freely vary between 0 to 1. Here, we should stress that the flavor ratios are actually energy-dependent according to the standard MSW solution. Nevertheless, in this section, for demonstration purposes, we take them as energy-independent in the fit and investigate how well the solar neutrino flux components can be measured. This contrasts to the following section in which we study δ CP sensitivity in the framework of the standard MSW solution. To date, only R e has been successfully measured, via combination of eES, neutrino-nucleus charged current (CC) and neutral current (NC) scattering. Among the three channels, eES and neutrino-nucleus CC scattering have different cross sections for ν e and ν x (x = µ or τ ) while neutrino-nucleus NC scattering is flavor independent at the leading order. When combining the three channels, R e is actually overconstrained 1 but the flavor composition ϕ νµ : ϕ ντ still cannot be resolved. To measure this flavor composition, one has to include the aforementioned radiative corrections which induce small differences between ν µ and ν τ cross sections. With the NLO corrections included and by taking (R e , R µ ) as free parameters, we perform a χ 2 -fit analysis to evaluate the potential of next-generation neutrino experiments to measure ϕ νe : ϕ νµ : ϕ ντ . The next-generation neutrino detectors will feature complementary advantages. The HK detector, which will be a 187 kt water Cherenkov detector, will have the highest statistics in the eES channel at energies above a certain threshold. We assume that such threshold is the same as for Super-Kamiokande, namely T ≳ 3.49 MeV [31]. The far detector of JUNO will be a 20 kt liquid scintillator (LS) detector and will also have high statistics in the eES channel. Despite the smaller fiducial mass, its detection threshold will be much lower than the one at HK due to the high light yield in LS. In our analysis, we assume that the threshold will be T ≳ 0.1 MeV 2 . The far detector of the DUNE experiment will contain 40 kt of liquid Ar and it is anticipated to measure solar neutrinos in both CC (ν e + 40 Ar → e − + 40 K, denoted by ArCC) and eES channels [33]. We assume that the threshold of ArCC process at DUNE will be E ν ≳ 7 MeV. Very recently, the THEIA experiment has been proposed [26], with a 100 kt water-based liquid scintillator (WbLS) detector placed deep underground. A high (low) percentage of LS in WbLS would decrease (increase) its capability of measuring the direction of interacting neutrinos, but it would lead to a lower (higher) energy threshold. As the percentage of LS is not determined yet, we consider the plan of using pure LS so that the energy threshold reaches the level of JUNO, at the expense of giving up on the directional sensitivity. Dark matter detectors could contribute in the NC channel by collecting coherent elactic neutrino-nucleus scattering (CEνNS) events induced by solar neutrinos. However, the statistics of such NC events in ton-scale detectors is low (in particular, they have not been measured to date), compared to the achieved SNO (kt-scale) observations of ν + 2 H → ν + n + p which is also a NC channel. The contribution of dark matter detectors for distinguishing between solar fluxes of ϕ µ and ϕ τ is therefore expected not to be competitive to the above introduced experiments and channels. In our analysis, the eES channel at HK, DUNE, JUNO and THEIA is studied and combined with the ArCC channel at DUNE. The eES event rate reads 3 dN dT = N e ∆t α dσ να dT (T, E ν ) ϕ να (E ν )dE ν ,(3) where N e denotes the number of electrons in a detector and ∆t is the exposure time and we assume ∆t = 10 years for all the considered experiments. For ArCC, in which T is also the main observable, the event rate is computed in a similar way, except that N e should be replaced by the number of argon nuclei. In our analysis, we also assume that the uncertainties on solar neutrino flux components [34] will improve and reach ∼ 1%. The treatment of such uncertainties is incorporated by ϕ total → (1 + a)ϕ total where the normalization factor, a, is included with an uncertainty of σ a = 1% and marginalized over in the χ 2 analysis. For each channel and each experiment, we perform a binned χ 2 analysis to assess the sensitivity to the flux ratios (R e , R µ ) assuming the true value is (0.3, 0.5). In Fig. 3, we show the results by considering several experiments individually as well as their combinations. The left panel shows how well the fluxes can be measured by using eES in HK (light blue) and ArCC in DUNE (light brown) as well as their combination (green). In the other two panels we add more experiments; the "pre.comb." label in the legend of a given panel refers to the combination (green region) from the panel to the left. If all nextgeneration experiments are combined, we expect that R µ could be measured to the precision of 0.5 ± 0.15. Solar neutrinos as a probe of CP violation. In Fig. 4, we show P νe→να (E ν ) for various δ CP values, obtained using the adiabatic approximation (for the range of its validity, see detailed discussion in [15]). As shown in the figure, P νe→νµ and P νe→ντ vary significantly for δ CP ∈ [0, π]; the variation can be as large as ∼ 50%. In making the figure, we have also included the matter effect for neutrinos propagating inside Earth; specifically, we have averaged over the day and the night values of P νe→να . Given the previous conclusion that the full flavor composition of solar neutrinos can be measured using NLO cross sections (see again Fig. 3), we expect that precision measurements of solar neutrinos should exhibit sensitivity to δ CP . To demonstrate that, we performed a χ 2 analysis and the result is presented in Fig. 5. As shown in the figure, the full combination of next-generation detectors allows δ CP = 0 and δ CP = π (π/2) to be differentiated at 1.2σ (0.6σ) CL. We also found that, for σ a ≲ 0.1%, 2σ can be reached. P νe→ντ ν e → ν τ δ CP 0 • 20 • 40 • 60 • 80 • 100 • 120 • 140 • 160 • 180 • In making Fig. 5, we took δ CP as the only fitting parameter and we fixed all other oscillation parameters at their present best fit values [35]. The reduction of uncertainties on the mixing angles, especially θ 23 , is anticipated across relatively short time scales [36]. Currently, varying θ 13 , θ 12 , and θ 23 within their 1σ ranges [35] would cause P eµ to change by 0.2%, 2%, 4%, respectively. We leave a dedicated investigation on the correlation between the mixing angles and δ CP to future work. Summary and Conclusions. The survival probability of solar ν e has been measured at a number of experiments in the ∼ 1-10 MeV energy range. On the other hand, directly measuring solar ν µ and ν τ fluxes is much more challenging and has not been performed to date. In this work, we propose a viable method to measure the flavor composition of solar neutrinos by utilizing differences between ν µ and ν τ eES cross sections. While these cross sections are identical at tree level, radiative corrections involving µ and τ leptons in loop diagrams induce an O(1%) difference. This allows one to probe the flavor composition of solar neutrinos through the observation of electron recoil spectrum. We have quantified such an effect by studying the potential of several forthcoming neutrino detectors including HK, DUNE and JUNO, together with the proposed THEIA experiment. We find that the O(1%) difference in the cross sections allows the combination of these experiments to effectively resolve the full flavor composition, see Fig. 3. Furthermore, since P νe→νµ and P νe→ντ depend on δ CP , the leptonic CP violation could be probed with solar neutrinos if the flavor-dependent radiative corrections are taken into consideration. We have assessed the sensitivity to δ CP and found that ∼ 1σ CL can be reached in differentiating between δ CP = 0 and δ CP = π. This result would improve to ∼ 2σ provided that the uncertainties in solar neutrino flux reach sub-percent level. Note Added. As we were finalizing this work, Ref. [37] appeared on arXiv. There, the authors scrutinize the prospects for measuring NLO effects with CEνNS at next-generation dark matter experiments. While both CEνNS and neutrino-electron scattering feature very small uncertainties in the cross section, the latter process has already been measured with large statistics while the coherent elastic neutrino-nucleus scattering is yet to be recorded for solar neutrinos. Hence, we regard neutrinoelectron scattering channel as more promising for measuring the flavor composition of solar neutrinos at nextgeneration experiments. Figure 1 . 1Feynman diagrams for νµ,τ scattering on electrons. Fig. 1 . 1This effect can be accounted for via redefinition s Figure 3 . 3The capability of next-generation neutrino detectors for measuring solar neutrino flavor compositions. All contours are at 1σ CL and the assumed true value is marked by ⋆. Figure 4 . 4The solar neutrino transition probabilities Pν e→να (Eν ) for various values of δCP. Figure 5 . 5δCP sensitivity in solar neutrino measurements at next-generation neutrino experiments. The data from the three channels turns out to be compatible with each other; see e.g.,Fig. 29of Ref.[30]. This is a reasonable assumption, given that KamLAND reached about 0.2 MeV according to Ref.[32] and JUNO will have a significantly higher yield of photoelectrons with respect to Kam-LAND. In this work we are only concerned with the electron energy spectrum and we do not consider the angular spectrum which can be measured at HK and THEIA. Including the angular spectrum could in principle further improve the results. 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[ "Classification of Histopathology Images of Lung Cancer Using Convolutional Neural Network (CNN)", "Classification of Histopathology Images of Lung Cancer Using Convolutional Neural Network (CNN)" ]
[ "Neha Baranwal ", "Preethi Doravari ", "Renu Kachhoria " ]
[]
[]
Cancer is the uncontrollable cell division of abnormal cells inside the human body, which can spread to other body organs. It is one of the non-communicable diseases (NCDs) and NCDs accounts for 71% of total deaths worldwide whereas lung cancer is the second most diagnosed cancer after female breast cancer. Cancer survival rate of lung cancer is only 19%. There are various methods for the diagnosis of lung cancer, such as X-ray, CT scan, PET-CT scan, bronchoscopy and biopsy. However, to know the subtype of lung cancer based on the tissue type H and E staining is widely used, where the staining is done on the tissue aspirated from a biopsy. Studies have reported that the type of histology is associated with prognosis and treatment in lung cancer. Therefore, early and accurate detection of lung cancer histology is an urgent need and as its treatment is dependent on the type of histology, molecular profile and stage of the disease, it is most essential to analyse the histopathology images of lung cancer. Hence, to speed up the vital process of diagnosis of lung cancer and reduce the burden on pathologists, Deep learning techniques are used. These techniques have shown improved efficacy in the analysis of histopathology slides of cancer. Several studies reported the importance of convolution neural networks (CNN) in the classification of histopathological pictures of various cancer types such as brain, skin, breast, lung, colorectal cancer. In this study tri-category classification of lung cancer images (normal, adenocarcinoma and squamous cell carcinoma) are carried out by using ResNet 50, VGG-19, Inception_ResNet_V2 and DenseNet for the feature extraction and triplet loss to guide the CNN such that it increases inter-cluster distance and reduces intra-cluster distance.
null
[ "https://arxiv.org/pdf/2112.13553v1.pdf" ]
245,502,440
2112.13553
ec3e162fcb7dca8d76edcbdb3804fe7482c7133f
Classification of Histopathology Images of Lung Cancer Using Convolutional Neural Network (CNN) Neha Baranwal Preethi Doravari Renu Kachhoria Classification of Histopathology Images of Lung Cancer Using Convolutional Neural Network (CNN) Page 1 of 25 Page 1 of 25ResNet 50CNNVGG-19Inception_ResNet_V2 and DenseNetHistopathology Images Cancer is the uncontrollable cell division of abnormal cells inside the human body, which can spread to other body organs. It is one of the non-communicable diseases (NCDs) and NCDs accounts for 71% of total deaths worldwide whereas lung cancer is the second most diagnosed cancer after female breast cancer. Cancer survival rate of lung cancer is only 19%. There are various methods for the diagnosis of lung cancer, such as X-ray, CT scan, PET-CT scan, bronchoscopy and biopsy. However, to know the subtype of lung cancer based on the tissue type H and E staining is widely used, where the staining is done on the tissue aspirated from a biopsy. Studies have reported that the type of histology is associated with prognosis and treatment in lung cancer. Therefore, early and accurate detection of lung cancer histology is an urgent need and as its treatment is dependent on the type of histology, molecular profile and stage of the disease, it is most essential to analyse the histopathology images of lung cancer. Hence, to speed up the vital process of diagnosis of lung cancer and reduce the burden on pathologists, Deep learning techniques are used. These techniques have shown improved efficacy in the analysis of histopathology slides of cancer. Several studies reported the importance of convolution neural networks (CNN) in the classification of histopathological pictures of various cancer types such as brain, skin, breast, lung, colorectal cancer. In this study tri-category classification of lung cancer images (normal, adenocarcinoma and squamous cell carcinoma) are carried out by using ResNet 50, VGG-19, Inception_ResNet_V2 and DenseNet for the feature extraction and triplet loss to guide the CNN such that it increases inter-cluster distance and reduces intra-cluster distance. Introduction Cancer is the uncontrollable cell division of abnormal cells inside the human body, which can spread to other body organs. The process of transformation of normal cells into cancerous cells due to genetic alteration is known as Carcinogenesis as shown in Figure 1. The process of carcinogenesis occurs in three phases. The first is the Initiation phase, where any alterations that occur in the normal cell due to gene mutation can cause a change in gene expression and even deletion of a part of Deoxyribonucleic acid (DNA) sometimes. If these changes skip the repair mechanism during the cell cycle, then the cell with altered genes remains as it is. In the Promotion phase, which is the second phase, the altered cell starts proliferation. In the final stage, the Progressive phase the cells start proliferating aggressively by number, size, and form primary tumors. In this stage, the cells become invasive and metastatic. Phases of carcinogenesis is shown in Figure 2 (Chegg.com, 2021). <Figure 1 here> The name for a cancer type is given based on the body organ or the cell type from which cancer originates. So far, there are more than 100 types of cancer found. There are various types of cancer such as breast, brain, lung, colon cancer, etc. Cancer is one of the noncommunicable diseases (NCDs) and NCDs account for 71% of total deaths worldwide (World Health Organization, 2019). Whereas lung cancer is the second most diagnosed cancer after female breast cancer (Ferlay, 2021). According to GLOBOCAN 2018, 2.09 million new lung cancer cases have been reported and accounted for 1.76 million deaths globally, resulting in the highest mortality rate in both males and females when compared to other cancer types (Bray, 2018). The incidence of lung cancer is higher among young women when compared to young men in the United States (Jemal, 2018). Approximately 63,000 lung cancer cases are recorded each year in India (Noronha, 2012). The cancer survival rate of lung cancer is only 19% (Siegel, Miller, and Jemal, 2019). <Figure 2 here> Lung cancer is divided into two major types based on histology, biological behaviour, prognosis and treatment. They are non-small cell lung cancer (NSCLC) and Small cell lung cancer (SCLC). NSCLC is the most common cancer type, which accounts for 85% and the remaining 15% is SCLC. NSCLC is again sub-divided into adenocarcinoma, squamous cell carcinoma and large cell carcinoma. As shown in Figure 3, adenocarcinoma is the most common cancer type and it is formed in epithelial cells that secrete mucus or fluids. Whereas in squamous cell carcinoma, cancer originates from squamous cells that line many organs such as the lung, bladder, kidney, intestines, and stomach (Pêgo-Fernandes, 2021; Cancer.gov, 2007). <Figure 3 here> There are various methods for the diagnosis of lung cancer, such as X-ray, CT scan, PET-CT scan, bronchoscopy and biopsy. However, to know the subtype of lung cancer based on the tissue type H and E staining is widely used, where the staining is done on the tissue aspirated from a biopsy. Hematoxylin (H) has a deep purple colour, stains nucleic acids in the cells and Eosin (E) have pink colour, and it stains proteins. (Fischer et al, 2008). Studies have reported that the type of histology is associated with prognosis and treatment in lung cancer (Hirsch et al, 2008;Itaya et al, 2007;Weiss et al, 2007). Recent advances in genomic studies paved the path to personalized medicine for lung cancer patients (Travis et al, 2021; Galli and Rossi, 2020). Therefore, early and accurate detection of lung cancer histology is an urgent need and as its treatment is dependent on the type of histology, molecular profile and stage of the disease, it is most essential to analyse the histopathology images of lung cancer. However, manual analysis of histopathology reports is time-consuming and subjective. With the advent of personalized medicine, pathologists are finding it difficult to manage the workload of dealing with a histopathologic cancer diagnosis. Hence, to speed up the vital process of diagnosis of lung cancer and reduce the burden on pathologists, Deep learning techniques (Baranwal et Analysis of Previous Research In the next few decades, cancer is expected to be the leading cause of death and is one of the biggest threats to human life (Tang et al, 2009). To improve the efficiency and speed of cancer diagnostics, Computer-aided diagnosis (CAD) was applied to the analysis of clinical data. There has been vast development in the field of CAD and many machine learning techniques are developed for the diagnosis purpose. Among all machine learning techniques, neural networks have shown increased performance in the detection of medical images. In the classification of lung cancer images, different CNN algorithms are used to improve the accuracy of the prediction and classification. Such accurate predictions aid doctors by reducing the workload and prevent human errors in the process of diagnosis. a) Computer aided diagnosis in medicine: Computer-aided diagnosis (CAD) is cuttingedge technology in the field of medicine that interfaces computer science and medicine. CAD systems imitate the skilled human expert to make diagnostic decisions with the help of diagnostic rules. The performance of CAD systems can improve over time and advanced CAD can infer new knowledge by analysing the clinical data. To learn such capability the system must have a feedback mechanism where the learning happens by successes and failures. During the last century, there is a dramatic improvement in human expertise and examination tools such as X-ray, MRI, CT, and ultrasound. With the discovery and study of new diseases and their progression, the diagnosis has become difficult and more complex. Various factors such as complex medical diagnosis, availability of vast data pertinent to conditions and diseases in the field of medicine, increasing knowledge on diagnostic rules, and the emergence of new areas such as AI, machine learning, and data mining in the field of computer science has led to the development of CAD (Yanase and Triantaphyllou, 2019a). Quantitative analysis of pathology images has gained importance among researchers in the field of pathology and image analysis. There is clearly a need for quantitative image-based evaluation of pathological slides as the diagnosis is based on the opinion of pathologists. CAD can reduce the burden on pathologists by filtering out the benign cancer images so that the pathologists can focus on more complicated images that are difficult to diagnose and suspicious. Quantitative analysis of pathology images not only helps in diagnosis but also in medical research (Gurcan and Boucheron, 2019 Hence there is a need to explore different techniques to improve the model performance other than increasing parameters. In the classification of images of cancer, there should be immense effort to differentiate cancer images from non-cancer images. The accuracy of the model needs to be high in such cases and the model should be able to detect both intra-class diversity and inter-class similarity. To consider such factors and guide the model accordingly, FaceNet introduced triplet loss (Schroff, Kalenichenko, and Philbin, 2015). Proposed Research Work To classify the lung cancer images, the dataset is obtained from LC25000 Lung and colon histopathological image dataset which is already augmented data having 5000 images in each class of lung cancer image set comprising three classes. This dataset is pre-processed using python tools and features are extracted by CNN techniques, later the model is created and evaluated. Various CNN techniques are used to compare and classify the images. Complete flow of proposed method is shown in Figure 4. Data pre-processing is an essential step, which helps in improving the quality of the images and it includes data preparation, data normalization, data cleaning, and data formatting. Data preparation aids in the transformation of data by modifying it into the appropriate format. Whereas data normalization makes a different image format into a regular format where all the images are uniform while in data transformation, the data is compressed (Zubi and Saad, 2011). As the images are already augmented, ImageDataGenerator which is imported from Keras. Preprocessing, image class used for the preprocessing of the image dataset. A total of 15000 images are used for the train-test split, in which 80% of the images are used for training and 20% for validating the data. D) Loss function: For a machine learning model to fit better while training the neural networks, loss function acts as a major key for adjusting the weights of the network. During the back propagation while training, loss function penalizes the model if there is any deviation between the label predicted by model and the actual target label (https://ieeexplore.ieee.org/abstract/document/8943952). Hence the use of loss function is very critical to achieve better model performance. Triplet loss is used as loss function in this study. Triplet loss: Triplet loss is first developed for face recognition by Schroff et al, 2015 by mapping Euclidean distance to find the similarities in the face images. Although the images are blurred with the help of the distances between faces of similar and different identities this method can be used . To increase the inter-cluster similarity and intra-cluster diversity, triplet loss is used as a cost function to guide the learning of Convolutional neural networks. It can increase the inter-class distance and decrease the intra-class aiding the classification process of the model. In equation 1, a and p are the vectors that belong to the same category, whereas n is a vector that belongs to another category. = ( ( , ) − ( , ) + , 0) (1) From the above formula, we can say that the triplet loss guides the model to shorten the distance between images of the same category and increases the distance between images that belong to different categories (Zhang et al, 2020). It has been reported that the use of triplet loss shown improved accuracy in binary classification when compared to using the base model (Agarwal, Balasubramanian and Jawahar, 2018). e) Model and evaluation metrics A CNN is created using a stack of layers for image recognition and classification. Before passing through the fully connected layer, the training and testing data is passed through parameters such as max-pooling and kernel filters. Activation function ReLU is used in all three hidden layers and a softmax function is applied to classify the images. In order to evaluate the performance of the model the following metrics are measured: Accuracy: Over the total number of data instances accuracy represents the correctly classified data. Equation (2) represents the formula to calculate accuracy. However, accuracy alone may not be a good measure to decide the performance of the model. Precision: This is used to measure the positive predictive observations. It represents the correctly predicted positive observations of total predicted positive observations. Equation (3) is the formula to calculate the precision. High precision relates to a low false-positive rate. Recall (Sensitivity): Recall represents the correctly predicted positive observations of total actual positive observations. The formula to calculate recall is given in Equation (4). It is also known as sensitivity or true positive rate. F1 score: Ideally, a good evaluation should consider both precisions and recall to seek balance. A weighted average of precision and recall is the F1 score. Equations (5) is the formula to calculate the F1 score. For uneven class distribution, the F1 score is more useful to evaluate the model. Accuracy= (TP + TN)/((TP + FP + FN + TN)) (2) Precision= TP/((TP + FP)) ( Recall= TP/((TP+FN)) (4) F1 Score= (2* (Recall * Precision))/((Recall + Precision)) (5) Result and analysis All four CNN architecture models have been trained using specific and fine-tuned parameters to achieve better model performance. Initially pre-trained CNN architecture is used to classify the lung cancer cells. In these models' cross entropy is used as loss function. VGG19 model is trained by adding two hidden layers with embeddings 256 and 128 with ReLU as an activation function and for the final output layer softmax is used as activation function. Table 2 for comparison. All the four CNN architecture Inception-ResNetv2 model has shown improved performance and classified benign tissue images from cancer images without any misclassifications. The only misclassification happened is between the subclasses of lung cancer images as shown in Figure 6. Even validation loss is also very minimum for this model as shown in Figure 7. To compare the pre-trained model with triplet neural network, again the four CNN architectures are trained using triplet neural network. In these models after train, test split, the data is divided into three images where first is the anchor, second is positive image which has same class label as anchor and the third is the negative image where the class label of this is different from anchor. For such triplet selection the loss function is introduced such that the distance between anchor and positive image should be always less than the distance between anchor and negative image. For such triplet loss function margin/alpha is added to calculate the distance. In these models this margin is set to 0.4 as while analysis, the model did not perform better at higher or lower margin other than 0.4. The batch size of input image is set to 16 and data type of each input is changed to float16 because of GPU memory constraints. After training the four models with introducing triplet selected, the learning rate of the Adam is also finely tuned to fit the model as shown in Table 5.2. For all the models Global Average pooling layer and L2 Normalization is used. Evaluation of triplet model is done by using KNN approach, where the model embeddings from training dataset are taken and trained using Nearest Neighbors. Later the nearest neighbor for test data embeddings are predicted using the trained model. Using this class label of the predicted test data is considered for evaluating the model. <Figure 8 here> It is observed that DenseNet121 has shown least validation loss of all the four networks. After the evaluation of all models, highest accuracy is reported by DenseNet121 and the least by ResNet50. The evaluation metrics of the models are given in table 4. As shown in Figure 9 when the test data embeddings are plotted the DensNet121 model showed defined clusters when compared to other models. al, 2019, Tripathi et al, 2013, Kumud et al., 2015 and singh et al, 2020) are used. These techniques have shown improved efficacy in the analysis of histopathology slides of cancer (Litjens et al, 2016). Data is drawn from the LC25000 Lung and colon histopathological image dataset, which consists of 5000 images each in three classes of benign (normal cells), adenocarcinoma and squamous carcinoma cells (both are cancerous cells). The dataset is HIPAA compliant and validated (Borkowski et al, 2019). The original images obtained are only 750 images in total and the size of the images are 1024 x 768 pixels, where each category gets 250 each. These images are cropped to 768 x 768 pixels using python and expanded using the augmentor software package. Thus, the expanded dataset contains 5000 images in each category. Augmentation is done by horizontal and vertical flips and by the left and right rotations (Borkowski et al, 2019). The sample images for each category are shown in Figure 5. <Figure 5 here> b) Data Pre-processing: Feature extraction is used to decrease the model complexity where important features are recognized from the images. For the knowledge extraction from images, not all the features provide interesting rules for the problem. This is the major step where the model performance and effectiveness are dependent. To extract such features as color, texture, and structure, image-processing techniques are used. This can be achieved by localizing the extraction to small regions and ensuring to capture all areas of the image (Zubi and Saad, 2011). For feature extraction, ResNet 50(He et al, 2016), VGG19(Munir et al, 2019), Inception_ResNet_V2 (Xie et al, 2019; Kensert, Harrison and Spjuth, 2019), DenseNet121(Huang, Liu, Van Der Maaten, and Weinberger, 2017; Chen, Zhao, Liu and Lin, 2021) is used. All four models are trained for 10 epochs using 150 steps in each epoch and validation steps of 50. Validation loss of all the four models is mentioned in theFigure 8. [ 3 ] 3Agarwal, N., Balasubramanian, V. N., & Jawahar, C. V. (2018). Improving multiclass classification by deep networks using DAGSVM and Triplet Loss. Pattern Recognition Letters, 112, 184-190. [4] Akkus, Z., Ali, I., Sedlar, J., Agrawal, J. P., Parney, I. F., Giannini, C., & Erickson, B. J. (2017). Predicting deletion of chromosomal arms 1p/19q in low-grade gliomas from MR images using machine intelligence. Journal of Digital Imaging, 30(4), 469-476. [5] Albarqouni S, Baur C, Achilles F, Belagiannis V, Demirci S, Navab N (2016) Aggnet: deep learning from crowds for mitosis detection in breast cancer histology images. IEEE Trans Med Imaging 35(5):1313-1321 [6] Bashiri, A., Ghazisaeedi, M., Safdari, R., Shahmoradi, L., & Ehtesham, H. (2017). Improving the prediction of survival in cancer patients by using machine learning techniques: Experience of gene expression data: A narrative review. Iranian Journal of Public Health, 46(2), 165-172 [7] Bijaya Kumar Hatuwal, H.C.T., (2021) Lung Cancer Detection Using Convolutional Neural Network on histopathological images. [online] Ijcttjournal.org. Available at: http://www.ijcttjournal.org/archives/ijctt-v68i10p104 [Accessed 19 Jun. 2021]. [8] Borkowski, A.A., Bui, M.M., Thomas, L.B., Wilson, C.P., DeLand, L.A. and Mastorides, S.M., (2019) Lung and Colon Cancer Histopathological Image Dataset (LC25000 [ 27 ] 27Fatima, M., & Pasha, M. (2017). Survey of machine learning algorithms for disease diagnostic. Journal of Intelligent Learning Systems and Applications, 09(01), 1-16. [28] Ferlay, J., Colombet, M., Soerjomataram, I., Parkin, D.M., Piñeros, M., Znaor, A. and Bray, F., (2021) Cancer statistics for the year 2020: An overview. International journal of cancer. Journal international du cancer. [29] Fischer, A.H., Jacobson, K.A., Rose, J. and Zeller, R., (2008) Hematoxylin and eosin staining of tissue and cell sections. CSH protocols, 20086, p.db.prot4986. [30] Fogel, A.L. and Kvedar, J.C., (2018) Artificial intelligence powers digital medicine. npj digital medicine, 11, p.5. [31] Freer, T. W., & Ulissey, M. J. (2001). Screening mammography with computer-aided detection: prospective study of 12,860 patients in a community breast center. Radiology, 220(3), 781-786. [32] Galli, G. and Rossi, G., (2020) Lung cancer histology-driven strategic therapeutic approaches. Shanghai chest, 40, pp.29-29. [33] Garg, S. and Garg, S., (2021) Prediction of lung and colon cancer through analysis of histopathological images by utilizing Pre-trained CNN models with visualization of class activation and saliency maps. arXiv [cs.CV]. [34] Ghoneim A, Muhammad G, Hossain MS (2020) Cervical cancer classification using convolutional neural networks and extreme learning machines. Future Gener Comput Syst 102:643-649 [35] Giger, M. L., Chan, H.-P., & Boone, J. (2008). Anniversary paper: History and status of CAD and quantitative image analysis: the role of Medical Physics and AAPM: History of CAD and quantitative image analysis. Medical Physics, 35(12), 5799-5820. [36] Glaab E, Bacardit J, Garibaldi JM, Krasnogor N (2012) Using rule-based machine learning for candidate disease gene prioritization and sample classification of cancer gene expression data. PLoS ONE 7(7):e39932 [37] Gurcan, M. N., Boucheron, L. E., Can, A., Madabhushi, A., Rajpoot, N. M., & Yener, B. (2009). Histopathological image analysis: a review. IEEE Reviews in Biomedical Engineering, 2, 147-171. [38] Hameed, Z., Zahia, S., Garcia-Zapirain, B., Javier Aguirre, J. and María Vanegas, A., (2020) Breast cancer histopathology image classification using an ensemble of deep learning models. Sensors (Basel, Switzerland), 2016, p.4373. [39] He, K., Zhang, X., Ren, S. and Sun, J., (2016) Deep residual learning for image recognition. In: 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp.770-778. [40] Hirsch, F.R., Spreafico, A., Novello, S., Wood, M.D., Simms, L. and Papotti, M., (2008) The prognostic and predictive role of histology in advanced non-small cell lung cancer: a literature review. Journal of thoracic oncology: official publication of the International Association for the Study of Lung Cancer, 312, pp.1468-1481. Figure 1 . 1Process of carcinogenesis (Chegg.com, 2021). Figure 2 .Figure 3 .Figure 4 .Figure 5 . 2345Three phases of carcinogenesis (Chegg.com, 2021) Types of Non-Small Cell Lung Cancer (NSCLC) (Lynne Eldridge, 2021) Proposed Sample images of three classes present in the dataset. (a) lung_n (lung normal cells), (b) lung_aca (lung adenocarcinoma cells) and (c) lung_scc (lung squamous cell carcinoma. Figure 6 .Figure 7 .Figure 8 . 678Confusion matrix of test data of Inception-ResNetv2ple images Validation loss obtained after training four CNN architectures Validation loss of four CNN architectures trained on triplet neural network. Figure 9 : 9Clusters obtained when test embeddings are plotted. (2D array of embeddings are plotted along x and y axis) Table 1. Summary of classification of various cancer types using machine learning techniques(Sharma and Rani, 2021).). At many hospitals in the United States CAD has become a part of routine clinical work for screening mammograms for the detection of breast cancer (Freer and Ulissey, 2001; Doi, 2007). In the fields of radiology and medical imaging, CAD has become the major research subject (Doi, 2007). These are cost-effective and can be used for the early detection of disease. Diseases like cancer are very aggressive when detected at later or advanced stages, hence screening and detection of such disease can avoid unnecessary invasive procedures for the treatment of the disease. Moreover, these models can eliminate human errors such as the detection of microcalcifications and help to improve the workflow of diagnostic screening procedures (Nishikawa et al, 2012; Yanase and Triantaphyllou, 2019a). b) CNN and cancer image detection: In the field of medicine to improve the quality of patient care machine learning-based approaches are used. These approaches are used to analyze and evaluate the complex data. The applications of artificial intelligence can speed up support delivery, be cost-effective, and at the same time can reduce medical errors (Jia et al, 2016). Recent studies revealed that advances in Artificial Intelligence (AI) have exceeded human performance in various fields and domains (Fogel and Kvedar, 2018) such as human robot interaction (Baranwal et al, 2017 and Singh et al, 2021), face recognition(Baranwal et al, 2019, Baranwal et al, 2016 and Singh et al, 2017) etc. Several studies reported the importance of convolution neural networks in the classification of histopathological pictures of various cancer types such as brain, skin, breast, lung, colorectal cancer (Garg and Garg, 2021; Mobadersany et al, 2018). Convolution neural networks have exceeded even human performance on ImageNet Large-Scale Visual Recognition Challenge (ILSVRC) and performed well in classification with second best error rate (Lundervold and Lundervold, 2019). Deep convolutional neural network (DCNN) models for the classification of lung cancer images showed increased accuracy and reduced model overfitting by using various data augmentation techniques (Teramoto et al, 2017). A study that used LC25000 and Colorectal Adenocarcinoma Gland (CRAG) datasets to train and classify the histopathology images reported the highest sensitivity using ResNet-50 (96.77%) which is followed by ResNet-30 and ResNet-18 with 95.74% and 94.79% sensitivity respectively (Bukhari et al, 2020). Another study used low dose computed tomography (LDCT) images for the detection of early lung cancer, where they reported 97.5 % of sensitivity where the SVM model was used for classification where VGG19 was used for feature extraction. As the image dataset small, they used transfer- learning methods to obtain better prediction results (Elnakib, Amer, and Abou-Chadi, 2020). Satvik Garg and Somya Garg developed eight Pre-trained CNN models that include various feature extraction tools such as VGG16, InceptionV3, ResNet50, etc. for the classification of lung and colon cancer images and achieved accuracies ranging from 96% to 100%. To boost the performance of the model better augmentation technique, an imaging library was used (Garg and Garg, 2021). Homology-based image processing (HI) model for the multicategory classification of lung cancer images achieved better accuracy when compared to conventional texture analysis (TA). For feature extraction in the HI model, Betti numbers are the important metrics (Nishio et al, 2021). A convolution neural network model with cross-entropy as a loss function achieved training and validation accuracy of 96.11% and 97.2% for the classification of lung cancer images (Bijaya Kumar Hatuwal and H.C.T, 2021). The combination of Deep Learning and Digital Image Processing for the classification of lung and colon cancer histopathology images obtained maximum accuracy of 96.33% (Masud et al, 2021). In the classification of histopathological images of colorectal cancer ResNet-50 along with transfer, learning reported an accuracy of 97.7% which is so far the best when compared to all previous results in the literature (Shawesh and Chen, 2021). In the classification of histopathology images of breast cancer, Inception_ResNet_V2 has proved to be the best deep learning architecture (Xie et al, 2019). Cancer type Contribution Technique used Leukemia, Colon cancer Gene Selection for Cancer Classification SVM technique based on Recursive Feature Elimination (RFE) (Guyon et al, 2002) Lymphoma Data, SRBCT, Liver Cancer, Different tumor types Finding the smallest set of genes Gene Importance Ranking, Support Vector Machines (SVMs) (Wang, Chu and Xie, 2007) Leukemia, Colon, and Lymphoma Cancer classification Ensemble of neural networks (Cho and Won, 2007) Ovarian cancer Ovarian cancer diagnosis Fuzzy neural network (Tan, Quek, Ng and Razvi, 2008) Prostate cancer, lymphoma, Breast cancer Gene Prioritization and Sample Classification Rule-Based Machine Learning (Glaab, Bacardit, Garibaldi and Krasnogor, 2012) Microarray data of six cancer types (leukemia, lymphoma, prostate, colon, breast, CNS embryonal tumor) Gene selection and classification Recursive Feature Addition, Supervised learning (Liu et al, 2011) Microarray data of multiple cancer types Cancer classification Particle swarm optimization, Decision tree classifier (Chen, Wang, Wang and Angelia, 2014) Multiple cancer types Cancer Classification Ensemble-based Classifiers (Margoosian and Abouei, 2013) breast cancer Cancer Classification Deep Belief Networks (Zaher and Eldeib, 2016) Leukemia Cancer classification Artificial neural network (ANN) (Dwivedi, 2018) Gene expression data from multiple cancer types. Molecular Cancer Classification Transfer Learning, Deep Neural Networks (Sevakula et al, 2018) Breast Cancer Breast Cancer Classification Convolutional Neural Network (Ting, Tan and Sim, 2019) Cervical cancer Cervical cancer classification Convolutional neural networks & extreme learning machines (Ghoneim, Muhammad and Hossain, 2020) Melanoma Automated Melanoma Recognition Deep Residual Networks (Yu et al, 2016) Breast Cancer Breast Cancer Detection Deep Learning From Crowds for Mitosis (Albarqouni et al, 2016) Cervical cancer Classification of cervical Pap smear images Mean-Shift clustering algorithm and mathematical morphology (Wang et al, 2019) Cervical cancer Cervical Cell Classification Deep Convolutional Networks (Zhang et al, 2017) Table 2 . 2Evaluation metrics for all the four CNN architectures<Figure 6 here> <Figure 7 here> Table 3 : 3Fine tuning of learning rate for different CNN models Table 4 : 4Abdel-Zaher AM, Eldeib AM (2016) Breast cancer classification using deep belief networks. Expert Syst Appl 46:139-144.Evaluation metrics for all the four CNN architectures <Figure 9 here> Conclusion CNN models have shown to increase accuracy with fine tuning of hyper parameters. Various CNN architectures are compared in the study to get better accuracy and to compare which architecture gives better performance for this dataset. Model performance of all four CNN models such as VGG19, ResNet50, Inception-ResNetv2 and DenseNet121 have shown increased accuracy. Although the pre-trained models are available, fine-tuning of these models Bron, E. E., Smits, M., van der Flier, W. M., Vrenken, H., Barkhof, F., Scheltens, P., Bukhari, S.U.K., Syed, A., Bokhari, S.K.A., Hussain, S.S., Armaghan, S.U. Chegg.com (2021) Learn About Carcinogenesis. [online] Available at: https://www.chegg.com/learn/biology/introduction-to-biology/carcinogenesis-inintroduction-to-biology [Accessed 16 Jun. 2021]. 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[ "DetarNet: Decoupling Translation and Rotation by Siamese Network for Point Cloud Registration", "DetarNet: Decoupling Translation and Rotation by Siamese Network for Point Cloud Registration" ]
[ "Zhi Chen [email protected] \nNational Key Laboratory of Science and Technology on Multispectral Information Processing School of Artifical Intelligence and Automation\nHuazhong University of Science and Technology\nChina\n", "Fan Yang [email protected] \nNational Key Laboratory of Science and Technology on Multispectral Information Processing School of Artifical Intelligence and Automation\nHuazhong University of Science and Technology\nChina\n", "Wenbing Tao [email protected] \nNational Key Laboratory of Science and Technology on Multispectral Information Processing School of Artifical Intelligence and Automation\nHuazhong University of Science and Technology\nChina\n" ]
[ "National Key Laboratory of Science and Technology on Multispectral Information Processing School of Artifical Intelligence and Automation\nHuazhong University of Science and Technology\nChina", "National Key Laboratory of Science and Technology on Multispectral Information Processing School of Artifical Intelligence and Automation\nHuazhong University of Science and Technology\nChina", "National Key Laboratory of Science and Technology on Multispectral Information Processing School of Artifical Intelligence and Automation\nHuazhong University of Science and Technology\nChina" ]
[]
Point cloud registration is a fundamental step for many tasks. In this paper, we propose a neural network named DetarNet to decouple the translation t and rotation R, so as to overcome the performance degradation due to their mutual interference in point cloud registration. First, a Siamese Network based Progressive and Coherent Feature Drift (PCFD) module is proposed to align the source and target points in highdimensional feature space, and accurately recover translation from the alignment process. Then we propose a Consensus Encoding Unit (CEU) to construct more distinguishable features for a set of putative correspondences. After that, a Spatial and Channel Attention (SCA) block is adopted to build a classification network for finding good correspondences. Finally, the rotation is obtained by Singular Value Decomposition (SVD). In this way, the proposed network decouples the estimation of translation and rotation, resulting in better performance for both of them. Experimental results demonstrate that the proposed DetarNet improves registration performance on both indoor and outdoor scenes. Our code will be available in https://github.com/ZhiChen902/DetarNet.
10.1609/aaai.v36i1.19917
[ "https://arxiv.org/pdf/2112.14059v1.pdf" ]
245,537,496
2112.14059
ad488e939cb9d31a25435d98987f570ef1f6a9c0
DetarNet: Decoupling Translation and Rotation by Siamese Network for Point Cloud Registration Zhi Chen [email protected] National Key Laboratory of Science and Technology on Multispectral Information Processing School of Artifical Intelligence and Automation Huazhong University of Science and Technology China Fan Yang [email protected] National Key Laboratory of Science and Technology on Multispectral Information Processing School of Artifical Intelligence and Automation Huazhong University of Science and Technology China Wenbing Tao [email protected] National Key Laboratory of Science and Technology on Multispectral Information Processing School of Artifical Intelligence and Automation Huazhong University of Science and Technology China DetarNet: Decoupling Translation and Rotation by Siamese Network for Point Cloud Registration Point cloud registration is a fundamental step for many tasks. In this paper, we propose a neural network named DetarNet to decouple the translation t and rotation R, so as to overcome the performance degradation due to their mutual interference in point cloud registration. First, a Siamese Network based Progressive and Coherent Feature Drift (PCFD) module is proposed to align the source and target points in highdimensional feature space, and accurately recover translation from the alignment process. Then we propose a Consensus Encoding Unit (CEU) to construct more distinguishable features for a set of putative correspondences. After that, a Spatial and Channel Attention (SCA) block is adopted to build a classification network for finding good correspondences. Finally, the rotation is obtained by Singular Value Decomposition (SVD). In this way, the proposed network decouples the estimation of translation and rotation, resulting in better performance for both of them. Experimental results demonstrate that the proposed DetarNet improves registration performance on both indoor and outdoor scenes. Our code will be available in https://github.com/ZhiChen902/DetarNet. Introduction Point cloud registration is one of the fundamental problems in computer vision, which is widely applied to 3D reconstruction, robotics, autonomous driving and medical tasks. It aims to establish correspondences between two point clouds, and estimate the rigid transformation (translation t and rotation R). The most commonly used way is first establishing coarse correspondences, and then recovering rigid transformation. The main challenge is that there always exist wrong correspondences (outliers). Although some methods attempt to generate more accurate correspondences through handcrafted (Rusu, Blodow, and Beetz 2009;Rusu et al. 2008) or deep-learning technique based descriptors Yew and Lee 2018;Choy, Park, and Koltun 2019), it is hard to be totally outlier-free when dealing with complicated scenarios. Thus, it is worth studying how to better perform point cloud registration in the scenarios when the initial correspondences contain outliers. Recently, some methods have studied how to use a classification neural networks (Pais et al. 2020;Choy, Dong, and (b), the consensus between inliers is easier to be mined. Koltun 2020; Bai et al. 2021;Lee et al. 2021) to find correct 3D correspondences between two point clouds, and then estimate the translation t and rotation R by weighted Singular Value Decomposition (SVD) (Choy, Dong, and Koltun 2020). The core of these methods is to learn the consensus (Bai et al. 2021) of correct correspondences (inliers). Each correspondence can be abstracted as an arrow between a pair of points, as shown in Fig. 1. The consensus of inliers is that the length and direction of inliers are satisfied with some consistency. As shown in Fig. 1 (a), due to the coupling of translation and rotation, the consistency between inliers is difficult to be mined. However, as illustrated in Fig. 1 (b), if we can first eliminate the translation t and only the rotation is left, it is much easier for us to find out the correct correspondences. Inspired by this observation, we expect to decouple the whole rigid transformation into separate translation and rotation estimation. Considering the non-linearity of the rotation space (Peng et al. 2019;Li et al. 2018), it is more feasible to first recover the translation t because it is linear and easy to handle. However, decoupling t and solving it accurately in advance is still challenging in two aspects: 1) It is hard for traditional geometric optimization methods to only recover t without considering R. These methods usually need to jointly optimize t and R. Although centroid alignment (Arun, Huang, and Blostein 1987) can yield a rough t, it can only be used to assist the optimization of the whole rigid transformation due to the existence of outliers. 2) Although deep learning networks have made remarkable progress in point clouds reg-istration, translation transformation is still hard to be separately modeled in the neural networks while excluding the influence of R. Most of the methods try to regress both t and R (Pais et al. 2020;Aoki et al. 2019) together. Based on the above analysis, we propose a Siamese Network based Progressive and Coherent Feature Drift (PCFD) module to decouple t from the whole transformation and solve it accurately. PCFD module converts the registration into an alignment process of two point clouds in highdimensional feature space. First, the features of the two point clouds are respectively extracted by a Siamese Network with shared parameters. Then a global feature offset is learned by establishing global interaction between the two point clouds. The global feature offset forces the source points to move towards the target points coherently as a group to preserve the topological structure of point sets. Thus, the transformation is explicitly encoded by the alignment process, which is named as Coherent Feature Drift (CFD) operation. The whole PCFD module is composed of multiple CFD operations, which progressively align the source and target points to obtain the optimal estimation of t. The formulation of PCFD module has three advantages: 1) CFD operations explicitly encode the transformation by the global features in the network. Thus, the coupling between R and t can be disentangled by introducing the supervision on the middle layers. We supervise the alignment process by using only the ground truth translation t gt , so that the global features tend to encode the translation transformation. 2) When putative correspondences are given, previous methods usually (Choy, Dong, and Koltun 2020;Pais et al. 2020;Bai et al. 2021) concatenate the two points of a correspondence and form a virtual point to process together. Different from them, our PCFD module adopts a Siamese Network to retain the features of two point clouds. Since the two point clouds are handled respectively, it is easier to establish interaction between them, which benefits the regression for transformation. 3) The network adopts a progressive alignment approach to regress t and gradually eliminates t by using a multi-layer CFD operation. The multiple layers of CFD constitute an iterative optimization structure, so t can be more accurately estimated. Since we obtain the accurate estimation of t, the correspondences between two point clouds are more obvious and easier to be decided, as shown in Fig. 1 (b). Then we follow the previous works (Moo Yi et al. 2018;Pais et al. 2020) and build a correspondence classification network to prune outliers. Specifically, a Consensus Encoding Unit (CEU) is proposed to remove t when encoding the consensus to make the feature more distinguishable. It combines the spatial and feature consistency items as the feature for each correspondence. Furthermore, we design a Spatial and Channel Attention (SCA) block for the construction of classification network. It simplifies the current spatial attention module (Sun et al. 2020;Chen, Yang, and Tao 2021) and combines it with an instance-unique channel attention. Thus, the network can capture more complex context to better find the consensus of inliers. Finally, according to the established correspondences, R is obtained by Singular Value Decomposition (SVD) (Arun, Huang, and Blostein 1987). The above modules are integrated into an end-to-end registration network named DetarNet. In a nutshell, our main contributions are threefold: 1. We propose a Progressive and Coherent Feature Drift (PCFD) module to gradually align the source and target points in feature space. With this process, the t vector can be accurately recovered. 2. We propose a Consensus Encoding Unit (CEU) to construct a feature for each correspondence and a Spatial and Channel Attention (SCA) block to find correct correspondences. They can establish accurate matches for R estimation. 3. The above modules are integrated to build decoupling solutions for R and t. Experiments show that the proposed network achieves state-of-the-art performance on both indoor and outdoor datasets. Related Works Feature-Based 3D Matching. A common way to establish correspondences between 3D point clouds is by extracting local descriptors. Traditional hand-crafted descriptors are usually generated by extracting the local information, such as histograms of spatial coordinates (Frome et al. 2004;Johnson and Hebert 1999;Tombari, Salti, and Di Stefano 2010) and geometric attributes (Chen and Bhanu 2007;Salti, Tombari, and Di Stefano 2014). Some other methods (Rusu et al. 2008;Rusu, Blodow, and Beetz 2009) aim to design rotation invariant descriptors. Recently, deep learning techniques are explored to learn 3D descriptors. Many of these methods (Su et al. 2015;Zeng et al. 2017;Deng, Birdal, and Ilic 2018) take the point cloud patches as input to learn local features. Some other methods (Choy, Gwak, and Savarese 2019;Choy, Park, and Koltun 2019;Yew and Lee 2018;Bai et al. 2020;Huang et al. 2021) use point clouds as input to generate dense feature descriptors on point clouds. Although the methods above can usually establish good initial correspondences, it is hard to be totally outlier-free in the application. Our method is to address the challenge of registration when there are outliers in the correspondences. Outlier Removal. Given a putative correspondence set that contains outliers, one can use outlier removal methods to remove outliers. The most widely used method is the RANdom SAmple Consensus (RANSAC) (Fischler and Bolles 1981), and its variants (Chum and Matas 2005;Fragoso et al. 2013;Brahmachari and Sarkar 2009;Goshen and Shimshoni 2008).Recently, some methods start adopting deep learning techniques to find good 2D-2D correspondences. Liu et al. 2021) aim to capture more context to enhance the performance. Besides, recent attempts try to use deep learning networks for finding 3D correspondences, such as 3DReg-Net (Pais et al. 2020), DGR-Net (Choy, Dong, and Koltun 2020) and PointDSC (Bai et al. 2021). Our work aims to better find correct 3D correspondences and align point clouds through decoupling translation and rotation transformations. Pose Estimation. Pose estimation is the final goal of rigid 3D point registration, i.e., estimating a rigid transformation Source X ... Figure 2: Overview of our network. 1. A Progressive and Coherent Feature Drift (PCFD) module progressively aligns the source and target points in feature space, and recovers t vector from the alignment process. 2. An Consensus Encoding Unit (CEU) constructs feature for each correspondence by combining the spatial and feature consistency. 3. Several Spatial and Channel Attention (SCA) blocks are adopted to find correct correspondences, and followed with weighted SVD for estimating R matrix. ... K-1 K-1 1 ( ) K f y  1 ( ) K f x  Translation Regression t vector Max Pooling Y ... ( ) K f x 1 ( ) f x ... 1 ( ) f y ( ) K f y 3. R matrix estimation Weighted SVD R Matrix Spatial Attention Batch Norm M× SCA Block Channel Attention t vector Target Y Concat t vector X Concat MLP 1D Convolution L Classification results ReLU Global Context Global Interaction Global Context ' 1 ( ) f y ' 1 ( ) f x 1 ( ) f y 1 ( ) f x Shared Coherent Feature Drift Global Context Global Interaction Global Context Shared Coherent Feature Drift Progressive Alignment ' ( ) K f y ' ( ) K f x K  1  2  1 K   K  Consensus Encoding Unit t Vector estimation ( ) K f y ( ) K f x 1  to align point clouds. Besl and McKay (Besl and McKay 1992) propose the iterative closest point (ICP) algorithm to align point cloud through iteratively establishing point correspondence and performing least squares optimization. The variants of ICP (Rusinkiewicz and Levoy 2001;Segal, Haehnel, and Thrun 2009;Bouaziz, Tagliasacchi, and Pauly 2013) are proposed to address the challenges existing in ICP, such as efficiency, partiality and sparsity. Recently, some methods adopt end-to-end frameworks for directly estimating the rigid transformation between point clouds. Deep Closest Point (DCP-Net) (Wang and Solomon 2019a) uses deep global features to form correspondences and estimate relative pose. Later works (Yew and Lee 2020; Wang and Solomon 2019b) aim to address the problem of partial visibility to further improve the performance of registration. Methods Given two point clouds to be registered: X = {x i ∈ R 3 | i = 1, ..., N x } and Y = {x j ∈ R 3 | j = 1, . .., N y }, we first form N pairs of correspondences as follows: C = x 1 x 2 ... x n y 1 y 2 ... y n ∈ R 2×N ×3 ,(1) where x i and y i (1 < i < N ) are a pair of matched points. These putative correspondences are established by extracting local descriptors and matching. Limited by the distinctiveness of descriptors, many of these correspondences are wrong (outliers) . The goal of the network is to recover the rigid transformation from these noisy correspondences. It takes the coordinates of these correspondences as input, and outputs the probability of being correct (inliers) for each correspondence and rigid transformation as follows: t, R, L = Φ(C); t ∈ R 3 , R ∈ R 3×3 , L ∈ R N ×1 ,(2) where Φ(·) is the network with trained parameters. t and R are the estimated translation and rotation respectively. L is the logit value of each correspondence being inlier. In this paper, we propose a decoupling solution for the t and R. The pipeline of our method is shown in Fig. 2. We will explain the details of each module in the following sections. Translation Estimation Progressive and Coherent Feature Drift. The PCFD module transforms point cloud registration into a process of coherently moving source points to target points. Since deep neural network can extract more informative feature for each point, we convert the coherent drift operation to highdimensional feature space. As shown in Fig. 2, the PCFD module is composed of K Coherent Feature Drift (CFD) operations. Each CFD tries to align the features of source and target points generated by the previous CFD layer, so it is a progressive process. Specifically, the CFD first encodes feature for each point in a Siamese architecture. We use the CN Block (Moo Yi et al. 2018), which is a variant of PointNet (Qi et al. 2017), for encoding global context. Formally, let f l−1 (x) and f l−1 (y) be the output of l − 1 layer, then l-th CFD encodes the features as follows: f l (x) = CN(f l−1 (x)), f l (y) = CN(f l−1 (y)),(3) where f l (x) and f l (y) are the extracted features for source and target points. Note that the CN operations for source and target points are parameter-shared. In this way, the feature difference between a pair of correspondences is completely caused by the rigid transformation between them. Then we perform coherent drift by moving the features of source points to target points. A core of coherent drift is to force the source points to move coherently as a group to preserve the topological structure of the point sets (Myronenko and Song 2010). To ensure coherent constraints, a global feature offset (δ l , 1 ≤ l ≤ K) shared by all the source points is learned the l-th CFD. After that, we hold the target points and move the source points are moved by the δ l to the target points. f l (x i ) = f l (x) + δ l , f l (y i ) = f l (y i ); 1 ≤ i ≤ N,(4) where x i and y i are the i-th point in the source and target points. f l (x i ) and f l (y i ) are the output feature. An important issue of CFD is how to learn the global feature offset δ l . Each δ l needs to make the source points gradually approach the target points in the feature space. In the CFD operation, a global interaction is adopted to learn it as in Fig. 3. It first computes the feature difference between the feature of source and target points, as follows: d l = f l (y) − f l (x).(5) As mentioned before, x i and y i are a pair of putative correspondences. So d l is the feature offset between putative correspondences. We then learn a weight (w l ) for each correspondence by a convolution and sigmoid function: w l = sigmoid(Conv(d l ))(6) Finally, we use an average pooling to integrate the feature difference of all correspondences to produce the global feature offset δ l . The 1D convolution plays two roles in the learning of δ l : 1) Since there are many outliers in the putative correspondences, the weights produced by the convolution and sigmoid function is expected to suppress the outliers. 2) There are learnable parameters in the convolution operation to increase the flexibility of global interaction. Supervising Drift Process. In order to reduce the interference of R for better encoding the translation transformation, we introduce supervision in the middle of the network. Supervision is applied to the global feature offset δ l (1 < l < N ). Specifically, we first add up all the previous offsets, then use a 1D convolution to regress a temporary t l ∈ R 3 vector as follows: t l = Conv(Sum(δ 1 , ..., δ l−1 , δ l )).(7) Then a drift loss is used to supervise all layers of t l : L align = 1 K K l=1 { 1 N N i=1 m i · ρ(y i , R gt x i + t l )},(8) where ρ(., .) is the distance metric function. R gt is the ground truth rotation. m i is the mask for correspondence i. m i is set to 1 if the ground-truth of correspondence i is inliers. Otherwise, it will be set to be 0. It is a semi-alignment loss that uses the estimated translation t l in l-th layer and ground-truth rotation R gt to align the two point cloud and penalizes the alignment error. Thus, it expects all alignment to approximate the accurate t vector. Translation Regression. As mentioned before, every time a CFD is performed, the source point cloud is globally aligned to the target point cloud by the global feature offset, and δ l encodes the alignment process. We can naturally solve the t matrix by integrating the offsets of all layers. We concatenate all of the δ l (1 ≤ l ≤ K) and then adopt a 1D convolution to regress t vector. Consensus Encoding Unit An important issue for finding correct correspondences from putative correspondences is to mine consensus of inliers (Pais et al. 2020;Choy, Dong, and Koltun 2020;Bai et al. 2021), so that outliers can be distinguished from inliers. As introduced in Introduction Section and Fig. 1, it becomes easier to mine the consensus when removing translation t and remaining only rotation R between two point clouds. Since our PCFD module can regress t vector in advance, the Consensus Encoding Unit (CEU) tries to remove translation for better encoding consensus. The architecture of CEU is shown in Fig. 2, it combines the consensuses in coordinate and feature space. For the consensus in coordinate space, it is intuitive to remove the t vector. We subtract the estimated t vector from the target point cloud, so that the translation t between the source point cloud and the target point cloud is removed. Then the coordinate offset between the source point cloud and the target point cloud is followed with a 1D convolution to be as a feature. Meanwhile, CEU also tries to mine feature consistency between the correct correspondences. It utilizes the feature produced by the previous PCFD module to construct feature for correspondence to integrate more information. As introduced before, in PCFD module, source points are aligned to target points in feature space. By introducing the supervision of the intermediate layer, the t transformation between the source points and the target points is removed in feature space. Thus, we use the feature difference of these layers to construct the feature for the correspondence, which can encode consensus without translation. In order to make full use of the context of shallow and deep networks, all the layers of PCFD module are used to form a multi-layer correlation feature. Then a max-pooling, which performs the best with other choices based on our experiments, is adopted to integrate multi-layer context. The consensus feature in coordinate and feature space is combined by a concatenation operation. Rotation Estimation After Consensus Encoding Unit constructs a feature for each correspondence, a classification network is adopted to finding correct correspondences, and followed with weighted SVD to recover R matrix. Classification Network. As shown in Fig. 2, the classification network is composed of M Spatial and Channel Attention (SCA) Blocks. Each SCA block integrates a spatial attention, batch normalization (Ioffe and Szegedy 2015), ReLU and a channel attention in a ResNet architecture. The spatial attention is already proposed for finding 2D-2D correspondences (Sun et al. 2020;Chen, Yang, and Tao 2021) operation. They integrate local, global and prior information to learn weight for learning global context to ignore outliers. In this paper, since the CEU can construct more distinguishable features, we simplified the spatial attention to reduce the parameters of the network as Fig. 4 (a). For the input feature, it first learns a weight vector (w s ∈ R B×N ×1 ) by means of a 1D convolution and Softmax function. Then the weight vector is utilized as guidance to perform a weighted context normalization (Moo Yi et al. 2018) for encoding global context. The weight vector is to allow outliers to be ignored when performing context normalization. Meanwhile, interdependencies between feature channels are proved to be helpful for feature learning (Hu, Shen, and Sun 2018). So we also introduce a channel attention operation as Fig. 4 (b). Different from the commonly used SE-Net, we use instance-unique channel attention. It learns an independent weight vector for each instance instead of a shared one. Thus, it can capture more complex channel information for each correspondence. In order to reduce network computation for learning the instance-unique weight map, the group convolution (Cohen and Welling 2016) is used instead of the regular one. Weighted SVD. We use weighted SVD (Choy, Dong, and Koltun 2020), which reformulates the traditional SVD (Arun, Huang, and Blostein 1987) into a weighted version, to recover R matrix. Specifically, x i and y i (1 ≤ i ≤ N ) are the points in source and target point clouds respectively. We first use the estimated t vector to process the target points: y i = y i − t, 1 ≤ i ≤ N. (9) Then a weighted matrix H for SVD is computed as follows: H = i∈I w i x i y i T , H ∈ R 3×3 ,(10) where the weight w i is computed by the logit value of classification as follows: (Fischler and Bolles 1981), FGR (Zhou, Park, and Koltun 2016) and ours. The alignment areas with large errors are marked with red boxes. Our method achieves the best alignment result among these methods. w i = tanh(ReLU(L i )), 1 ≤ i ≤ N,(11) Loss Function We formulate our training objective as a combination of four types of loss functions, including translation loss (l trans ), classification loss (l cls ), alignment loss (l align ) and drift loss (l drif t ) as follows: loss = λ 1 l trans + λ 2 l cls + λ 3 l align + λ 4 l drif t (13) l trans is the L2 loss between the ground-truth and estimated t vector. l cls is the cross entropy loss. l align penalizes the wrong alignment between correct correspondences as follows: l align = 1 N N i=1 m i · ρ(y i , Rx i + t),(14) where ρ(., .) is Euclidean distance. t and R are the estimated translation and rotation transformation. N is the number of correspondences. m i is also the mask for correspondence i to label inliers, as introduced in Eq. 8. l drif t is to supervise the middle layer as Eq. 8. Experiments Experimental Setup Outdoor Dataset. We use the KITTI (Geiger, Lenz, and Urtasun 2012) odometry dataset, which contains 11 outdoor driving scenarios of points clouds. We follow the splitting way of previous works (Bai et al. 2020;Choy, Park, and Koltun 2019) and use scenario 0 to 5 for training, 6 to 7 for validation and 8 to 10 for testing. Then for each point cloud, we construct 30cm voxel grid to downsample the point cloud (Choy, Park, and Koltun 2019). Indoor Datasets. We use the SUN3D dataset (Xiao, Owens, and Torralba 2013) to generate the dataset for training and testing. Sun3D is composed of 268 sequences of RGBD videos. We randomly select 115 sequences for training and validation, and 20 sequences for testing. For each video sequence, we first subsample the videos by a factor of 10. Then for each frame, we recover the point cloud by depth map, and construct 5cm voxel grid to downsample the point cloud (Zhou, Park, and Koltun 2018). The 7scenes (Shotton et al. 2013) dataset contains 46 RGBD sequences under various camera motion statuses, we follow the official split to use the 18 sequences of them as test dataset. It is adopted for generalization experiments. Data Processing. Following 3DReg-Net (Pais et al. 2020), we use FPFH descriptors (Rusu, Blodow, and Beetz 2009) to generate 2560 pairs of correspondences between adjacent frames as input. Then we generate the ground-truth rotation and translation according to the offered camera pose of each frame and label the correspondences as inlier/outlier (1 refers inliers and 0 refers outliers) by a predefined distance threshold. Evaluation Metrics. For a pair of point clouds, we evaluate the results by computing the errors between the estimated and ground-truth rigid transformation. The errors of rotation (RE) are evaluated by the isotropic error (Ma et al. 2012). The errors of translation (TE) are evaluated by the L2 error (Choy, Dong, and Koltun 2020). For the whole test dataset, we first report the mean of rotation (MRE) and translation (MTE) errors. Then, given an error threshold of R and t, we can determine whether each estimated pose is accurate or not. We build a normalized cumulative precision curve of pose estimation in the whole test set. After that, we use (5 • , 15 cm) as threshold to figure the recall (Choy, Dong, and Koltun 2020) and the area under the curve as mean average precision (mAP) (Moo Yi et al. 2018). Implementation Details. In the PCFD module, we use 10 layers of CFD to progressively align the two point clouds (K = 10 in Fig. 2). In the classification module, 4 SCA blocks are utilized to build classification network (M = 4 in Fig. 2). The number of channels in all layers of the network is set to 128. During training, λ 1 , λ 2 , λ 3 and λ 4 in loss function (Eq. 13) are set to 2, 1, 1 and 0.05 respectively. The network is trained by Adam optimizer (Kingma and Ba 2015) with a learning rate being 10 −3 and batch size being 16. All the experiments are conducted on a machine with an INTEL Xeon E5-2620 CPU and a single NVIDIA GTX1080Ti. For time-consuming, to do a fair comparison for all the methods, all computation timings are obtained using CPU. Comparison to Other Baselines We compare our method with other baselines, including ICP (Besl and McKay 1992), FGR (Zhou, Park, and Koltun 2016), RANSAC (Fischler and Bolles 1981), GCRANSAC (Barath and Matas 2018), DGR (Choy, Dong, and Koltun 2020), PointLK (Aoki et al. 2019), PointDSC (Bai et al. 2021) and 3DRegNet (Pais et al. 2020). ICP, FGR, RANSAC and GCRANSAC are classical methods while DGR, PointLK, PointDSC and 3DRegNet are learning based methods. All the learning based networks are retrained with the same dataset. For ICP, RANSAC and FGR, we use the version Open3D implemented, while the released codes are adopted for other methods. We present the quantitative results on the KITTI, SUN3D and 7Scenes Datasets. The results on 7Scenes are obtained by the model trained on SUN3D dataset as generalization experiments. As shown in Tab. 1, the recall and mAP of our method are higher than other methods. It shows the overall performance of our methods. More specifically, the t error of our method is much smaller than other methods, especially on indoor scenes. The t error of our method is 7.81cm and 7.47cm smaller than that of our baseline network (3DRegNet) on SUN3D and 7Scenes datasets. It implies that the proposed Progressive and Coherent Feature Drift (PCFD) module can boost the performance of t estimation. For time-consuming, since our network can output the results without repeated sampling as RANSAC (Fischler and Bolles 1981) and postprocessing, it is faster than other methods except for 3DReg-Net. Finally, in order to visually demonstrate the registration performance, we present the visualized alignment results in Fig. 5. We select multiple point clouds and calculate the relative pose between each point cloud and its neighbor. Then we transform these point clouds into the same coordinate frame. The results of 3DRegNet (Pais et al. 2020), RANSAC (Fischler and Bolles 1981), FGR (Zhou, Park, and Koltun 2016) are presented as comparison. Our method achieves the best alignment results with fewer errors. Registration Robustness So far, we have demonstrated the overall performance of the proposed network. In order to further analyze the registra- Figure 6: The error curves of R (MRE) and t (MTE) under different inlier ratios of initial correspondences on SUN3D dataset. tion robustness anti-noise, we test the performance under the scenarios with the different inlier ratios of initial correspondence set. Specifically, we divide the test set of SUN3D dataset into several subsets according to the inlier ratio, and respectively compute the mean errors of R (MRE) and t (MTE) estimation at each inlier ratio, as shown in Fig. 6. As we can see, our method has obtained results with smaller errors under the scenarios of different inlier ratios for both R and t. It demonstrates that our method is robust to outliers. Besides, when the inlier ratio changes, the error range of our method is also smaller, which shows that the performance of our method is relatively stable. Method Analysis In this section, we will analyze our method in detail. Expect the previously introduced four evaluation metrics, we also report the classification accuracy (Acc in Tab. 2 and 3) for better understanding the effect of each module. Regression or SVD -Tab. 2. In our network, t is estimated by regression while R is solved by SVD. We discuss these two estimation heads for t and R. The regression and weighted SVD are adopted as the estimation heads for t and R, respectively. Through permutation and combination, we can generate four alternatives. For each alternative, we use the proposed network as the backbone for feature extraction. By analyzing these four groups of control experiments, we can get the following observations: 1) When the estimation heads of R are consistent, direct regression will get better results than the SVD method for t estimation. When the estimated head of t is the same, the R result obtained by SVD is better. This proves that regressing t in advance, which adopted in our method is a good choice. 2) We further compare the results of the 2-th group and the 4-th group (ours). We can find that our method obtains more accurate t. Meanwhile, although the 2-th and the 4-th group use the same estimation head for R estimation, the 4-th group still achieves better results for R estimation. This can be explained as our method can estimate and remove t in advance before finding the correspondence. Thus, there exists only rotation between correspondence, which helps the classification of inliers and outliers. In fact, the 4-th group dose achieve a better classification result than that of the 2-th group. The above results prove the effectiveness of the idea of decoupling t and R. Ablation Study -Tab. 3. Finally, we perform ablation studies on SUN3D dataset to further analyze the effect of the proposed modules, including Progressive and Coherent Feature Drift (PCFD), Consensus Encoding Unit (CEU) and Spatial and Channel Attention (SCA) block. The 3DRegNet (Pais et al. 2020) is adopted as our baseline model. Since we have already proved that regression for t and SVD for R is the most suitable combination of estimation head, we use 3DRegNet with this alternative instead of the vanilla version. We gradually add the proposed modules into the baseline model. First, we use the PCDF module to replace the CN Blocks (Moo Yi et al. 2018) of 3DRegNet. The error of t estimation significantly has decreased, which confirms the effectiveness of PCDF for regressing t. Then we adopt CEU to construct features for correspondence classification. As we can see, the classification accuracy is improved by 10% compared with only using PCFD, leading to a better result of R. It shows that the proposed CEU can construct better classification features. Finally, we replace the CN blocks in 3DRegNet with SCA blocks, the performance of correspondences classification and R estimation are further enhanced. Conclusion In this work, we develop a point cloud registration network named DetarNet, which decouples the estimation of rotation and translation. Specifically, we first propose a Progressive and Coherent Feature Drift (PCFD) module. It transforms the point cloud alignment process into a coherent drift operation in high-dimensional feature space and gradually estimates the translation. Then, we adopt a classification module to perform outlier pruning. It uses the proposed Consensus Encoding Unit (CEU) to construct feature for each correspondence, and adopts a Spatial and Channel Attention (SCA) for classification. Thus, the network can establish correct matches by taking advantages of the estimated t. Finally, R matrix is obtained by performing weighted SVD. Extensive experiments on real scenes demonstrate the effectiveness of the proposed DetarNet. Figure 1 : 1(a): A toy example shows the challenge when translation and rotation are coupled. When accurately estimating translation in advance and there only exists rotation as shown in The CN-Net (Moo Yi et al. 2018) proposes a Context Normalization (CN) operation for finding correct correspondences. Later works (Plötz and Roth 2018; Zhang et al. 2019; Zhao et al. 2019; Sun et al. 2020; Brachmann and Rother 2019; Figure 3 : 3The global interaction operation. Figure 4 : 4The Spatial Attention and Channel Attention in SCA block. Figure 5 : 5Finally, R can be obtained by performing SVD on H matrix as follows:R = U diag(1, 1, det(U V T ))V T , H = U V T .(12) The visualized alignment results of four different methods. From top row to bottom: 3DRegNet (Pais et al. 2020), RANSAC Table 1 : 1Quantitative results on the KITTI, SUN3D and 7Scenes Datasets. The mean rotation error (MRE), mean translation error (MTE), mAP and recall under the threshold of (5 • , 15 cm) are reported. KITTI SUN3D 7Scenes (Generalization ) MRE MTE mAP recall MRE MTE mAP recall MRE MTE mAP recall Time ICP 1.208 90.21 0.54 1.41 6.178 15.40 19.6 49.7 5.504 12.44 27.8 50.3 0.28 RANSAC 0.759 29.65 43.1 80.9 3.580 15.16 43.9 80.5 2.107 12.41 32.2 68.5 2.79 GCRANSAC 0.152 70.41 1.89 3.12 1.920 9.672 37.6 75.6 1.946 10.43 30.7 78.1 0.82 FGR 0.298 12.13 31.4 72.0 2.895 10.85 39.2 73.6 2.913 13.52 31.5 66.3 0.32 DGR 0.157 9.773 41.6 82.0 2.239 9.663 41.3 82.8 2.166 13.54 30.0 63.4 0.76 PointLK 5.352 43.84 4.01 7.25 7.732 27.64 17.2 32.0 26.49 32.37 5.12 9.61 0.13 PointDSC 0.152 8.966 46.9 91.5 1.913 7.283 50.1 89.7 1.902 11.31 38.6 78.4 0.09 3DRegNet 0.752 31.62 12.3 28.7 2.889 13.13 31.2 68.6 6.424 15.21 26.7 58.2 0.03 Ours 0.148 8.126 48.1 88.1 1.840 5.317 56.3 93.1 2.011 7.739 42.7 84.9 0.04 Table 2 : 2The registration result of using different estimation heads. t R Acc MRE MTE mAP recall 1 Reg Reg 65.7 2.45 6.22 49.7 86.5 2 SVD SVD 69.1 2.29 9.17 43.2 80.0 3 SVD Reg 60.2 2.69 9.01 35.7 62.3 4 Reg SVD 72.9 1.84 5.32 56.3 93.1Tag Table 3 : 3Ablation studies of proposed modules.Baseline PCDF CEU SCA Acc MRE MTE mAP recall63.2 2.66 13.1 33.1 70.2 60.0 2.92 6.02 41.7 78.8 70.1 2.02 5.99 52.1 90.5 72.9 1.84 5.32 56.3 93.1 AcknowledgementsThis work was supported by the National Natural Science Foundation of China under Grants 62176096 and 61991412. 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[ "https://github.com/ZhiChen902/DetarNet." ]
[ "The Strongly Asynchronous Massive Access Channel", "The Strongly Asynchronous Massive Access Channel" ]
[ "Sara Shahi [email protected] \nUniversity of Illinois at Chicago\n60607ChicagoILUSA\n", "Daniela Tuninetti [email protected] \nUniversity of Illinois at Chicago\n60607ChicagoILUSA\n", "Natasha Devroye [email protected] \nUniversity of Illinois at Chicago\n60607ChicagoILUSA\n" ]
[ "University of Illinois at Chicago\n60607ChicagoILUSA", "University of Illinois at Chicago\n60607ChicagoILUSA", "University of Illinois at Chicago\n60607ChicagoILUSA" ]
[]
This paper considers a Strongly Asynchronous and Slotted Massive Access Channel (SAS-MAC)where K n :" e nν different users transmit a randomly selected message among M n :" e nR ones within a strong asynchronous window of length A n :" e nα blocks, where each block lasts n channel uses. A global probability of error is enforced, ensuring that all the users' identities and messages are correctly identified and decoded. Achievability bounds are derived for the case that different users have similar channels, the case that users' channels can be chosen from a set which has polynomially many elements in the blocklength n, and the case with no restriction on the users' channels. A general converse bound on the capacity region and a converse bound on the maximum growth rate of the number of users are derived.July 27, 2018 DRAFT ‚ We define a new massive identification paradigm in which we allow the number of sequences in a classical identification problem to increase exponentially with the sequence blocklength (or sample size). We find asymptotically matching upper and lower bounds on the probability of identification error for this problem. We use this result in our SAS-MAC model to recover
10.3390/e25010065
[ "https://arxiv.org/pdf/1807.09934v1.pdf" ]
50,785,861
1807.09934
202210096eef03f6f3830b9c5f2427ff7adbd403
The Strongly Asynchronous Massive Access Channel Sara Shahi [email protected] University of Illinois at Chicago 60607ChicagoILUSA Daniela Tuninetti [email protected] University of Illinois at Chicago 60607ChicagoILUSA Natasha Devroye [email protected] University of Illinois at Chicago 60607ChicagoILUSA The Strongly Asynchronous Massive Access Channel 1 This paper considers a Strongly Asynchronous and Slotted Massive Access Channel (SAS-MAC)where K n :" e nν different users transmit a randomly selected message among M n :" e nR ones within a strong asynchronous window of length A n :" e nα blocks, where each block lasts n channel uses. A global probability of error is enforced, ensuring that all the users' identities and messages are correctly identified and decoded. Achievability bounds are derived for the case that different users have similar channels, the case that users' channels can be chosen from a set which has polynomially many elements in the blocklength n, and the case with no restriction on the users' channels. A general converse bound on the capacity region and a converse bound on the maximum growth rate of the number of users are derived.July 27, 2018 DRAFT ‚ We define a new massive identification paradigm in which we allow the number of sequences in a classical identification problem to increase exponentially with the sequence blocklength (or sample size). We find asymptotically matching upper and lower bounds on the probability of identification error for this problem. We use this result in our SAS-MAC model to recover I. INTRODUCTION In the Internet of Things (IoT) paradigm it is envisioned that many types of devices will be wirelessly connected. A foundational study to understand the fundamental tradeoffs and thus enable the successful deployment of ubiquitous, interconnected wireless network is needed. This new paradigm imposes new traffic patterns on the wireless network. Moreover, devices within such a network often have strict energy consumption constraints, as they are often battery powered sensors transmitting bursts of data very infrequently to an access point. Finally, as the name suggests, these networks must support a huge number of inter-connected devices. Due to these new network characteristics, we propose a novel communication and multipleaccess model: the Strongly Asynchronous Slotted Massive Access (SAS-MAC). In a SAS-MAC, the number of users K n :" e nν increases exponentially with blocklength n with occupancy exponent ν ě 0. Moreover, the users are strongly asynchronous, i.e., they transmit in one randomly chosen time slot within a window of length A n :" e nα blocks, each block of length n, where α ě 0 is the asynchronous exponent. In addition, when active, each user can choose from a set of M n :" e nR messages to transmit.All transmissions are sent to an access point and the receiver is required to jointly decode and identify all users. The goal is to characterize the set of all achievable pR, α, νq triplets. A. Past work Strongly asynchronous communications were first introduced in [1] for synchronization of a single user, and later extended in [2] for synchronization with positive transmission rate. In [3] the authors of [2] made a brief remark about a "multiple access collision channel" extension of their original single-user model. In this model, any collision of users (i.e., users who happen to transmit in the same block) is assumed to result in output symbols that appear as if generated by noise. The error metric is taken to be the per user probability of error, which is required to be vanishing for all but a vanishing fraction of users. In this scenario, it is fairly easy to quantify the capacity region for the case that the number of users are less than the square root of the asynchronous window length (i.e., in our notation ν ă α{2). However, finding the capacity of the "multiple access collision channel" for global / joint probability of error, as opposed to per user probability of error, is much more complicated and requires novel achievability schemes and novel analysis tools. This is the main subject and contribution of this paper. Recently, motivated by the emerging machine-to-machine type communications and sensor networks, a large body of work has studied "many-user" versions of classical multiuser channels as pioneered in [4]. In [4] the number of users is allowed to grow linearly with blocklength n. A full characterization of the capacity of the synchronous Gaussian (random) many access channel was given [4]. In [5], the author studied the synchronous massive random access channel where the total number of users increases linearly with the blocklength n. However, the users are restricted to use the same codebook and only a per user probability of error is enforced. In the model proposed here, the users are strongly asynchronous, the number of users grow exponentially with blocklength, and we enforce a global probability of error. Training based synchronization schemes (the use of pilot signals) was proven to be suboptimal for bursty communications in [2]. Rather, one can utilize the users' statistics at the receiver for synchronization or user identification purposes. The identification problem (defined in [6]) is a classic problem considered in hypothesis testing. In this problem, a finite number of distinct sources each generates a sequence of i.i.d. samples. The problem is to find the underlying distribution of each sample sequence, given the constraint that each sequence is generated by a distinct distribution. Studies on identification problems all assume a fixed number of sequences. In [7], authors study the Logarithmically Asymptotically Optimal (LAO) Testing of identification problem for a finite number of distributions. In particular, the identification of only two different objects has been studied in detail, and one can find the reliability matrix, which consists of the error exponents of all error types. Their optimality criterion is to find the largest error exponent for a set of error types for given values of the other error type exponents. The same problem with a different optimality criterion was also studied in [8], where multiple, finite sequences were matched to the source distributions. More specifically, the authors in [8] proposed a test for a generalized Neyman-Pearson-like optimality criterion to minimize the rejection probability given that all other error probabilities decay exponentially with a pre-specified slope. In this paper, we too allow the number of users to increase in the blocklength. We assume that the users are strongly asynchronous and may transmit randomly anytime within a time window that is exponentially large in the blocklength. We require the receiver to recover both the transmitted messages and the users' identities under a global/joint probability of error criteria. By allowing the number of sequences to grow exponentially with the number of samples, the number of different possibilities (or hypotheses), would be doubly exponential in blocklength and the analysis of the optimal decoder becomes much more challenging than classical (with constant number of distributions) identification problems. These differences in modeling the channel require a number of novel analytical tools. B. Contribution In this paper, we consider the SAS-MAC whose number of users increase exponentially with blocklength n. In characterizing the capacity of this model, we require its global probability of error to be vanishing. More specifically our contributions are as follows: ‚ We propose a new achievability scheme that supports strictly positive values of pR, α, νq for identical channels for the users. ‚ We propose a new achievability scheme for the case that the channels of the users are chosen from a set of conditional distributions. The size of the set increases polynomially in the blocklength n. In this case, the channel statistics themselves can be used for user identification. ‚ We propose a new achievability scheme without imposing any restrictive assumptions on the users' channels. We show that strictly positive pR, α, νq are possible. ‚ We propose a novel converse bound for the capacity of the SAS-MAC. ‚ We show that for ν ą α, not even reliable synchronization is possible. These results were presented in parts in [9], [10], [11]. C. Paper organization In Section II we introduce our massive identification model and present a technical theorem (Theorem 1) that will be needed later on in the proof of Theorem 3. In Section III we introduce the SAS-MAC model and in Section IV we present our main results. More specifically, we introduce different achievability schemes for different scenarios and a converse technique to derive an upper bound on the capacity of the SAS-MAC. Finally, Section V concludes the paper. Some proofs may be found in the Appendix. D. Notation Capital letters represent random variables that take on lower case letter values in calligraphic letter alphabets. The notation a n . " e nb means lim nÑ8 log an n " b. We write rM : N s, where M, N P Z, M ď N , to denote the set tM, M`1, . . . , N u, and rKs :" r1 : Ks. We use y n j :" ry j,1 , ..., y j,n s, and simply y n instead of y n 1 . The binary entropy function is defined by hppq :"´p logppq´p1´pq logp1´pq. II. MASSIVE IDENTIFICATION PROBLEM We first introduce notation specifically used in this Section and then introduce our model and results. July 27, 2018 DRAFT A. Notation When all elements of the random vector X n are generated i.i.d according to distribution P , we denote it as X n i.i.d " P . We use S n , where |S n | " n!, to denote the set of all possible permutations of a set of n elements. For a permutation σ P S n , σ i denotes the i-th element of the permutation. txu r is used to denote the remainder of x divided by r. K k´a1 , . . . , a p k 2 q¯i s the complete graph with k nodes with edge index i P r`k 2˘s and edge weights a i , i P r`k 2˘s . We may drop the edge argument and simply write K k when the edge specification is not needed. A cycle c of length r in K k may be interchangeably defined by a vector of vertices as c pvq " rv 1 , . . . , v r s or by a set of edges c peq " ta 1 , . . . , a r u where a i is the edge between pv i , v i`1 q, @i P rr´1s and a r is that between pv r , v 1 q. With this notation, c pvq piq is then used to indicate the i-th vertex of the cycle c. C prq k is used to denote the set of all cycles of length r in the complete graph K k´a1 , . . . , a p k 2 q¯. The cycle gain, denoted by Gpcq, for cycle c " ta 1 , . . . , a r u P C prq k is the product of the edge weights within the cycle c, i.e., Gpcq " ś r i"1 a i , @a i P c. The Bhatcharrya distance between P 1 and P 2 is denoted by BpP 1 , P 2 q :" ř xPX a P 1 pxqP 2 pxq. B. Problem Formulation Let P :" tP 1 , . . . , P A u, P i P P X , @i P rAs consist of A distinct distributions and also let Σ be uniformly distributed over S A , the set of permutations of A elements. In addition, assume that we have A independent random vectors tX n 1 , X n 2 , . . . , X n A u of length n each. For σ, a realization of Σ, assign the distribution P n σ i to the random vector X n i , @i P rAs. After observing a sample x nA " rx n 1 , . . . , x n A s of the random vector X nA " rX n 1 , . . . , X n A s, we would like to identify P σ i , @i P rAs. More specifically, we are interested in finding a permutationσ : X nA Ñ S A to indicate that X n i i.i.d " Pσ i , @i P rAs. LetΣ "σpX nA q. The average probability of error for the set of distributions P is given by P pnq e " P "Σ ‰ Σ ı " 1 pAq! ÿ σPS A P "Σ ‰ σ|X n i i.i.d " P σ i , @i P rAs ı " P "Σ ‰ p1, 2, . . . , AqˇˇX n i i.i.d " P i , @i P rAs ı . We say that a set of distributions P is identifiable if lim nÑ8 P pnq e Ñ 0. July 27, 2018 DRAFT C. Condition for Identifiability In Theorem 1 we characterize the relation between the number of distributions and the pairwise distance of the distributions for reliable identification. Moreover, we introduce and use a novel graph theoretic technique in the proof of Theorem 1 to analyze the optimal Maximum Likelihood decoder. Theorem 1. A sequence of distributions P " tP 1 , . . . , P An u is identifiable iff lim nÑ8 ÿ 1ďiăjďAn e´2 nBpP i ,P j q " 0.(1) The rest of this section contains the proof. To prove Theorem 1, we provide upper and lower bounds on the probability of error in the following subsections. D. Upper bound on the probability of identification error We use the optimal Maximum Likelihood (ML) decoder, which minimizes the average probability of error, given byσ px n 1 , . . . , x n An q :" arg max σPS An An ÿ i"1 log pP σ i px n i qq ,(2) where P σ i px n i q " ś n t"1 P σ i px i,t q. The average probability of error associated with the ML decoder can also be written as log Pσ i pX n i q P i pX n i q ě 0ˇˇp H ff ,(4) where p H :" ! X n i i.i.d " P i ,1 tσ i ‰iu " r ÿ j"1 1 tσ i j ‰i j u " r + . These two sequences in (4) in fact indicate the event that we have (incorrectly) identified X n i j i.i.d " Pσ i j instead of the (true) distribution X n i j i.i.d " P i j , @j P rrs. For a complete graph K An , the set of edges between ppi 1 ,σ i 1 q, . . . , pi r ,σ ir qq in K An would produce a single cycle of length r or a set of disjoint cycles with total length r. However, we should note that in the latter case where the sequence of edges construct a set of (lets say of size L) disjoint cycles (each with some length r l forr l ă r such that ř L l"1r l " r), then those cycles and their corresponding sequences are already taken into account in the (union of) set ofr l error events. As an example, assume A n " 4 and consider the error event log P 2 pX n 1 q P 1 pX n 1 q`l og P 1 pX n 2 q P 2 pX n 2 q`l og P 4 pX n 3 q P 3 pX n 3 q`l og P 3 pX n 4 q P 4 pX n 4 q ě 0, which corresponds to the (error) event of choosing rσ 1 ,σ 2 ,σ 3 ,σ 4 s " r2, 1, 4, 3s over r1, 2, 3, 4s with r " 4 errors. In the graph representation, this gives two cycles of length 2 each, which correspond to " log P 2 pX n 1 q P 1 pX n 1 q`l og P 1 pX n 2 q P 2 pX n 2 q ě 0 * X " log P 4 pX n 3 q P 3 pX n 3 q`l og P 3 pX n 4 q P 4 pX n 4 q ě 0 * , and are already accounted for in the events trσ 1 ,σ 2 ,σ 3 ,σ 4 s " r2, 1, 3, 4su Y trσ 1 ,σ 2 ,σ 3 ,σ 4 s " r1, 2, 4, 3su with r " 2. As the result, in order to avoid double counting, in evaluating (4) for each r we should only consider the sets of sequences which produce a single cycle of length r. Before proceeding further, we define the edge weights for a complete weighted graph K An pa p1,2q , . . . a pKn,1q q. In particular, we define a pi,jq :" e´n BpP i ,P j q to be the edge weight between vertices pi, jq in the complete graph K An shown in Fig. 1. Gpcq, where r enumerates the number of incorrect matchings and where cpiq is the i-th vertex in the cycle c. In (6), we have leveraged the fact that e´n BpP i ,P j q is the edge weight between vertices pi, jq in the complete graph K An and hence Gpcq " e´n / - " exp #´n r ÿ i"1 BpP c pvq piq , P c pvq pti`1urq q + . The fact that we used t " 1{2 in (7) instead of finding the exact optimizing t, comes from the fact that t " 1{2 is the optimal choice for r " 2 and as we will see later, the rest of the error events are dominated by the set of only 2 incorrectly identified distributions. This can be seen as follows for X n 1 i.i.d " P 1 , X n 2 i.i.d " P 2 P " log P 1 pX n 2 q P 2 pX n 2 q`l og P 2 pX n 1 q P 1 pX n 1 q ě 0  " ÿ P 1 ,P 2 : ř xPXP 1 pxq log P 2 pxq P 1 pxqP 2 pyq log P 1 pxq P 2 pxq ě0 exp ! nD´P 1 P 1¯´n D´P 2 P 2¯) . " e´n DpP P 1q´n DpP P 2q " e´2 nBpP 1 ,P 2 q , whereP in the first equality in (8), by using the Lagrangian method, can be shown to be equal toP pxq " ? P 1 pxqP 2 pxq ř x 1 ? P 1 px 1 qP 2 px 1 q and subsequently the second inequality in (8) is proved. In order to calculate the expression in (6), we use the following graph theoretic Lemma, the proof of which is given in Appendix A. Lemma 1. In a complete graph K k pa 1 , . . . , a n k q and for the set of cycles of length r, C prq k " tc 1 , . . . c N r,k u, we have 1 N r,k`G pc 1 q`. . . Gpc N r,k q˘ďˆa 2 1`. . .`a 2 n k n k˙r 2 where N r,k , n k are the number of cycles of length r and the number of edges in the complete graph K k , respectively. e´2 nBpP i ,P j q " 0. As a result of Lemma 1, it can be seen from (88) that the sum of probabilities that r ě 3 distributions are incorrectly identified is dominated by the probability that only r " 2 distributions are incorrectly identified. This shows that the most probable error event is indeed an error event with two wrong distributions. E. Lower bound on the probability of identifiability error For our converse, we use the optimal ML decoder, and as a lower bound to the probability of error in (4), we only consider the set of error events with only two incorrect distributions, i.e., the set of events with r " 2. In this case we have P pnq e ě P « ď 1ďiăjďAn log P i pX n j q P j pX n j q`l og P j pX n i q P i pX n i q ě 0| p H ff ě´ř 1ďiăjďAn P rξ i,j s¯2 ř pi,jq,pj,kq pi,jq‰pl,kq i‰j,l‰k Prξ i,j , ξ k,l s ,(10) where (10) is by [12] and where ξ i,j :" " log P i P j pX n j q`log P j P i pX n i q ě 0| p H * .(11) We prove in Appendix C that a lower bound on P pnq e is given by P pnq e ě´ř 1ďiăjďAn e´2 nBpP i ,P j q¯2 ř i,j,k e´n BpP i ,P j q´nBpP i ,P k q´nBpP k ,P j q`˜ř i,j e´2 nBpP i ,P j q¸2 (12) ě´ř i,j e´2 nBpP i ,P j q¯2 8˜ř 1ďiăjďAn e´2 nBpP i ,P j q¸3 2`˜ř 1ďiăjďAn e´2 nBpP i ,P j q¸2 (13) " b ř 1ďiăjďAn e´2 nBpP i ,P j q 8`b ř 1ďiăjďAn e´2 nBpP i ,P j q ,(14) where (13) is by Lemma 1. As it can be seen from (14), if lim nÑ8 ř 1ďiăjďAn e´2 nBpP i ,P j q ‰ 0, the probability of error is bounded away from zero. As a result, we have to have lim nÑ8 ÿ 1ďiăjďAn e´2 nBpP i ,P j q " 0, which also matches our upper bound on the probability of error in (89). Remark 1. As it is clear from the result of Theorem 1, when A n is a constant or grows polynomially with n, the sequence of distributions in P are always identifiable and the probability of error in the identification problem decays to zero as the blocklength n goes to infinity. The interesting aspect of Theorem 1 is in the regime that A n increases exponentially with the blocklength. Having proved the criterion for identifiability of a massive number of distributions in Theorem 1, we move on to the SAS-MAC problem. We use the result of Theorem 1 to identify the massive number of users by their induced probability distribution at the receiver. III. SAS-MAC PROBLEM We first introduce the special notation used in the SAS-MAC and then formally define the problem. A. Special Notation A stochastic kernel / transition probability / channel from X to Y is denoted by Qpy|xq, @px, yq P XˆY, and the output marginal distribution induced by P P P X through the channel Q as rP Qspyq :" ÿ xPX P pxqQpy|xq, @y P Y,(15) where P X is the space of all distributions on X . We define the shorthand notation Q x pyq :" Qpy|xq, @y P Y.(16) For a MAC channel Qpy|x 1 , . . . , x K q, @px 1 , . . . , x K , yq P X 1ˆ. . .ˆX KˆY , we define the shorthand notation Q S py|x S q :" Qpy|x S , ‹ S c q, @S Ď rKs,(17) to indicate that users indexed by S transmit x i , @i P S, and users indexed by S c transmit their respective idle symbol ‹ j , @j P S c " rKszS. When |S| " 1, we use Q i py|x i q :" Q tiu py|x i q " Qpy|‹ 1 , . . . , ‹ i´1 , x i , ‹ i`1 , . . . , ‹ K q, and when |S| " 0, we use Q ‹ pyq :" Qpy|‹ 1 , . . . ‹ K q. The empirical distribution of a sequence x n is p P x n paq :" 1 n N pa|x n q " 1 n n ÿ i"1 1 tx i "au , @a P X ,(18) where N pa|x n q denotes the number of occurrences of letter a P X in the sequence x n ; when using (18) the target sequence x n is usually clear from the context so we may drop the subscript x n in p P x n p¨q. The P -type set and the V -shell of the sequence x n are defined, respectively, as T pP q :" tx n : N pa|x n q " nP paq, @a P X u , T V px n q :" " y n : N pa, b|x n , y n q N pa|x n q " V pb|aq, @pa, bq P pX , Yq * ,(19) where N pa, b|x n , y n q " ř n i"1 1 t x i "a y i "b u is the number of joint occurrences of pa, bq in the pair of sequences px n , y n q. We use DpP 1 P 2 q to denote the Kullback Leibler divergence between distribution P 1 and P 2 , and DpQ 1 Q 2 |P q :" ř x,yPXˆY P pxqQ 1 py|xq log Q 1 py|xq Q 2 py|xq for the conditional Kullback Leibler divergence. We let IpP, Qq " DpQ rP Qs|P q denote the mutual information between random variable pX, Y q with joint distribution P X,Y px, yq " P pxqQpy|xq. B. SAS-MAC Problem Formulation Let M be the number of messages, A be the number of blocks, and K be the number of users. An pM, A, K, n, q code for the SAS-MAC consists of: ‚ A message set rM s, for each user i P rKs, from which messages are chosen uniformly at random and are independent across users. ‚ An encoding function f i : rM s Ñ X n , for each user i P rKs. We define x n i pmq :" f i pmq.(21) Each user i P rKs choses a message m i P rM s and a block index t i P rAs, both uniformly at random. It then transmits r‹ npt i´1 q i x n i pm i q ‹ npA´t i q i s, where ‹ i P X is the designated 'idle' symbol for user i. July 27, 2018 DRAFT ‚ A destination decoding functioǹ p p t 1 , p m 1 q, . . . , p p t K , p m K q˘" gpY nA q,(22) such that its associated probability of error, P pnq e , satisfies P pnq e ď where P pnq e :" 1 pAM q K ÿ pt 1 ,m 1 q,...,pt K ,m K q P " gpY n q ‰ ppt 1 , m 1 q, . . . , pt K , m K qq |H ppt 1 ,m 1 q,...,pt K ,m K qq ‰ ,(23) where the hypothesis that user i P rKs has chosen message m i P rM s and block t i P rAs is denoted by H ppt 1 ,m 1 q,...,pt K ,m K qq . A tuple pR, α, νq is said to be achievable if there exists a sequence of codes`e nR , e nα , e nν , n, nw ith lim nÑ8 n " 0. The capacity region of the SAS-MAC at asynchronous exponent α, occupancy exponent ν and rate R, is the closure of all possible achievable pR, α, νq triplets. IV. MAIN RESULTS FOR SAS-MAC In this Section we first introduce an achievable region for the case that different users have identical channels (in Theorem 2). We then move on to the more general case where the users' channels belong to a set of conditional probability distributions of polynomial size in n (in Theorem 3). In this case, we use the output statistics to distinguish and identify the users. We then remove all restrictions on the users' channels and derive an achievability bound on the capacity of the SAS-MAC (in Theorem 4). After that, we propose a converse bound on the capacity of general SAS-MAC (in Theorem 5). We then provide a converse bound on the number of users (in Theorem 6). A. Users with Identical Channels The following theorem is an achievable region for the SAS-MAC for the case that different users have identical channels toward the base station when they are the sole active user. In this scenario, users' identification and decoding can be merged together. Theorem 2. For a SAS-MAC with asynchronous exponent α, occupancy exponent ν and rate R, assume that Q tiu py|xq " Qpy|xq (recall definition (17)) for all users. Then, the following pR, α, νq region is achievable ď P PP X λPr0,1s $ ' ' ' ' ' & ' ' ' ' ' % ν ă α 2 ν ă DpQ λ Q|P q α`R`ν ă DpQ λ Q ‹ |P q R`ν ă IpP, Qq , / / / / / . / / / / / - ,(24) where Q λ py|xq :" Qpy|xq λ Q ‹ pyq 1´λ ř y 1 PY Qpy 1 |xq λ Q ‹ py 1 q 1´λ , @px, yq P XˆY.(25) Proof. Before starting the proof, we note that for ν ă α 2 (first bound in (24)), with probability approaching one as the blocklength n grows to infinity, the users transmit in distinct blocks. Hence, in analyzing the joint probability of error of our achievability scheme, we can safely condition on the hypothesis that users do not collide. The probability of error given the hypothesis that collision has occurred, which may be large, is then multiplied by the probability of collision and hence is vanishing as the blocklength goes to infinity, regardless of the achievable scheme. The probability of error for this two-stage decoder can be decomposed as P rErrors " P rSynchronization errors (26) P rDecoding error|No synchronization errors . Codebook generation: Let K n " e nν be the number of users, A n " e nα be the number of blocks, and M n " e nR be the number of messages. Each user i P rK n s generates a constant composition codebook with composition P by drawing each message's codeword uniformly and independently at random from the P -type set T pP q (recall definition in (19)). The codeword of user i P rK n s for message m P rM n s is denoted as x n i pmq. Probability of error analysis: A two-stage decoder is used, to first synchronize and then decode (which also identifies the users' identities) the users' messages. We now introduce the two stages and bound the probability of error for each stage. Synchronization step. We perform a sequential likelihood test as follows. Fix a threshold T P r´DpQ ‹ Q|P q, DpQ Q ‹ |P qs.(28) For each block j P rA n s if there exists any message m P rM n s for any user i P rK n s such that Lpx n i pmq, y n j q :" 1 n log Qpy n j |x n i pmqq Q ‹ py n j q ě T,(29) then declare that block j is an 'active' block, and an 'idle' block otherwise. Let H p1q :" H pp1,1q,p2,1q,...,pKn,1qq ,(30) be the hypothesis that user i P rK n s is active in block i and sends message m i " 1. The average probability of synchronization error, averaged over the different hypotheses, is upper bounded by P rSynchronization errors " P " Synchronization error|H p1q ‰ (31) ď Kn ÿ j"1 P « Kn č i"1 Mn č m"1 Lpx n i pmq, Y n j q ă T |H p1q ff`A n ÿ j"Kn`1 P « Kn ď i"1 Mn ď m"1 Lpx n i pmq, Y n j q ě T |H p1q ff (32) ď Kn ÿ j"1 P " Lpx n j p1q, Y n j q ă T |H p1q ‰`A n ÿ j"Kn`1 Kn ÿ i"1 Mn ÿ m"1 P " Lpx n i pmq, Y n j q ě T |H p1q ‰ (33) ď e nν e´n DpQ λ Q|P q`enpα`ν`Rq e´n DpQ λ Q‹|P q ,(34) where (31) Decoding stage. In this stage, by conditioning on no synchronization error, we have a superblock of length nK n , for which we have to distinguish between K n !pM n q Kn . " e nKnpR`νq different messages. We note that all the codewords for this superblock also have constant composition P (since they are formed by the concatenation of constant composition codewords). We can hence use a Maximum Likelihood (ML) decoder for random constant composition codes, introduced and analyzed in [14], on the super-block of length nK n to distinguish among e nKnpR`νq different messages with vanishing probability of error if R`ν ă IpP, Qq. This retrieves the last bound in (24). B. Users with Different Choice of Channels We now move on to a more general case in which we remove the restriction that the channels of all users are the same. Theorem 3 finds an achievable region when we allow the users' channels to be chosen from a set of conditional distributions of polynomial size in the blocklength. Theorem 3. For a SAS-MAC with asynchronous exponent α, occupancy exponent ν and rate R, assume that Q i py|xq " W cpiq py|xq is the channel for user i P rK n s, for some cpiq P rS n s where S n " polypnq. Then, the following region is achievable ď ně1 ď P j PP X λ j Pr0,1s č jPrSns $ ' ' ' ' ' & ' ' ' ' ' % ν j ă α 2 ν j ă DprP j W j s λ j rP j W j sq α ă DprP j W j s λ j Q ‹ q R`ν j ă IpP j , W j q , / / / / / . / / / / / - ,(35) where ν j :" 1 n logpN j q,(36)N j :" Kn ÿ i"1 1 tQ i "W j u : Sn ÿ j"1 N j " K n .(37) Proof. Before starting the proof, we should note that with similar arguments as the ones in Theorem 2, by imposing the first bound in (35), different users transmit in distinct blocks with a probability which goes to one as blocklength goes to infinity; thus we can assume no user collision in the following. We now propose a three-stage achievability scheme. The three stages perform the task of synchronization, identification and decoding, respectively. The joint probability of error for this three-stage achievable scheme can be decomposed as Codebook generation: Let K n " e nν be the number of users, A n " e nα be the number of blocks, M n " e nR be the number of messages, and S n " polypnq be the number of channels. Each user i P rK n s generates a random i.i.d codebook according to distribution P cpiq where the index cpiq P rS n s is chosen based on the channel Q i " W cpiq . For each user i P rK n s, the codeword for each message m P rM n s is denoted as x n i pmq. Probability of error analysis: A three-stage decoder is used. We now introduce the three stages and bound the probability of error for each stage. Synchronization step. We perform a sequential likelihood ratio test for synchronization as follows. Recall Q i p¨|¨q " W cpiq p¨|¨q for all user i P rK n s. Fix thresholds T cpiq P "´D`Q ‹ rP cpiq W cpiq s˘, D`rP cpiq W cpiq s Q ‹˘‰ , i P rK n s.(41) For each block j P rA n s if there exists any user i P rK n s such that L i py n j q :" 1 n log rP cpiq W cpiq spy n j q Q ‹ py n j q ě T cpiq ,(42) where a dot, as in p.q, is used instead of specifying the messages to emphasize that the decoder finds the location of the users, irrespective of their transmitted messages. The average probability of synchronization error, averaged over the different hypotheses, is upper bounded by P rSynchronization errors " P " Synchronization error|H p2q ‰ (44) ď P « Kn ď i"1 L i pY n i q ă T cpiq |H p2q ff`P « An ď j"Kn`1 Kn ď i"1 L i pY n j q ě T cpiq |H p2q ff (45) ď Kn ÿ i"1 P " L i pY n i q ă T cpiq |H p2q ‰`p A n´Kn qP « Kn ď i"1 L i pY n q ě T cpiq |H p2q ff (46) ď Kn ÿ i"1 e´n DprP cpiq W cpiq s λ i rP cpiq W cpiq sq`enα Sn ÿ j"1 e´n DprP j W j s λ j Q‹q (47) " Sn ÿ j"1 N j e´n DprP j W j s λ j rP j W j sq`enα e´n DprP j W j s λ j Q‹q ,(48) where rP j Q j s λ pyq :" prP j Q j spyqq λ pQ ‹ pyqq 1´λ ř y 1 PY prP j Q j spy 1 qq λ pQ ‹ py 1 qq 1´λ . The probability of error in this stage will decay to zero if for all j P rS n s ν j :" 1 n logpN j q ă D`rP j W j s λ j rP j W j s˘, α ă D`rP j W j s λ j Q ‹˘.(50) This retrieves the second and third bounds in (35). Identification step. Having found the location of the 'active' blocks, we move on to the second stage of the achievability scheme to identify which user is active in which block. We note that, by the random codebook generation and the memoryless property of the channel, the output of the block occupied by user i P rK n s is i.i.d distributed according to the marginal distribution rP cpiq Q cpiq spyq :" ÿ xPX P cpiq pxqQ cpiq py|xq. We leverage this property and customize the result in Theorem 1 to identify the different distributions of the different users. Note that at this point, we only distinguish the users with different channels from one another. In Theorem 1, it was assumed that all the distributions are distinct, while in here, our distributions are not necessarily distinct. The only modification that is needed in order to use the result of Theorem 1 is as follows. We need to consider a graph in which the edge between every two similar distributions have edge weights equal to zero (as opposed to e BpP,P q " e 0 " 1). By doing so, we can easily conclude that the probability of identification error in our problem using an ML decoder goes to zero as blocklength n goes to infinity since PrIdentification error|No synchronization errors ď ÿ 1ďiăjďSn e´2 nBprP i Q i s,rP j Q j sq Ñ 0, and since S n " polypnq by assumption. Decoding stage. After finding the permutation of users in the active blocks, we can go ahead with the third stage of the achievable scheme to find the transmitted messages of the users. In this stage, we can group the users who have similar channel Q j to get superblocks of length nN j , j P rS n s. For each superblock, we have to distinguish pM n q N j pN j q N j « e nN j pR`ν j q different message permutations. By using a typicality decoder, we conclude that the probability of decoding error for each superblock will go exponentially fast in blocklength to zero if R`ν j ă IpP j , W j q, @j P rS n s. This retrieves the last bound in (35) and concludes the proof. C. Users with no restriction on their channels Now we investigate a SAS-MAC with no restriction on the channels of the users. The key ingredient in our analysis is a novel way to bound the probability of error reminiscent of Gallager's error exponent. We show an achievability scheme that allows a positive lower bound on the rates and on ν. This proves that reliable transmission with an exponential number of users in an exponential asynchronous exponent is possible. We use an ML decoder sequentially in each block to identify the active user and its message. In our results, we use the following notation. The Chernoff distance between two distributions is defined as CpP, Qq :" sup 0ďtď1´l og˜ÿ x P pxq t Qpxq 1´t¸.(55) We extend this definition and introduce the quantity CpP i , Q i , P j , Q j q :" sup 0ďtď1 µ i,j ptq,(56) where µ i,j ptq :"´log ÿ x i ,x j ,y P i px i qP j px j qQ i py|x i q 1´t Q j py|x j q t(57) is a concave function of t. We also define Cp., Q ‹ , P j , Q j q :" sup 0ďtď1´l og˜ÿ x j ,y P j px j qQ ‹ pyq 1´t Q j py|x j q tţ o address the special case with i " 0 and hence all users are idle. Theorem 4. For a SAS-MAC with asynchronous exponent α, occupancy exponent ν and rate R, the following region is achievable ď ně1 ď P i PP X č iPrKns $ ' ' ' ' ' & ' ' ' ' ' % ν ă α 2 ν`R ă BpP i , Q i q, 2ν`R ă inf j‰i CpP j , Q j , P i , Q i q, α`ν`R ă Cp . , Q ‹ , P i , Q i q, , / / / / / . / / / / / -(58) Proof: Codebook generation: Each user i P rK n s generates an i.i.d. random codebook according to the distribution P i . Probability of error analysis: The receiver uses the following block by block decoder: for each block s P rA n s, the decoder outputs log Q i pY n i | x n i pmqq Q i pY n i | x n i p1qq ą 0|H p1q  ÿ iPrKns ÿ jPr0:Kns j‰i ÿ mPrMns P " log Q j pY n i |x n j pmqq Q i pY n i |x n i p1qq ą 0|H p1q  ÿ sPrKn`1:Ans ÿ jPrKns ÿ mPrMns P " log Q j pY n s |x n j pmqq Q ‹ pY n s q ą 0|H p1q  ď ÿ iPrKns e nR e´n sup t´l og E «ˆQ i pY i |X i q Q i pY i |X i q˙t f ÿ iPrKns ÿ jPr0:Kns j‰i e nR e´n sup t´l og E «ˆQ j pY i |X j q Q i pY i |X i q˙t f e nα ÿ jPrKns e nR e´n sup t´l og E «ˆQ j pYs|X j q Q‹pYsq˙t ff , where P X,X px, x 1 q " P X pxqP X px 1 q. The last inequality is due to the Chernoff bound. In order for each term in the probability of error upper bound to vanish as n grows to infinity, we find the conditions stated in the theorem. Remark 3. Note that (see Appendix D): BpP, Qq :" CpP, Q, P, Qq "´log ÿ x,x 1 ,y P pxqP px 1 q a Qpy|xqQpy|x 1 q,(59a) Cp . , Q ‹ , P j , Q j q ď IpP j , Q j q`D prP j Q j s Q ‹ q , (59b) CpP i , Q i , P j , Q j q ď IpP j , Q j q`D pP i rP j Q j s P i Q i q ,(59c) where, due to symmetry, in CpP, Q, P, Qq the supremum is achieved at the midpoint t " 1 2 , and hence BpP, Qq " CpP, Q, P, Qq " µp 1 2 q. The bounds in (59) show that the achievable rates in Theorem 4 are less than the corresponding point-to-point bounds. Example 1. Consider the SAS-MAC with asynchronous exponent α, occupancy exponent ν, and rate R with input-output relationship Y " ř iPrKns X i 'Z with Z " Berpδq for some δ P p0, 1{2q. In our notation Qpy|xq " PrX i ' Z " y|X i " xs " PrZ " x ' ys (60) " $ ' & ' % 1´δ x ' y " 0 pi.e., x " yq δ x ' y " 1 pi.e., x " yq .(61) Assume that the input distribution used is P " Berppq for some p P p0, 1{2q. The achievability region of this example, based on Theorem 2, includes the following region ď pPr0, 1 2 s λPr0,1s $ ' ' ' ' ' & ' ' ' ' ' % ν ă α{2 ν ă p¨dp λ δq α`R`ν ă p¨dp λ 1´δq R`ν ă hpp˚δq´hppq , / / / / / . / / / / / - ,(62) where dpp qq :" p log p q`p 1´pq log 1´p 1´q , λ :" δ λ p1´δq p1´λq δ λ p1´δq p1´λq`p 1´δq λ δ p1´δq , p˚q :" pp1´qq`p1´pqq. Moreover, by assuming P i " Berpp i q for all i P rK n s, we can show that the optimal t in CpP i , Q i , P j , Q j q " sup t µ i,j ptq is equal to t " 1{2 and hence the achievability region for this channel based on Theorem 4 is given by ď ně1 ď P i PP X č iPrKns $ ' ' ' ' ' & ' ' ' ' ' % ν ă α 2 ν`R ă BpP i , Qq " gpp i˚pi , δq, 2ν`R ă inf j‰i CpP j , Q, P i , Qq " inf i‰j gpp i˚pj , δq, α`ν`R ă Cp . , Q ‹ , P i , Qq " gpp i , δq, , , / / / / / . / / / / / - where gpa, bq :"´log´1´a`2a a bp1´bq¯. (a) Achievable region in (62). Finally, by symmetry, we can see that the optimal p i " 1 2 , @i P rK n s and hence gp 1 2 , δq " log´1{2`aδp1´δq¯ą 0. So on the BSC(δ) strictly positive rates and ν are achievable. In this regrard, the region in Theorem 4 reduces to α`ν`R ă´log´1{2`aδp1´δq¯. (63) The achievable region in (62) for pα, ν, Rq is shown in Fig. 2(a). In addition, the achievable region for pα, ν, Rq with the achievable scheme in [9] is also plotted in Fig. 2(b) for comparison. Fig. 1 shows that the achievable scheme in Theorem 2 indeed results in a larger achievable region than the one in Theorem 4. Note that the fact that the achievability region for Theorem 2 is larger than the achievability region of Theorem 4 for identical channels is not surprising. In Theorem 2 we separated the synchronization and decoding steps, whereas in Theorem 4 synchronization and codeword decoding was done the same time, sequentially for each block. The sequential block decoding step result in smaller achievability region in Theorem 4. D. Converse on the capacity region of SAS-MAC Thus far, we have provided achievable regions for the SAS-MAC for the cases that different users have identical channels; the case that their channels belong to a set of size that grows polynomially in the blocklength, and the case without any restriction on the users' channels. Theorem 5 next provides a converse to the capacity region of SAS-MAC that holds for any choice of the users' channels. Theorem 5. For the SAS-MAC with asynchronous exponent α, occupancy exponent ν and rate R, such that ν ă α{2, the following region is impermissible ď ně1 ď iPrKns P i PP X λ i Pr0,1s $ & % $ & % ν ą 1 Kn ř Kn i"1 DpQ i λ i Q i |P i q, α ą 1 Kn ř Kn i"1 DpQ i λ i Q ‹ |P i q´p1´r n qpν`Rq , . - ď tR ą IpP i , Q i qu , . - ,(64) wherer n is the infimum probability of error, over all estimators T , in distinguishing different hypothesis Q i λ i py n |x n i pmqq, i P rK n s, m P rM n s, i.e., r n :" inf T 1 K n M n Kn ÿ i"1 Mn ÿ m"1 Q i λ i pT ‰ i, m|x n i pmqqq.(65) Proof. We first define the following special shorthand notations that we will use throughout this proof F n :" M n K n , Q n i,m py n q :" Q i py n |x n i pmqq, Q n i,m λ i py n q,`Ě P Y n˘t py n q :"`Ě P Y n py n q˘t pQ n ‹ py n qq 1´t ř y n`Ě P Y n py n q˘t pQ n ‹ py n qq 1´t , Q n ‹ py n q :" n ź i"1 Q ‹ py i q.(69) We use the optimal Maximum Likelihood (ML) decoder to find the location of the 'active' blocks. In this stage, we are not concerned about the identity or the message of the users. In this regard, the decoder output is determined via arg max pl 1 ,...,l Kn q l i ‰l j ,@i‰j l i PrAns,iPrKns Kn ÿ i"1 log Ě P Y n pY n l i q Q n ‹ pY n l i q .(70) Given the hypothesis that the users are active in blocks rK n s, denoted by H p2q in (43), the corresponding error events to the ML decoder are given by error|H p2q ( " ď pl 1 ,...,l Kn q ‰p1,...,Knq # Kn ÿ i"1 log Ě P Y n pY n l i q Q n ‹ pY n l i q ą Kn ÿ i"1 log Ě P Y n pY n i q Q n ‹ pY n i q + Ě ď iPrKns jPrKn`1:Ans # log Ě P Y n pY n j q Q n ‹ pY n j q ě log Ě P Y n pY n i q Q n ‹ pY n i q + Ě $ & % ď jPrKn`1:Ans log Ě P Y n pY n j q Q n ‹ pY n j q ě T , . - č $ & % ď iPrKns T ě log Ě P Y n pY n i q Q n ‹ pY n i q , . - ,(71) for any T P R. We restrict ourselves to a subset of T 's and we take T to be T :" 1 F n Kn ÿ i"1 Mn ÿ m"1´D´Q n i,m λ i Q n ‹¯´D´Q n i,m λ i Ě P Y n¯¯,(72) for different λ i P r0, 1s, i P rK n s. We also find the following lower bounds, which are proved in Appendix E, Q n ‹ " log Ě P Y n Q n ‹ pY n q ě T  ě e´n Kn p ř Kn i"1 DpQ iλ i Q‹|P i q´pR`νqp1´s rnq`h ps rnq n q ,(73)Ě P Y n " log Ě P Y n Q n ‹ pY n q ď T  ě e´n Kn ř Kn i"1 DpQ iλ i Q i |P i q .(74) By using the inequalities in (73) and (74), we find a lower bound on the probability of (71) as follows: P » - ď jPrKn`1:Ans log Ě P Y n pY n j q Q n ‹ pY n j q ě T X ď iPrKns T ě log Ě P Y n pY n i q Q n ‹ pY n i q |H p2q fi fl (75) " P » - ď jPrKn`1:Ans log Ě P Y n pY n j q Q n ‹ pY n j q ě T |H p2q fi fl P » - ď iPrKns T ě log Ě P Y n pY n i q Q n ‹ pY n i q |H p2q fi fl (76) ": P rZ 1 ě 1s PrZ 2 ě 1s (77) ěˆ1´V arrZ 1 s E 2 rZ 1 s˙ˆ1´V arrZ 2 s E 2 rZ 2 s˙( 78) ě˜1´1 ř An j"Kn`1 Prξ j " 1s¸˜1´1 ř Kn i"1 Prζ i " 1s¸( 79) ě´1´e´n α`n Kn p ř Kn i"1 DpQ iλ i Q‹|P i q´pR`νqp1´s rnq`h ps rnq n q¯´1´e´nν`n Kn ř Kn i"1 DpQ iλ i Q i |P i q¯,(80) where (76) follows by independence of Y n i and Y n j whenever i ‰ j, @i, j P rA n s and the inequality in (78) is by Chebyshev's inequality, where we have defined Z 1 :" An ÿ j"Kn`1 ξ j , ξ j :" BerˆQ n ‹ " log Ě P Y n Q n ‹ pY n j q ě T ˙,(81)Z 2 :" Kn ÿ i"1 ζ i , ζ i :" BerˆĚ P Y n " log Ě P Y n Q n ‹ pY n i q ď T ˙,(82) where tξ j , ζ i u are mutually independent. We can see from (80) that if ν ą lim nÑ8 # 1 K n Kn ÿ i"1 DpQ i λ i Q i |P i q + ,(83)α ą lim nÑ8 # 1 K n Kn ÿ i"1 DpQ i λ i Q ‹ |P i q´p1´r n qpν`Rq + ,(84) then the probability of error is strictly bounded away from zero and hence it is impermissible. Moreover, the usual converse bound on the rate of a synchronous channel also applies to any asynchronous channel and hence the region where R ą IpP, Qq is also impermissible. This concludes the proof. It should be noted that even though the expression (64) involves a union over all blocklengths n, in order to compute this bound, we only need to optimize with respect to P i , i P rK n s (as opposed to P n in the conventional n-letter capacity expressions). However, since we still have exponential (in blocklength n) number of users, and in theory we have to optimize all of their distributions, we need to take the union with respect to all blocklengths. E. Converse on the number of users in a SAS-MAC In previous sections and in our achievability schemes, we restricted ourselves to the region where ν ă α 2 to be able to simplify the analysis. However, an interesting question is that irrespective of the achievability scheme and the decoder, how large a ν we can have in the network. Theorem 6 provides a converse bound on the value of ν such that for ν ą α, not even reliable synchronization is not possible. Theorem 6. For a SAS-MAC with asynchornous exponent α and occupancy exponent ν : ν ą α, reliable synchronization is not possible, i.e., even with M " 1, one has P pnq e ą 0. Fig. 3: Extended codebook. 1 2 A n M n x n i (M n ) x n i (M n ) x n i (M n ) x n i (1) x n i (1) ⋆ n i 1 Proof: User i P rK n s has a codebook with M n " e nR codewords of length n. Define for i P rK n s an 'extended codebook' consisting of A n M n codewords of length nA n constructed such that @m P rM n s and @t i P rA n s r X nAn i pm i , t i q :" " ‹ npt i´1 q i f i pm i q ‹ npAn´t i q i ‰ , as depicted in Fig. 3. By using Fanno's inequality, i.e., HpX nAn 1 , . . . , X nAn Kn |Y nAn q ď n n : n Ñ 0 as n Ñ 8, for any codebook of length nA n we have , . . . , X nAn Kn ; Y nAn q ď n n`n e nα |Y| ðñ ν`l og´1`1 αKn ř iPrKns R in ď α`l og´1` n e nα |Y|n , where logp1`1 αKn ř iPrKns R iq n ě 0 and logp1` n e nα |Y| q n ě 0 vanish as n goes to infinity. This implies that ν ď α is a necessary condition for reliable communications. In other words, for ν ą α not even synchronization (i.e., M n " 1, @i P rK n s) is possible. V. DISCUSSION AND CONCLUSION In this paper we studied a Strongly Asynchronous and Slotted Massive Access Channel (SAS-MAC) where K n :" e nν different users transmit a randomly selected message among M n :" e nR ones within a strong asynchronous window of length A n :" e nα blocks of n channel uses each. We found inner and outer bounds on the pR, α, νq tuples. Our analysis is based on a global probability of error in which we required all users messages and identities to be jointly correctly decoded. Our results are focused on the region ν ă α 2 , where the probability of user collisions in vanishing. We proved in Theorem 6 that for the region ν ą α, not even synchronization is possible. Hence, we would like to take this chance to discuss some of the difficulties that one may face in analyzing the region α 2 ď ν ď α. As we have mentioned before, for the region ν ă α 2 , with probability p An Kn q pAnq Kn which approaches to one as blocklength n goes to infinity, the users transmit in distinct blocks. Hence, in analyzing the probability of error of our achievable schemes, we could safely condition on the hypothesis that users are not colliding. For the region α 2 ď ν ď α, we lose this simplifying assumption. In particular, based on Lemma 2 (proved in the Appendix F), for the region α 2 ď ν ď α, the probability of every arrangement of users is itself vanishing in the blocklength. Lemma 2. For the region α 2 ď ν ď α the non-colliding arrangement of users has the highest probability among all possible arrangements, yet, the probability of this event is also vanishing as blocklength n goes to infinity. As a consequence of Lemma 2, one needs to propose an achievable scheme that accounts for every possible arrangement and collision of users and drives the probability of error in all (or most) of the hypothesis to zero. It is also worth noting that the number of possible hypotheses is doubly exponential in the blocklength. Finally, it is worth emphasizing the reason why the authors in [15] can get to ν ď α. In [15] the authors require the recovery of the messages of a large fraction of users and they also require the per-user probability of error to be vanishing. To prove whether or not strictly positive pR, α, νq are possible in the region α 2 ď ν ď α, with vanishing global probability of error, is an open problem. APPENDIX A. Proof of Lemma 1 We first consider the case that r is an even number and then prove rpn k q r 2´1`G pc 1 q`. . . Gpc N r,k q˘ď N r,k r n k`a 2 1`. . .`a n k 2˘r 2 . July 27, 2018 DRAFT We may drop the subscripts and use N :" N r,k and n :" n k in the following for notational ease. Our goal is to expand the right hand side (RHS) of (85) such that all elements have coefficient 1. Then, we parse these elements into N different groups (details will be provided later) such that using the AM-GM inequality (i.e., n p ś n i"1 a i q 1 n ď ř n i"1 a i ) on each group, we get one of the N terms on the LHS of (85). Before stating the rigorous proof, we provide an example of this strategy for the graph with k " 4 vertices shown in Fig. 4. In this example, we consider the Lemma for r " 4 cycles (for which N " 3). ( . It can be easily seen that if we use the AM-GM inequality on Θ 1 , Θ 2 and Θ 3 , we can get the lower bound equal to 24pa 1 a 2 a 3 a 4 q, 24pa 1 a 6 a 3 a 5 q and 24pa 4 a 5 a 2 a 6 q, respectively where rn r 2´1 " 24 and hence (85) holds in this example. We proceed to prove Lemma 1 for arbitrary k and (even) r ě 2. We propose the following scheme to group the elements on the RHS of (85) and then we prove that this grouping indeed leads to the claimed inequality in the Lemma. Grouping scheme: For each cycle c i " ta i 1 . . . , a ir u, we need a group of elements, Θ i , from the RHS of (85). In this regard, we consider all possible subsets of the edges of cycle c i with 1 : r 2 elements (e.g. ta i 1 u, . . . ta i 1 , a i 2 u, . . . ta i 1 . . . , a i r{2 u, . . . ( ). For each one of these subsets, we find the respective elements from the RHS of (85) that is the multiplication of the elements in that subset. For example, for the subset ta i 1 , a i 2 , a i 3 u, we consider the elements like a n i 1 i 1 a n i 2 i 2 a n i 3 i 3 for all possible n i 1 , n i 2 , n i 3 ą 0 from the RHS of (85). However, note that we do not assign all such elements to cycle c i only. If there are l cycles of length r that all contain ta i 1 , a i 2 , a i 3 u, we should assign 1 l of the elements like a n i 1 i 1 a n i 2 i 2 a n i 3 i 3 , n i 1 , n i 2 , n i 3 ą 0 to cycle c i (so that we can assign the same amount of elements to other cycles with similar edges). We state some facts, which can be easily verified: Fact 1. In a complete graph K k , there are N " N r,k "`k r˘p r´1q! 2 cycles of length r. where ś r t"1 a n i t it is the product of the elements in Θ i . By symmetry of the grouping scheme for different cycles, it is obvious that @t P rrs, n it " n jt . Hence n it " n jt " p t , @i, j P rN s. i.e., we By symmetry of the grouping scheme over the elements of each cycle, we also get that n i k " n i l " q i , @k, l P rrs. i.e. Θ i ě rn r 2´1`a q i i 1ˆ. . .ˆa q i ir˘ˆ1 rn r 2´1˙.(87) It can be seen from (86) and (87) that all the elements of all groups have the same power n it " p, @i P rN s, t P rrs. i.e., Θ i ě rn r 2´1`a p i 1ˆ. . .ˆa p ir˘ˆ1 rn r 2´1˙. Since each element on the RHS of (85) is assigned to one and only one group and since ś r t"1 a n i t it " ś r t"1 a p it is the product of the elements of each group Θ i , the product of all elements in Θ 1`. . .`Θ N (which is equal to product of the elements in the expanded version of the RHS of (85)) is ś N i"1 ś r t"1 a p it . In addition, since each a i appears in exactly N r n of the cycles, by Fact 3 and a double counting argument, we have For odd values of r, the problem that may arise by using the grouping strategy in its current form, is when r ă k 2 . In this case, some of the terms on the RHS of (85) may contain multiplication of a i 's that are not present in any of the Gpc i q's. To overcome this, take both sides to the power of 2m for the smallest m such that rm ą k 2 . Then the RHS of (85) is at most the multiplication of rm different a i 's and on the LHS of (85), there are 2m cycles of length r multiplied together. By our choice of 2m, now, all possible combinations of a i 's on the RHS are present in at least one cycle multiplication in the LHS. Hence, it is straightforward to use the same strategy as even values of r to prove the theorem for the odd values of r. 4 r˜ÿ 1ďiăjďAn e´2 nBpP i ,P j q¸r {2 (88) ď 16´ř 1ďiăjďAn e´2 nBpP i ,P j q1´4 b ř 1ďiăjďAn e´2 nBpP i ,P j q ,(89) where (88) As the result, (89) will go to zero as n goes to infinity if lim nÑ8 ÿ 1ďiăjďAn e´2 nBpP i ,P j q " 0. C. Proof of (12) We upper bound the denominator of (10) by Prξ i,j , ξ i,k s " P " log P i pX n j q P j pX n j q`l og P j pX n i q P i pX n i q ě 0 X log P i pX n k q P k pX n k q`l og P k pX n i q P i pX n i q ě 0  ď P " log P i pX n j q P j pX n j q`l og P j pX n i q P i pX n i q`l og P i pX n k q P k pX n k q`l og P k pX n i q P i pX n i q ě 0  ď exp # n inf t log˜E «ˆP i pX n j q P j pX n j q¨P j pX n i q P i pX n i q¨P i pX n k q P k pX n k q¨P k pX n i q P i pX n i q˙t ff¸+ ď exp # n log E «ˆP i pX n j q P j pX n j q¨P j pX n i q P i pX n i q¨P i pX n k q P k pX n k q¨P k pX n i q P i pX n i q˙1 2 ff+ " exp t´nBpP i , P j q´nBpP j , P k q´nBpP i , P k qu .(90) An upper bound for P rξ i,j , ξ k,l s can be derived similarly. D. Proof of (59b) We find an upper bound on Cp . , Q ‹ , P i , Q i q by noting that µ 0,i ptq defined in (57) is concave in t with µ 0,i p1q " 0 and Bµ 0,i ptq Bt | t"1 "´IpP i , Q i q´DprP i Q i s Q ‹ q ď 0. Hence µ 0,i ptq is always less than pIpP i , Q i q`DprP i Q i s Q ‹ qqp1´tq and that for 0 ď t ď 1 it is always less than IpP i , Q i q`DprP i Q i s Q ‹ q. The inequality in (59c) follows similarly. E. Proof of (73) and (74) Before deriving lower bounds on (73) and (74), we note that by the Type-counting Lemma [16], at the expense of a small decrease in the rate (which vanishes in the limit for large blocklength) we may restrict our attention to constant composition codewords. Henceforth, we assume that the composition of the codewords for user i P rK n s is given by P i . Moreover, to make this paper self-contained, we restate the following Lemmas that we use in the rest of the proof. Lemma 3 (Compensation Identity). For arbitrary π i : ř K i"1 π i " 1 and arbitrary probability distribution functions P i P P X , i P rKs, we defineP pxq " ř K i"1 π i P i pxq. Then for any probability distribution function R we have: D`P R˘`K ÿ i"1 π i D`P i P˘" wherer :" inf T 1 N ÿ θPF P θ tT ‰ θu(93) in which the infimum is taken over all possible estimators T . We now continue with the proof of (73). Using the Chernoff bound we can write " e´s up t Aptq . The Chernoff bound exponent, sup t Aptq, is expressed and simplified as follows `Ě P Y n˘t" 1 K n Kn ÿ i"1 nD`Q i λ i Q ‹ |P i˘´1 F n Kn ÿ i"1 Mn ÿ m"1 D´Q n i,m λ i `Ě P Y n˘t¯,(95) where (95) is the result of constant composition structure of the codewords. As a result, sup t Aptq " 1 K n Kn ÿ i"1 nD`Q i λ i Q ‹ |P i˘´i nf t # 1 F n Fn ÿ i"1 D´Q n i,m λ i `Ě P Y n˘t¯+ .(96) Moreover, Ě P pλq Y n¯ě p1´s r n q log pF n p1´s r n qq`s r n logˆF n s r n F n´1˙( 102) ě p1´s r n q log F n´h pr n q, where hp.q is the binary entropy function. As a result sup t Aptq ď 1 K n Kn ÿ i"1 nDpQ i λ i Q ‹ |P i q´npR`νqp1´s r n q`hpr n q. Now we continue with the proof of (74). Again, using the Chernoff bound we have Ě P Y n « log P Y n Q n ‹ pY n q ď 1 F n Kn ÿ i"1 Mn ÿ m"1 D´Q n i,m λ i Q n ‹¯´D´Q n i,m λ i Ě P Y n¯ff (105) " Ě P Y n « log Q n ‹ P Y n pY n q ě 1 F n Kn ÿ i"1 Mn ÿ m"1 D´Q n i,m λ i Ě P Y n¯´D´Q n i,m λ i Q n ‹¯ff . " e´s up t Bptq , py n q log Q n ‹ py n q Ě P Y n py n q Ě P Y n py n q´l og ÿ y n pQ n ‹ py n qq t`Ě P Y n py n q˘1´t " sup t ÿ y n Ě P pλq Y n py n q log`Ě P Y n˘1´t py n q Ě P Y n py n q " sup t D´Ě P pλq Y n Ě P Y n¯´D´Ě P pλq Y n `Ě P Y n˘1´tď D´Ě P pλq Y n Ě P Y n¯" and where the inequality in (107) is by Log-Sum inequality. F. Proof of Lemma 2 We will prove the Lemma by contradiction. Define t i fi Number of users in block i. Assume that the arrangement with highest probability (let us call it A) has at least two blocks, say blocks 1, 2, for which t 1´t2 ą 1. This assumption means that the arrangement with the highest probability is not the non-overlapping arrangement. Fig. 1 : 1Complete graph K An with edge weight e´n BpP i ,P j q for every pair of vertices i ‰ j P rK n s.Hence, we can upper bound the probability of error in (BpP c pvq piq ,P c pvq pti`1ur q q BpP c pvq piq ,P c pvq pti`1ur q q is the gain is by the symmetry of different hypothesis and (34) can be derived as in [13, Chapter 11]. The upper bound on the probability of error for the synchronization error in (34) vanishes as n goes to infinity if the second and third bound in (24) hold. P rErrors " P rSynchronization errors (38) P rIdentification error|No synchronization errors (39) P rDecoding error|No synchronization and No identification errors . Remark 2 . 2The achievability proof of Theorem 2 relies on constant composition codes whereas the achievability proof of Theorem 3 relies on i.i.d. codebooks. The reason for these restrictions is that in 3 we also need to distinguish different users and in order to use the result of[11], we focused our attention on i.i.d. codebooks.July 27, 2018 DRAFT We now find an upper bound on probability of error given the hypothesis H p1q in ( Fig. 2 : 2Comparison of the achievable region in Theorem 2 and Theorem (4), for the Binary Symmetric Channel with cross over probability δ " .11. m py n q˘λ i pQ n ‹ py n qq 1´λ i ř y n`Q n i,m py n q˘λ i pQ n ‹ py n qq 1´λ i HpX nAn 1 , 1. . . , X nAn Kn q " Hpm 1 , t 1 , . . . , m Kn , t Kn q " nK n pα`Rq " HpX nAn 1 , . . . , X nAn Kn |Y nAn q`IpX nAn 1 Fig. 4: A complete graph with 4 vertices p " rn r 2´1 . Hence, the lower bound of the AM-GM inequality on the Θ 1`. . .`Θ N , will result in rn r 2´1 Gpc 1 q`. . .`rn r 2´1 Gpc Nr q, and the Lemma is proved for even r. DpP i ||Rq.(91)Lemma 4 (Fano). Let F be an arbitrary set of size N . ForP " P˘ě p1´rq log pN p1´rqq`r logˆNr N´1˙, P Y n py n q˘t pQ n ‹ py n qq 1´t m, i's (m P rM n s, i P rK n s) Q i λ i Q|P i˘, denote the fact that we have identified r distributions incorrectly. Consider the two sequences ri 1 , . . . , i r s and rσ i 1 , . . . ,σ ir s for which we have #@i P rA n s ) and where (3) is due to the requirement that each sequence is distributed according to a distinct distribution and hence the number of incorrect distributions ranges from r2 : A n s. In order to avoid considering the same set of error events multiple times, July 27, 2018 DRAFT we incorporate a graph theoretic interpretation of ! ř An i"1 1 tΣ i ‰iu " r ) in (4) which is used to An ÿ i"1 then declare that block j is an 'active' block. Else, declare that block j is an 'idle' block. Notethat were able to calculate the probabilities of error corresponding to (29) by leveraging the constant composition construction of codewords in Theorem 2. In here, we can leverage the i.i.d. constructure of the codewords and calculate the probability of error corresponding to (42). We now find an upper bound on the average probability of error for this scheme over different hypotheses. Before doing so, we should note that by the symmetry of different hypotheses, the average probability of error over different hypothesis is equal to probability of error given the hypothesis that user i P rK n s transmits in block i; this hypothesis is denoted by H p2q :" H pp1,.q,p2,.q,...,pKn,.qq , Fact 2. By expanding the RHS of (85) such that all elements have coefficient 1, we end up with`N r n˘n r 2 elements.Fact 3. Expanding the RHS of (85) such that all elements have coefficient 1, and finding their product yields pa 1ˆ. . .ˆa n q p N r n qrn Fact 4. In above grouping scheme each element on the RHS of (85) is summed in exactly one group. Hence, by symmetry and Fact 2, each group is the sum of rn 2´1 elements. Now, consider any two cycles c peq i " ta i 1 , . . . , a ir u, c peq j" ta j 1 , . . . , a jr u. Assume that using the above grouping scheme, we get the group of elements Θ i , Θ j (where by fact 3 each one is the sum of rn 2´1 elements). If we apply the AM-GM inequality on each one of the two groups,r 2´1 . r r we get Θ i ě rn r 2´1`a n i 1 i 1ˆ. . .ˆa n 1r ir˘ˆ1 rn r 2´1˙, Θ j ě rn r 2´1`a n j 1 j 1ˆ. . .ˆa n jr jr˘ˆ1 rn r 2´1˙, July 27, 2018DRAFT The probability of this arrangement, PpAq, is proportional toWe now consider a new arrangement, A new , in which t 1,new " t 1´1 and t 2,new " t 2`1 and all other blocks remain unchanged. This new arrangement is also feasible since we have not changed the number of users. The probability of this new arrangement is proportional toComparing PpAq and PpA new q we see that PpAq ă PpA new q which is a contradiction to our primary assumption that A has the highest probability among all arrangements. Hence there do not exist two blocks which differ more than one in the number of active users within them in the arrangement with the highest probability. Optimal sequential frame synchronization. V Chandar, A Tchamkerten, G Wornell, IEEE Transactions on Information Theory. 548V. Chandar, A. Tchamkerten, and G. Wornell, "Optimal sequential frame synchronization," IEEE Transactions on Information Theory, vol. 54, no. 8, pp. 3725-3728, Aug 2008. Asynchronous communication: Capacity bounds and suboptimality of training. A Tchamkerten, V Chandar, G W Wornell, IEEE Transactions on Information Theory. 593A. Tchamkerten, V. Chandar, and G. W. Wornell, "Asynchronous communication: Capacity bounds and suboptimality of training," IEEE Transactions on Information Theory, vol. 59, no. 3, pp. 1227-1255, March 2013. Energy and sampling constrained asynchronous communication. A Tchamkerten, V Chandar, G Caire, IEEE Transactions on Information Theory. 6012A. Tchamkerten, V. Chandar, and G. Caire, "Energy and sampling constrained asynchronous communication," IEEE Transactions on Information Theory, vol. 60, no. 12, pp. 7686-7697, Dec 2014. Capacity of gaussian many-access channels. X Chen, T.-Y. Chen, D Guo, IEEE Transactions on Information Theory. 636X. Chen, T.-Y. Chen, and D. Guo, "Capacity of gaussian many-access channels," IEEE Transactions on Information Theory, vol. 63, no. 6, pp. 3516-3539, 2017. A perspective on massive random-access. Y Polyanskiy, 2017 IEEE International Symposium on Information Theory (ISIT). Y. Polyanskiy, "A perspective on massive random-access," in 2017 IEEE International Symposium on Information Theory (ISIT), June 2017, pp. 2523-2527. Sequential identification and ranking procedures, with special reference to Koopman-Darmois populations. J Kiefer, M Sobel, University of Chicago PressJ. Kiefer and M. Sobel, Sequential identification and ranking procedures, with special reference to Koopman-Darmois populations. University of Chicago Press, 1968. On logarithmically asymptotically optimal testing of hypotheses and identification. R Ahlswede, E Haroutunian, General Theory of Information Transfer and Combinatorics. SpringerR. Ahlswede and E. Haroutunian, "On logarithmically asymptotically optimal testing of hypotheses and identification," in General Theory of Information Transfer and Combinatorics. Springer, 2006, pp. 553-571. Asymptotically optimal matching of multiple sequences to source distributions and training sequences. J Unnikrishnan, IEEE Transactions on Information Theory. 611J. Unnikrishnan, "Asymptotically optimal matching of multiple sequences to source distributions and training sequences," IEEE Transactions on Information Theory, vol. 61, no. 1, pp. 452-468, 2015. On the capacity of strong asynchronous multiple access channels with a large number of users. S Shahi, D Tuninetti, N Devroye, IEEE International Symposium on Information Theory (ISIT). S. Shahi, D. Tuninetti, and N. Devroye, "On the capacity of strong asynchronous multiple access channels with a large number of users," in IEEE International Symposium on Information Theory (ISIT), July 2016, pp. 1486-1490. On the capacity of the slotted strongly asynchronous channel with a bursty user. 2017 IEEE Information Theory Workshop (ITW). --, "On the capacity of the slotted strongly asynchronous channel with a bursty user," in 2017 IEEE Information Theory Workshop (ITW), Nov 2017, pp. 91-95. On identifying a massive number of distributions. IEEE International Symposium on Information Theory (ISIT). --, "On identifying a massive number of distributions," in IEEE International Symposium on Information Theory (ISIT), June 2018. On the application of the borel-cantelli lemma. K L Chung, P Erdos, Transactions of the American Mathematical Society. 721K. L. Chung and P. Erdos, "On the application of the borel-cantelli lemma," Transactions of the American Mathematical Society, vol. 72, no. 1, pp. 179-186, 1952. 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[ "Self-interference of a toroidal Bose-Einstein condensate Self-interference of a toroidal Bose-Einstein condensate 2", "Self-interference of a toroidal Bose-Einstein condensate Self-interference of a toroidal Bose-Einstein condensate 2" ]
[ "L A Toikka [email protected] \nTurku Centre for Quantum Physics\nDepartment of Physics and Astronomy\nUniversity of Turku\n20014TurkuFinland\n" ]
[ "Turku Centre for Quantum Physics\nDepartment of Physics and Astronomy\nUniversity of Turku\n20014TurkuFinland" ]
[]
We demonstrate the role played by a ring trap and the associated ring geometry in inducing self-interference of a toroidal Bose-Einstein condensate. We start by showing how the existence of the self-interference can be inferred from the Wigner function representation of the ring. Then, using analytical expressions for the time-evolution of a freely expanding ring condensate with and without a persistent current, we show that the self-interference of the ring condensate is possible only in the absence of the persistent current. We conclude by proposing an experimental protocol for the creation of ring dark solitons using the toroidal self-interference.
10.1088/1367-2630/16/4/043011
[ "https://arxiv.org/pdf/1309.5935v3.pdf" ]
118,570,177
1309.5935
adebb5de138ac7eb001d4a29d22e17b640577e4b
Self-interference of a toroidal Bose-Einstein condensate Self-interference of a toroidal Bose-Einstein condensate 2 12 Dec 2013 L A Toikka [email protected] Turku Centre for Quantum Physics Department of Physics and Astronomy University of Turku 20014TurkuFinland Self-interference of a toroidal Bose-Einstein condensate Self-interference of a toroidal Bose-Einstein condensate 2 12 Dec 2013PACS numbers: 0375Kk, 0375Lm arXiv:13095935v3 [cond-matquant-gas] We demonstrate the role played by a ring trap and the associated ring geometry in inducing self-interference of a toroidal Bose-Einstein condensate. We start by showing how the existence of the self-interference can be inferred from the Wigner function representation of the ring. Then, using analytical expressions for the time-evolution of a freely expanding ring condensate with and without a persistent current, we show that the self-interference of the ring condensate is possible only in the absence of the persistent current. We conclude by proposing an experimental protocol for the creation of ring dark solitons using the toroidal self-interference. Introduction Following the pioneering demonstration of interference between two freely expanding Bose-Einstein condensates (BECs) [1], also self-interference of a single condensate has been experimentally and numerically observed in hard-wall reflections [2,3]. Later generalisations to optical lattices [4] have led to interference between as many as 30 separate BECs [5,6]. In a self-interference setting, the wavefunction splits up into two (or more) pieces, which travel different paths; subsequently the self-interference arises when both parts are later spatially recombined. This process can happen next to a hard-wall potential, as mentioned, whereby the condensate interferes with its reflection. However, a natural setting to consider the self-interference of a BEC in the absence of any boundary effects is a non-simply connected geometry such as a torus. A BEC in a ring trap forms an extended quantum object that winds back to itself, and by controlling the size of the central forbidden region, one might expect the separate parts of the same condensate to show self-interference phenomena. As the phase itself is not an observable, the appearance of interference fringes reflects the relative long-range coherence between the separate condensates, or different parts of the same condensate. Even in the case of well-defined atom numbers and fully uncertain phases, interference is produced by the action of the quantum measurement, which assigns instantaneous relative phase correlations [7]. In the case of self-interference, this action is equivalent to the condensate encountering itself. The phase coherence of BECs also leads to many other important properties such as quantised vortices [8] and dark solitons [9]. In general, ring traps [10][11][12][13][14][15][16][17][18] have attracted significant interest in the form of e.g. persistent currents [19,20], atomic-phase interference devices [21], vortex dynamics [22,23], and ring dark solitons [24][25][26][27], but their role in inducing selfinterference is an open question. Also, given the high experimental relevance of ring traps, observation of toroidal self-interference offers one way to probe the spatial phase coherence over the extent of the condensate in the ring, and it can also serve as a mechanism for the creation of ring dark solitons, as we show in Section 3.3. In the work presented here, we investigate in detail how the self-interference of a Bose-Einstein condenstate can appear in a ring trap. We consider analytically the Wigner function representation of a Gaussian ring condensate, showing that if the radius of the ring trap is brought to zero, all the self-interference phenomena disappear, as expected. For non-zero radii, on the other hand, we demonstrate a possible experimental way to observe the self-interference by letting the ring expand freely and overlap with itself to produce circular fringes. Interestingly, we find that in the presence of a persistent current, there cannot be any self-interference because the central vortex is enforcing a density hole at r = 0, preventing the condensate from overlapping with itself. Insofar as the centrifugal potential barrier acts as a circular hard wall, we will not consider the separate case of self-interference arising from reflection of the condensate. Finally, based on the toroidal self-interference, we propose a protocol for the experimental creation of ring dark solitons. Theoretical background We consider a scalar order parameter ψ, representing the macroscopic wavefunction of a Bose-Einstein condensate trapped in a potential given by V trap , which is a solution to the Gross-Pitaevskii equation (GPE): iψ t = −∇ 2 ψ + V trap ψ + C 2D |ψ 2 |ψ. (1) Here we have assumed a two-dimensional condensate whereby the z-direction is tightly trapped to the corresponding harmonic oscillator ground state (ω z ω x = ω y ≡ ω r ) and has been projected onto the xy-plane. Then C 2D = 4 √ πN a/a (z) osc , where N , a, and a (z) osc are the number of atoms in the cloud, the s-wave scattering length of the atoms, and the characteristic trap length in the z-direction respectively. We have obtained dimensionless quantities by measuring time, length and energy in terms of ω −1 r , a osc ≡ a osc = /(2mω r ) and ω r respectively, where ω r is the angular frequency of the trap in the r-direction. This basis is equivalent to setting ω r = = 2m = 1. All the units in this work are expressed in this dimensionless basis. Using the experimental parameters of the toroidal BEC of [28] and 87 Rb, we obtain a osc = 0.8 µm, C 2D ∼ 200 − 2000, and one unit of time is ∼ 1.5 ms. We consider the toroidal condensate to be in the ground state of the harmonic ring trap given by V trap = 1 4 ω 2 (r − r 0 ) 2 ,(2) where we take r 0 = 4.5, and ω = 20. To a very good approximation, the ground state of the potential (2) with a weak nonlinear interaction of C 2D = 50 is given by a Gaussian, ϕ G (r) = N e − (r−r 0 ) 2 2σ 2 ,(3) where N is the normalisation, and σ = 2/ω. Self-interference in the torus Wigner function of the ring The effect of the ring trap in inducing self-interference can be seen in the Wigner function [29,30]. There exists a one-to-one mapping between the vector r | ψ = ψ(r) and the corresponding Wigner function W ψ (r, p) ∈ R, defined for our two-dimensional states by W ψ (r, p) = 1 (2π) 2 R 2 du ψ * r − u 2 ψ r + u 2 e −iu·p .(4) The Wigner function can be thought of as being another name or representation for the state |ψ , containing all the possible information of the state, with the important property that the marginals of the Wigner function recover the position and momentum distributions: R 2 dp W ψ (r, p) = | r | ψ | 2 ,(5a)R 2 dr W ψ (r, p) = | p | ψ | 2 .(5b) A necessary and sufficient condition for the Wigner function to be a true phase space density [31] states that this happens only for coherent and squeezed vacuum states (Gaussian states). However, the Wigner function representation is often quite suitable for presenting nonclassical states, and in our case, the negativity of the Wigner function is associated with the emergence of toroidal matter-wave interference. (6)] of a toroidal condensate of radius r 0 at the centre of the ring (r = 0). Self-interference fringes arising from the toroidal geometry appear for r 0 > 0. The light blue line shows the case corresponding to the left panel (within numerical accuracy). Red (blue) color corresponds to positive (negative) values. (4), and setting r = 0 as we are interested in the neighbourhood of the origin, we obtain Substituting Equation (3) into Equation W ϕ G (0, p) = N 2 ∞ 0 du upJ 0 (up)e − (u/2−r 0 ) 2 σ 2 ,(6) where J 0 is the Bessel function of order 0. We can already see that W ϕ G (0, p)| r0=0 = 2N 2 pσ 2 e −p 2 σ 2 , which is manifestly positive, i.e. there are no fringes. The interference effects due to the toroidal geometry vanish when the radius of the torus is taken to zero. To consider the case of r 0 > 0, the integral in Equation (6) can be evaluated analytically with a small correction that must be calculated numerically (see the Appendix). The interference effects induced by the toroidal geometry (of radius r 0 ) are shown in Fig. 1 in terms of the Wigner function. Considering the case of r 0 = 4.5 (left panel), the fringes in the middle of the torus show that the opposite parts of the ring are interfering with each other. As expected, for r 0 = 0 there can be no toroidal self-interference, and the Wigner function is manifestly positive (right panel). As r 0 is being increased from zero and therefore the ring gets a non-zero radius, we can see the appearance of interference fringes in p (at the origin of the ring r = 0), with the fringe separation decreasing as r 0 increases. The fringes of Fig. 1 do not appear in the position distribution as long as the trapping potential forbids occupation in the central region. Still, these results suggest that if we start with a toroidal condensate at some finite radius and let the ring expand towards the origin, we should observe circular fringes when the opposite and in general different parts of the ring overlap with each other. Self-interference of an expanding ring To gain preliminary understanding, we integrate the GPE (1) numerically in a timedependent Mexican hat potential after the central region of the hat has been suddenly removed and the toroidal condensate allowed to expand freely towards the centre. Direct numerical simulation of the GPE (1) with Vtrap given by Equation (7), and (a) C 2D = 50 and (b) C 2D = 400. The left panel shows the initial density distribution at t = 0, and the middle panel shows the same distribution at t = 1.5. The top and bottom insets show a slice along the xaxis through the cylindrically symmetric position distributions (solid line), and the initial ring is also shown (dashed line). The right panel shows the phase distributions corresponding to the densities of the middle panels. In both cases, the ring expands towards the centre, and once the opposite parts of the ring overlap, circular long-lived soliton-like fringes associated with phase steps appear in the position distribution. The central peak for the C 2D = 50 case is 5 times higher than for C 2D = 400, giving rise to a higher kinetic energy density. The colouring is the same in all figures. The ground state is first found by propagating the GPE in imaginary time, and the propagation is done in real time. We use a split-operator Fourier method [32]. The potential we choose has the following form: V trap (r, t) = 1 4 r 2 + 30H(−t)e −0.1r 2 ,(7) where H is the Heaviside step function. The simulation reveals that once the opposite parts of the ring start to overlap around the origin at t ≈ 1.5, circular fringes develop around a bright central spot (see Fig. 2). This is the self-interference of the ring induced by the toroidal geometry. We can also study more analytically the time-evolution of the wavefunction after the ring is allowed to expand inwards. We neglect the (weak) nonlinear interaction and also assume that the external potential vanishes after the toroidal trapping is released so that the time evolution is given by the free-particle linear Schrödinger equation. To get the time evolution, we multiply the zeroth order Fourier-Bessel transform ϕ G (k) of ϕ G (r) by e ik 2 t in k-space, and take the inverse transformation, viz.: ϕ G (r, t) = ∞ 0 dk kJ 0 (kr)φ G (k)e ik 2 t(8)= N ∞ 0 dk dr kr J 0 (kr)J 0 (kr )e − (r −r 0 ) 2 2σ 2 e ik 2 t .(9) We note that whileφ G (k) can be evaluated exactly similarly to Equation (6), it is a good approximation to consider only the m = 0 term (see the Appendix): ϕ G (r, t) = N σ 2 ∞ 0 dk kJ 0 (kr)J 0 (kr 0 )e − k 2 σ 2 2 e ik 2 t .(10) If r 0 = 0, then Equation (10) becomes exact and equivalent to Equation (8), and can be evaluated in closed form to obtain an expanding radial Gaussian with no further dynamics; ϕ G (r, t)| r0=0 = N σ 2 σ 2 − 2it exp − r 2 2σ 2 − 4it .(11) For r 0 > 0, on the other hand, the extra factor of J 0 (kr 0 ) will give rise to oscillations, i.e. the origin of the self-interference fringes of a ring condensate in a cylindrically symmetric system is a Bessel function as opposed to the typical cosine modulation that gives rise to interference in planar systems. Evaluating Equation (10) gives ϕ G (r, t) = N σ 2 ∞ m=0 2 −r 2 0 m m! (2σ 2 − 4it) m+1 1 F 1 1 + m, 1, − r 2 2σ 2 − 4it (12) = N σ 2 σ 2 − 2it exp − r 2 + r 2 0 2σ 2 − 4it I 0 rr 0 σ 2 − 2it ,(13) where the second equality follows after some algebra and has the form of a Skellam distribution for complex arguments. In the limit as r 0 → 0, Equation (11) is a special case of Equation (13), which in general describes an expanding ring condensate of initial radius r 0 . We note that Equation (13) is general given that C 2D = 0, and it has the form of an exponential envelope modulated by I 0 , the modified Bessel function of order 0. The self-interference fringes arise from I 0 as its argument is complex, but an exhaustive mapping of fringe periods, time scales, and contrasts as functions of r 0 and t is beyond the scope of this work. Figure 3 (a-c) show the time evolution given by Equation (13) for various r 0 . All the essential features of Fig. 2 are reproduced; the central bright peak and the circular self-interference fringes around it can be seen to emerge once the opposite parts of the ring overlap. As an interesting generalisation, we conclude this section by considering a vortex state (i.e. a persistent current) with winding number ζ, given by ϕ v (r, θ) = ϕ G (r)re iζθ . Taking the m = ζ term in a similar fashion as above, but setting ζ = 1 for definitess and assuming r 0 1 to avoid cumbersome expressions, we get ϕ v (r, θ, t) = N σ 4 e iθ ∞ 0 dk k 2 J 1 (kr)J 0 (kr 0 )e − 1 2 k 2 σ 2 e ik 2 t(14) = N σ 4 e iθ r There is no self-interference because the condensate cannot expand to r = 0 to overlap with itself. We have numerically confirmed that direct integration of the GPE with C 2D = 0 gives the same results as the analytical evolutions shown here. The colouring is not to scale. = N σ 4 e iθ (σ 2 − 2it) 2 exp − r 2 + r 2 0 2σ 2 − 4it rI 0 rr 0 σ 2 − 2it − r 0 I 1 rr 0 σ 2 − 2it(16) The vortex at r = 0 must be accompanied by a hole in the density to avoid a diverging kinetic energy. This makes the vortex state unstable in an unrotated simply-connected condensate [33], but on toroidal geometry, the vortex line can exist in the central forbidden region with the result that the persistent current state is stable. The presence of the phase singularity means that the opposite parts of the ring condensate cannot overlap in the free expansion, and hence their self-interference is not possible [see Figure 3 (d)]. In [34], it was experimentally observed that this central hole in the atomic density is present even after a long period of free expansion. (1) with Vtrap given by Equation (17) and C 2D = 400. The red solid line shows a slice of the potential along the positive x-axis. In the first frame, the dashed line shows the potential for negative t. In the last frame, also the phase steps of ∼ 0.9π over the RDSs are shown along the negative x-axis. From left to right, the times correspond to t = 0, 1.5, 3.5, 4.8, 10.7, and 13.1. Experimental creation of ring dark solitons by toroidal self-interference Earlier, we have proposed a protocol for the controlled creation of ring dark solitons (RDSs) by means of a time-dependent double-well trap [27]. Ring dark solitons are a cylindrically symmetric extension of planar dark solitons, whose existence and stability depend on the nonlinearity and quasi-one-dimensionality of the system, respectively [26]. In [35], RDSs are identified with the nodes of numerically found radial solutions of the GPE (with cylindrical symmetry) that approach the Bessel functions in the linear limit, which, as we have shown, also give rise to the selfinterference fringes. To date, RDSs have not been experimentally observed in cold atomic quantum liquids, however, and here, we propose an alternative protocol that involves the use of the toroidal self-interference as a density imprinting mechanism for their creation. In other words, we let the BEC produce an interference pattern, which then evolves into ring dark solitons. We note that density imprinting has been experimentally demonstrated for the production of planar dark solitons [36], and that interference fringes have been shown in general to evolve into stable solitonlike structures in the case of two colliding condensates if the kinetic energy of the condensates during the collision does not dominate over the nonlinear self-energy [37]. At least some of the fringes shown in Figure 2 are soliton-like rings that are long-lived and oscillate back and forth in the harmonic trap. When they are at the turning points of their oscillation, the associated phase step has a magnitude close to π, which characterises dark solitons in general. As a proof-of-concept demonstration for the creation of ring dark solitons through toroidal self-interference, we consider the potential (7), but after the self-interference has taken place, we adiabatically ramp the central region back up. Because the condensate obtains kinetic energy upon the removal of the central region, we must replace it by a higher barrier to prevent the condensate from sloshing back and forth over it. As an example potential, we consider a linear ramp starting at t = 1 and lasting until t = 8: V trap (r, t) = 1 4 r 2 + 30 [H(−t)+ 3H(t − 1)H(−t + 8) t − 1 7 + 3H(t − 8) e −0.1r 2(17) with C 2D = 400. The time evolution given by the potential (17) above is shown in Figure 4. Initially as the central region is removed, the condensate starts expanding and obtains kinetic energy. When the potential blocking the central region is later turned back on, the condensate is bound on the toroidal geometry again, albeit with some kinetic energy. As is evident in Figure 4, in this case the self-interference prints two clear long-lived ring dark solitons. We have not observed the snake instability [26] in the time scales associated here. Apart from the resulting motion of the condensate, we have demonstrated that it is possible to generate ring dark solitons in toroidal (and harmonic) traps by starting from a toroidal condensate that is let to interfere with itself. We note that such a protocol is easy to implement experimentally. For example, in [28], a condensate is prepared in a ring of radius 4 µm using time-averaged painted potentials, comparable to the first frame in Figure 4. The spatial resolution of the potential is stated to be ∼ 1.5 µm, which is more than enough to paint the central barrier in our case. As stated in Section 2, the physical scales corresponding to the experiment [28] of our dimensionless units are a osc = 0.8 µm, C 2D ∼ 200 − 2000, and one unit of time is ∼ 1.5 ms. Our choice of C 2D in Figure 4 falls within this regime. Conclusions In conclusion, we have studied the self-interference of a toroidal condensate both with and without a persistent current. We investigated the Wigner function representation of the ring condensate to see how the self-interference arises from the toroidal geometry. In particular, for a ring of zero radius, all interference phenomena vanished. We predicted both numerically and analytically the appearance of circular fringes if a toroidal condensate is allowed to freely expand towards the origin. In contrast, in the presence of a toroidal persistent current, we showed that there cannot be any ringinduced self-interference because the opposite parts of the condensate cannot overlap in the free expansion. Furthermore, we gave a proof-of-principle demonstration of how ring dark solitons can be experimentally created using the toroidal self-interference. The results are important because they open new possibilities for demonstrating the quantum wave-nature of matter on toroidal geometry. This enables probing the phase-coherence over the ring, for example, and also to experimentally create ring dark solitons in ring traps. Letting the ring dark solitons decay into necklaces consisting of vortex-antivortex pairs leads eventually to quantum turbulence, but before that they have been shown to recombine back into the ring dark solitons [26,27]. Experimental study of the coherent vortex dynamics before the onset of quantum turbulence forms another interesting area for future work. Figure 1 . 1Left panel: The Wigner function W ψ (r, p) of the ground state of the potential (2) with r 0 = 4.5. Taking the marginal over p recovers the original spatial density (solid green line). Right panel: The Wigner function Wϕ G (0, p)|r 0 [see Equation Figure 2 . 2Figure 2. Direct numerical simulation of the GPE (1) with Vtrap given by Equation (7), and (a) C 2D = 50 and (b) C 2D = 400. The left panel shows the initial density distribution at t = 0, and the middle panel shows the same distribution at t = 1.5. The top and bottom insets show a slice along the xaxis through the cylindrically symmetric position distributions (solid line), and the initial ring is also shown (dashed line). The right panel shows the phase distributions corresponding to the densities of the middle panels. In both cases, the ring expands towards the centre, and once the opposite parts of the ring overlap, circular long-lived soliton-like fringes associated with phase steps appear in the position distribution. The central peak for the C 2D = 50 case is 5 times higher than for C 2D = 400, giving rise to a higher kinetic energy density. The colouring is the same in all figures. (2σ 2 − 4it) m+2 1 F 1 2 + m, 2, − r 2 2σ 2 − 4it(15) Figure 3 . 3Density plot of a ring-shaped Gaussian condensate of initial radius (a) r 0 = 1.5, (b) r 0 = 4.5, and (c) r 0 = 20 expanding freely as a function of time [see Equation (13)]. Self-interference fringes appear when opposite parts of the rings meet at t ≈ 0.1, t ≈ 0.4, and t ≈ 1.1 respectively. Here σ = 2/20. (d) Density plot of a ring-shaped persistent current (r 0 = 0.4) expanding freely as a function of time [see Equation (16)]. Figure 4 . 4Density plots of the self-interference of the initial torus (first frame), and the subsequent generation of two ring dark solitons as given by numerically integrating the GPE F 1 is the confluent hypergeometric function. If r 0 = 0, Equation (A.5) reduces to 2N 2 pσ 2 e −p 2 σ 2 , as required. In the right panel ofFig. 1, we have cut the infinite summation over m with m max = 10. AcknowledgmentsWe acknowledge the support of the Academy of Finland (grant 133682) and Jenny and Antti Wihuri Foundation. We thank Kalle-Antti Suominen for fruitful discussions, and Vladimir S. Ivanov for obtaining the simplified Eqs.(13)and(16).Appendix A. Evaluation of Equation(6)Equation(6),is a non-standard integral involving the zeroth order Bessel function J 0 . Let us first make the substitution t = u/2 − r 0 in Equation (A.1):where we have assumed that r 0 is small enough that the lower limit of the integral can be set to 0. The error thus introduced could be corrected by numerically evaluating the integral in Equation (A.2) from t = −r 0 to t = 0. Let us then make use of the Bessel function addition theoremtogether with the symmetry property J −m (z) = (−1) m J m (z) to obtainAfter evaluating the integrals in Equation (A.4), we obtainwhere I 0 is the modified Bessel function of order 0, Γ is the Gamma function, and Observation of interference between two Bose condensates. 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[ "From Molecular to Multiasperity Contacts: How Roughness Bridges the Friction Scale Gap", "From Molecular to Multiasperity Contacts: How Roughness Bridges the Friction Scale Gap" ]
[ "Lucas Frérot ", "Alexia Crespo ", "Jaafar A El-Awady ", "Mark O Robbins ", "Juliette Cayer-Barrioz ", "Denis Mazuyer " ]
[]
[]
The tangential force required to observe slip across a whole frictional interface can increase over time under a constant load, due to any combination of creep, chemical, or structural changes of the interface. In macroscopic rate-andstate models, these frictional aging processes are lumped into an ad hoc state variable. Here we explain, for a frictional system exclusively undergoing structural aging, how the macroscopic friction response emerges from the interplay between the surface roughness and the molecular motion within adsorbed monolayers. The existence of contact junctions and their friction dynamics are studied through coupled experimental and computational approaches. The former provides detailed measurements of how the friction force decays, after the stiction peak, to a steady-state value over a few nanometers of sliding distance, while the latter demonstrates how this memory distance is related to the evolution of the number of cross-surface attractive physical links, within contact junctions, between the molecules adsorbed on the rough surfaces. We also show that roughness is a sufficient condition for the appearance of structural aging. Using a unified model for friction between rough adsorbed monolayers, we show how contact junctions are a key component in structural aging and how the infrajunction molecular motion can control the macroscopic response.
10.1021/acsnano.2c08435
[ "https://export.arxiv.org/pdf/2111.13588v2.pdf" ]
244,709,743
2111.13588
889ef7caf0cdd4a3b6a052cb9836ec16dbf3ba81
From Molecular to Multiasperity Contacts: How Roughness Bridges the Friction Scale Gap Lucas Frérot Alexia Crespo Jaafar A El-Awady Mark O Robbins Juliette Cayer-Barrioz Denis Mazuyer From Molecular to Multiasperity Contacts: How Roughness Bridges the Friction Scale Gap 10.1021/acsnano.2c08435Cite This: ACS Nano 2023, 17, 2205−2211 Read Online ACCESS Metrics & More Article Recommendations * sı Supporting Informationfrictiontransientresponseroughnesscontact junctionfatty acid monolayers The tangential force required to observe slip across a whole frictional interface can increase over time under a constant load, due to any combination of creep, chemical, or structural changes of the interface. In macroscopic rate-andstate models, these frictional aging processes are lumped into an ad hoc state variable. Here we explain, for a frictional system exclusively undergoing structural aging, how the macroscopic friction response emerges from the interplay between the surface roughness and the molecular motion within adsorbed monolayers. The existence of contact junctions and their friction dynamics are studied through coupled experimental and computational approaches. The former provides detailed measurements of how the friction force decays, after the stiction peak, to a steady-state value over a few nanometers of sliding distance, while the latter demonstrates how this memory distance is related to the evolution of the number of cross-surface attractive physical links, within contact junctions, between the molecules adsorbed on the rough surfaces. We also show that roughness is a sufficient condition for the appearance of structural aging. Using a unified model for friction between rough adsorbed monolayers, we show how contact junctions are a key component in structural aging and how the infrajunction molecular motion can control the macroscopic response. F riction is a phenomenon that affects the behavior of virtually every mechanical system: from the movement of geological faults that can cause earthquakes to the sliding of atomic force microscopy tips. In these systems, friction is intimately linked to contacting asperities: 1,2 the inevitable roughness of natural and manufactured surfaces implies that the true contact interface is made up of a sparse set of contact junctions 3 which govern the frictional response, 2 as well as other tribological phenomena. 4−6 At macroscopic scales, the static friction force has been observed to increase logarithmically with resting contact time for amorphous materials, including woods, 7 rocks, 8 and polymers. 3 In general, this can attributed to any combination of the following effects: an increase of the true contact area due to a mechanical creeping of contact spots 3 (geometrical aging), a change in interaction energy between the surfaces 9 via chemical changes 10,11 (chemical aging), and structural/physical changes 12 (structural aging). Upon sliding, the contact interface rejuvenates over a characteristic sliding distance D 0 . 13,14 Such behavior is widely modeled using rate-and-state friction, 8,15,16 which describes the friction force in terms of a state variable ϕ, which represents the average age of the microcontacts and whose evolution equation encompasses aging and rejuvenation. Despite recent efforts to relate rate-and-state parameters to the physics of rough surfaces, 11,17−20 in practice D 0 remains a phenomenological variable fitted to laboratory experiments. Geometrical aging, due to creep, has been extensively discussed in its effects on the macroscopic friction response, 8,15 and both chemical and structural (or physical) agings have been shown to increase the "contact quality": i.e., the junction shear strength. 9,10,12 To our knowledge, no attempt has been made to explain the latter's underlying molecular mechanisms and how they interact with surface roughness to produce structural aging and rejuvenating. Our aim here is therefore 2-fold: elucidating the influence of roughness on these nanoscale friction mechanisms and integrating the physical contribution of these mechanisms into a macroscopic friction description. We focus on a model system representative of structural aging: two rough cobalt surfaces coated with a stearic acid (C 17 H 35 COOH, commonly used as an environmentally friendly lubricant) in dodecane (C 12 H 26 ) dilute solution. After deposition of the solution, the stearic acid adsorbs on the surfaces and forms a monolayer. 21,22 These two rough monolayer-covered surfaces are brought into contact in our molecular tribometer 23 at a constant normal force. A slide− hold−slide protocol is applied with constant velocity and varied hold times. Molecular dynamics (MD) simulations reproducing the experimental protocol (at shorter time scales) are used to probe the details of the contact interface, for which we combine the two surfaces roughness profiles into a single rough-on-flat setting (roughness profiles are generated using measurements of the experimentally used surfaces). Nanoscale mechanisms uncovered with MD and experimental results are used to establish a unifying friction model that we show reproduces the transient friction behavior observed in experiments. Figure 1a illustrates the multiscale aspect of friction of surfaces coated with fatty acid monolayers: the inevitable roughness of the surfaces in contact partitions the apparent contact interface into contact junctions, 3 where the fatty acid molecules are close enough to interact. This occurs, as we show in this work, even with a root-mean-square (RMS) roughness as small as 0.6 nm, as measured in the current experiments with atomic force microscopy (AFM) over 1 μm 2 . Figure 1b,d shows the transient friction response of stearic acid monolayers for a slide−hold− slide (SHS) protocol, where t 0 is the start time of the holding stage and μ exp and μ MD are the ratios of tangential to normal force for the experiment and simulation, respectively. During the holding stage, relaxation occurs and the tangential force decreases to a nonzero value. 9 After rest, when the sliding resumes at the velocity prior to the hold phase, the friction force overshoots the steady-state value by Δμ exp (respectively Δμ MD ). This overshoot is observed in both the experiments and MD simulations of rough-on-flat (cf. Figure 1d) and rough-on-rough (cf. Figure S1 in the Supporting Information) and is consistent with previous observations of frictional aging in experiments 7,8,12,13 and simulations 10 at a macroscopic scale. Figure 1c,d shows that for both the experiments and the simulations the magnitude of the overshoot increases with the waiting time, t w , for times longer than the relaxation times τ exp = 2.2 s and τ MD = 0.8 ns of the experiment and simulation, respectively. We have defined τ exp directly from Figure 1c, but τ MD is defined from scaling regime changes in the mean-square displacement of monomers in an equilibrium simulation (see Figure S2), hence our interpretation of τ exp and τ MD as relaxation time scales. An independent simultaneous measurement of the tangential stiffness in the SHS experiment 24 shows a reversible increase of the stiffness during the hold step (see Figure S3). This, combined with previous observations of the decreasing film thickness at rest, 14 confirms the presence of structural aging at the molecular scale during rest. With both our experimental and computational systems showing evidence of aging, we investigate the role of roughness in the observed transient friction response. RESULTS We compare in Figure 2a the MD transient friction response (as a function of sliding distance, δ, normalized by the molecular length, L 0 = 2.14 nm) of the rough-on-flat system shown in Figure 1 to a system with identical monolayers on two atomically flat surfaces. Unlike the rough-on-flat system, this flat-on-flat system does not overshoot the steady-state friction level, regardless of waiting time t w , indicating that the system does not age. To quantify the aging difference between the flat and rough systems, we plot in Figure 2b the total number, N, of attractive interactions that atoms of one surface have with the other. We call these interactions cross-surface links: they are van der Waals bonds between molecules belonging to different surfaces. The relative change in N, compared to the steady-state value N ss , while the system is at rest gives a metric for the aging process: it is apparent from Figure 2b that in the flat system N stays constant (with thermal noise) while the rough system sees its number of cross-surface links increase over the resting period (300 τ MD ). While an increase, e.g. due to creep, of the true contact area could cause this, we show in Figure S4 that the footprint of the contact does not evolve during rest in our simulations, and the increase in tangential stiffness measured in experiments ( Figure S3), in conjunction with the relative compliance of the monolayers compared to the substrate, also excludes this mechanism in our empirical observations. We are therefore quantifying the structural age of the system. The difference between the rough and flat systems can be explained by the dynamics of the contact junctions that occur in the rough system: at the edges of these junctions the surfaces are close enough that molecules in the vicinity of the contact can have their tail move in and out of the contact, thereby providing degrees of freedom for the system to evolve. This does not occur in the flat case, where all molecules are already participating in the contact. The similarity between the rough-on-flat curve in Figures 2 and 1e seems to reinforce the relationship between age, friction, and cross-surface links, analogously to how entanglement density controls the interface strength in polymer welding. 25 We now investigate this relationship in the sliding phase to understand how the interface rejuvenates. We show in Figure 3 how the number of cross-surface links N returns to its steady-state value as a function of the sliding distance δ. Symbols show N for different sliding velocities (vτ MD /L 0 = 0.7, 1.2, 1.4, 1.9, from light to dark shades). We compare N to another memory function, the survival contact fraction α. It defines, as a function of sliding distance, how much of the contact interface between two rough surfaces is common to the interface when the system was at rest: i.e., α = | A(δ)∩A(0)|/|A(0)| with A(δ) being the set of contact points at a given sliding distance. This memory function postulates that the rejuvenation of the contact comes from the geometric renewal of the microcontact population. 3 We define ΔN = N(δ) − N ss (respectively Δα = α(δ) − α ss ) and ΔN w = N(0) − N ss (idem for Δα w ). The quantity −ln(ΔN/ΔN w ) gives a measure of the rate at which the system rejuvenates: rate-and-state models that use the aging law ϕ = 1 − vϕ/D 0 predict that ϕ(δ) − ϕ ss ∝ exp(−δ/ D 0 ) . In Figure 3, axes are chosen so that an exponential decay is a straight line with slope 1/D 0 . We show that α, which represents a memory definition based on the geometry of the contact, decays to a steady state at a much lower rate than the number of cross-surface links. The latter decays in good agreement with an exponential decay having D 0 = 3.5 nm. This is consistent, in magnitude, with the distance needed for the experimental friction force to return to steady state, measured to be 4.8 ± 1.4 nm. Although the number of cross-surface links and the friction force should not be directly compared, the MD simulations still reproduce a value of D 0 independent of sliding velocity and in the same order of magnitude as the experiment, despite the 10 orders of magnitude difference in sliding velocity between simulations and experiments. Furthermore, no simulation parameter was adjusted to match the experimental data. This decay holds well for N even for several times D 0 , regardless of sliding velocity (symbol shapes, vτ MD /L 0 = 0.7, 1.2, 1.4, 1.9, from light to dark shades, all with t w /τ MD = 300), suggesting that N is a representative quantity of the interface state, since the experimental decay distance to steady state is 4.8 ± 1.4 nm and the memory function based on the renewal of the microcontact geometry, α, decays more slowly. Dispersion of the data on long sliding distances is expected due to the natural noise of the systems, which introduces uncertainty in the steady-state estimate (gray area). ACS Nano www.acsnano.org Article Uncertainty (due to noise) in the measurement of the steadystate value of an exponential decay causes deviations from the straight line. The gray area in Figure 3 shows the deviation extent based on the noise in N measured in the simulations. The rejuvenation difference between N and α indicates that D 0 is not an intrinsic property of the junction sizes, as is commonly interpreted, 3 but rather a velocity-independent system property 11,20 that combines the surface roughness and the molecular organization of the fatty-acid molecules. Note that in systems where two or more aging mechanisms contribute to the transient friction response, D 0 may be the result of more complex interactions that are absent from our experiments. Our experiments and simulations show that taking the surface roughness into account, even at the nano scale, and by extension the formation of contact junctions, is a key component in structural frictional aging, although junctions could arise from other sources of heterogeneity, such as imperfect surface coverage of the adsorbed layer. 9 We also demonstrate that the cross-surface link formation and destruction within contact junctions govern key aspects of the aging process and of the transient frictional response. We combine these two ideas into a model that links the nanoscopic and macroscopic scales and analytically reproduces the steady-state friction response as well as the transient overshoot in the presence of roughness. This model unifies existing approaches at two length scales: the macro scale where the true contact area is made up of monolayer junctions due to the presence of surface roughness and the scale of molecular interactions within a junction. At the molecular scale, we use the theory developed in ref 26 for adhesive friction of polymer chains. This study postulated that chains at the interface are in either a bound state or free state and that the transition from bound to free can occur by thermal fluctuations or an external force. Three characteristic times govern the state transitions: the time to break a molecular link (i.e., cross-surface link), the time to (re)activate a molecular link, and the delay time related to the withdrawal of a link from the contact zone. 27 Chernyak and Leonov assumed a stationary stochastic process and constant surface separation to compute steady-state values of the shear stress within a contact junction, as a function of sliding velocity, σ ss (v). At the macro scale, we define the interface age ϕ, which follows the state law mentioned above, and the aging factor 28 f a (ϕ) = 1 + ω ln(1 + ϕ/τ 1 ) . The contact is made up of contact junctions totaling a true contact area A r . This is sufficient for a calculation of the macroscopic steady-state friction force F t,ss . In the inset of Figure 4, we show our fit of F t,ss to the steady-state experimental values of the friction force at different sliding velocities (values of the model parameters, both measured and fitted, are given in Methods). For the characteristic detachment time, we find values in the same order of magnitude as the relaxation time measured in Figure 1, and as values measured for different organic monolayers with the same thickness. 12 To account for transient effects at the onset of sliding, i.e. on sliding distances shorter than D 0 , the elastic tangential response of the asperities in contact is approximated with Mindlin's theory of elastic spheres in frictional contact, 29,30 extended in ref 31 to a Greenwood−Williamson approach, 5 which approximates the junction distribution. The resulting friction force is expressed as F t (v,t) = f a (ϕ(v, t))(1 − exp(− vt/ δ*))A r σ ss (v), where the exponential term models the transition from elastic tangential response (stick) to the slip regime and δ* is the ratio of the steady-state friction force to the measured tangential stiffness of the interface. The hypotheses leading to the full derivation of this equation are given in Methods. These ingredients provide a good fit to the experimental data in the stationary and transient regimes, by accounting for the time, sliding velocity, surface roughness, and elastic properties of the monolayers, as illustrated in Figure 4. Without introducing contact junctions, the proposed approach predicts no overshoot of the stationary friction value, in agreement with our MD simulations. This demonstrates that the physics of friction between the monolayers is well captured by the coupling of link formation inside contact junctions, governed by the aforementioned characteristic times, and the sliding dynamics of the junctions themselves all over the contact area. Thus, the interface accommodates the shearing through a combined effect of roughness and molecular interactions. This approach can readily be generalized to other systems with adsorbed organic layers and rough surfaces, which are commonplace in microelectromechanical systems 12 and biomechanics: 32 e.g., for natural or artificial joints where proteins can form a protective layer on a hard substrate. 33 CONCLUSIONS Combining experiments and simulations of friction between fatty-acid monolayers deposited on rough surfaces, complemented by a multiscale theoretical approach, we were able to probe the molecular mechanisms underlying structural frictional aging and its transient friction response and bridge the scale gap to the macroscopic friction behavior. We have shown, for monolayers adsorbed on rough, stiff surfaces, that the stiction peak and its decay are controlled by molecular mechanisms within contact junctions, not the sizes of the junction themselves. To uncover these molecular mechanisms, we have demonstrated that in the absence of contact junctions (e.g., due to the absence of roughness), structural aging disappears. The system aging can then be explained by the capacity of molecule tails to come in and out of contact junctions, which cannot happen when surfaces are flat at the atomic scale. This sheds light on the memory length scale D 0 with important implications on a broad range of frictional systems. Our findings highlight the ACS Nano www.acsnano.org Article importance of both surface roughness at the molecular scale and molecular mechanisms at the macroscopic scale. We combined these aspects into a multiscale theoretical model that correctly reproduces the transient friction overshoots observed in experiments and whose principles are generalizable to a wide variety of frictional systems with surface roughness and coatings, such as biomechanical systems, e.g. cartilaginous or artificial joints, for which roughness can dramatically alter the proper function and lifetime. 34 METHODS Experimental Friction Measurement. Using stearic acid (99.0% purity, from Sigma-Aldrich) with dehydrated and filtered dodecane, a dilute solution was prepared at a concentration of 0.002 mol/L. The surfaces consisted of a fused silicate glass sphere of radius 2.030 ± 0.005 mm and a ⟨100⟩ silicon wafer. The latter was cleaned with isopropanol and deionized water using a spin-coater at 8000 rpm and then dried under a nitrogen flow. Both surfaces were then coated with a 40 nm thin cobalt layer by means of a cathodic sputtering system under low argon pressure (10 −6 mbar). Experiments were conducted by sliding the sphere over the plane using an ATLAS molecular tribometer. 23 A typical friction experiment was performed by approaching the sphere toward the plane, confining the stearic acid monolayers on each surface until a constant normal force of 0.70 ± 0.01 mN: i.e., a corresponding maximum Hertzian contact pressure of 27 MPa (at a normal velocity of 0.1 nm/s). Then without breaking contact, a slide−hold−slide procedure was used: the sphere slid over the plane over a few hundreds of nanometers at a constant sliding velocity of 0.5 nm/s and then was held stationary for a time, t w , before resuming the lateral displacement. Hold times were varied between 1 and 120 s. During the experiment, the response to a superimposed oscillating sphere displacement in both directions, normal and tangential, of amplitude 0.1 nm and 38 Hz (respectively 0.03 nm and 70 Hz) provided, without disturbing the friction process, the stiffness and the viscous damping of the confined interface in both directions. 23 All measurements were carried out in a sealed chamber with a relative humidity lower than 1% and T = 23.0 ± 0.5°C under an argon atmosphere. Surface Topography Characterization. Multiscale characterization of the surface topography was performed before and after the experiment to ensure no surface damage. AFM measurements of the surface topography over an area of 1 μm × 1 μm provided an RMS of surface heights of 0.6 nm and a radially averaged power-spectrum density (PSD) as shown in Figure S6. At a larger scale, a Bruker interferometry profilometer provided an RMS value of 0.5 nm on both surfaces in Phase Shift Interferometry mode over an area of 63 μm × 47 μm. To generate synthetic rough surfaces from the measured surface profile, we used the PSD computed from the AFM data. We cut off long-wavelength modes as necessary to generate a smaller surface, i.e. for MD simulations that were 200 nm × 200 nm, and used uniformly distributed phases 35 to produce surfaces with the same (or reduced) spectral content as the surface used in experiments. Full-size (1 μm × 1 μm) surfaces were used for continuum simulations of dry elastic contact, while a reduced size (200 nm × 200 nm) was used for MD simulations. Molecular Dynamics. Molecular dynamics simulations were conducted using coarse-grained potentials 36 adjusted for alkane chains with one bead corresponding to two CH 2 groups. Stearic acid chains consisted of nine beads. Head groups were positioned on a hexagonal lattice with spacing 5.5 Å. The top lattice was rotated 90°to avoid commensurate effects in the flat/flat friction response. The applied normal pressure was p ̅ = 27 MPa. Roughness was applied to the head group lattice by vertical displacement of the beads and their connected chain. The system was initially equilibrated at T = 300 K, with the surfaces separated, using a Langevin thermostat and a time step of Δt = 1 fs. Surfaces were brought together with the applied normal pressure and equilibrated again. Sliding of the top head group lattice was done via a spring attached to its center of mass. The stiffness of the spring was such that the period of the mass-spring system was 3.5 ps. In the initial sliding phase, the free end of the spring slid at velocity v for 600 Å and Δt = 1.25 fs. The system was then allowed to rest by setting v to zero for 30 ns with Δt = 3 fs. Restart of the sliding was done by setting v back to its original value with Δt = 1.25 fs. The friction force was measured as the force in the spring, and the temperature was controlled at 300 K with a Langevin thermostat acting on the degrees of freedom normal to the sliding direction. The number of cross-surface links was computed with a radius cutoff of 10 Å for attractive links and 5 Å for compressive links, which corresponded to the potential cutoff and equilibrium distance, respectively. All simulations were conducted with the opensource software LAMMPS. 37,38 Figures were generated with Blender, Ovito, 39 Matplotlib, 40 Scipy 41 and Numpy. 42 The source code of simulations and figures is available. 43 Continuum Elastic Rough Contact. A Fourier-based boundary integral approach 44,45 was used with a projected conjugate gradient algorithm 46 to solve the elastic rough contact problem. The linear elastic material properties used were determined from the experiments: 21 the contact Young's modulus E*= E/(1 − ν 2 ) was set to 48 GPa and the average pressure was set to p ̅ = 27 MPa. The contact problem was solved with a compound roughness 30 h = h 2 − h 1 from two generated surfaces h 1 and h 2 , the latter of which was shifted by δ, the sliding distance. The true contact area was the area where contact pressure was strictly positive. The survival fraction at δ was the normalized magnitude of the area in common with the initial contact area. All simulations were conducted with the open-source library Tamaas. 47,48 Junction-Based Friction Model. The friction force per junction in our model was expressed as the product of a velocity and age-dependent shear stress σ, by the junction area A j : F j (v,t) = σ(v,ϕ(t))A j , where ϕ is the junction age typically defined in rate-and-state models 15,28 and obeys the state equation ϕ = 1 − vϕ/D 0 . Its contribution to the shear stress comes in the form of an aging factor, i.e. σ(v,ϕ) = f a (ϕ)σ ss (v), with f a (ϕ) = 1 + ω ln(1 + ϕ/τ 1 ) from ref 28 and σ ss given in ref 26. This decomposition of σ is due to Chernyak and Leonov's assumption of a stationary stochastic process for the attachment and detachment of molecules at the interface to compute the velocity-dependent shear stress, which excludes aging. The macroscopic friction force F t is given by the sum of F j over all contact junctions. For the sake of simplicity, we assumed all junctions to have the same age ϕ. We approximated, for each junction, the transition from stick to slip with the model derived by Mindlin 29,30 for the frictional contact of elastic spheres. The application of Mindlin's model 5 to a multiasperity approach proposed in ref 31 was used here, but more accurate models for rough surface contact, such as boundary integral simulations, could be employed. To summarize the provenance of each contribution to the macroscopic tangential force: • The velocity-dependent steady-state shear stress was computed using the friction model devised in refs 26 and 27 based on the dynamics of molecular links breaking/formation during sliding • Rate-and-state models 8,15,18,28 gave the aging contribution. • The elastic response of asperities for short tangential displacements was approximated with a spherical frictional contact model developed in ref 29 and extended in ref 31 to a ref 5 approach. We now go through the procedure we used to determine each parameter of the model. In the Steady-State Regime. The Chernyak−Leonov theory is used to describe the friction between the stearic acid molecules, according to three elementary times: 27 τ 0 , the time necessary to break a link, τ, the time necessary to form a link, and τ, the time for a molecule to withdraw from the interpenetration zone. According to this model, the interfacial shear strength σ ss can be written as In the expression of σ ss , only τ 0 , γ, and χ are free parameters that require fitting: m can be computed using Figure S2, while G and L H are measured 23 and L 0 can be found in the literature. 49 The coefficients ω and τ 1 in f a can be independently fitted from Figure 1, and the values are given in Table 1. Assuming a an average pressure within contacts of 700 MPa, lower than the cobalt hardness, we find that both τ 1 and τ 0 have values in the same order of magnitude as τ exp = 2.2 s and values reported in the literature for different organic compounds but similar monolayer thicknesses, 12 confirming that they relate to a chain relaxation mechanism as postulated in ref 26. We also find that γ is effectively zero, indicating that the retraction time τ̂is very large compared to the attachment time τ. Onset of Sliding. The macroscopic transient friction force for an interface transitioning from rest to a sliding velocity v is calculated from the extension of Mindlin's theory to rough surfaces. 31 This is possible because the transient friction behavior is observed over sliding distances much smaller than the characteristic diameter of the contact junctions (see Figure S5). In a multiasperities interface, this means some contact spots remain in partial sliding while others are moved in total sliding. For a single spherical junction, Mindlin determined that the elementary tangential force f required to move a microcontact in partial sliding is simply f = μ f n [1 − (1−δ/δ*) 3/2 ], where f n is the normal load applied on one microcontact and δ* is the applied tangential displacement necessary for full sliding. We can obtain it from our in situ tangential stiffness measurements: 50 δ* = F t,ss /K x . The Greenwood−Williamson model applied to the multicontact interface 31 gives the global force contribution F t = A r (1−exp(−vt/δ*))f a (ϕ(t))σ ss (v) . The remaining model parameter is D 0 , which we can estimate from experiments. The measured distance, denoted D ss , required for the friction force to return to its steady-state value can be used to identify the distance necessary in the model for the force to be within 90% of its steady-state value, thereby estimating D 0 . This threshold corresponds to the error in the steady-state response of the experiment. ASSOCIATED CONTENT * sı Supporting Information The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsnano.2c08435. Presence of structural aging and exclusion of other aging modes and additional data on surface roughness, relaxation time scales, and memory length scale (PDF) Figure 1 . 1Transient friction behavior of stearic acid monolayers. (a) Schematic of a ball-on-flat contact experiment for fatty-acid monolayers showing the multiscale nature of friction. The apparent contact area has a radius of 3.52 μm, while the true contact area is made up of sparse junctions where the adsorbed monolayers interact due to the surface roughness. Friction response of (b, c) the experiments (±10% error) and (d, e) simulations, respectively. (b) and (d) show the transient friction behavior in a slide−hold−slide protocol, with hold highlighted in gray (t 0 being the start time of the hold stage). An overshoot of the steady-state friction force can be observed at the onset of the second slide stage. (c) and (e) show that the magnitude of the overshoot increases with the hold time, t w , if it is greater than a relaxation time of τ exp = 2.2 s in the experiment and τ MD = 0.8 ns in the simulations. Figure 2 . 2Effects of roughness on the transient friction response. (a) Comparison of the transient friction response of rough-on-flat and flat-onflat systems after a hold time t w (darker curves have longer t w ).The flat/flat system shows that the friction force recovers a steady-state value without overshooting, unlike the rough/flat system, which exhibits a friction force peak above μ ss for large t w . (b) Comparison of the increase in the number of cross-surface links (ΔN) in the holding stage. While ΔN increases markedly for the rough/flat system, which indicates structural aging, it remains constant in the flat/flat system. Figure 3 . 3Evolution to steady state of the number of cross-surface links (N, symbols), and contact survival fraction (α, dashed line with circles) in slide-after-hold as a function of the sliding distance δ. The dashed (red) line shows a decay to steady state of the form exp(−δ/ D 0 ) with D 0 = 3.5 nm. Figure 4 . 4Transient friction derived from a multiscale friction model. The theoretical response (black line) is compared to the experimental friction transient (v = 0.5 nm/s) and steady-state (inset) responses (circles). The characteristic times of the nanoscale contribution to the friction force are consistent with measured relaxation times (cf.Figure 1), and the model is capable of reproducing the transient friction behavior as well as the stationary response. u = tan χvτ 0 /(2L 0 ), m = τ/τ 0 , γ = τ/τ, and σ 0 = (2G/tan χ)(L 0 /L H ) deduced from ref 27, where χ is the angle made by the stretched ACS Nano www.acsnano.org Article molecule in sliding. Table 1 . 1Numerical Values Used for the Junction-Based Friction Modelmeasd value steady-state param aging param https://doi.org/10.1021/acsnano.2c08435 ACS Nano 2023, 17, 2205−2211 ACKNOWLEDGMENTS The Area of Contact between Stationary and between Moving Surfaces. F P Bowden, D Tabor, 10.1098/rspa.1939.0005Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 169Bowden, F. P.; Tabor, D. The Area of Contact between Stationary and between Moving Surfaces. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 1939, 169, 391−413. Mechanism of Metallic Friction. F P Bowden, D Tabor, 10.1038/150197a0Nature. 150Bowden, F. P.; Tabor, D. Mechanism of Metallic Friction. Nature 1942, 150, 197−199. Direct Observation of Frictional Contacts: New Insights for State-Dependent Properties. Pure and applied geophysics. J H Dieterich, B D Kilgore, 10.1007/BF00874332143Dieterich, J. H.; Kilgore, B. D. Direct Observation of Frictional Contacts: New Insights for State-Dependent Properties. Pure and applied geophysics 1994, 143, 283−302. Contact and Rubbing of Flat Surfaces. J F Archard, 10.1063/1.1721448J. Appl. Phys. 24Archard, J. F. Contact and Rubbing of Flat Surfaces. J. Appl. Phys. 1953, 24, 981−988. Contact of Nominally Flat Surfaces. 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[]
[ "Hunting scalar partners of the Higgs boson at the LHC Hunting scalar partners of the Higgs boson at the LHC", "Hunting scalar partners of the Higgs boson at the LHC Hunting scalar partners of the Higgs boson at the LHC" ]
[ "Werner Porod [email protected] \nInstitut für Theoretische Physik und Astrophysik\nUni Würzburg\nEmil-Hilb-Weg 22D-97074WürzburgGermany\n", "Werner Porod \nInstitut für Theoretische Physik und Astrophysik\nUni Würzburg\nEmil-Hilb-Weg 22D-97074WürzburgGermany\n" ]
[ "Institut für Theoretische Physik und Astrophysik\nUni Würzburg\nEmil-Hilb-Weg 22D-97074WürzburgGermany", "Institut für Theoretische Physik und Astrophysik\nUni Würzburg\nEmil-Hilb-Weg 22D-97074WürzburgGermany" ]
[ "School and Workshops on Elementary Particle Physics and Gravity" ]
Composite Higgs models with a fermionic ultraviolet completion predict in general additional pseudo Nambu Goldstone bosons beside the Higgs multiplet. In this contribution we discuss their LHC signatures and present first bounds in simplified models which can also be applied to generic models like multi-Higgs models. We then demonstrate how these can be combined taking a concrete model based on the SU(5)/SO(5) coset as an example. We use this to show how a proper combination of different channels can lead to an improved bound compared to a single channel analysis.
null
[ "https://export.arxiv.org/pdf/2304.10753v1.pdf" ]
258,291,390
2304.10753
112e065b9747e0eb51b8eb9e28a4aae6eeb0d292
Hunting scalar partners of the Higgs boson at the LHC Hunting scalar partners of the Higgs boson at the LHC 2022. 28 August -1 October, 2022 Werner Porod [email protected] Institut für Theoretische Physik und Astrophysik Uni Würzburg Emil-Hilb-Weg 22D-97074WürzburgGermany Werner Porod Institut für Theoretische Physik und Astrophysik Uni Würzburg Emil-Hilb-Weg 22D-97074WürzburgGermany Hunting scalar partners of the Higgs boson at the LHC Hunting scalar partners of the Higgs boson at the LHC School and Workshops on Elementary Particle Physics and Gravity Corfu, Greece2022. 28 August -1 October, 2022* Speaker Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ Composite Higgs models with a fermionic ultraviolet completion predict in general additional pseudo Nambu Goldstone bosons beside the Higgs multiplet. In this contribution we discuss their LHC signatures and present first bounds in simplified models which can also be applied to generic models like multi-Higgs models. We then demonstrate how these can be combined taking a concrete model based on the SU(5)/SO(5) coset as an example. We use this to show how a proper combination of different channels can lead to an improved bound compared to a single channel analysis. Introduction The Standard Model (SM) of particle physics features a single scalar field being a doublet of weak isospin SU(2) that is responsible for the breaking of the electroweak (EW) symmetry [1,2]. Once it acquires a vacuum expectation value (vev), a massive physical scalar particle arises, the Higgs boson [2] discovered in 2012 at the Large Hadron Collider (LHC) experiments [3,4]. In contrast, most models of new physics contain extended Higgs sectors: for instance, minimal supersymmetric models [5] and two Higgs doublet models [6] feature a second doublet and the type-II seesaw models [7,8] a zero hypercharge triplet. Scalar triplets appear also in the Georgi-Machacek model [9]. In all these scenarios, the scalar fields acquire sizeable couplings to the SM gauge bosons and fermions via vevs and/or via mixing with the SM Higgs boson. Consequently, they are dominantly singly-produced at colliders, and most current searches focus on these channels. Single production of a scalar is always model dependent and it can be suppressed by tuning the single-scalar couplings. By contrast, pair production only depends on the gauge quantum numbers of the scalars and cannot be tuned to be small. The couplings of two scalars from SU(2) × U(1) multiplets to the EW gauge bosons stem from the covariant derivatives in the scalar kinetic terms and are always present. Their presence imply dominant pair production channels via Drell-Yan processes, where two initial state quarks merge via an s-channel gauge boson. There is some model dependence if their is mixing in the scalar sector but this mixing cannot reduce Drell-Yan pair production cross sections of all scalars at the same time and some channels are guaranteed to remain sizeable. In this contribution, we focus on models where pair production is the dominant mode for scalars charged under SU(2) × U(1) . Such scenarios appear naturally in composite Higgs models, where the Higgs boson is accompanied by additional light states, protected by parities internal to the strong sector [10]. The Higgs boson emerges in these models as a pseudo-Nambu-Goldstone boson (pNGB) [11] following the dynamical breaking of the EW symmetry triggered by misalignment in a condensing strong dynamics at the TeV scale [12,13]. A minimal model SO(5)/SO(4) based only on the global symmetries with exactly four pNGBs matching the Higgs doublet components can be constructed [14]. However, it is not easy to obtain this symmetry pattern in an underlying gauge/fermion theory à la QCD. A fermion condensate can only generate the following patterns [15,16]: SU(2 )/Sp(2 ), SU( )/SO( ) or SU( ) 2 /SU( ) depending on whether the representation of under the confining gauge symmetry is pseudo-real, real or complex, respectively. The minimal model with custodial symmetry [17,18] features SU(4)/Sp(4) [19,20], and has one additional gauge singlet pNGB besides the Higgs doublet. The next-to-minimal cases contain significantly more pNGBs: 14 for SU(5)/SO(5) [21,22] and 15 for SU(4) 2 /SU(4) [23,24]. The departure from minimality does not contradict the null results of direct searches for physics beyond the SM (BSM) at colliders. The pNGBs are typically heavier than the Higgs boson and have only EW interactions, hence being difficult to discover at hadron colliders and are too heavy for past + − colliders such as LEP or SLC. The dominant channel is pair production via Drell-Yan and the vector boson fusion (VBF) pair production via gauge couplings is found to be subleading to Drell-Yan [25,26]. VBF single production is generated via topological anomalies, hence it is suppressed by a small anomaly coupling. Drell-Yan single production could also be present if the pNGBs couple to quarks. q q W + S − S ++ W + W + W − γ/Z p p ¯ t /b S (with decays to SM states) S (with decays to SM states) QCD Figure 1: Examples of di-scalar channels from pair production via Drell-Yan processes with subsequent decays into SM particles. q q /Z S ++ S W W W + W + q q 0 W + S S ++ S +(⇤) W + b t t b ⌘ 0 3 ⌘ ± W ⌥ W ± /Z ⌘ 0 3 ⌘ 0 3/5 Z /Z /Z However, it is expected that pNGBs in models with partial compositeness couple only very weakly to light quark flavours as the couplings are roughly proportional to the quark mass. Consequently, the dominant couplings involve third generation quarks and the neutral pNGBs can be singly-produced via gluon fusion similar to the Higgs boson. Moreover, both neutral and charged pNGBs can be singly-produced in association with either or . This can provide a relevant contribution if the couplings are large enough. We will present here recent results from recent investigations presented in refs. [26,27] to which we refer for further details. We will first discuss simplified models and then focus on a specific model based on SU(5)/SO (5) to present the interplay between various channels. This class of models can be realised in the context of four-dimensional models with a microscopic description [28][29][30] and emerges as the minimal symmetry pattern from the condensate of two EW-charged fermions if is in a real irreducible representation of the confining gauge group, e.g. (4). We note for completeness, that the properties of the confining gauge dynamics, based on Sp(4), has been studied on the Lattice with promising results [31][32][33][34]. Complementary information on the mass spectrum and decay constants of the composite states can also be obtained using holographic techniques [35][36][37]. Bounds on Drell-Yan pair-produced scalars in simplified models We start with bounds in simplified models where we focus on pair production, via the dominant Drell-Yan channels. We use parts of a simplified model which has been introduced in refs. [26,27] to which we refer for the underlying Lagrangian. We extend the SM by colourless scalar states 0 , 0 , ± , ±± that are physical mass eigenstates labelled by their electric charge. We include the minimal set of states with charge up to two that have all the possible couplings to the EW gauge bosons. In case of the neutral we include two neutral states with opposite parity and assume that none of the BSM scalars obtains a non-zero vev. We investigate all combinations of scalar pairs produced at the LHC through the Drell-Yan processes: → ±± ∓ , ± 0( ) , ++ −− , + − , 0 0 .(1) Together with the first tier decays of the scalar pairs into SM particles, these production processes yield many di-scalar channels, see for example Fig. 1. Charge-conjugated states belong to the same channel. We consider two complementary scenarios for the decays of the scalars fermiophobic Table 1: Classification of the 24 di-scalar channels in terms of the 5 pair production cases (columns) and the 15 combinations of gauge bosons (rows) from decays. In the channels, the first two and second two bosons are resonantly produced. The notation { } = + indicates the two permutations. Charge-conjugated states belong to the same di-scalar channel. ++ −− ±± ∓ + − ± 0( ) 0 0 / 0 0 + + − − - - - + − + − - ± ± ∓ - ± + − - - ± ± ∓ - ± + − - - - + − - + − - - ± ∓ - + − - - + − - + − - - - ± - - - - ± { } - - - - ± { } - - - - ± - - - - - - - - - - - - - { } - - - - - - - - 1. The fermiophobic case, where couplings to SM fermions are absent at leading order and the dominant decays are into EW gauge bosons: ++ → + + , (2a) + → + , + , (2b) 0( ) → + − , , , .(2c) Combining the different Drell-Yan scalar pairs with the above decay channels leads to 24 di-scalar channels, each containing four gauge bosons. One sample process is shown in the left diagram of Fig. 1, while a complete list of all channels is shown in Table 1. 2. The fermiophilic case, where the scalars decay dominantly into a pair of third generation quarks: ++ → +¯, (3a) + →¯, (3b) 0( ) →¯or¯.(3c) Note that doubly charged scalars cannot decay to two quarks due to their charge, but if they are part of an SU(2) multiplet, the three-body decay ++ → + + * → +¯i s allowed. Consequently, this yields 8 possible di-scalar channels for pair-produced scalars in this scenario. One sample process is shown in the right diagram of Fig. 1 and a complete list in Tab. 2. fermiophilic Table 2: Classification of the 8 di-scalar channels in terms of the 5 pair production cases (columns) and the 5 combinations of top and bottom from decays (rows). In cases with one or two doubly charged scalars, one always obtains with one or two additional 's, respectively. The charge-conjugated states are not shown. ++ −− ++ − + − + 0( ) 0 0 / 0 0 - - - -¯- - -¯¯- - -¯¯-¯- - -¯¯- - - - -¯- +¯¯- - - +¯−¯- - - - For the simulation of signal events we use the publicly available eVLQ model presented in ref. [26], which implements the simplified models as a FeynRules [38] model at next-to-leading order in QCD. All events are generated at a centre-of-mass energy of 13 TeV in collisions. For each di-scalar channel, we perform a scan over the scalar mass ; for channels involving two different scalars, we assume them to be mass degenerate. We generate 10 5 [49] and the anti-algorithm [50] implemented in FastJet [51]. The exclusion associated with the events is calculated with the CL prescription [52]. Moreover, we run the events against the SM measurements implemented in Rivet [53] version 3.1.5 and extract exclusions from the respective YODA files using Contur [54, 55] version 2.2.1. To present simplified model bounds, we determine the signal cross section 95 which is excluded at 95% CL. We note for completeness, that the procedure differs somewhat between the tools and we refer to ref. [27] for the details. For each channel and each parameter point, we take the minimal value for 95 obtained from MadAnalysis5, CheckMATE, and Contur as the final bound. We do not attempt to combine them. We display results in Fig. 2, where we present the simplified model bounds on the cross section for various di-scalar channels, i.e. bounds on the production cross section of the scalar pair times both branching ratios. We show in Fig. 2a the bounds on the 8 di-scalar channels for the fermiophilic scenario, consisting of third generation quarks plus one additional boson per doubly charged scalar due to the 3-body decay of ±± . In channels with multiple top quarks, the dominant bounds stem from a search for -parity violating supersymmetry [60], while various supersymmetric searches [61][62][63][64] and the generic search of ref. [65] are relevant for the multi-bottom channels. Figures 2b to 2d show bounds for various channels of the fermiophobic scenario which are split into three figures for the sake of readability. In Fig. 2b we display di-scalar channels with at least one doubly-charged scalar, leading to at least 3 bosons plus a , , or photon. The photon channel can be constrained using measurements of the production cross section [68,69]. The most relevant searches for the and channels look for multi-lepton final states [57,70]. For these channels, the results of the ATLAS search [56] apply, and they are shown as blue and orange dashed lines. The bounds from this dedicated search are obviously stronger compared the bounds obtained from our recasts of a large number of BSM searches which target different signatures and scenarios. This had to be expected and Figure 2c shows the di-scalar channels from + − production. We note, that the bounds on are by far the strongest, stemming from a search for gauge-mediated supersymmetry in final states containing photons and jets [58]. The main bounds for the channels and stem from a multi-lepton search [65] and the cross section measurements [68,69], respectively. Finally, in Fig. 2d we present the 0 0 channels containing at least 2 photons. The generic search [62] and the measurement of the -production cross section [59] constrain the channel. For the remaining channels, the most important analysis is a (multi-)photon search [58]. S ± S 0 tbbb S ± S 0 tbtt S + S tbtb S + + S WtbWtb S ± ± S Wtbtb S 0 S 0 tttt S 0 S 0 ttbb S 0 S 0 bbbb(S 0 S 0 S 0 S 0 Z S 0 S 0 ZZ S 0 S 0 Z Z S 0 S 0 WW (d) 0 0 with di-boson decays with ≥ 2 photons Bounds on the SU(5)/SO(5) pNGBs Investigating simplified model is very useful approach as the limits can be applied to a broad class of models, at least to a certain extent. We investigate now a specific full model with an extended EW scalar sector, study the bounds on the full model and compare the results to estimates one can very quickly obtain by using the simplified model approach of the previous section. We take the SU(5)/SO(5) coset [25] as an example as it features a doubly charged scalar. We first summarise some key elements and discuss some details of the underlying LHC phenomenology. For detailed discussions and the underlying couplings we refer to refs. [26,27]. The pNGBs of the EW sector form a 14 of SO(5) which decomposes with respect to the custodial SU(2) × SU(2) as 14 → (3, 3) + (2, 2) + (1, 1) . We identify the (2, 2) with the Higgs doublet. The bi-triplet can be decomposed under the custodial SU(2) ⊂ SU(2) × SU(2) as [25] (3, 3) → 1 + 3 + 5 ≡ 1 + 3 + 5 , with 1 = 0 1 , 3 = ( + 3 , 0 3 , − 3 ), 5 = ( ++ 5 , + 5 , 0 5 , − 5 , −− 5 ).(5) This basis is suggested by the fact that the vacuum of the strong sector preserves the custodial SU(2) . In the following we neglect a possible mixing and assume that the three multiplets have common masses 1 , 3 and 5 , respectively, to simplify the analysis. Mass differences are due to the EW symmetry breaking, hence one naively expects a relative mass split of the order / ( = 1, 3, 5) with being the vev of the Higgs boson. The LHC signatures of pNGB pair production depend strongly on whether the pNGBs are fermiophilic or fermiophobic as already mentioned above. We start with a brief discussion of the fermiophobic case and refer to ref. [27] for further details. The singly charged states decay as + 3,5 → + , + ,(7) with dominant photon channel as Br( + 3,5 → + ) ≈ cos 2 ≈ 78% [25] for both multiplets in case of a small mass split between the multiplets. The neutral singlet and quintuplet decay dominantly as 0 1,5 → , , ,(8) with comparable branching ratios, again for small mass split. The only available decay channel for the doubly charged pNGB in the quintuplet is ++ 5 → + + .(9) Finally, the 0 3 is CP-even and thus undergoes three-body decays via off-shell pNGBs: The discussion so far applies to the lightest multiplet and also covers scenarios where the multiplets are very close in mass. However, there could be a sizeable mass split. In such a case, cascade decays from one multiplet into a lighter one and a (potentially off-shell) vector boson become important. Taking the case 5 > 3 > 1 as an example, we have ++ 5 → +( * ) + 3 , + 5 → ( * ) + 3 , +( * ) 0 3 , 0 5 → ±( * ) ∓ 3 , ( * ) 0 3 ,(11a)+ 3 → +( * ) 0 1 , 0 3 → ( * ) 0 1 . (11b) One finds that both classes of decays are of similar importance once the mass split is between 30 and 50 GeV [27]. We note for completeness, that the 5 multiplet does not couple to 0 1 in the model considered. One expects in the fermiophilic case that the couplings scale like the quark masses, e.g. 0 = , 0 = and + = ,(12) where is the decay constant of the 14-plet and the coefficients are of order one. In this case the decays to third generation quarks dominate over the loop-level anomaly-induced decays into two vector bosons or the three-body decays discussed above and, thus, we consider the decays + 3,5 →¯, 0 1,3,5 →¯,¯.(13) From Eq. (12), the¯channel dominates over¯above threshold. In the case of ++ 5 , the three-body decay ++ 5 → +¯ ( 14) via an off-shell + 3,5 dominantes over the decay to + + . In case of 5 > 3 also the decay ++ 5 → +( * ) + 3 becomes important [27]. We consider in a first step only the quintuplet 5 and apply the simplified model bounds from the previous section. In Fig. 3a we compare the cross section times branching ratio of all multiphoton final states (solid lines) with the corresponding bounds from Fig. 2 (dashed lines). From the individual channels one finds that masses below 340 GeV are excluded, with the strongest bound coming from the channel ± 5 0 5 → . We perform in addition a full simulation in which all states contained in the quintuplet are pair-produced and decayed. The solid green line denotes the sum over all pair production cross sections of the quintuplet and the dashed green line the corresponding bound, i.e. the sum of scalar pair production cross sections that would be needed in order to exclude the convolution of all decay channels from quintuplet states. As can be seen, one obtains a bound of 485 GeV on the mass which is significantly higher than the bounds obtained from individual channels. The apparent discrepancy between simplified models and the full simulation can be understood from the fact that all multi-photon channels populate the same signal region of the search presented in ref. [58]. Adding up the various signal cross sections with two or more photons gives the blue line in Fig. 3b. This summed cross section yields an estimated bound on of 460 − 500 GeV when compared with the bounds from different multi-photon channels (shaded area in Fig. 3b). This is in agreement with the result of the full simulation. This example shows the usefulness and limitations of the simplified model bounds and demonstrate how they can be combined in the context of a particular model. In a second step, we take all multiplets into account and consider two scenarios which are characterised by varying a single mass scale : S-eq: 3 = − 2 GeV, 5 = , 1 = + 2 GeV ; (15a) S-135: 1 = − 50 GeV, 3 = , 5 = + 50 GeV .(15b) The choice of 50 GeV is motivated by the fact that the mass splits are expected to be a fraction of the Higgs vev. The phenomenology differs in the two cases: In scenario S-eq, all particles decay via the anomaly and 0 3 exhibits three-body decays. We introduce a small mass split of 2 GeV to avoid a cancellation for some 0 3 decays as discussed in ref. [27]. In scenario S-135 the heavier states decay into the next lighter states or di-bosons, while the lightest state only has anomaly induced decays. We present the bounds on the mass parameter for the two scenarios in Fig. 4. In orange the sums over all scalar pair production cross sections 95 is shown that would be needed to exclude the model at 95% CL at each parameter point. The strongest bounds come from multi-photon channels, with ref. [58] being the dominant analysis. Note, that the kink in 95 is due to a change in dominant signal region within the same analysis. In blue, the actual sum over all pair production cross sections in this model is drawn. The different bounds for these scenarios considered are due to the relative size of the cross section for the triplet and quintuplet. We turn now to the scenarios in which the pNGBs couple dominantly to quarks. In these scenarios, one has single scalar production via the processes → 0¯a nd → ±(16) induced by strong interactions. In addition, the couplings of the neutral scalars to quarks induce couplings to gluons and photons at the one-loop level leading to processes like It turns out that presently available searches can constrain these channels for masses of up to 500 GeV only if the -factors in Eq. (12) are close to 5 and a decay constant of 1 TeV which is only a very small part of the available parameter space. We now turn to Drell-Yan pair production and display our results in Fig. 5. We have assumed that all pNGBs are mass degenerate and that all -factors are 1. Note, that neither branching ratios nor production cross sections depend on . The blue line gives the total cross section summing over all pNGBs irrespective of their decay modes. The orange lines give the exclusion when considering all possible channels. The exclusion is dominated by the results of ref. [60] implemented in CheckMATE. CheckMATE uses the signal region with the strongest expected bound and reports the corresponding observed bound as final result. One obtains the bound given by the solid orange line using this standard procedure. This can lead to difficulties in cases for which observed and expected bounds differ significantly. This is the reason for the kinks at = 350 GeV and 450 GeV. However, modifying the procedure such that always the strongest observed bound is taken, one obtains a smoother curve shown by the dashed orange line, see [27] for further details. → 0 →¯and → 0 → .(17) We note for completeness, that models with and gauge/fermionic underlying dynamics [10,28,29] also predict pNGBs which are strongly interacting. The SU(5)/SO(5) coset is for example realized in the model M5 of ref. [30] which predicts beside the electroweak pNGBs discussed above also a color triplet pNGB 3 and a color octet pNGB 8 . The 8 decays either dominantly into or depending on whether we are in a fermiophilic or a fermiophobic scenarios, respectively. In both cases currently available analyses give a bound of about 1.1 TeV on its mass [71]. The 3 has the same quantum numbers as a right-handed stop, the supersymmetric partner of the right-handed top-quark, and its phenomenology depends on the mass spectrum of the baryonic bound states [72]. This baryonic bound states contain two color neutral fermions denoted by˜andh in [72] where the first one is a gauge singlet and the second an (2) doublet with hypercharge 1/2. It is possible that this states, being color singlets, can be lighter than 3 . In such a scenario the 3 decays into one of these fermions and an SM-fermion: In addition we have taken as a concrete example a composite Higgs model based on the coset SU(5)/SO (5), which features a custodial bi-triplet. We show that while the limits on individual channels lead to relatively weak bounds on the scalar masses, significantly stronger bounds can be obtained by combining various pair production channels. Considering various benchmark scenarios, limits on the scalar mass scale around 500 − 700 GeV have been established in the fermiophobic case. The bounds are close to 500 GeV in scenarios in which decays into top and bottom quarks dominate. Figure 2 : 2Upper limits on the cross section of the di-scalar channels from Drell-Yan pair production. The scalars decay to: (a) third generation quarks or (b)-(d) two vector bosons. Both scalars are assumed to have the same mass. The analyses contributing to the bounds are refs. [57-70]. Bounds from sum of multiphoton channels Figure 3 : 3Application of the model-independent bounds to a specific model, the custodial quintuplet 5 from the SU(5)/SO(5) coset. In (a) we determine the bounds from the dominant individual channels by comparing the cross section time branching ratio from the model (solid) with the upper limits from Fig. 2 (dashed). In green we show the results of a full simulation. The blue line in (b) is the sum of the individual multi-photon cross sections shown in (a). Further details are given in the text. Figure 4 : 4Bounds on the pNGB masses for the Drell-Yan production of the full bi-triplet for the benchmark mass spectra defined in Eq.(15). In (a) all masses are approximately equal and in (b) there is a 50 GeV mass split between the multiplets. Figure 5 : 5Bounds on the pNGB masses for the Drell-Yan production of the full bitriplet with decays to third-generation quarks. events of Drell-Yan scalar pairs with decay into the target channel for each scan point. We use MadGraph5_aMC@NLO [39] version 3.3.2 at NLO, in association with the parton densities in the NNPDF 2.3 set [40, 41]. Afterwards, we interface the events with Pythia8 [42] for the decays of the SM particles, showering and hadronisation. The resulting signal events are then analysed with MadAnalysis5 [43-46] version 1.9.60 and CheckMATE [47, 48] version 2.0.34. Both tools reconstruct the events using Delphes 3 →˜,h 0 ,h + .(18)This resembles the decays of stops in supersymmetric models. In principle, decays into lighter families, like˜and˜, are also possible, but in the spirit of composite Higgs models we expect those to be strongly suppressed. Consequently, LHC bounds from stop searches can be directly applied[73][74][75]. For large mass differences this gives a bound of about 1.3 TeV. We note fore completeness, that in case of small mass difference between 3 and˜three-body decays via an off-shell top-quark would become important similar to supersymmetric models[76][77][78]. In case that 3 is lighter than these baryons, lepton or baryon number violating interactions need to be included in order to avoid a stable 3 which require an extension of this model[72]. The simplest possibilities are 3 →¯¯with = , , and ≠(19)or 3 → , with = , , and = , , .(20)The former violates baryon number whereas the latter violates lepton number. Note, that only one of the two interaction types can be present as otherwise the proton could decay at a rate incompatible with experiment. 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[]
[ "Demonstration of a bright and compact source of tripartite nonclassical light", "Demonstration of a bright and compact source of tripartite nonclassical light" ]
[ "Alessia Allevi \nDipartimento di Fisica, Università degli Studi di Milano and C\nItaly and ISI Foundation\nDipartimento di Fisica e Matematica\nNational Laboratory for Ultrafast and Ultraintense Optical Science -C.N.R.-I.N.F.M. and C.N\nU.d.R. Como\nUniversità degli Studi dell'Insubria and CC.N.I.S.MI-22100, 22100, 20133, I-10133, 22100Como, Como, Milano, Torino, Como.I.S.M., U.d.R. Como, I-.N.I.S.M., U.d.R. Milano, I-.N.I.S.M., U.d.R. Como, IItaly, Italy, Italy, Italy\n", "Maria Bondani \nDipartimento di Fisica, Università degli Studi di Milano and C\nItaly and ISI Foundation\nDipartimento di Fisica e Matematica\nNational Laboratory for Ultrafast and Ultraintense Optical Science -C.N.R.-I.N.F.M. and C.N\nU.d.R. Como\nUniversità degli Studi dell'Insubria and CC.N.I.S.MI-22100, 22100, 20133, I-10133, 22100Como, Como, Milano, Torino, Como.I.S.M., U.d.R. Como, I-.N.I.S.M., U.d.R. Milano, I-.N.I.S.M., U.d.R. Como, IItaly, Italy, Italy, Italy\n", "Matteo G A Paris \nDipartimento di Fisica, Università degli Studi di Milano and C\nItaly and ISI Foundation\nDipartimento di Fisica e Matematica\nNational Laboratory for Ultrafast and Ultraintense Optical Science -C.N.R.-I.N.F.M. and C.N\nU.d.R. Como\nUniversità degli Studi dell'Insubria and CC.N.I.S.MI-22100, 22100, 20133, I-10133, 22100Como, Como, Milano, Torino, Como.I.S.M., U.d.R. Como, I-.N.I.S.M., U.d.R. Milano, I-.N.I.S.M., U.d.R. Como, IItaly, Italy, Italy, Italy\n", "Alessandra Andreoni \nDipartimento di Fisica, Università degli Studi di Milano and C\nItaly and ISI Foundation\nDipartimento di Fisica e Matematica\nNational Laboratory for Ultrafast and Ultraintense Optical Science -C.N.R.-I.N.F.M. and C.N\nU.d.R. Como\nUniversità degli Studi dell'Insubria and CC.N.I.S.MI-22100, 22100, 20133, I-10133, 22100Como, Como, Milano, Torino, Como.I.S.M., U.d.R. Como, I-.N.I.S.M., U.d.R. Milano, I-.N.I.S.M., U.d.R. Como, IItaly, Italy, Italy, Italy\n" ]
[ "Dipartimento di Fisica, Università degli Studi di Milano and C\nItaly and ISI Foundation\nDipartimento di Fisica e Matematica\nNational Laboratory for Ultrafast and Ultraintense Optical Science -C.N.R.-I.N.F.M. and C.N\nU.d.R. Como\nUniversità degli Studi dell'Insubria and CC.N.I.S.MI-22100, 22100, 20133, I-10133, 22100Como, Como, Milano, Torino, Como.I.S.M., U.d.R. Como, I-.N.I.S.M., U.d.R. Milano, I-.N.I.S.M., U.d.R. Como, IItaly, Italy, Italy, Italy", "Dipartimento di Fisica, Università degli Studi di Milano and C\nItaly and ISI Foundation\nDipartimento di Fisica e Matematica\nNational Laboratory for Ultrafast and Ultraintense Optical Science -C.N.R.-I.N.F.M. and C.N\nU.d.R. Como\nUniversità degli Studi dell'Insubria and CC.N.I.S.MI-22100, 22100, 20133, I-10133, 22100Como, Como, Milano, Torino, Como.I.S.M., U.d.R. Como, I-.N.I.S.M., U.d.R. Milano, I-.N.I.S.M., U.d.R. Como, IItaly, Italy, Italy, Italy", "Dipartimento di Fisica, Università degli Studi di Milano and C\nItaly and ISI Foundation\nDipartimento di Fisica e Matematica\nNational Laboratory for Ultrafast and Ultraintense Optical Science -C.N.R.-I.N.F.M. and C.N\nU.d.R. Como\nUniversità degli Studi dell'Insubria and CC.N.I.S.MI-22100, 22100, 20133, I-10133, 22100Como, Como, Milano, Torino, Como.I.S.M., U.d.R. Como, I-.N.I.S.M., U.d.R. Milano, I-.N.I.S.M., U.d.R. Como, IItaly, Italy, Italy, Italy", "Dipartimento di Fisica, Università degli Studi di Milano and C\nItaly and ISI Foundation\nDipartimento di Fisica e Matematica\nNational Laboratory for Ultrafast and Ultraintense Optical Science -C.N.R.-I.N.F.M. and C.N\nU.d.R. Como\nUniversità degli Studi dell'Insubria and CC.N.I.S.MI-22100, 22100, 20133, I-10133, 22100Como, Como, Milano, Torino, Como.I.S.M., U.d.R. Como, I-.N.I.S.M., U.d.R. Milano, I-.N.I.S.M., U.d.R. Como, IItaly, Italy, Italy, Italy" ]
[]
We experimentally demonstrate the nonclassical photon number correlations expected in tripartite continuous variable states obtained by parametric processes. Our scheme involves a single nonlinear crystal, where two interlinked parametric interactions take place simultaneously, and represents a bright and compact source of a sub-shot-noise tripartite light field. We analyze the effects of the pump intensities on the numbers of detected photons and on the amount of noise reduction in some details, thus demonstrating a good agreement between the experimental data and a single-mode theoretical description.
10.1103/physreva.78.063801
[ "https://arxiv.org/pdf/0810.4063v1.pdf" ]
119,208,937
0810.4063
047539096ef2bf2b0bd836af917e81a1b7714684
Demonstration of a bright and compact source of tripartite nonclassical light 22 Oct 2008 Alessia Allevi Dipartimento di Fisica, Università degli Studi di Milano and C Italy and ISI Foundation Dipartimento di Fisica e Matematica National Laboratory for Ultrafast and Ultraintense Optical Science -C.N.R.-I.N.F.M. and C.N U.d.R. Como Università degli Studi dell'Insubria and CC.N.I.S.MI-22100, 22100, 20133, I-10133, 22100Como, Como, Milano, Torino, Como.I.S.M., U.d.R. Como, I-.N.I.S.M., U.d.R. Milano, I-.N.I.S.M., U.d.R. Como, IItaly, Italy, Italy, Italy Maria Bondani Dipartimento di Fisica, Università degli Studi di Milano and C Italy and ISI Foundation Dipartimento di Fisica e Matematica National Laboratory for Ultrafast and Ultraintense Optical Science -C.N.R.-I.N.F.M. and C.N U.d.R. Como Università degli Studi dell'Insubria and CC.N.I.S.MI-22100, 22100, 20133, I-10133, 22100Como, Como, Milano, Torino, Como.I.S.M., U.d.R. Como, I-.N.I.S.M., U.d.R. Milano, I-.N.I.S.M., U.d.R. Como, IItaly, Italy, Italy, Italy Matteo G A Paris Dipartimento di Fisica, Università degli Studi di Milano and C Italy and ISI Foundation Dipartimento di Fisica e Matematica National Laboratory for Ultrafast and Ultraintense Optical Science -C.N.R.-I.N.F.M. and C.N U.d.R. Como Università degli Studi dell'Insubria and CC.N.I.S.MI-22100, 22100, 20133, I-10133, 22100Como, Como, Milano, Torino, Como.I.S.M., U.d.R. Como, I-.N.I.S.M., U.d.R. Milano, I-.N.I.S.M., U.d.R. Como, IItaly, Italy, Italy, Italy Alessandra Andreoni Dipartimento di Fisica, Università degli Studi di Milano and C Italy and ISI Foundation Dipartimento di Fisica e Matematica National Laboratory for Ultrafast and Ultraintense Optical Science -C.N.R.-I.N.F.M. and C.N U.d.R. Como Università degli Studi dell'Insubria and CC.N.I.S.MI-22100, 22100, 20133, I-10133, 22100Como, Como, Milano, Torino, Como.I.S.M., U.d.R. Como, I-.N.I.S.M., U.d.R. Milano, I-.N.I.S.M., U.d.R. Como, IItaly, Italy, Italy, Italy Demonstration of a bright and compact source of tripartite nonclassical light 22 Oct 2008(Dated: October 22, 2008)numbers: 4250-p4250Dv4250Ar4265Lm * Electronic address: mariabondani@uninsubriait 1 We experimentally demonstrate the nonclassical photon number correlations expected in tripartite continuous variable states obtained by parametric processes. Our scheme involves a single nonlinear crystal, where two interlinked parametric interactions take place simultaneously, and represents a bright and compact source of a sub-shot-noise tripartite light field. We analyze the effects of the pump intensities on the numbers of detected photons and on the amount of noise reduction in some details, thus demonstrating a good agreement between the experimental data and a single-mode theoretical description. I. INTRODUCTION Multimode light beams endowed with nonclassical correlations, as those exhibited by multipartite entangled states, represent a resource for quantum technology. They are at the heart of enhanced quantum imaging, either ghost imaging or ghost diffraction [1,2], and represent a building block for the development of an integrated quantum network. In turn, nonlinear interactions involving multimode beams of radiation have attracted much attention in the recent years, either to realize all-optical information processing [3] or to generate nonclassical states of light [4]. Several experimental schemes to generate multimode entangled states have been suggested and demonstrated. The first example is provided by the original continuous variable (CV) teleportation experiments in Ref. [5], where one mode of a twin beam was mixed with a coherent state, although no specific analysis was made on the entanglement properties besides the verification of teleportation. A similar scheme, where one mode of a twin beam is mixed with the vacuum, has been demonstrated and applied to controlled dense coding [6]. Moreover, a fully inseparable three-mode entangled state has been generated and verified by mixing three independent squeezed vacuum states in a network of beam splitters [7]. Recently we suggested and demonstrated a compact scheme to realize three-mode entanglement by means of two interlinked χ (2) interactions occurring in a single nonlinear crystal in a type-I non-collinear phase-matching geometry [8,9]. Other schemes involving cascaded interactions have been also analyzed either in periodically poled crystals [10] or in second-order nonlinear ones [11,12,13]. Notice, however, that the use of a single nonlinear medium makes the system more compact and robust compared to the other schemes that have been suggested and demonstrated so far, in which additional parametric sources and linear devices, such as beam splitters, introduce unavoidable losses. Finally, parametric oscillators have been suggested as a source of tripartite signal-idler-pump entanglement in triply resonant cavities [14]. In this paper we experimentally demonstrate the nonclassical photon correlations exhibited by tripartite states generated by a single nonlinear crystal, where two interlinked parametric interactions take place simultaneously. Our scheme realizes a bright and compact source of sub-shotnoise three-mode light beams and allows the implementation of simultaneous ghost imaging and ghost diffraction protocols with enhanced sensitivity. The paper is structured as follows: in the next Section we provide a theoretical description of our system and evaluate correlations and noise reduction as a function of the coupling parameters. In Section III we describe our experimental apparatus, illustrate the results with focus on nonclassical photon-number correlations, and analyze the sources of noise in some details. Section IV closes the paper with some remarks. II. THEORETICAL DESCRIPTION In our scheme two interlinked interactions, namely a spontaneous parametric downconversion process and a sum-frequency generation, take place simultaneously in a single nonlinear crystal. In principle, five modes a j are involved in the interactions, two of which, say a 4 and a 5 , are non-evolving undepleted pumps and thus are included in the coupling coefficients (parametric approximation). The effective Hamiltonian describing the interaction is thus given by H int = g 1 a † 1 a † 3 + g 2 a † 2 a 3 + h.c. ,(1) where g 1 and g 2 are coupling coefficients linearly dependent on the pump fields a 4 and a 5 , respectively. The earliest studies on the dynamics and the quantum properties of the states realized via this Hamiltonian can be traced back to the works in Refs. [15,16]. The relevance of studying the dynamics generated by the above Hamiltonian in details lies in the fact that H int can be realized in a variety of different contexts, from quantum optics [10,12,17,18,19] to condensate physics [20,21]. The coupling between two optical modes and one vibrational mode of a macroscopic object, such as a mirror, has been considered [22] and also ions trapped in a cavity have been demonstrated to realize the Hamiltonian in Eq. (1) for a suitable configuration [23]. The Hamiltonian admits the constant of motion ∆(t) ≡ N 1 (t) − N 2 (t) − N 3 (t) ≡ ∆(0). If we take the vacuum |0 ≡ |0 1 ⊗ |0 2 ⊗ |0 3 as the initial state, we have N 1 (t) = N 2 (t) + N 3 (t) ∀t, being N j (t) = a † j (t)a j (t) the mean number of photons in the j-th mode. Under these hypotheses the evolved state |T = exp{−iH int t}|0 may be written as |T = mr N m/2 2 N r/2 3 (1 + N 1 ) (1+m+r)/2 (m + r)! m!r! |m + r, m, r ,(2) where we omitted the time dependence of N j . As a matter of fact the state in Eq. (2) is a fully inseparable three-mode Gaussian state [24], i.e. a state that is inseparable with respect to any grouping of the modes, thus permitting realizations of truly tripartite quantum protocols such as conditional twin-beam generation and telecloning [17,18]. The mean numbers of photons N j that appear in Eq. (2) can be obtained by the Heisenberg evolution of the field operators. In particular, by introducing Ω = |g 2 | 2 − |g 1 | 2 we have N 1 = N 2 + N 3 and N 2 = |g 1 | 2 |g 2 | 2 Ω 4 [cos Ωt − 1] 2 N 3 = |g 1 | 2 Ω 2 sin 2 (Ωt) .(3) We see that when |g 2 | 2 > |g 1 | 2 the dynamics is oscillatory; viceversa, when |g 1 | 2 > |g 2 | 2 we find an exponential behavior. The above description of the system has been derived under the hypothesis of perfect frequencymatching and phase-matching conditions among single-mode fields and the time t appearing in Eqs. (3) represents the interaction time inside the crystal. In this case we did not need to take into account the existence of temporal modes and spatial coherence areas. On the other hand, if the pump fields are pulsed, the generated fields are temporally multimode [25]. Moreover, in a non-collinear interaction geometry, the momentum conservation in the transverse direction can be satisfied in more than one way. Thus coherence areas exist [24,26], whose angles of divergence depend on several parameters, such as the pumps intensities, the distance from the collinear interaction geometry and the wavelengths of the generated fields. It is interesting to point out that in the CV regime the demonstration of the entangled nature of the state in Eq. (2) critically depends on the correct collection of these coherence areas [27]. In fact, collecting light from more than a single coherence area corresponds to the introduction of spurious light, while collecting less than a coherence area determines a loss of information, which is detrimental to the investigations of the nonclassical properties. In addition, we have to select a triplet of areas as there is a one-to-one correspondence between the coherence areas in each field. To achieve such a selection we can apply a criterion which represents a necessary but not sufficient condition, based on the study of the correlation in the number of photons. In fact, due to the constant of motion, the state in Eq. (2) is endowed with perfect correlations in the number of photons. The three-mode photon-number distribution is given by P T (n, m, r) = δ n,m+r N m 2 N r 3 (1 + N 1 ) 1+m+r (m + r)! m!r! ,(4) from which we can derive the photon-number correlation coefficients between the components of the entangled state. In particular, due to the conservation law, we expect the existence of strong intensity correlations between the number of photon n 1 and the sum of the other two, say n 2 + n 3 . In order to quantify correlations we denote by γ(n j , n k ) = n j n k − n j n k and σ 2 (n j ) = n j 2 − n j 2 the covariance and the variance of the number of photons, respectively, and introduce the correlation coefficients as follows ǫ j,k = γ(n j , n k ) σ(n j )σ(n k ) .(5) Upon exploiting Eq. (4) we have that the correlation coefficient ǫ 1,2+3 is identically equal to one, independently of the number of photons generated by the interlinked interactions. On the other hand, for the partial photon-number correlations we obtain expressions that do depend on the mean number of photons involved. Upon writing N k = β k N where β 1 = β 2 + β 3 and N is the total number of photons of the state we have ǫ 1,k = N k (1 + N 1 ) N 1 (1 + N k ) N ≫1 ≃ 1 − β 1 − β k 2β 1 β k N (6) ǫ 2,3 = N 2 N 3 (1 + N 2 )(1 + N 3 ) N ≫1 ≃ 1 − β 2 + β 3 2β 2 β 3 N(7) where from now on k = 2, 3. As the detectors we used to perform the correlation measurements are not ideal, we have to rewrite the expressions of the correlation coefficients by taking into account the non-unit quantum efficiency of the detection apparatus. The probability operatorvalued measure (POVM) of each detector, describing the statistics of detected photons, is given by a Bernoullian convolution of the ideal number operator spectral measurê Π m j = η j m j ∞ n j =m j (1 − η j ) n j −m j   n j m j   |n j n j |(8) with j = 1, 2, 3. Equation (8) can be exploited to calculate the expressions of mean number, m j , and variance, σ 2 (m j ), of the detected photons m j in terms of the mean number of the photons n j and of its variance σ 2 (n j ) [31] M j ≡ m j = η j n j = η j N j (9) σ 2 (m j ) = η 2 j σ 2 (n j ) + η j (1 − η j )N j We notice that, in general, the statistical distribution of the number of detected photons is different from that of the number of photons. Nevertheless, the correlation coefficients, ǫ m , calculated for the detected photons can also assume high values; in particular, the correlation coefficient calculated between m 1 and the sum m 2 + m 3 reads as follows ǫ m 1,2+3 = η(1 + N 1 ) (1 + ηN 1 ) N ≫1 ≃ 1 − 1 − η η 1 β 1 N(10) where we have assumed that all the detectors have the same quantum efficiency η. In turn, the partial correlations are given by ǫ m 1,k N ≫1 ≃ 1 − β 1 + β k − 2ηβ k 2β 1 β k N (11) ǫ m 2,3 N ≫1 ≃ 1 − β 2 + β 3 2ηβ 2 β 3 N ,(12) and approach unit value for large N values. As a matter of fact a large value of the correlation indices is not sufficient to discriminate between quantum and classical correlations [31]. A trivial example is given by the mixture ̺ = nmr P T (n, m, r)|n n| ⊗ |m m| ⊗ |r r|, which, with P T (n, m, r) given as in Eq. (4), exhibits the same correlations of the state |T . A more realistic example is provided by the tripartite state generated by sending a thermal state on two subsequent beam-splitters, whose second port is unexcited: the state is classical and shows large intensity correlations, approaching unit value for large mean photon numbers [28]. In order to obtain a proper marker of nonclassicality we may take into account the difference photocurrents d j,k = m j − m k [27] and build the so-called noise reduction factor R j,k = σ 2 (d j,k ) m j + m k ,(13) which is smaller than one for nonclassically correlated states. Note also that, for states generated by the Hamiltonian in Eq.(1), the existence of sub-shot noise photon-number correlations is a sufficient condition for entanglement, i.e. the condition of negative partial transpose is subsumed by the condition of sub-shot noise correlations [18]. By using Eqs. (9) we may write R j,k = 1 − η + η [σ 2 (n j ) + σ 2 (n k ) − 2γ(n j , n k )] n j + n k ,(14) for the noise reduction of bipartite correlations whereas, for the difference photocurrent between the mode a 1 and the sum of the other two modes, we have R ≡ R 1,2+3 R = 1 − η + η p σ 2 (n p ) + 2Γ({n k }) p n p(15) where Γ({n k }) = γ(n 2 , n 3 ) − γ(n 1 , n 2 ) − γ(n 1 , n 3 ) .(16) For the state in Eq. (2) we have R = 1 − η ,(17) which shows that state |T exhibits nonclassical tripartite correlations for any value of the mean number of photons. Besides, Eq. (17) says that the noise reduction can be detected for any value of the quantum efficiency η. The corresponding bipartite quantities read as follows R 1,k = 1 + η [(N 1 − N k ) 2 − 2N k ] N 1 + N k N ≫1 ≃ ηN (β 1 − β k ) 2 β 1 + β k (18) R 2,3 = 1 + η(N 2 − N 3 ) 2 N 2 + N 3 N ≫1 ≃ ηN (β 2 − β 3 ) 2 β 2 + β 3 ,(19) and say that the correlations between modes a 2 and a 3 are always classical whereas the correlations between mode a 1 and either mode a 2 or mode a 3 may be nonclassical in certain regimes. More specifically, we have R 1,k < 1 if N 1 < N k + √ 2N k . Since N 1 = N 2 + N 3 we may have both the noise reduction parameters below the classical threshold only for an overall energy of the state N 1 + N 2 + N 3 < 4. III. EXPERIMENT The experimental scheme used to generate the nonclassical state of Eq. (2) is depicted in Fig. 1. The harmonics of a continuous-wave mode-locked Nd:YLF laser regeneratively amplified at a repetition rate of 500 Hz (High Q Laser Production, Hohenems, Austria) provide the two pump fields. In particular, the third harmonic pulse at 349 nm (∼ 4.45 ps pulse-duration) is exploited as the pump field a 4 in the downconversion process, whereas the fundamental pulse at 1047 nm (∼ 7.7 ps pulse-duration) is used as the pump field a 5 in the upconversion process. The two processes must simultaneously satisfy energy-matching (ω 4 = ω 1 + ω 3 , ω 2 = ω 3 + ω 5 ) and phase- matching (k e 4 = k o 1 + k o 3 , k e 2 = k o 3 + k o 5 ) conditions, in which ω j are the angular frequencies, k j are the wavevectors and suffixes o, e indicate ordinary and extraordinary field polarizations. As depicted in Fig. 2, we set the pump-field a 4 direction so that the wavevector k 4 is normal to the crystal entrance face and propagates along the z-axis of the medium. We also align the wavevector With reference to Fig. 2, we indicate as ϑ j the angles in the plane (y, z) formed by each wavevector with k 4 and as β j the angles of each wavevector with respect to this plane. For the experimental realization of the interaction scheme we choose the solutions in the plane (y, z), thus β j = 0 for j = 1 − 3: in particular, we sent the pump field a 5 at an external angle ϑ 5,ext = −24.47 deg with respect to the other pump field a 4 . Under these hypotheses, for λ 1 = 632.8 nm, λ 2 = 446.4 nm and λ 3 = 778.2 nm, we calculated the following external interaction angles with respect to the pump field a 4 : ϑ 1,ext = −9.78 deg, ϑ 2,ext = −3.25 deg and ϑ 3,ext = +12.06 deg [9]. The preliminary use of a He:Ne laser as the seed field allowed us to position three pin-holes on the path of the three generated fields in such a way that then, when operating the system from vacuum (i.e. in the absence of any seed fields), we could collect a triplet of coherence areas. Distances and sizes of the pin-holes were chosen by searching for the condition of maximum intensity correlations between the generated fields [28]. In fact, as shown in Section II, we expect strong intensity correlations not only between the number of detected photons m 1 and the sum of the other two, but also between m 1 and m 2 , m 2 and m 3 and m 1 and m 3 . By applying this criterion, we finally decided to put two pin-holes of 30 µm diameter at distances d 1 = 60 cm and d 3 = 49 cm from the BBO along the path of the signal beam at 632.8 nm and of the idler beam at 778.2 nm, respectively. The two different distances were chosen to compensate for the difference in the divergence of signal and idler due to their wavelengths [24]. Moreover, as the beam at 446.4 nm has a divergence smaller than those of the other two fields, we selected it by means of a 50 µm diameter pin-hole placed at a distance d 2 = 141.5 cm from the crystal. The light, suitably filtered by means of bandpass filters located in front of each pin-hole, was focused on each detector by a lens (f 1 = f 3 = 25 mm, f 2 = 10 mm). Since we performed measurements in the macroscopic intensity regime (more than 1000 photons per coherence area), we used three p-i-n photodiodes (two, D 1,2 in Fig. 1, S5973-02 and one, D 3 , S3883, Hamamatsu, Japan) as the detectors. In order to obtain the same overall detection efficiency (bandpass filter plus detector) on the three arms, we put two adjustable neutral-density filters in the pathways of a 2 and a 3 , thus obtaining the same value η = 0.28 on the three arms. The current output of the detectors was amplified by means of two low-noise charge-sensitive pre-amplifiers (CR-110, Cremat, Watertown, MA) followed by two amplifiers (CR-200-4 µs, Cremat). We connected the detectors D 2 and D 3 to the same amplifier device by means of a T-adapter. The two amplified outputs were then integrated by synchronous gated-integrators (SGI in Fig. 1 In the following we discuss the measurements of the intensities of field a 1 and of the sum a 2 + a 3 as, according to Eqs. (17)- (19), we expect a nonclassical behavior. Partial measurements performed by alternatively blocking the light impinging on the detectors D 2 and D 3 were not very reliable as the numbers of detected photons on the two fields separately were too close to the electronic noise of the detection chain. This is an important drawback as, for all calculations that follow based on experimental data, we must take into account the electronic noise that we measured in the absence of light [27]. As the pump fields are pulsed and their duration is longer than the characteristic time of the nonlinear processes, the distributions of the detected photons collected by the pin-holes are temporally multimode [25]. The same is also true for the statistical distribution of the sum m 2 + m 3 . Moreover these distributions should be characterized by the same number of modes [24]. From the experimental point of view, the main difficulty to be overcome was the correct selection of a triplet of coherence areas. In fact, in the CV domain we have to avoid spurious light that could be detrimental to the experimental results; moreover, the interaction scheme presented here involves not only two generated fields, but also a third one, which obviously makes the detection more critical. Finally, we have two pump fields instead of one and in particular we are not able to exactly measure the effective portions of them that interact into the crystal. In spite of all these difficulties, we characterized the state produced by the interlinked interactions and in particular we proved its quantum nature by performing sub-shot noise photon-number correlation measurements as a function of the pump fields intensities. In fact, as remarked in Section II, the evaluation of the noise reduction factor R for the distribution of the difference photocurrent d = m 1 − (m 2 + m 3 ) provides a sufficient condition in order to test the quantum nature of the generated state. We firstly investigated the evolution of the mean number of photons as a function of the intensity of one of the two pumps by keeping fixed the intensity of the other one [29]. In fact, if on one hand this analysis allows us to verify that the mean number of photons does not depend on the correct selection of the coherence areas, on the other one it is essential for the determination of the effective values of the pump fields intensities from the fitting curves. As a first check we studied the evolution of the mean number of detected photons, M 1 and M 2 + M 3 , as a function of the intensity of field a 4 for a fixed value of the intensity of field a 5 . Note that temporal evolution in Eqs. (3) is transformed into spatial evolution by identifying |g 2 | 2 − |g 1 | 2 t with |γ 2 | 2 − |γ 1 | 2 z, z being the effective interaction length [30]. In the experimental condition each M j represents the total mean number of photons detected beyond each pin-hole; actually, it can be expressed as M j = µ m j , where µ is the number of temporal modes and m j the average population of each mode. To vary the intensity of field a 4 , we changed its energy by means of an adjustable neutral-density filter. For each energy value, measured by means of a movable thermal detector (D 4 in Fig. 1, mod. 03A-P-CAL-SH, Ophir Optronics Ltd., Jerusalem, Israel), we measured the mean number of photons by averaging over 50000 subsequent laser shots. In We found η 1 = 0.31 and η sum = 0.28. The difference is within the error justified by the tolerance of the pin-holes sizes (∅ 1 = ∅ 3 = 30 ± 2 µm and ∅ 2 = 50 ± 3 µm) and it is also justified by possible imperfections in the positioning of the pin-holes at the right distances from the crystal. As a second check, we studied the evolution of M 1 and M 2 + M 3 as a function of the intensity of field a 5 , by keeping the intensity of field a 4 fixed. To change the energy of field a 5 we placed a half-wave plate on the pathway of the infrared pump field. A movable thin-film plate polarizer was used to measure the energy fraction corresponding to the ordinarily polarized component of the field for each step of rotation of the λ/2 plate. For each energy value, measured by means of the thermal detector (D 5 in Fig. 1), we measured the mean number of photons by averaging over 50000 subsequent laser shots. In Fig. 3(b), we show the measured values of M 1 and M 2 + M 3 as functions of |γ 2 | 2 , for a fixed value of |γ 1 | 2 . Also in this case, the experimental data are plotted together with the fitting curve of the two sets of data obtained from Eqs. (3). Obviously, we have to interchange the roles of the pumps: in fact, here |γ 1 | 2 is treated as the parameter and |γ 2 | 2 as the variable. In particular, we obtained |γ 1 | 2 = 1.52 × 10 6 m −2 and |γ 2 | 2 in the range (1.97 × 10 4 − 1.27 × 10 5 ) m −2 . Even in this case, the experimental data satisfy the conservation law as they are almost superimposed and the optimization of the quantum efficiencies still gives very small corrections: η 1 = 0.283 and η sum = 0.28. By exploiting the values of the pump fields intensities obtained from the fitting curves, we can investigate the behavior of the correlation coefficient ǫ m 1,2+3 (see Eq. (10)) and of the noise reduction R (see Eq. (17)). First of all, in Fig. 4 we show the intensity correlation coefficient, (a), and the noise reduction, (b), as functions of |γ 1 | 2 by keeping fixed the value of |γ 2 | 2 . During these measurements the collection areas were kept fixed (same pin-holes located at the same distances as above). The variation of the correlation coefficient as a function of the pump field intensity through |γ 1 | 2 is indeed not so strong, but the noise reduction factor is critically dependent on the changes in the intensity value. In fact, very much as in the case of the twin-beam state [27], there is an optimum condition at which R is minimum and, correspondingly, the value of the correlation coefficient is maximum. Note that |γ 1 | 2 is larger than |γ 2 | 2 in the entire range of variation. Moreover, we note that increasing the pump intensity, hence |γ 1 | 2 , also increases the size of the coherence areas so that they are only partially transmitted by the pin-holes. On the other hand, lowering the pump intensity reduces the size of the coherence areas and allows uncorrelated light to pass the pin-holes. Note that the values of R corresponding to the selection of more than a single coherence area remain quite close to the shot-noise limit as the information contained in the area is not lost, but only made more noisy. On the contrary, the selection of only a part of the areas causes a loss of information that determines a more remarkable increase of R above the shot-noise limit (note the axis break). This result represents an indication of the need of a perfect matching of the pin-hole areas in order to obtain sub-shot noise correlations. Secondly, we investigated the intensity correlation coefficient and the noise reduction as functions of |γ 2 | 2 by keeping fixed the value of |γ 1 | 2 . Also in this case the collection areas were kept fixed by using the same pin-holes as before located at the same distances. The intensity regime in which these measurements were performed is different from the previous one as the absolute values of the two pump fields intensities are smaller than in the other case. However, |γ 2 | 2 is again smaller than |γ 1 | 2 in all its range of variation. For all these reasons, the variations in the experimental values of the correlation coefficient and of the noise reduction are smaller (see Fig. 5). Moreover, the minimum value of R is quite near to the lower limit R min = 0.72 fixed by the quantum efficiency. In fact, the use of less intense pumps reduces the quantity of spurious light that can be revealed by the detectors; in addition, the fluctuations of the laser source and the possible discrepancy of its photon-number distribution with respect to the ideal Poissonian statistics play a less important role [31]. As a further investigation, we performed other measurements in order to study how critical is the sub-shot noise condition with respect to a slight change in the values of the intensities of the two pump fields. In particular, we verified that it is always possible to choose the pumps in such a way that only micro-metric adjustments of the pin-holes positions are necessary to select the coherence areas. In Fig. 6 we show a number of sub-shot noise measurements obtained for different pairs of pump values. As we can see, not all the measurements reach the optimum minimum value, R ∼ 0.72, due to residual imperfections in the selection of the coherence areas. In particular, as we remarked above, this operation is more critical when the intensity values are higher because other noise sources become important [31]. However, we want to emphasize that there are many possible choices of the pumps values that allow us to perform sub-shot noise measurements thus demonstrating that our scheme is particularly versatile and useful for several applications in different photon-number regimes. IV. CONCLUSIONS AND OUTLOOKS In conclusion, we have presented the experimental realization of an entangled state that involves three modes of radiation in the macroscopic regime. We verified the quantum nature of the state produced by our all-optical interaction scheme by means of sub-shot noise photon-number correlations, which also subsumes the inseparability condition. In particular, we investigated how critical is the sub-shot noise condition by studying its dependence on the intensities of the two pump fields. In spite of the difficulties in measuring the light of triplet coherence areas correctly and in avoiding the detection of spurious light, we obtained quite relevant results that could be further optimized. In the immediate future we plan to use three acquisition chains instead of only two to separately and simultaneously detect the three fields. Moreover, in order to reduce the noise that is detrimental to the shot-noise reduction factor we still intend to operate in the macroscopic regime, but with lower numbers of photons and to use hybrid photodetectors, which are endowed with a reasonable quantum efficiency (η ≃ 0.4) and a linear response in the mesoscopic regime (up to a few hundreds of detected photons), instead of the p-i-n photodiodes. We also plan to modify our collection system by using optical fibers in order to avoid spurious light and to minimize the uncertainty in the collection areas. The experimental improvements would make the whole system more easily controllable and suitable for several applications, such as the production of conditional twin-beam states and the generation of quasi-Fock states with a number of photons sensibly greater than one. Overall, our system represents a robust and tunable scheme to obtain nonclassical photon number correlations in tripartite CV systems thus allowing the simultaneous realization of ghost-imaging and ghost-diffraction with enhanced sensitivity. This work has been supported by MIUR projects PRIN-2005024254-002 and FIRB- k 5 of the other pump field a 5 in the plane (y, z) containing the optical axis (OA) of the crystal and the wavevector k 4 . The nonlinear medium is a β-BaB 2 O 4 crystal (BBO, Fujian Castech Crystals, China, 10 mm × 10 mm cross section, 4 mm thickness) cut for type-I interaction (ϑ cut = 38.4 deg), into which both pumps are strongly focused. Typical intensity values of the pumps were ∼ 5 GW/cm 2 for a 4 and ∼ 2 GW/cm 2 for a 5 . The required superposition in time of the two pumps is obtained by a variable delay line. ) operating in external trigger modality (SR250, Stanford Research Systems, Palo Alto, CA). The voltage outputs were then sampled, digitized by a 12-bit converter (AT-MIO-16E-1, DAQ National Instruments) and recorded by a computer. Fig. 3 3(a), we show the measured values of M 1 and M 2 + M 3 as functions of |γ 1 | 2 , for a fixed value of |γ 2 | 2 . Note that |γ 1 | 2 ∝ E 4 /(πr 2 4 ω 4 τ 4 ), E 4 being the pulse energy of field a 4 , τ 4 the pulse duration and r 4 the beam radius. The experimental data are displayed together with their common fitting curve, obtained from Eqs. (3) with |γ 2 | 2 as the parameter and |γ 1 | 2 as the variable. In this case we get |γ 2 | 2 = 8.17 × 10 5 m −2 and |γ 1 | 2 in the range 1.86 × 10 6 − 2.17 × 10 6 m −2 . Note that, as expected, the experimental data satisfy the photon-number conservation law as they are almost superimposed. The best fitting curve has been obtained allowing a slight difference in the quantum efficiency values of the detection chains and finding the values from the conservation law. FIG. 1 : 1Scheme of the experimental setup: BBO, nonlinear crystal; NF, variable neutral-density filter; λ/2, half-wave plate; TFP, thin-film plate polarizer; P 1−3 , pin-holes; f 1−5,5 ′ , lenses; D 1−3 , p-i-n photodiodes; D 4,5 , thermal detectors; M, Aluminum mirrors; PRE+AMP, low-noise charge-sensitive pre-amplifiers followed by amplifiers; SGI, synchronous gated-integrator; ADC+PC, computer integrated digitizer. FIG. 2: Scheme of the phase-matched interlinked interactions: (x, y)-plane coincides with the crystal entrance face; α, tuning angle; β j 's, angles to (y, z)-plane; ϑ j 's, angles on the (y, z)-plane; ϕ, angle to the optical axis (OA). FIG. 3 :FIG. 4 :FIG. 5 :FIG. 6 : 3456(a) Evolution of the mean numbers of detected photons as a function of |γ 1 | 2 which is proportional to the intensity of field a 4 for |γ 2 | 2 = 8.17 × 10 5 m −2 . Black circles: measured values of M 1 ; grey triangles: measured values of M 2 + M 3 ; solid straight line: fitting curve. (b) Evolution of the mean numbers of detected photons as a function of |γ 2 | 2 which is proportional to the intensity of field a 5 for |γ 1 | 2 = 1.52 × 10 6 m −2 . Black circles: measured values of M 1 ; grey triangles: measured values of M 2 + M 3 ; solid straight line: fitting curve. (a) Intensity correlation coefficient and (b) quantum noise reduction R (note the axis break) as functions of |γ 1 | 2 for |γ 2 | 2 = 8.17 × 10 5 m −2 . (a) Intensity correlation coefficient and (b) quantum noise reduction R (note the axis break) as functions of |γ 2 | 2 for |γ 1 | 2 = 1.52 × 10 6 m −2 . Noise reduction, R, as a function of |γ 1 | 2 and |γ 2 | 2 . . M Angelo, Y Shih, Laser Phys. Lett. 12567M. D'Angelo, Y. Shih, Laser Phys. Lett. 12, 567 (2005). . A Gatti, E Brambilla, L A Lugiato, Progress in Optics. 51251A. Gatti, E. Brambilla, L. A. Lugiato, Progress in Optics 51, 251 (2008). . T Kartaloglu, Z G Figen, O Aytür, J. Opt. Soc. Am. B. 20343and references thereinT. Kartaloglu, Z.G. 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[ "Toward Better Physics Labs for Future Biologists", "Toward Better Physics Labs for Future Biologists" ]
[ "K Moore \nDepartment of Physics\nUniversity of Maryland\n20742College ParkMD\n", "J Giannini \nBiophysics Program\nUniversity of Maryland\n20742College ParkMD\n", "&amp; W Losert \nDepartment of Physics\nUniversity of Maryland\n20742College ParkMD\n\nBiophysics Program\nUniversity of Maryland\n20742College ParkMD\n" ]
[ "Department of Physics\nUniversity of Maryland\n20742College ParkMD", "Biophysics Program\nUniversity of Maryland\n20742College ParkMD", "Department of Physics\nUniversity of Maryland\n20742College ParkMD", "Biophysics Program\nUniversity of Maryland\n20742College ParkMD" ]
[]
We have developed a set of laboratories and hands on activities to accompany a new two-semester interdisciplinary physics course that has been successfully developed and tested in two small test classes of students at the University of Maryland, College Park (UMD) in 2012-2013. We have designed the laboratories to be taken accompanying a reformed course in the student's second year, with calculus, biology, and chemistry as prerequisites. This permits the laboratories to include significant content on physics relevant to cellular scales, from chemical interactions to random motion and charge screening in fluids. We also introduce the students to research-grade equipment and modern physics analysis tools in contexts relevant to biology, while maintaining the pedagogically valuable open-ended laboratory structure of reformed laboratories. Preliminary student results from these two small test classes are discussed.
10.1119/1.4870388
[ "https://export.arxiv.org/pdf/1308.3882v1.pdf" ]
87,053,142
1308.3882
6f972a63c5abe64c10d51d24bd00c4edf6875b10
Toward Better Physics Labs for Future Biologists K Moore Department of Physics University of Maryland 20742College ParkMD J Giannini Biophysics Program University of Maryland 20742College ParkMD &amp; W Losert Department of Physics University of Maryland 20742College ParkMD Biophysics Program University of Maryland 20742College ParkMD Toward Better Physics Labs for Future Biologists Moore, Giannini, & Losert Better Physics Labs for Biologists 1 We have developed a set of laboratories and hands on activities to accompany a new two-semester interdisciplinary physics course that has been successfully developed and tested in two small test classes of students at the University of Maryland, College Park (UMD) in 2012-2013. We have designed the laboratories to be taken accompanying a reformed course in the student's second year, with calculus, biology, and chemistry as prerequisites. This permits the laboratories to include significant content on physics relevant to cellular scales, from chemical interactions to random motion and charge screening in fluids. We also introduce the students to research-grade equipment and modern physics analysis tools in contexts relevant to biology, while maintaining the pedagogically valuable open-ended laboratory structure of reformed laboratories. Preliminary student results from these two small test classes are discussed. I. INTRODUCTION The increasingly urgent calls for changes to the undergraduate introductory physics curriculum for life sciences majors (IPLS) 1,2,3,4,5 have spurred much activity in the physics community. The physics education community has been re-examining our current curricular methods and choices, and biological physicists have been working to make significant changes to the physics content covered in introductory classes, with both communities striving to provide a better physics foundation for aspiring life scientists. The NEXUS/Physics project, 6 described in detail elsewhere in this volume, 7 has brought together physics education researchers and biological physicists to tackle a reform of both physics content and pedagogy, with significant input and contributions from the biomedical community. As part of this project, we have endeavored to create a new IPLS laboratory curriculum focused on physics relevant to living systems that blends epistemic commitments 8 with authentic biological contexts 7,9 as a first step toward better physics labs for budding biologists. Our laboratory development team, which includes both biological physicists and physics education researchers, zeroed in on two main aspects of the physics foundation for future life scientists that stand out as particularly relevant to new laboratories. First, from a physics education research perspective, physics provides a platform to develop modeling skills, the ability to frame observations in the context of models, and in particular mathematical, quantitative models, as we will describe in detail later. This is a well-recognized foundation where well-designed laboratories may make a significant contribution. Indeed quantitative modeling, which traditionally had been used in only a few areas of biology such as ecology and neurosci-ence, is now becoming part of virtually all biological and biomedical research, driven by a rapid surge in quantitative biological information. Second, from a biological physics perspective, the physics needed to gain insights into living systems is different from the physics taught in many introductory courses. 10,11 Physics at the cellular scale, for example, involves Brownian motion and entropy, as well as fluid flow, ionic charges, and charge screeningtopics not usually covered in physics classes for life scientists. Similarly, the physics topics relevant at the scale of organisms or populations, such as scaling relations, are not usually the focus of introductory classes or labs. Many of the physical principles relevant to living systems, especially physics relevant to nanoscopic systems such as proteins and DNA, and microscopic systems such as cells, are very distinct from the physics of macroscopic objects that students encounter in everyday life. Thus the introductory physics laboratories provide the unique opportunity to build from scratch hands-on experience with forces and motion at the nanoscopic and microscopic scales. Before describing our proposed IPLS lab curriculum, it is worthwhile to consider the styles of curricula currently available. At most institutions, introductory physics students are given either a "traditional" or a "reform" laboratory curriculum. While these curricula have their advantages, we argue that they are inadequate when facing the challenge of educating this new generation of life sciences students. "Traditional" laboratory curricula, of the explicitlydirected, "cookbook" variety, have the advantages of covering a wide range of physics content and employing analysis methods that can be quite sophisticated. These labs are often used to demonstrate theoretical principles already presented in the lecture portion of the course. The disadvantages directly resulting from the breadth of topic coverage and the intricate mathematical analysis are that these labs require heavyhanded guidance that often has the down-side of stifling student attempts to engage in sense-making. 12 Students often view these types of labs as a sequence of disconnected steps that must be followed, which have no relation to each other or to the rest of the course. 8 Moreover, the physics content covered by these labs, intended to lay the groundwork for physics investigations in advanced physics courses, fails to meet the needs of life sciences students, for whom the introductory physics class is likely to be the only physics course they take. These traditional labs may also employ lab techniques and tools that are hopelessly outdated: ticker-tape and stopwatches are not the tools of modern science. In recognition of the limitations of traditional lab curricula, several "reform" lab curricula have been created, e.g. UMD's Scientific Community Labs (SCL) 13,14 and Rutgers' Investigative Science Learning Environment (ISLE) 15,16,17 labs. These lab curricula share an emphasis on developing scientific reasoning through experimental design and thoughtful data and error analysis, with an explicit focus on epistemological considerations: what it means to "know" and what counts as evidence. Through explicit engagement in sense-making, group work, and communication, these reformed labs show dramatic gains in student reasoning and scientific processing. 18,19,20 Yet the physics contexts of these labs still reflects traditional content selection and employs the same kinds of lowtechnology, outdated equipment. While these reformed labs succeed in demonstrating epistemic gains, they are not successful in helping students see how the physics labs can help prepare them for their future careers. 21 Thus we argue that a new kind of lab curriculum needs to be developed for IPLS courses. II. OUR VISION FOR A NEW IPLS LAB CURRICULUM We aim to retain the epistemic gains sought by the various reformed lab curricula while seeking changes that particularly benefit our target student demographic: • a focus on physics relevant to microscopic and living systems; • the use of 21st century tools and software; • the ability to engage with data-rich environments; and • preparation for future contributions to biomedical research. Presenting the students with physics relevant to microscopic and living systems allows us to engage their interest and build a stronger conceptual under-standing. By accessing, challenging, and revising their grasp of scientific concepts learned in previous biology and chemistry classes, it is possible to forge a stronger connection than would be formed by presenting traditional physics topics. For example, students engage in investigations of randomness and Brownian motion, charges and motion in fluids, and fluorescence. Though they have encountered many of these topics in their previous biology and chemistry courses, their prior knowledge often reflects a cursory exposure with limited depth of understanding. In using 21st century tools and software, we hope to convey to students that physics is a modern science. We also aim to show the applicability of physical measurement in analyzing biological processes and to provide students with modern analysis tools that will enable them to engage with the data-rich environments they will encounter in their research. The use of modern equipment such as video capture (through digital cameras in the microscopes) and realtime data collection probes (like the spectrometer) generate a wealth of data that most students find overwhelming. By presenting students with sophisticated analysis methods for large(r) datasets (such as log(r 2 ) vs. log(t) plots, discussed below) and with flexible software packages (such as NIH's ImageJ for image and video analysis), students have the opportunity to learn cutting-edge skills with immediate applicability in biomedical research. In addition to these benefits and the lab skills called for in Competency E2 of the HHMI-NEXUS project (i.e., Demonstrate understanding of the process of scientific inquiry, and explain how scientific knowledge is discovered and validated), 3 the development of the lab curriculum focuses on five main threads: modeling; experimental design/protocol development; error analysis; technical lab skills; and interdisciplinary thinking. A. Modeling Crucial to a scientific mode of thinking and to the development of deep and meaningful understanding of science content is the ability to engage in modeling. 22,23,24,25,26 Models come in many forms (conceptual, physical, mathematical, diagrammatic, graphical, et al.), but all share some basic elements. Students must learn to choose which aspects of a system to model, to choose appropriate representations of these aspects, to make predictions based on their model, and to determine the limitations of their models (where and why a model breaks down). B. Experimental Design / Protocol Development The ability to design and carry out an experiment is a hallmark of scientific thinking, whether or not one becomes an experimental scientist. 27 In the biomedical sciences, experimental design is often referred to as protocol development and is a highly prized skill both for future biologists and for medical researchers. 1, 3 Yet the skills required to design an experiment/protocol are often ignored in lab curricula across the sciences, leaving undergraduate science majors with no specific training in these skills. By presenting students with a question to answer and minimal guidance (mostly in the form of how to use the hightechnology equipment), these open-ended labs encourage students to build these crucial skills. C. Error Analysis As Kung, one of the developers of the SCL reform labs, notes: "Many laboratory courses teach students the mathematics of uncertainty analysis such as the arithmetic mean, standard deviation, and percent error, but students are rarely able to use these constructs to make a strong argument from their data. Even worse, using such tools without understanding may be detrimental to future development of understanding." 14 Thus we have made an effort to raise issues of error analysis in contexts that clearly demonstrate how this skill can help choose amongst competing models of a phenomenon and also translate the challenges of measurement into meaningful interpretation of results. D. Technical Lab Skills Alongside the epistemological framework we present to our students, it is important that they gain practical, technical skills that they will be able to apply in professional contexts. To this end, half of our labs employ an inverted microscope with CCD camera for collecting high-resolution, real-time videos of microscale phenomena. Analysis of such data-rich videos requires an image analysis software, such as NIH's ImageJ (used widely in biomedical research labs), and sophisticated facility with spreadsheet programs. Students are given explicit, extensive support on the acquisition and application of these technical skills, leaving them more time and energy to devote to the openended experimental design and data interpretation (for which very few protocols are provided). The other significant high-tech equipment is a USB Infrared to UV spectrometer that collects and displays data in real-time, which is used in labs 10 and 11 in the second semester (see appendix). E. Interdisciplinary Thinking As with other aspects of the NEXUS/Physics project, interdisciplinary thinking is an important aspect of these labs. 7 We aim to foster this connection by carefully choosing laboratory topics and measurement contexts. We also reinforce this thinking by connecting labs where the focus is on physical systems and measurements (e.g. for random and directed motion of beads in lab 3, described in appendix) with labs where physics is measured directly on living systems (e.g. measurements of organelle motion in cells in lab 5, reviewed in depth below). With this scaffolding, we are in a position to ask students to consider what biology can be learned from a physical measurement. They are also asked to seek the biological authenticity in each scenario and to recall, reconsider, and revise their understanding of science content learned in previous courses. It is this thread, together with the specific high-tech tools and modern analysis methods, that most distinguish this new curriculum from the traditional and reform lab curricula. III. DETAILED DESCRIPTION OF A NEXUS/PHYSICS LAB Here we give a detailed illustration of one example laboratory, showing how we have tried to accomplish our goals. A brief introduction to all of the other laboratories for both semesters can be found in the Appendix. As can be seen in the sample laboratory, the skills learned in one lab are applied in other labs, both in new contexts and to encourage interdisciplinary transfer. Lab 5, Fall Semester: Motion and Work in living systems How much work is involved in Active Transport? Classifying Motion and Examining Work in Onion Cells. In this two-week lab, which is the capstone lab for the Fall semester, students examine the intracellular motion of organelles, small functional compartments that are visible inside living cells. The example we have chosen is one of the simplest living systems: a thin layer of freshly cut onion skin is actually alive, with many microscopic organelles visible under magnification (Fig. 1a). In order to examine the motion inside of cells, students must be able to successfully operate the microscope with high precision. Students design an experiment to examine the motion of organelles in the cell and consider the different types of transport taking place. Building on tracking skills learned early in the first semester, they make choices as to which areas to track, how many organelles to track, and how to analyze their tracking data. The student's choices and experimental design affect their ability to draw strong conclusions from the data. Ideally, students will track a large number of organelles from a variety of parts of the cell over a long period of time; this gives them a large amount of data from which to draw conclusions about the motion of vesicles inside the cells. This management of large sets of data has been developed throughout previous labs, first introduced in Lab 1, Part 2 (see appendix). The goal is for students to examine trends in the data they collect and use them to relate to different models of motion. By analyzing the scaling relations between the distances organelles travel in known times, students can determine if they are moving in a directed manner, in a random manner, or are confined (trapped) in some way. 28 As they learned and practiced in labs 3 and 4 (see appendix), students use a log(r 2 ) vs. log(t) plot of the type of motion that is observed (Fig. 2) to draw conclusions about the method of cellular transport being utilized in that area. Students are asked to connect the quantitative physical analysis they have done with the underlying biological processes. Finally, the students connect motion to energy and further explore biological implications of the physical measurements. From the work needed to achieve the observed directed motion in the cell, students estimate the rate of ATP hydrolysis involved in the process. This allows them to further realize how the behavior observed inside the cell reveals the underlying biological phenomena at work. IV. STUDENT RESPONSE These labs have been run with two small IPLS test classes at the University of Maryland, College Park, in the 2012-2013 academic year, with the graduate student lab developers (Moore and Giannini) as the teaching assistants (TAs). In the coming academic year, these labs will be run with large enrollment courses, currently scheduled for 240 students, and TAs newlyexposed to the lab curriculum. Throughout the course, multiple assessments were administered to the students to gauge their reaction to the new lab curriculum and to gather information needed to revise the lab curriculum. Though the data gathered thus far represents only a small sample group (maximum N = 31), we have some promising results. A. Comparison of NEXUS/Physics labs to ISLE and SCL reform labs: Open-ended prompt In an open-ended response survey question, researchers for all three curricula (Rutgers' ISLE labs, UMD's SCL, and our NEXUS/Physics labs) asked students to list what they had learned from the labs. Due to the open-ended nature of such a question, students respond with a wide variety of things learned (e.g., physics content, lab equipment used, software programs used) in addition to the more abstract lab skill goals, such as experimental design and analysis and interpretation of data. The frequency of occurrence of abstract lab skill goals can be taken as an indication of student orientation to the value and effectiveness of the curriculum in achieving such goals. For the goals of learning to design experiments and learning to analyze and interpret data, common to all three curricula, we compared data from SCL 18 and ISLE 21 (both N ~ 200, student demographics comparable to NEXUS/Physics test classes at UMD) measured at the end of the respective courses to our students (max N = 31, measured halfway through the 1st (Fall) semester and at the end of the 2nd (Spring) semester). The results, summarized in Table 1, give us hope that we are headed in the right direction. It remains to be seen whether we can replicate these results with a larger course enrollment and different TAs. B. Comparison of ISLE and NEXUS/Physics labs: Likert-style survey of goals Etkina and Murthy employed a student perception diagnostic tool at the end of the ISLE curriculum in 2005 that we decided to use with our students. 21 This diagnostic tool aims to determine if the goals valued by instructors and curriculum-developers are perceived and valued by the students, and also to gauge students' opinions of the effectiveness of the lab curriculum in achieving these goals. Students' perceptions are important because of the strong interaction between expectation and motivation in conceptual change, especially for complex concepts. 21,29,30 Students were asked to consider a list of learning goals that an introductory physics course could have: • Learn to design your own experiment, • Learn to interpret experimental data, • Understand concepts better , • Learn to work with other people, • Learn to communicate ideas in different ways, and • Prepare for your future professional career; and were then asked two questions: How important is each goal for you? and How successful were the labs in terms of achieving each goal? Students gave their responses on a scale from 1 (Not Important/Not Successful) to 5 (Very Important/Very Successful). Fol-lowing Etkina and Murthy's lead, a response of 1 or 2 was classified as "Low", a response of 3 was classified as "Medium," and a response of 4 or 5 was classified as "High." The relative importance placed on each goal by the two student populations (ISLE and NEXUS) were very similar for all six goals. For the first five goals on the list, the level of importance was also strongly correlated with the level of success; thus, this importance data will not be displayed for the first five goals. Instead, Figure 3 shows a comparison of the students' ratings for how successful the lab curriculum was in achieving each goal. Our NEXUS/Physics labs show comparable levels of success to the ISLE labs. The one area where Etkina and Murthy noted a troublesome mismatch between levels of importance and success in the student response was for the goal of preparing for a future professional career. Though their students placed medium to high value on this goal students felt that the labs had medium to low success in achieving this goal. Etkina and Murthy provide plausible explanations for this mismatch, including that students may not be aware of what skills they will need in their future career. The new lab curriculum we have created shows a closer match between student perceptions of importance and success in achieving this goal as highlighted in Fig. 4. C. Other student results for NEXUS/Physics labs In the many surveys used to gain student feedback on the labs (max N = 31), we gathered both encouraging and discouraging comments. As with most new curricula, the students expressed a resistance to change and took a while to adapt to this new way of engaging in learning. 8,31,32 When asked to respond to the openended prompt "Are you learning lab skills relevant to biology?" in the middle of the Fall semester and at the end of the Spring semester, approximately 2/3 of students in each survey indicated an affirmative response. While some students remained neutral throughout the course, by the end of the Spring semester there were no students responding in the negative. One student remarked: Critical thinking and communication between peers are definitely valuable skills in the realm of biology. Using tracking programs, Excel, and also using microscopes are definitely beneficial skills to have in biology. When asked if the labs were interesting, another student remarked: Yes! Labs such as the neuron lab which relate to background knowledge we already know are especially interesting because we are exploring and proving an area of a subject we did not completely understand! Students were also asked to respond to an openended prompt regarding the experimental design/protocol development thread. They were asked to discuss whether this emphasis was a beneficial or a hindrance, and to describe the advantages and disadvantages of having to create their own protocol. This prompt was administered halfway through the Fall semester and at the end of the Spring semester. The percentage of students finding this emphasis beneficial increased from 64% to 86%. Most strikingly, there were no students in either survey indicating that this emphasis was purely a hindrance. The disadvantage mentioned most frequently by the students was that designing their own experiment takes time and can cause the lab to feel rushed. Advantages noted by the students are illustrated in these two representative comments by different students: I find that I am paying attention to every step and can explain the experiment from beginning to end, with an understanding of why things happened and how the results change when a variable is manipulated. and I definitely see it as beneficial because a large part of science and especially biology is how to create an experiment so this is great practice for the future. These results affirm our belief that these labs are headed in the right direction and that future iterations of this new lab curriculum may have similar success, even in large-enrollment courses. V. CONCLUSION The promising preliminary response to our new IPLS laboratories with reformed content and pedagogy highlight that it is possible to sustain the successes of reform pedagogy with a significant topical change, modern equipment, and an interdisciplinary angle. The significant changes to the lab material and needed equipment are guided by the topical shift of the course, but are more focused on the physics on micro-and nanoscales. By explicitly addressing situations where students cannot draw from everyday experience for foothold principles, such as random motion in the first semester, these labs guide students to confront and analyze concepts that are not intuitive. One significant accomplishment of these pilot labs is the shift of student perception. A majority of our mostly pre-med students considered the physics laboratories as valuable for their future career. We feel that the explicit focus on interdisciplinary connections played a role in this outcome, and hope to gain significantly more insights into this attitude shift when the revised labs will be implemented for large enrollment classes. ACKNOWLEDGMENTS The authors would like to thank the UMD Physics Education and Biology Education research groups, and well as the UMD Physics Department, the Biophysics Program, and the College of Mathematical and Natural Sciences, for their support and feedback throughout this development process. Special thanks are extended to Mark Reeves (George Washington University, Physics), Ben Geller (UMD, PERG), Sergei Sukharev (UMD, Biology), Eric Anderson (UMBC, Physics), Lili Cui (UMBC, Physics), Catherine Crouch (Swarthmore College, Physics), and Karen Carleton (UMD, Biology), for ideas, suggestions, and (where possible) springboard materials. The material presented in this paper is based upon work supported by the Howard Hughes Medical Institute NEXUS grant, and the US National Science Foundation under Award DUE 11-22818. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of HHMI or the National Science Foundation. APPENDIX: BRIEF DESCRIPTIONS OF NEXUS/PHYSICS LABS NEXUS/Physics Labs (Fall Semester): [11 weeks] Lab1: Quantifying motion from Images and Videos. Part 1: How do you quantify motion? Excel Analysis of the 1-D Motion of an Amoeba. In this one-week lab, students explore the concepts of position, displacement, velocity and acceleration while learning basic Excel skills for data presentation and analysis. The students are given a stop-motion sequence of outlines of an amoeba undergoing roughly linear motion. The students must determine what should be measured, how to measure it, and how to manipulate the resulting data to make statements about the amoeba's motion. Part 2: Can you learn any biology from physical measurements? Analysis of Cell Motion Using ImageJ. In this one-week lab, students are introduced to digital image analysis. We utilize ImageJ, a freeware program widely used in biomedical research, to analyze short video clips of biological motion. We use the question "For a simple cut, should a doctor prescribe antibiotics?" to open a discussion of the biological context of this lab: the immunological response of a wound to bacterial infections. Students analyze videos of wound-closure, bacterial motion, and neutrophils (white blood cells) to determine their relative speeds. The surprising result (that bacteria are ~30X faster than neutrophils) prompts students to think about both the validity and the implications of their numerical results. The students re-examine their underlying assumptions about how neutrophils operate and hypothesize what mechanisms enable the human body to defend itself in the absence of antibiotics. Lab 2: Inferring force characteristics from motion analysis. How can information about forces be derived from a video? Introduction to Video Capture & Analysis of Directed Motion and Resistive Forces. In this two-week lab, students examine how various resistive forces scale with the speed of a macroscopic object. Students examine both viscous resistive forces (laminar flow) and drag forces (turbulent flow). Students capture and analyze their own video samples for coffee filters falling through air and for plastic and metal spheres falling through various concentrations of glycerol solution. The students begin examining error analysis and how experimental design and data collection decisions affect the uncertainty in an experimental 8 result. We use macroscopic objects to allow students to tap into their intuitive physical experiences and to begin the process of connecting their everyday experiences with physical laws. Students are introduced to the process of (1) representing observations in multiple ways (graphs, equations, diagrams) and (2) using these multiple representations to make statements about a physical model (e.g. via fitting or via scaling arguments)-a process that is a core component of the rest of the labs. Lab 3: Observing Brownian motion at a microscopic scale. 33 What does 'Random' motion look like? Describing Diffusion and Random Motion. In this three-week lab, students explore the nature of random motion generated by the rapidly varying (i.e. random) forces exerted by thermal fluctuations. Brownian motion is ubiquitous in living systems at a molecular and cellular level and is distinct from the types of motion in students' everyday experiences. They look at how diffusive motion scales with time for microspheres (beads) suspended in solution. Using the microscopes, students gather their own videos and use ImageJ to collect data on the positions of the beads. They then make histograms of the motion at various times. Throughout this lab, each student group has been working with intentionally varied materials: different bead masses, bead sizes, and fluid viscosities, such that each lab group has one changing parameter and two controlled parameters. Finally, using the collective data produced by all the groups, students determine which factors (mass, size, or viscosity) affect the diffusion constant for random motion and determine a mathematical model for how the diffusion constant will change with each parameter. Building on their experience in Lab 2, scaling and modeling arguments are used extensively to make sense of physical observations. Lab 4: The competition between Brownian motion and directed forces. 34 How large a force is needed to transition from random to directed motion? Random vs. Directed Motion. Machines on microscopic scales, such as molecular motors or ion pumps in living systems, constantly compete against and take advantage of the thermal forces that create random Brownian motion. In this two-week lab, students use beads of different sizes suspended in solution to explore the crossover from random to directed motion as an external force is applied. By tilting the microscope and thus inclining the slide, students can gather and analyze videos to determine the motion of the different sized beads under a condition where the gravitational force is affecting bead motion in addition to thermal forces. Stu-dents can directly observe the crossover from random motion where thermal forces dominate to directed motion where gravity dominates and analyze how it depends on the size of the bead. Due to differences in how displacements scale with time, even small directed forces lead to directed motion on long enough timescales; thus students also examine the effect of the time frame over which the bead is followed. In this twoweek lab, students examine the relations between pressure, flow rate, and pipe-width for a fluid containing microspheres flowing through channels arranged in parallel or in series. Students model the expected results using the H-P equation and then develop an experimental protocol to test their predictions. Students discuss connections of their observations to biological fluid channel networks (e.g. the circulatory system) and to the design of health interventions (e.g. cardiac bypass). Lab 7: Analyzing electric forces in a fluid. How do charged objects in a fluid interact with each other and respond to external electric fields? Electrophoresis and Charge Screening in Fluids. In this two-week lab, students explore how microspheres that are suspended in solution move in an electric field. Students discuss that beads charge spontaneously when placed in solution. By varying fluid salinity, the effective charge is changed due to screening of charged objects by the surrounding ions (Debye screening). Using the basic vocabulary of colloidal chemistry, students model the mechanism of electrophoresis, examining how the terminal velocity of the bead's motion in an electric field is related to the effective charge of the bead. Lab 8: Modeling signal transmission along nerve axons. What affects the distance over which an electrical signal is transmitted and the speed of transmission? Testing Models of Signal Transmission Along Nerve Axons. In this two-week lab, students model signal transmission along nerve axons and explore the mechanisms by which signal speed and the distance traveled by the signal can be increased. Students explore conceptual, mathematical, diagrammatic, and graphical models for signal transmission in order to design and test physical models of electrical circuits using resistors, a breadboard, and voltage probes with the aid of Logger Pro software. Students explore what biological insights can be gained from the measurements on these physical models-in particular, insights about the role of myelination and issues related to nerve malformation. (Adapted from Eric Anderson and Lili Cui at UMBC and Catherine Crouch at Swarthmore College.) Lab 9: Introducing geometric optics through experimental observations. 35 How can microscopes magnify objects? Exploring Light and Lenses. In this two-week lab, students explore basic optics components and principles as a first exposure to light and lenses using a vertically positioned optical rail. While the study of magnification in traditional geometric optics labs involves the movement of an object and analysis of the corresponding image position, this is not reflective of the optics systems in microscopes and other biomedical imaging devices. We designed this optics lab with some realistic constraints found in real microscopes, such as a fixed total optical length, which we enforce via the vertical arrangement of the optics. Students collect information about the focal lengths, as well as object and image distances. This data sets the stage for the core lab activity, where students develop and assess their own mathematical models relating the image and object distances to the focal lengths. Students discover that even for a fixed total optical length, each lens can form a clear image by adjusting the lens position, and that each resulting image has a different magnification. Lab 10: Analyzing light spectra and exploring implications for living systems. 36 How can measurements of light spectra provide insights into the nature of matter and the characteristics of living systems? Spectroscopy-Exploring Emission, Absorption & Evolutionary Adaptation. In this two-week lab, students use a variety of spectroscopes (rudimentary to high-tech) to explore emission and absorption of light. Examining the Hydrogen spectrum, students investigate the Bohr model of the atom-in particular the electron level transitions that result in the Lyman, Balmer, and Paschen series. Students then explore how different combinations of sources and filters affect observed spectra, including everyday filters such as water or sunglasses. Using this accumulated knowledge, students hypothesize which environmental factors may have constrained the evolution of the spectrum that is visible to the human eye and gather evidence to support the plausibility of their hypotheses. Lab 11: Exploring complex absorption and emission in molecules. Must 'what goes in' be the same as 'what comes out'? Spectroscopy & Fluorescence in Chlorophyll. In this one-week lab, students examine the fluorescence of a chlorophyll solution when exposed to a variety of light sources in order to gain a foundational understanding of the concepts of how light is absorbed and emitted by fluorescent molecules. Students then consider the physical implications of the ways in which fluorescence is presented in other science venues, such as fluorophore excitation and emission spectra. Figure 1 : 1Imaging Organelles in an Onion Skin. a) Image of organelles, small dots, in an onion cell (above)-two of these organelles have been tracked over the course of the video, showing light and dark paths; b) Students working with a microscope to image the onion cell (below). Figure 2 : 2Student plot of log(r 2 ) vs. log(t). The slope of ~1 for the line of best fit indicates random motion. Fig. 3 : 3Comparison between student rating of success of ISLE and NEXUS labs in achieving goals. Fig. 4 : 4Comparison of ISLE and NEXUS labs in Importance to Student and Success of Labs in Preparing for Future Career. Lab 5 : 5Motion and Work in living systems. How much work is involved in Active Transport? Classifying Motion and Examining Work in Onion Cells. (See example lab description, Section III.) NEXUS/Physics Labs (Spring Semester): [11 weeks] Lab 6: Modeling fluid flow. How do channel geometries and arrangements affect fluid flow? Exploring Fluid Dynamics and the Hagen-Poiseuille (H-P) Equation. Table 1 : 1Student Response to open-ended "What have you learned from these labs?" prompt. Comparison of SCL, ISLE, and NEXUS/Physics lab curricula. A New Biology for the 21 st Century. Nat'l Academy PressNational Research CouncilNational Research Council, A New Biology for the 21 st Century (Nat'l Academy Press, 2009). 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Directed motion, where r is proportion- al to t, and thus r 2 is proportional to t 2 , will have a log(r 2 ) vs. log(t) plot with a slope of 2. Confined (trapped) motion, which is sub-diffusive, will have a correspondingly lower slope, less than 1. Student Expectations in introductory physics. E F Redish, J M Saul, R N Steinberg, Am. J. Phys. 663E. F. Redish, J. M. Saul, and R. N. Steinberg, "Stu- dent Expectations in introductory physics," Am. J. Phys., 66(3), 212-224 (1998). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. A Schoenfeld, Handbook of Research in Mathematics Teaching and Learning. D. A. GrouwsNew York, NYMacMillanA. Schoenfeld, "Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics," in Handbook of Research in Mathematics Teaching and Learning, edited by D. A. Grouws (MacMillan, New York, NY, 1992) pp. 334-370. Reformed Physics Instruction Through the Eyes of Students. E Etkina, M Ruibal-Villasensor, AIP Conf. Proc. 883E. Etkina and M. Ruibal-Villasensor, "Reformed Physics Instruction Through the Eyes of Students," AIP Conf. Proc., 883, 105-108 (2007). When and How Do Students Engage in Sense-Making In A Physics Lab?. A Karelina, E Etkina, AIP. Conf. Proc. 883A. Karelina and E. Etkina, "When and How Do Stu- dents Engage in Sense-Making In A Physics Lab?," AIP. Conf. Proc., 883, 93-96 (2007). With thanks to Ben Geller (UMD, Physics Education Research Group) for extensive development help. George Washington UniversityMotivated by Mark Reeves at the Department of PhysicsMotivated by Mark Reeves at the Department of Physics, George Washington University. With thanks to Ben Geller (UMD, Physics Education Research Group) for extensive development help. Motivated by Sergei Sukharev at the Department of Biology. College ParkUniversity of MarylandMotivated by Sergei Sukharev at the Department of Biology, University of Maryland, College Park. Motivated by Catherine Crouch at the Department of Physics. Swarthmore CollegeMotivated by Catherine Crouch at the Department of Physics, Swarthmore College. Motivated by Karen Carleton at the Department of Biology. College ParkUniversity of MarylandMotivated by Karen Carleton at the Department of Biology, University of Maryland, College Park.
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[ "DeepNVM++: Cross-Layer Modeling and Optimization Framework of Non-Volatile Memories for Deep Learning", "DeepNVM++: Cross-Layer Modeling and Optimization Framework of Non-Volatile Memories for Deep Learning" ]
[ "MehmetAhmet Inci ", "Meric Isgenc ", "Fellow, IEEEDiana Marculescu " ]
[]
[]
Non-volatile memory (NVM) technologies such as spin-transfer torque magnetic random access memory (STT-MRAM) and spin-orbit torque magnetic random access memory (SOT-MRAM) have significant advantages compared to conventional SRAM due to their non-volatility, higher cell density, and scalability features. While previous work has investigated several architectural implications of NVM for generic applications, in this work we present DeepNVM++, a framework to characterize, model, and analyze NVM-based caches in GPU architectures for deep learning (DL) applications by combining technology-specific circuit-level models and the actual memory behavior of various DL workloads. We present both iso-capacity and iso-area performance and energy analysis for systems whose last-level caches rely on conventional SRAM and emerging STT-MRAM and SOT-MRAM technologies. In the iso-capacity case, STT-MRAM and SOT-MRAM provide up to 3.8× and 4.7× energy-delay product (EDP) reduction and 2.4× and 2.8× area reduction compared to conventional SRAM, respectively. Under iso-area assumptions, STT-MRAM and SOT-MRAM provide up to 2× and 2.3× EDP reduction and accommodate 2.3× and 3.3× cache capacity when compared to SRAM, respectively. We also perform a scalability analysis and show that STT-MRAM and SOT-MRAM achieve orders of magnitude EDP reduction when compared to SRAM for large cache capacities. Our comprehensive cross-layer framework is demonstrated on STT-/SOT-MRAM technologies and can be used for the characterization, modeling, and analysis of any NVM technology for last-level caches in GPUs for DL applications.Index Terms-non-volatile memory, deep learning, GPU architectures, convolutional neural networks (CNNs), deep neural networks (DNNs), SRAM, magnetic random access memory (MRAM), spin-transfer-torque MRAM (STT-MRAM), spinorbit-torque MRAM (SOT-MRAM).
10.1109/tcad.2021.3127148
[ "https://arxiv.org/pdf/2012.04559v1.pdf" ]
227,746,663
2012.04559
7ce954ac9d596811051633673d6bc8cc0312f851
DeepNVM++: Cross-Layer Modeling and Optimization Framework of Non-Volatile Memories for Deep Learning MehmetAhmet Inci Meric Isgenc Fellow, IEEEDiana Marculescu DeepNVM++: Cross-Layer Modeling and Optimization Framework of Non-Volatile Memories for Deep Learning 1 Non-volatile memory (NVM) technologies such as spin-transfer torque magnetic random access memory (STT-MRAM) and spin-orbit torque magnetic random access memory (SOT-MRAM) have significant advantages compared to conventional SRAM due to their non-volatility, higher cell density, and scalability features. While previous work has investigated several architectural implications of NVM for generic applications, in this work we present DeepNVM++, a framework to characterize, model, and analyze NVM-based caches in GPU architectures for deep learning (DL) applications by combining technology-specific circuit-level models and the actual memory behavior of various DL workloads. We present both iso-capacity and iso-area performance and energy analysis for systems whose last-level caches rely on conventional SRAM and emerging STT-MRAM and SOT-MRAM technologies. In the iso-capacity case, STT-MRAM and SOT-MRAM provide up to 3.8× and 4.7× energy-delay product (EDP) reduction and 2.4× and 2.8× area reduction compared to conventional SRAM, respectively. Under iso-area assumptions, STT-MRAM and SOT-MRAM provide up to 2× and 2.3× EDP reduction and accommodate 2.3× and 3.3× cache capacity when compared to SRAM, respectively. We also perform a scalability analysis and show that STT-MRAM and SOT-MRAM achieve orders of magnitude EDP reduction when compared to SRAM for large cache capacities. Our comprehensive cross-layer framework is demonstrated on STT-/SOT-MRAM technologies and can be used for the characterization, modeling, and analysis of any NVM technology for last-level caches in GPUs for DL applications.Index Terms-non-volatile memory, deep learning, GPU architectures, convolutional neural networks (CNNs), deep neural networks (DNNs), SRAM, magnetic random access memory (MRAM), spin-transfer-torque MRAM (STT-MRAM), spinorbit-torque MRAM (SOT-MRAM). I. INTRODUCTION O VER the last decade, the performance boost achieved through CMOS scaling has plateaued, necessitating sophisticated computer architecture solutions to gain higher performance in computing systems while maintaining a feasible power density. These objectives, however, are concurrently challenged by the limitations of the performance of memory resources [1]. In contrast to the initial insight of Dennard on [17] power density [2], deep CMOS scaling has exacerbated static power consumption, causing the heat density of ICs to reach catastrophic levels unless properly addressed [3]- [5]. As computers suffer from memory and power related limitations, the demand for data-intensive applications has been on the rise. With the increasing data deluge and recent improvements in GPU architectures, deep neural networks (DNNs) have achieved remarkable success in various tasks such as image recognition [6], [7], object detection [8], and chip placement [9] by utilizing inherent massive parallelism of GPU platforms. However, DNN workloads continue to have large memory footprints and significant computational requirements to achieve higher accuracy. Thus, DNN workloads exacerbate the memory bottleneck which degrades the overall performance of the system. To this end, while DL practitioners focus on model compression techniques [10]- [12], hardware designers augment memory capacities of GPU architectures to overcome the memory bottleneck problem and improve the overall system performance. We note the current trend of GPU architectures is towards increasing last-level cache capacity as shown in Figure 1. Our analysis shows that conventional SRAM technology incurs scalability problems as far as power, performance, and area (PPA) is concerned [13]- [15]. Non-volatile memory (NVM) technology is one of the most promising solutions to tackle memory bottleneck problem for data-intensive applications [16]. However, because much of emerging NVM technology is not available for commercial use, there is an obvious need for a framework to perform design space exploration for these emerging NVM technologies for deep learning (DL) workloads. In this work, we present DeepNVM++, an extended and improved framework [18] to characterize, model, and optimize NVM-based caches in GPU architectures for deep learning workloads. Without loss of generality, we demonstrate our framework for spin-transfer torque magnetic random access memory (STT-MRAM) and spin-orbit torque magnetic random access memory (SOT-MRAM), keeping in mind that it can be used for any NVM technology, GPU platform, or deep learning workload. Our cross-layer analysis framework incorporates both circuit-level characterization aspects and the memory behavior of various DL workloads running on an actual GPU platform. DeepNVM++ enables the evaluation of power, performance, and area of NVMs when used for lastlevel (L2) caches in GPUs and seeks to exploit the benefits of this emerging technology to improve the performance of deep learning applications. To perform iso-capacity analysis, we carry out extensive memory profiling of various deep learning workloads for both training and inference on existing GPU platforms. For the iso-area analysis, existing platforms cannot be used for varying cache sizes, so we rely on architecture-level simulation of GPUs to quantify and better understand last-level cache capacity and off-chip memory accesses. In both cases, our framework automatically combines resulting memory statistics with circuit and microarchitecture-level characterization and analysis of emerging NVM technologies to gauge their impact on DL workloads running on future GPU-based platforms. II. RELATED WORK AND PAPER CONTRIBUTIONS Although 16nm has become a commonplace technology for high-end customers of foundries, an intriguing inflection point awaits the electronics community as we approach the end of the traditional density, power, and performance benefits of CMOS scaling. To move beyond the computing limitations imposed by staggering CMOS scaling trends, MRAM has emerged as a promising candidate. The enabling technology of MRAM consists of magnetic tunnel junction (MTJ) pillars that can store data as a resistive state. An MTJ pillar consists of a thin oxide film sandwiched by two ferromagnetic layers. One of these ferromagnetic layers has a fixed magnetization which serves as a reference layer. The magnetization of the other layer can be altered by changing the direction of the current that flows through the pillar. If the magnetization of the free layer and the reference layer are in parallel, the device is in the low resistance state. If the magnetization of layers is in opposite directions, the device is in the high resistance state [19]. STT bitcells [20] use an MTJ pillar as their core storage element and an additional access transistor to enable read and write operations. Although STT bitcells offer non-volatility, low read latency, and high endurance [21], the write current is also high [22]- [24], which increases power consumption. To this end, SOT bitcells have been proposed to overcome the write current challenges by isolating the read and write paths [25]. Because the read disturbance errors are much less likely in SOT bitcells, both read and write access devices can be tuned in accordance with the lower current requirements [26], [27]. The read and write current requirements of STT and SOT bitcells can have a crucial impact on the eventual MRAM characteristics because they affect the CMOS access transistors, bitcell area, and peripheral logic. Thus, a comparison of these bitcells and the traditional SRAM merits a meticulous analysis that take these factors into account. Prior work has proposed effective approaches to overcome the shortcomings of emerging NVM technologies such as using hybrid SRAM and NVM-based caches that utilize the complementary features of different memory technologies [28]- [31], relaxing non-volatility properties to reduce the high write latency and energy [32]- [35], and implementing cache replacement policies [36]- [38] for higher level caches such as L1 caches and register files. However, NVM technology appear to be a better choice for lower level caches such as L2 or L3 caches due to its long write latency and high cell density. Higher level L1 caches are latency-sensitive and optimized for performance, whereas last-level caches are capacity-sensitive and optimized for a high hit rate to reduce off-chip memory accesses. Therefore, NVM-based caches provide a better use case for replacing SRAM in last-level caches due to their high cell density when compared to SRAM-based caches. To this end, we evaluate power, performance, and area of NVM technology when used for last-level caches in GPU platforms. While prior work has shown the potential of NVM technologies for generic applications to some extent, there is a need for a cross-layer analysis framework to explore the potential of NVM technologies in GPU platforms, particularly for DL workloads. The most commonly used modeling tool for emerging NVM technologies is NVSim [39], a circuit-level model for performance, energy, and area estimation. However, NVSim is not sufficient to perform a detailed cross-layer analysis for NVM technologies for DL workloads since it does not take architecture-level analysis and application-specific memory behavior into account. In this paper, we incorporate NVSim with our cross-layer modeling and optimization flow including novel architecture-level iso-capacity and iso-area analysis flow to perform design space exploration for conventional SRAM and emerging NVM caches for DL workloads. This paper makes the following novel contributions: 1) Circuit-level bitcell characterization. We perform detailed circuit-level characterization combining a commercial 16nm CMOS technology and prominent STT [40] and SOT [41] models from the literature to iterate through our framework in an end-to-end manner to demonstrate the flexibility of DeepNVM++ for future studies. 2) Microarchitecture-level cache design exploration. We use NVSim [39] to perform a fair comparison between SRAM, STT-MRAM, and SOT-MRAM by incorporating the circuit-level models developed in 1) using 16nm technology and choosing the best cache configuration for each of them. 3) Iso-capacity analysis. To compare the efficacy of MRAM caches to conventional SRAM caches, we perform our novel iso-capacity analysis based on actual platform profiling results for the memory behavior of various DNNs by using the Caffe framework [42] on a high-end NVIDIA 1080 Ti GPU (implemented in 16nm technology) for the ImageNet dataset [43]. 4) Iso-area analysis. Because of their different densities, we compare SRAM and NVM caches in an iso-area analysis to quantify the benefits of higher density of NVM technologies on DL workloads running on GPU platforms. Since existing platforms do not support resulting iso-area cache sizes, we extend the GPGPU-Sim [44] simulator to run DL workloads and support larger cache capacities for STT-MRAM and SOT-MRAM. 5) Scalability analysis. Finally, we perform a thorough scalability analysis and compare SRAM, STT-MRAM, and SOT-MRAM in terms of power, performance, and area to project and gauge the efficacy of NVM and SRAM-based caches for DL workloads as cache capacity increases. To the best of our knowledge, putting everything together, DeepNVM++ is the first comprehensive framework for crosslayer characterization, modeling, and analysis of emerging NVM technologies for deep learning workloads running on GPU platforms. Our results show that in the iso-capacity case, STT-MRAM and SOT-MRAM achieve up to 3.8× and 4.7× energy-delay product reduction and 2.4× and 2.8× area reduction compared to SRAM baseline, respectively. In the isoarea case, STT-MRAM and SOT-MRAM achieve up to 2× and 2.3× energy-delay product reduction and accommodate 2.3× and 3.3× cache capacity compared to SRAM, respectively. The rest of the paper is organized as follows. In Section III, we describe the details of our methodology from circuit to microarchitecture-level characterization, modeling, and analysis to obtain SRAM, STT-MRAM, and SOT-MRAM cache parameters. We also detail our iso-capacity and iso-area analysis methodology. In Section IV, we show experimental results demonstrating the efficiency of STT-MRAM and SOT-MRAM over the conventional SRAM for iso-capacity and iso-area cases. Furthermore, we perform a scalability analysis and show the PPA of SRAM, STT-MRAM, and SOT-MRAM. Next, we discuss the implications of the results shown in this paper in Section V. Finally, Section VI concludes the paper by summarizing the results. III. METHODOLOGY In this section, we present our cross-layer analysis framework, as shown in Figure 2. First, we show our detailed circuit-level characterization analysis using CMOS, STT, and SOT device models (Section III-A). After developing bitcell models, we present our microarchitecture-level cache design methodology to obtain cache area, latency, and energy results (Section III-B). Next, we describe our iso-capacity analysis flow in which we gather actual memory statistics through GPU profiling (Section III-C). Finally, we detail our iso-area analysis in which we extend GPGPU-Sim to run deep learning workloads and support larger cache capacities for STT-MRAM and SOT-MRAM (Section III-D). A. Circuit-level NVM Characterization A vast majority of work in the literature uses simple bitcell models [26] to assess the PPA of corresponding cache designs. Because bitcells are the core components of the memory, the methodology to calculate the bitcell latency, energy, and area is crucial for accurate comparisons. To this end, we use a commercial 16nm bitcell design as a baseline as we model the STT and SOT bitcells. This technology node also matches the fabrication technology of the GPU platform that we use to gather actual memory statistics in Section III-C. The key bitcell parameters needed for cache modeling are read and write currents and latency values for high-to-low and low-to-high resistive transitions. These parameters can be optimized by tuning the size of the access transistors. While larger access transistors enable faster reads and writes, they increase the energy consumption and the bitcell layout size. For our simulations, we used perpendicular to the plane STT [40] and SOT [41] models and a commercial 16nm FinFET model that takes post-layout effects into account. To find the latency and energy parameters, we used parametrized SPICE netlists wherein the read/write pulse widths were modulated to the point of failure. Furthermore, we swept a range of fin counts for the access devices to find the optimal balance between the latency, energy, and area. To calculate the bitcell area for the 16nm layout design rules, we used the bitcell area formulations provided in prior work [45]. We summarize the obtained bitcell parameters in Table I. The sensing delay is measured from wordline activation to the point where the bitline voltage difference reaches 25mV. The sense energy is the integration of the power consumed over the sensing time window. For both magnetic flavors, the sense delay is similar; however, SOT-MRAM is more energyefficient in terms of read operation owing to the separation of the read/write terminals. The write latency in this context refers to the time between the arrival of the write enable signal to the access transistor and a complete magnetization change for the MTJ. The write latencies for STT and SOT bitcells are significantly different, as expected. This difference can be seen in the energy values as well. A bigger access device is needed in order to enable a larger current flow to the STT bitcell. The isolation of the read and the write terminals in the SOT bitcell allows for a smaller write access device. The area values are normalized by the foundry bitcell area. We highlight the significant area difference and demonstrate its impact on the cache characteristics in Section III-B. We use these bitcell parameters for energy-delay-area product (EDAP) optimized cache design exploration as discussed in the next section. B. Microarchitecture-level Cache Design Exploration Algorithm 1: EDAP-Optimal Cache Tuning Algorithm mem ∈ M = {SRAM, ST T, SOT }; cap ∈ C = {1, 2, 4, 8, 16, 32}; opt ∈ O = {Read Latency , W rite Latency , Read Energy , ...W rite Energy , Read EDP , W rite EDP , Area, Leakage}; acc ∈ A = {N ormal, F ast, Sequential}; for each mem ∈ M do for each cap ∈ C do Q ← ∞; for each opt ∈ O do for each acc ∈ A do Q ← calculate(EDAP ); if Q < Q then Q ← Q; end end end T unedConf ig.append(argv(Q)); end end return T unedConf ig; In order to demonstrate the impact of using STT and SOT bitcells in L2 caches, we use NVSim [39], a circuit-level analysis framework that delivers energy, latency, and area results. After developing NVSim-compatible bitcell models as described in Section III-A, we analyzed a range of cache capacities (1MB to 32MB) for all possible configurations and cache access types to demonstrate the potential of STT-MRAM and SOT-MRAM as the cache capacity tends to grow. Such a scalability study will help in determining the benefits of switching from conventional SRAM to NVM-based caches in future GPU platforms as depicted by the trend in Figure 1. Algorithm 1 depicts the EDAP-optimal cache tuning algorithm. Based on the optimization target used in NVSim, the cache PPA values vary substantially. Therefore, we independently choose the best configuration for each type of memory technology in terms of EDAP metric to perform a fair comparison that encompasses all and not just one of the design constraint dimensions. As described in Section III-A, we use a commercial 16nm bitcell design. To ensure a correct analysis, we modified the internal technology file of NVSim to the corresponding 16nm technology parameters. Next, we compare SRAM, STT-MRAM, and SOT-MRAM for various cache capacities in terms of area, latency, and energy results. Based on these, we determine the EDAP for the cache (as denoted by calculate(EDAP) in Algorithm 1). Table II shows the latency, energy, and area results that correspond to the cache capacity of 1080 Ti GPU (3MB) and to the larger MRAM caches that fit into the same area of SRAM baseline. We convert read and write latencies to clock cycles based on 1080 Ti GPU's clock frequency for our calculations. For STT-MRAM and SOT-MRAM, we show parameters for both iso-capacity and iso-area when compared to SRAM. We use these parameters to evaluate the workload dependent impact of memory choices using DL workloads with diverse structures and multiply-accumulate operations (MACs) configurations. The energy and latency benefits of STT-MRAM and SOT-MRAM depend on the data characteristics of a given workload. To account for differences in the data-related read/write characteristics, we used a simple model where we multiply the number of read and write transactions by the corresponding latency and energy values for those operations. Implications in architecture-level analysis. To gauge the benefits of using MRAM technology, we consider two scenarios: (i) First, one could replace the SRAM cache in a GPU with the same capacity MRAM with a smaller area. (ii) Alternatively, by using the same area dedicated to the cache, one can increase the on-chip cache capacity, thereby reducing costly DRAM traffic. We analyze and discuss both approaches through platform profiling results for iso-capacity scenario and a set of architecture-level simulations for iso-area scenario. C. Architecture-level Iso-Capacity Analysis As the platform target to demonstrate our work, we use a high-end 1080 Ti GPU which is fabricated in a commercial 16nm technology node which also matches our bitcell and cache models. We use the Caffe [42] framework to run various DNNs such as AlexNet [46], GoogLeNet [47], VGG-16 [48], ResNet-18 [49], and SqueezeNet [50] for the ImageNet [43] dataset as shown in Table III. Our analysis is generalizable to other types of neural network architectures since we cover a wide range of DNN configurations with various workload characteristics. We use the NVIDIA profiler [51] to obtain the device memory and L2 cache read and write transactions to better understand both on-chip and off-chip memory behavior of DNN workloads. D. Architecture-level Iso-Area Analysis Since the iso-area larger capacities enabled by higher density NVM implementations do not exist in existing platforms, we use GPGPU-Sim [44] to explore power and performance implications of having these larger L2 caches in GPU architectures for DNN workloads. For comparison, we model the high-end GTX 1080 Ti GPU. The configurations for 1080 Ti GPU are shown in Table IV. We extend the GPGPU-Sim simulator to support the cache capacity of GTX 1080 Ti GPU. This GPU is built using a commercial 16nm technology node which matches our bitcell and cache models. In particular, for GPGPU-Sim compatibility, we set L2 cache capacity to 3MB. We use this capacity for our analysis in the rest of the paper. We measure the number of DRAM transactions to quantify and better understand the relationship between larger L2 caches and the overall system power and performance. As a DNN benchmark, we use AlexNet [46] with the ImageNet [43] dataset which is provided by the DarkNet [52] framework. We extend DarkNet source code to enable deep learning workloads on GPGPU-Sim. 1 IV. RESULTS We analyze STT-MRAM and SOT-MRAM in terms of energy, performance, and area results by using GPU profiling Fig. 5: Impact of batch size on energy-delay product (lower is better) normalized with respect to SRAM by using NVMs with iso-capacity (3MB) for AlexNet for training (left chart) and inference (right chart) results for both iso-capacity and iso-area cases in Section IV-A and Section IV-B, respectively. In Section IV-B, we use iso-area cache parameters as shown in Table II and we use GPGPU-Sim to quantify the DRAM access reduction in the iso-area case at larger cache capacities. We include DRAM accesses in our performance and energy calculations for isoarea case. In Section IV-C, we perform a scalability analysis to project the implications of the current GPU trend shown in Figure 1 on performance and energy results. A. Performance and Energy Results for Iso-Capacity By combining the actual technology-dependent latency and energy metrics from Table II, we can perform a performance and energy analysis for replacing conventional SRAM caches with MRAM caches. We choose batch size 4 for inference and 64 for training for our workloads as it is typically used in related work [53]. Figure 3 shows normalized dynamic energy and leakage energy breakdown results for 1080 Ti GPU based on actual platform memory statistics and our MRAM cache models at the same cache capacity. We use our cache parameters and profiling results to calculate results for various DNNs for both inference and training workloads. In Figure 3, we observe that STT-MRAM has 2.1× dynamic energy whereas SOT-MRAM has 1.3× dynamic energy on average when compared to SRAM baseline. Furthermore, our results show that 83% of the total dynamic energy of SRAM comes from read operations whereas write operations only make for 17% of all transactions on average across all workloads. Our profiling results also support these findings as read operations dominate write operations in DL workloads. On the other hand, Figure 3 also shows that STT-MRAM and SOT-MRAM provide 5.9× and 10× lower leakage energy on average when compared to SRAM, respectively. Based on this result, Figure 4 shows significant total normalized energy reduction of STT-MRAM and SOT-MRAM when compared to SRAM given that leakage energy dominates the total energy. In more detail, STT-MRAM and SOT-MRAM achieve 5.1× and 8.6× energy reduction on average across all workloads compared to SRAM baseline, respectively, due to their significantly low leakage energy. Moreover, Figure 4 shows that STT-MRAM and SOT-MRAM provide up to 3.8× and 4.7× EDP reduction and 2.4× and 2.8× area reduction, respectively. The impact of batch size on EDP. We perform this study to better understand the relationship between batch size and its implications for performance and energy results of SRAM, STT-MRAM, and SOT-MRAM. Figure 5 shows the impact of batch size on EDP results for AlexNet during training and inference stages based on 1080 Ti memory profiling statistics. We show that batch size significantly affects the improvement of STT-MRAM and SOT-MRAM for training. For training, STT-MRAM provides 2.3× to 4.6× EDP reduction as batch size increases. On the other hand, SOT-MRAM provides 7.2× to 7.6× EDP reduction when compared to SRAM baseline. For inference, STT-MRAM and SOT-MRAM achieve 4.1× to 5.4× and 7.1× to 7.3× EDP reduction, respectively. These results also confirm the different workload characteristics of training and inference. STT-MRAM provides higher EDP reduction for training workloads as batch size increases. On the other hand, SOT-MRAM follows the same pattern for inference workloads due to their different access characteristics as shown in Table II. We observe that training workloads become more read dominant whereas inference workloads have lower read/write ratio as batch size increases. As in the iso-capacity study, for iso-area analysis we use a batch size 4 for inference and 64 for training. Figure 6 shows the reduction in the total number of DRAM accesses as L2 cache capacity increases. We use GPGPU-Sim and start with the baseline configuration which is 3MB for GTX 1080 Ti and double its cache capacity up to 24MB to quantify the percentage of DRAM access reduction for STT-MRAM and SOT-MRAM at larger cache capacities. Figure 6 shows that replacing SRAM with STT-MRAM and SOT-MRAM equivalents that fit into the same area significantly reduces the total number of DRAM transactions by 14.6% and 19.8%, respectively for 1080 Ti GPU. Figure 7 shows normalized dynamic energy and leakage energy breakdown results for 1080 Ti GPU based on actual platform memory statistics and our MRAM cache models at the same area. We use our iso-area cache parameters in which STT-MRAM (7MB) and SOT-MRAM (10MB) have larger cache capacities for the same area budget with SRAM. We use these cache parameters and profiling results to calculate results for various DNNs for both inference and training workloads. B. Performance and Energy Results for Iso-Area In Figure 7, we observe that STT-MRAM has 2.5× dynamic energy whereas SOT-MRAM has 1.4× dynamic energy on average when compared to SRAM baseline. On the other hand, Figure 7 also shows that STT-MRAM and SOT-MRAM provide 2.1× and 2.3× lower leakage energy on average when compared to SRAM, respectively. Based on this result, STT-MRAM and SOT-MRAM achieve 2× and 2.3× lower energy when compared to SRAM. Furthermore, Figure 8 shows that STT-MRAM and SOT-MRAM provide 1.1× and 1.2× EDP reduction and 2.3× and 3.3× larger cache capacity on average across all workloads when compared to SRAM and off-chip DRAM accesses are not included in the calculations, respectively. When DRAM accesses are included in determining EDP, as shown in Figure 8, STT-MRAM and SOT-MRAM provide 2× and 2.3× EDP reduction on average across all workloads when compared to SRAM, respectively. We show that although the cache latency and energy results for STT-MRAM and SOT-MRAM do not outperform SRAM results at larger cache capacities as shown in Table II, they do outperform SRAM when costly off-chip DRAM accesses are also considered in EDP calculations. To this end, Chen et al. [54] showed that the normalized energy cost of a global buffer access relative to a MAC operation is 6×, whereas a DRAM access is 200× for a machine learning hardware accelerator. By the same token, the higher cell density of NVM can be exploited to shift the memory traffic from DRAM to L2 cache to further improve power and performance of the overall system. This approach can dramatically reduce the total number of costly DRAM accesses and reduce data movement, which is a daunting impediment for achieving energy-efficient machine learning hardware [53]- [57]. C. Scalability Analysis As shown in Figure 1, the current trend for NVIDIA GPUs is towards increasing L2 size with each new GPU generation. The most recent high-end NVIDIA GPUs have even up to 6MB L2 cache to further improve performance of the system by reducing costly off-chip memory accesses. However, SRAM has a scalability problem due to its high leakage and large bitcell area, which poses a significant challenge to further continue the current GPU trend. To this end, non-volatile memory technologies come to the rescue of future GPU architectures since their PPA scale better as cache Fig. 9: Cache capacity scaling results for SRAM, STT-MRAM, and SOT-MRAM for area, latency, and energy metrics capacity increases. Therefore, there is a need for a scalability analysis to project and quantify performance and energy gains that can be achieved by using more scalable memory solutions. To this end, we perform a scalability analysis by first comparing SRAM, STT-MRAM, and SOT-MRAM for various cache capacities in terms of area, latency, energy results following the DeepNVM++ framework methodology as described in Section III. Therefore, each memory technology is optimized for EDAP objective at each cache capacity independently to make a fair comparison among SRAM, STT-MRAM, and SOT-MRAM. Next, we evaluate and show how NVMbased caches behave in terms of performance and energy when compared to conventional SRAM-based caches for deep learning workloads in a scalability analysis. Area. Figure 9(a) demonstrates the impact of higher cell density of MRAMs on the area of caches compared to SRAM. The area difference between SRAM and the MRAM variants grow significantly as the cache capacity increases. The main reason of this difference comes from the bitcell area difference between SRAM and MRAMs as shown in the last row of Table I. Particularly for deeply scaled technology nodes wherein interconnects account for a significant portion of parasitics, bigger bitcells translate to longer wires, bigger buffers, and peripheral logic. Therefore, STT-MRAM and SOT-MRAM caches become more area-efficient when compared to SRAM caches as cache capacity increases. Latency. Figure 9(b) shows that for capacities smaller than 3MB SRAM offers lower read latency, whereas both MRAM variants have lower read latency than SRAM beyond 4MB. In terms of write latency, STT-MRAM has always the highest among all memory technologies due to its inherent device characteristic. In contrast, the write latency of SOT-MRAM becomes increasingly smaller than that of SRAM. Moreover, the write latency of SRAM almost matches that of STT-MRAM at 32MB. Energy. In terms of read access energy, Figure 9(c) shows that 7MB is a break even point where SOT-MRAM becomes more efficient than SRAM whereas STT-MRAM clearly has the highest read energy among all memories. Regarding write access energy, SOT-MRAM is the most efficient option whereas SRAM consumes the most energy for a write operation beyond 3MB. Based on these PPA results, we perform a detailed scal-ability analysis for SRAM, STT-MRAM, and SOT-MRAM. In Figure 10, we show the normalized energy, latency, and EDP results with respect to SRAM for STT-MRAM and SOT-MRAM for various cache capacities. As it can be seen, STT-MRAM and SOT-MRAM provide lower energy and latency results as cache capacity increases. In terms of energy, STT-MRAM and SOT-MRAM provide lower energy as cache capacity increases. Specifically, STT-MRAM and SOT-MRAM caches achieve up to 31.2× and 36.4× energy reduction as cache capacity increases, respectively. In terms of latency, STT-MRAM and SOT-MRAM have higher latency results for cache capacities up to 4MB, whereas both MRAM variants have lower latency results when compared to SRAM beyond that point. In more detail, SRAM provide up to 3.2× and 2× latency reduction for small cache capacities when compared to STT-MRAM and SOT-MRAM, respectively. However, STT-MRAM and SOT-MRAM achieve up to 2.1× and 2.6× latency reduction as cache capacity increases, respectively. In terms of EDP, we show that STT-MRAM and SOT-MRAM provide up to 65× and 95× EDP reduction when compared to SRAM, respectively. Therefore, we conclude that for latency-critical applications, SRAM-based caches become a more suitable option when compared to MRAM variants for small cache capacities whereas MRAMs provide more energy-efficient solutions. Although SRAM provide lower EDP results for smaller cache capacities, STT-MRAM and SOT-MRAM outperform SRAM by orders of magnitude for larger cache capacities due to their better PPA scalability when compared to SRAM. These results show that a significant portion of the overall system energy or latency is saved and can be used for additional on-chip resources or capabilities that are not available now. V. DISCUSSION In this section, we discuss the implications of the results shown in this paper. We also share the potential future directions to guide our community to better explore the use of non-volatile memories for deep learning workloads in different design spaces. Scalability is a major problem for SRAM. As we show in Figure 9 and Section IV-C, one of the key challenges for the current GPU architectures is the scalability problem of SRAM due to its significantly high leakage energy and large area when compared to STT-MRAM and SOT-MRAM. We observe that there is a current trend in GPU architectures towards increasing L2 cache capacity and we show that SRAM has significant scalability problems in terms of area, latency, and energy. We show that STT-MRAM and SOT-MRAM have promising solutions for larger cache capacities which can maintain the current trend shown in Figure 1 with increasing performance and energy benefits. Implications of dense NVM caches on logic usage. Figure 9(a) shows the area results for SRAM, STT-MRAM, and SOT-MRAM for various cache capacities. We note that STT-MRAM and SOT-MRAM provide increasingly smaller area than SRAM as cache capacity increases. For the same cache capacity, STT-MRAM and SOT-MRAM provide 58% and 65% area reduction on average, respectively. Therefore, the remaining whitespace can be utilized by cramming more processing elements, register files, or L2 cache on the die. This analysis is left for future work. As CMOS scaling issues limit the affordable improvement of computing systems, our results from device-level simulations to actual GPU profiling show that MRAMs are extremely promising candidates. Particularly, as STT-MRAM and SOT-MRAM fabrication processes become more mature, systemlevel benefits of STT-MRAM and SOT-MRAM can be maximized, enabling faster and more energy-efficient computation. Mobile design space exploration for NVM. In this work, we explore the GPU architecture design space to un-veil the potential of non-volatile memories for deep learning workloads. Having said that, we note that inference at the edge devices also becomes a common practice for many service providers such as Google [58], Amazon [59], and Facebook [60] to improve user experience by reducing latency and preserving the private user data on device [61]. To this end, Wu et al. [60] shows that majority of mobile inference for Facebook workloads run on mobile CPUs. Mobile platforms have various resource constraints such as energy, memory, and computing capabilities. Thus, last-level caches of mobile CPUs or hardware accelerators can also be replaced by STT-MRAM and SOT-MRAM to improve performance and energy by reducing leakage energy and costly off-chip memory accesses due to their non-volatility and higher cell density [62]- [65]. Therefore, the design space exploration of STT-MRAM and SOT-MRAM for mobile CPUs and hardware accelerators for inference workloads merits further research. VI. CONCLUSION In this paper, we present the first cross-layer analysis framework to characterize, model, and analyze various NVM technologies in GPU architectures for deep learning workloads. Our novel framework can be used to further explore the feasibility of emerging NVM technologies for DL applications for different design choices such as technology nodes, bitcell models, DL workloads, cache configurations, optimization targets, and target platforms. Our results show that in the iso-capacity case, STT-MRAM and SOT-MRAM provide up to 3.8× and 4.7× EDP reduction and 2.4× and 2.8× area reduction when compared to SRAM, respectively. In the iso-area case, STT-MRAM and SOT-MRAM achieve up to 2× and 2.3× EDP reduction and accommodate 2.3× and 3.3× cache capacity when compared to SRAM, respectively. Finally, we perform a scalability analysis and show that STT-MRAM and SOT-MRAM outperform their SRAM counterpart by orders of magnitude in terms of energydelay product for large cache capacities. The newly created energy or latency slack can be used for additional on-chip resources or capabilities that are currently not possible. Fig. 2 : 2Overview of the cross-layer analysis flow Fig. 3 : 3Dynamic energy (left c energy (right chart) (lower is better) normalized with respect to SRAM by using NVMs with iso-capacity (3MB) for inference (I) and training (T) stagesFig. 4: Iso-capacity (3MB) energy and energy-delay product for NVM-based caches (lower is better) normalized with respect to SRAM-based caches for inference (I) and training (T) stages. DRAM energy and latency are also included in EDP results. Fig. 6 : 6Simulation results for the reduction in the total number of DRAM accesses in percentage (%) Fig. 7 : 7Dynamic energy (left chart) and leakage energy (right chart) (lower is better) normalized with respect to SRAM by using STT-MRAM (7MB) and SOT-MRAM (10MB) with iso-area for inference (I) and training (T) stagesFig. 8: Iso-area energy-delay product results for STT-MRAM (7MB) and SOT-MRAM (10MB) (lower is better) normalized with respect to SRAM-based caches for inference (I) and training (T) stages without (left chart) and with (right chart) DRAM energy and latency. Fig. 10 : 10Mean energy (top), latency (middle), and energy-delay product (bottom) across all workloads (lower is better) normalized with respect to SRAM for various cache capacities for inference (left) and training (right) stages. Error bars show standard deviation across workloads. Ahmet Inci is with the Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213 USA e-mail: [email protected] Mehmet Meric Isgenc was with the Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213 USA e-mail: [email protected] Diana Marculescu is with the Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX, 78712 USA and Carnegie Mellon University, Pittsburgh, PA, 15213 USA e-mail: [email protected] Preprint Fig. 1: L2 cache capacity in recent NVIDIA GPUs TABLE I : ISTT-MRAM and SOT-MRAM bitcell parameters after device level characterizationSTT-MRAM SOT-MRAM Sense Latency (ps) 650 650 Sense Energy (pJ) 0.076 0.020 Write Latency (ps) 8400 (set) / 7780 (reset) 313 (set) / 243 (reset) Write Energy (pJ) 1.1 (set) / 2.2 (reset) 0.08 (set) / 0.08 (reset) Fin Counts 4 (read/write) 3 (write) + 1 (read) Area (normalized) 0.34* 0.29* *: Area is normalized with respect to the foundry SRAM bitcell TABLE II : IILatency, energy, and area results for SRAM, STT-MRAM, and SOT-MRAM caches for iso-capacity and iso-areaSRAM STT-MRAM SOT-MRAM Iso-Capacity Iso-Area Iso-Capacity Iso-Area Capacity (MB) 3 3 7 3 10 Read Latency (ns) 2.91 2.98 4.58 3.71 6.69 Write Latency (ns) 1.53 9.31 10.06 1.38 2.47 Read Energy (nJ) 0.35 0.81 0.93 0.49 0.51 Write Energy (nJ) 0.32 0.31 0.43 0.22 0.40 Leakage Power (mW) 6442 748 1706 527 1434 Area (mm 2 ) 5.53 2.34 5.12 1.95 5.64 TABLE III : IIIConfigurations for DNNs under considerationAlexNet GoogLeNet VGG-16 ResNet-18 SqueezeNet Top-5 error 16.4 6.7 7.3 10.71 16.4 CONV Layers 5 57 13 17 26 FC Layers 3 1 3 1 0 Total Weights 61M 7M 138M 11.8M 1.2M Total MACs 724M 1.43G 15.5G 2G 837M TABLE IV : IVGPGPU-Sim ConfigurationsGTX 1080 Ti Number of Cores 28 Number of Threads/Core 2048 Number of Registers/Core 65536 L1 Data Cache 48 KB, 128 B line, 6-way LRU L2 Data Cache 128 KB/channel, 128 B line, 16-way LRU Instruction Cache 8 KB, 128 B line, 16-way LRU Number of 4 Schedulers / Core Frequency (MHz): 1481, 2962, Core, Interconnect, 1481, 2750 L2, Memory We plan to open source our framework after publication to enable future research on GPU architectures for deep learning workloads. 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[Online]. . 10.1145/3352460.3358258Available: https://doi.org/10.1145/3352460.3358258 On-chip memory technology design space explorations for mobile deep neural network accelerators. H Li, M Bhargav, P N Whatmough, H. . Philip Wong, 2019 56th ACM/IEEE Design Automation Conference (DAC). H. Li, M. Bhargav, P. N. Whatmough, and H. . Philip Wong, "On-chip memory technology design space explorations for mobile deep neural network accelerators," in 2019 56th ACM/IEEE Design Automation Conference (DAC), 2019, pp. 1-6. He is currently a Ph.D. candidate at Carnegie Mellon University. His research interests include systems for ML, computer architecture, hardware-efficient deep learning, and HW/ML model co-design. Ahmet Inci received his B.Sc. degree in Electronics Engineering at Sabanci University, IstanbulAhmet Inci received his B.Sc. degree in Elec- tronics Engineering at Sabanci University, Istanbul, Turkey in 2017. He is currently a Ph.D. candidate at Carnegie Mellon University. His research inter- ests include systems for ML, computer architec- ture, hardware-efficient deep learning, and HW/ML model co-design. Mehmet Meric Isgenc was born in Izmir, Turkey in 1992. He received his B.S in Microelectronics Engineering from Sabanci University in 2014 and his M.Sc and Ph.D. degrees in Electrical and. Cupertino, CaliforniaComputer Engineering from Carnegie Mellon UniversityHis research interests are low-power SoC accelerator design implementation and automationMehmet Meric Isgenc was born in Izmir, Turkey in 1992. He received his B.S in Microelectronics Engineering from Sabanci University in 2014 and his M.Sc and Ph.D. degrees in Electrical and Computer Engineering from Carnegie Mellon University in 2016 and 2019 respectively. He is currently with Apple Inc. in Cupertino, California. His research interests are low-power SoC accelerator design im- plementation and automation. She is the Department Chair, Cockrell Family Chair for Engineering Leadership #5, and a Professor, Motorola Regents Chair in Electrical and. Los Angeles, CA, USADiana Marculescu (Fellow, IEEE) received the Dipl.Ing. degree in computer science from the Polytechnic University of Bucharest, Bucharest ; Computer Engineering #2, with the University of Texas at Austin. Prior to joining UT AustinRomania, in 1991, and the Ph.D. degree in computer engineering from the University of Southern California. she was theDiana Marculescu (Fellow, IEEE) received the Dipl.Ing. degree in computer science from the Poly- technic University of Bucharest, Bucharest, Roma- nia, in 1991, and the Ph.D. degree in computer engi- neering from the University of Southern California, Los Angeles, CA, USA, in 1998. She is the Depart- ment Chair, Cockrell Family Chair for Engineering Leadership #5, and a Professor, Motorola Regents Chair in Electrical and Computer Engineering #2, with the University of Texas at Austin. Prior to joining UT Austin in December 2019, she was the ) and has served as an Associate Department Head for Academic Affairs in Electrical and Computer Engineering (2014-2018), all with Carnegie Mellon University. Her research interests include energy-and reliability-aware computing, hardware aware machine learning, and computing for sustainability and natural science applications. and several best paper awards. She was the IEEE Circuits and Systems Society Distinguished Lecturer (2004-2005) and the Chair of the Association for Computing Machinery (ACM) Special Interest Group on Design Automation. David Edward Schramm Professor of Electrical and Computer Engineering, the Founding Director of the College of Engineering Center for Faculty Success ; Carnegie Institute of Technology George Tallman Ladd Research AwardShe was the recipient of the National Science Foundation Faculty Career AwardDavid Edward Schramm Professor of Electrical and Computer Engineering, the Founding Director of the College of Engineering Center for Faculty Success (2015-2019) and has served as an Associate Department Head for Academic Affairs in Electrical and Computer Engineering (2014-2018), all with Carnegie Mellon University. Her research interests include energy-and reliability-aware computing, hardware aware machine learning, and computing for sustainability and natural science applications. She was the recipient of the National Science Foundation Faculty Career Award (2000-2004), the ACM SIGDA Technical Leadership Award (2003), the Carnegie Institute of Technology George Tallman Ladd Research Award (2004), and several best paper awards. She was the IEEE Circuits and Systems Society Distinguished Lecturer (2004-2005) and the Chair of the Association for Computing Ma- chinery (ACM) Special Interest Group on Design Automation (2005-2009). She chaired several conferences and symposia in her area and is currently an Associate Editor for the IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS. She was selected as an ELATE Fellow. Barbara Lazarus Award from Carnegie Mellon Universityand is a recipient of an Australian Research Council Future Fellowship. She is a Fellow of ACMShe chaired several conferences and symposia in her area and is currently an Associate Editor for the IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS. She was selected as an ELATE Fellow (2013-2014), and is a recipient of an Australian Research Council Future Fellowship (2013-2017), the Marie R. Pistilli Women in EDA Achievement Award (2014), and the Barbara Lazarus Award from Carnegie Mellon University (2018). She is a Fellow of ACM.
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[ "Noncontact Haptic Rendering of Static Contact with Convex Surface Using Circular Movement of Ultrasound Focus on a Finger Pad", "Noncontact Haptic Rendering of Static Contact with Convex Surface Using Circular Movement of Ultrasound Focus on a Finger Pad" ]
[ "Tao Morisaki ", "Member, IEEEMasahiro Fujiwara ", "Yasutoshi Makino ", "Member, IEEEHiroyuki Shinoda " ]
[]
[]
A noncontact tactile stimulus can be presented by focusing airborne ultrasound on the human skin. Focused ultrasound has recently been reported to produce not only vibration but also static pressure sensation on the palm by modulating the sound pressure distribution at a low frequency. This finding expands the potential for tactile rendering in ultrasound haptics as static pressure sensation is perceived with a high spatial resolution. In this study, we verified that focused ultrasound can render a static pressure sensation associated with contact with a small convex surface on a finger pad. This static contact rendering enables noncontact tactile reproduction of a fine uneven surface using ultrasound. In the experiments, four ultrasound foci were simultaneously and circularly rotated on a finger pad at 5 Hz. When the orbit radius was 3 mm, vibration and focal movements were barely perceptible, and the stimulus was perceived as static pressure. Moreover, under the condition, the pressure sensation rendered a contact with a small convex surface with a radius of 2 mm. The perceived intensity of the static contact sensation was equivalent to a physical contact force of 0.24 N on average, which was 12 times the radiation force physically applied to the skin.
10.48550/arxiv.2301.11572
[ "https://export.arxiv.org/pdf/2301.11572v1.pdf" ]
256,358,325
2301.11572
c64c1b4a20337844592f65972970c307737f45ef
Noncontact Haptic Rendering of Static Contact with Convex Surface Using Circular Movement of Ultrasound Focus on a Finger Pad Tao Morisaki Member, IEEEMasahiro Fujiwara Yasutoshi Makino Member, IEEEHiroyuki Shinoda Noncontact Haptic Rendering of Static Contact with Convex Surface Using Circular Movement of Ultrasound Focus on a Finger Pad BLANK 1Index Terms-Static contact sensationconvex surfacemidair hapticsfocused ultrasound A noncontact tactile stimulus can be presented by focusing airborne ultrasound on the human skin. Focused ultrasound has recently been reported to produce not only vibration but also static pressure sensation on the palm by modulating the sound pressure distribution at a low frequency. This finding expands the potential for tactile rendering in ultrasound haptics as static pressure sensation is perceived with a high spatial resolution. In this study, we verified that focused ultrasound can render a static pressure sensation associated with contact with a small convex surface on a finger pad. This static contact rendering enables noncontact tactile reproduction of a fine uneven surface using ultrasound. In the experiments, four ultrasound foci were simultaneously and circularly rotated on a finger pad at 5 Hz. When the orbit radius was 3 mm, vibration and focal movements were barely perceptible, and the stimulus was perceived as static pressure. Moreover, under the condition, the pressure sensation rendered a contact with a small convex surface with a radius of 2 mm. The perceived intensity of the static contact sensation was equivalent to a physical contact force of 0.24 N on average, which was 12 times the radiation force physically applied to the skin. I. INTRODUCTION A IRBORNE ultrasound tactile display (AUTD), which can present a noncontact tactile stimulus, is a promising tool for haptics since it dose not require users to physically contact with any devices [1]. An AUTD is a device with an array of independently controllable ultrasound transducers [2], [3], [4]. AUTDs can focus ultrasound waves on arbitrary points in the air by controlling the phase of each transducer. At the focus, a nonnegative force called acoustic radiation force is generated [5], which conveys a noncontact tactile stimulus onto human skin. This has been used in various applications [1], such as human motion guidance [6], [7], [8], touchable midair image displays [9], [10], [11], and remote visual-haptic communication system [12], as the noncontact stimulus by AUTD does not obstruct a user's movement and vision. Recently, Morisaki et al. reported that AUTD can present not only vibratory sensations but also static pressure sensations [13]. A static pressure sensation is indispensable for Manuscript received xx; revised xx. This work was supported in part by JSPS KAKENHI Grant Number 21J12305 and JST CREST JPMJCR18A2. The authors are with the Graduate School of Frontier Sciences, the University of Tokyo, Kashiwa-shi, Chiba, 277-8561, Japan (e-mail: [email protected]; Masahiro [email protected]; yasutoshi [email protected]; hiroyuki [email protected]). tactile displays because the sensation is the main component of contact perception and is perceived with a higher resolution than vibratory sensations [14]. However, in the conventional ultrasound haptics technique, a static pressure sensation is excluded from the presentable sensation of the AUTD. Ultrasound radiation force must be spatiotemporally modulated as it is less than several tens of mN [15], [16], [17], [18], [19]. This modulation has limited the tactile stimulus presented by the AUTD to a vibratory sensation. Morisaki et al. addressed this limitation and found that AUTD can present a static pressure sensation by repeatedly moving an ultrasound focus along the human skin at 5 Hz with a 0.2 mm spatial step width of the focus movement [13]. The focal trajectory was a 6 mm line, and the presentation location was a palm only. In this study, we experimentally demonstrate that static pressure sensation by ultrasound can be evoked even at a finger pad. Moreover, we also show that by using a circular focal trajectory, the pressure sensation can render a static contact with a small convex surface on the finger pad. The radius of the rendered convex surface is varied from 2 to 4 mm. Rendering static contact with such a small convex surface has been difficult for conventional ultrasound haptics techniques because the perceptual resolution of vibratory sensations is lower than that of static pressure sensations [14]. This contact sensation rendering enables the noncontact tactile reproduction of fine corrugated surfaces with a minimal spot size of several millimeters, which is equivalent to a spatial resolution of 1 cm. Previous studies rendered an uneven surface (e.g., bumps and holes) using ultrasound. However, in these studies, the contact sensation was not static as the finger and palm must be moved to perceive the rendered surface. Howard et al. and Somei et al. rendered an uneven surface by dynamically changing the intensity or position of the ultrasound focus according to hand movement [20], [21]. In the experiment, an ultrasound focus rotating in a circle at 5 Hz is presented to a finger pad, and the radius of the trajectory is varied from 2 to 6 mm. We evaluate the intensity of the vibratory and movement sensations of the focus produced by the presented stimulus. We also evaluated curvature of the tactile shape (i.e., flat, convex, or concave) perceived on the finger pad. Moreover, we examine the optimal ultrasound focus shape for creating a perfect static pressure sensation. II. RELATED WORKS In this section, we summarize previous studies on point stimulation and haptic shape rendering using ultrasound to clarify the contribution of this study. A. Vibratory and Static Pressure Sensation by Ultrasound Two presentation methods have been employed to create a single point vibrotactile sensation: Amplitude Modulation (AM) [16] and Lateral Modulation (LM) [17], [18]. AM is a stimulation method wherein the amplitude of the presented radiation pressure is temporally modulated [16]. In LM, a vibratory stimulus is presented by periodically moving a single stimulus point (ultrasound focus) along the skin surface with constant pressure [17], [18]. Takahashi et al. presented an LM stimulus on the palm and showed that its perceptual threshold was lower than that of the AM stimulus [17], [18]. The focal trajectory used by Takahashi et al. was a line and circle with representative lengths of a few millimeters. Additionally, Spatiotemporal Modulation (STM) method have been used to create a larger trajectory of a moving focus [19], [22]. Frier et al. presented a circular STM stimulus with circumferences of 4-10 cm, which were larger than that of the LM stimulus presented by Takahashi et al [19], [17], [18]. A static pressure sensation can be produced by a lowfrequency LM stimulus with a fine spatial step width of the focal movement. Morisaki et al. presented a static pressure sensation using an LM stimulus at 5 Hz with a step width of 0.2 mm [13]. The focal trajectory was a 6 mm line. Under this condition, the vibratory sensation included in the LM stimulus was suppressed to 5% in a subjective measure, and the perceived intensity was comparable to 0.21 N physical pushing force on average. The pressure sensation by ultrasound has been presented only on the palm, and whether the pressure sensation can be evoked on a finger pad has not been confirmed. This study aims to present the static pressure sensation to a finger pad. Morisaki et al. and Somei et al. presented a low frequency-fine step LM stimulus to a finger pad. However, they did not evaluate its tactile feeling [11], [21]. B. Rendering Haptic Shape Using Ultrasound Several studies have presented symbolic two-dimensional haptic shapes, such as a line and circle on the palm using AUTD. To render them, Korres and Eid used AM with multiple foci [23]. Marti et al. and Hajas et al. used STM stimulus, wherein the focal trajectory is the perimeter of the target shape [24], [25]. Mulot et al. drew a curved line to the palm using STM stimulus and evaluated whether its curvature can be discriminated [26], [27]. Moreover, AUTD has been used for tactile reproduction of contact between 3D objects and hands. Inoue et al. presented a 3D static haptic image using an ultrasound standing wave [28]. Long et al. presented multiple ultrasound foci on a palm and rendered the contact shape with a virtual 3D object [29]. Matsubayashi calculated the contact area between a finger and a virtual 3D object and rendered this area to a finger pad by presenting an LM stimulus whose focal trajectory was the perimeter of the calculated contact area [30], [31]. These studies aimed to reproduce the macroscopic shape of a 3D object and did not reproduce contact shape with a fingertipsized small convex surface, as in this study. Moreover, static pressure sensations were not presented in these studies. Long et al. used AM at 200 Hz [29] and Matsubayashi et al. LM at 100 Hz [30], [31]. The static haptic image presented by Inoue et al. was not modulated, but the participants had to keep moving their hands to perceive its tactile sensations [28]. Several studies have reproduced uneven surfaces using AUTD. Howard et al. presented three tactile shapes to a palm: bump, hole, and flat, by dynamically changing the intensity of the ultrasound focus based on the hand position [20]. Somei et al. presented a convex surface sensation to a finger pad by changing the position of the ultrasound tactile stimulus according to finger position [21]. Perceiving tactile shapes using these methods require active finger or hand movement. However, this study aims to perceive a static convex shape while the fingers are stationary. III. AIRBORNE ULTRASOUND TACTILE DISPLAY (AUTD) In this study, we used Airborne Ultrasound Tactile Display (AUTD) to present noncontact tactile stimuli. AUTD comprises an array of ultrasound transducers [2], [3], [4]. An AUTD can focus ultrasound by controlling a phase of each transducer, and focused ultrasound generates a nonnegative pressure called acoustic radiation pressure. Ultrasound focus can be narrowed to the diffraction limit. Four AUTDs were used in the experiments. The one AUTD unit was equipped with 249 ultrasound transducers operating at 40 kHz (TA4010A1, NIPPON CERAMIC Co., Ltd.) [32], [33]. Fig. 1 shows the AUTD. Each AUTD communicated via the EtherCat protocol and was synchronously driven. IV. STIMULUS DESIGN A. Overview In this section, we propose and describe two stimulus methods: LM-single focus (LM-S) and LM-multi foci (LM-M). In the subject experiment, we compared and evaluated them to investigate whether they could render a static contact sensation with convex surface. Fig. 2 shows a schematic of these stimulus methods. In LM-S, a single ultrasound focus is periodically moved in a circle on the finger pad. The LM-S has been used in previous studies [19], [18], [30]; however, these studies have not evaluated whether this stimulus can produce static pressure and static contact sensations. In the LM-M stimulus, multiple ultrasound foci were simultaneously presented and periodically moved in a circle. The foci were placed along the circular focal trajectory so that they were in close proximity. The distance between foci d was fixed at 3 mm in the experiments. In the experiment, the amplitude of each transducer was set to maximum and the driving phase for presenting the LM-M stimulus was calculated using a linear synthesis scheme. Let φ i ∈ R Ntrans be the phase for presenting each focus in the LM-M, and the phase for simultaneously presenting multiple foci φ ∈ R Ntrans is expressed as follows: φ = N focus i φ i ,(1) where i ∈ {1, ...N focus } is the index number of multiple foci, N focus is the total number of multiple foci, and N trans is the total number of transducers. B. Formulation First, we formulated a focus movement for the LM-S stimulus. The focus position in LM-S r j ∈ R 3 is given by the following: r j = r cnt + A(cos θ j r a + sin θ j r b ) + z j r c ,(2)θ j = 2π N (j − 1),(3) where j ∈ {1, ...N } is the index of the focus position, N is the total number of focus positions in one cycle of the LM, r cnt ∈ R 3 is the center of the focal trajectory, and A is the radius of the trajectory. r a , r b , and r c are unit vectors whose origin is at r cnt and parallel to the x-, y-, and z-axis, respectively. The value of z j was determined using the measured finger depth position. Based on these definitions, the step width of the focus movement is d LM = 2πA N . The index of focus position j changes after the dwell time of focus t d . Dwell time was t d = 1 N f LM if the frequency of the LM stimulus is f LM . Second, we formulated the LM-M stimulus. Let r i,j ∈ R 3 be the focus position on the LM trajectory of the i-th focus among the foci presented simultaneously. r i,j is chosen from r j , which is the position discretized with d LM , such that the motion step width of the multi foci is fixed to d LM . The conversion from r j to r i,j is expressed as follows: r i,j = r j+(i−1)l ,(4)l = d d LM ,(5) where l is the index number calculated from the distance between the multi foci d. l is an integer, and the decimal point is rounded down. V. EXPERIMENTAL EQUIPMENT In this section, we describe the experimental equipment that presents a midair image with noncontact tactile feedback. This equipment was used in all the subject experiments conducted in this study. Fig. 3 shows the experimental equipment and its coordinate system. This system consists of the four AUTDs, a midair image display (ELF-SR1 Spatial Reality Display, SONY), and a depth camera (RealSense D435, Intel) used to measure the finger position. In the experiments, we used the midair image display to instruct participants where to put their fingers. The coordinate system is a right-handed system whose origin is the center of the surface of the image display. A. System Overview Throughout all the experiments, the system presented a 1 × 1 cm image marker at (0, 30, 30) mm. Ultrasound waves were output from the AUTDs when participants placed their fingertips on the marker. The presented ultrasound wave refracted on the surface of the image display and then focused on the finger pad. The position of the reflected ultrasound focus r ref ∈ R 3 can be calculated as the mirror image of the original focus position r org ∈ R 3 which is expressed as follows: r ref = r org + 2((r p − r org ) · n)n,(6) where n is the normal vector of the display surface (reflective surface), and r org is an arbitrary point on the display surface. Fig. 4. Algorithm for presenting LM stimulus. The size of the detection area is 1 × 1 × 2 cm. The part of the finger within this detection area is measured as the contact area, and the focal trajectory of the LM stimulus is calculated using this area. The center of the LM is the centroid of the contact area. deg Force gauge B. Algorithm for Presenting LM Stimulus In the system, there are three processes for presenting a circular LM stimulus to the finger pad of the participant. Fig. 4 illustrates the presentation process. First, the system detects the contact area between a participant's finger and midair image marker using a depth camera. The size of the image marker is 1×1×0.5 cm. However, to measure the contact position stably, we used the area from the surface of the image marker to 2 cm behind (1 × 1 × 2 cm) for the contact detection. Part of the finger within the detection area was measured as the contact area. Second, the system calculated the focal trajectory for the circular LM stimulus using eq. 2 or eq. 4. The center position of the LM stimulus r cnt was the centroid of the detected contact area. The measured depth map of the fingertip surface was used for the z-position of the focal trajectory. Third, the focus is presented and moved along with the calculated trajectory at a pre-specified frequency. In this algorithm, the r cnt is asynchronously updated with the focus position at 90 fps. A Gaussian filter was applied to the calculated r cnt of 10 frames to suppress the measurement error of the depth camera. C. Measurement of Radiation Force We measured the radiation force of the focus presented by the system and it was 0.02 N. Fig. 5 shows the measurement setup. In this experiment, the tip of a force gauge, to which a 1.5 cm diameter acrylic disk was attached, was placed at the focal point. This force gauge (IMADA ZTS-2N) can measure forces up to 2 N with a resolution of 0.001 N. The force gauge was tilted by 50 deg so that this disk opposes the propagation direction of the ultrasound wave. The size of the acrylic disk was determined based on the preliminary simulation such that the disk size was larger than the focus size. Fig. 6 shows the simulated radiation force distribution of a single focus and Fig. 7 shows the ultrasound transducer setup used for the simulation. The measurement range of the acrylic disk is superimposed on the simulated result as a white circle. In this simulation, the focus was generated at (0, 0, 250) mm, and the reflection of the sound waves was not considered. The focus position is shown in Fig. 7 as a cross mark. VI. EXPERIMENT1: STATIONARITY AND SURFACE CURVATURE In this experiment, we evaluated the intensity of vibratory and movement sensations in the LM stimulus and the perceived curvature of the surface of the object produced by the LM same stimulus (i.e., flat, convex, or concave). A. Stimulus Condition In this experiment, we presented the LM-M (LM-multi foci) and LM-S (LM-single focus) stimuli at 5 Hz (as described in Section IV). For comparison, an LM-S stimulus at 25 Hz was also presented. The radii of LM stimuli A were 2, 3, 4, 5, and 6 mm. The motion step width d LM of the LM stimulus at 5 Hz was as fine as 0.23 mm to elicit static pressure sensation [13]. Moreover, the step d LM at 25 Hz was 4 mm to avoid exceeding the AUTD update limits (1 kHz) [33]. For the 5 Hz LM-M stimuli, the number of simultaneously presented foci N focus was four, and their placement interval d was 3 mm. All stimuli were presented in random order. Each participant underwent two sets of experiments. Therefore, 30 experimental trials were conducted (i.e., 3 different LM stimuli × 5 stimulus radii × 2 sets = 30 experimental trials). B. Procedure Eight males (24-31 age) and two females (24 and 28 age) participated in this experiment. The experimental equipment was a visuo-tactile display (Fig 3 and Section V). Participants were instructed to place their index fingertips on the presented midair image marker. The tactile stimulus was always presented while the fingertip was touching the marker. First, to evaluate the tactile sensation of the presented stimulus, the participants answered the following two questions with a seven-point Likert scale: Q1. How intensely did you perceive a vibratory sensation in the presented stimulus? Q2. How intensely did you perceive the movement of the stimulus position? Participants were instructed to answer 1 if they perceived no vibration or movement. In Q2, we evaluated whether the participants noticed the circular focus movement of the LM. Second, the participants evaluated the curvature rendered by the LM stimulus on their finger pads. In this experiment, we provided three typical shapes as references (i.e., flat, convex, and concave). Three images corresponding to the three shapes ( Fig. 8) were presented to the participants as reference images. To evaluate the perceived curvature, the participants responded to Q3 with a seven-point Likert scale. Q3. Does the stimulus shape perceived at your finger pad match the situation illustrated in the reference images? For one stimulus condition, flat, convex, and concave reference images (Fig. 8) were presented successively in random order. Participants independently reported perceptual similarity to each reference image (i.e., flat, convex, and concave). We varied the radius of the illustrated object in the reference images to match that of LM stimulus A. Participants were instructed to ignore differences in the perceived size between the image and tactile stimulus to evaluate only the similarity of the perceived curvature (i.e., flat, convex, and concave). The overall size of the finger sketch, which was drawn in the reference image, was adjusted so that its nail size matches the average Japanese adult nail length (13.6 mm) [34]. C. Results and Analysis 1) Stationarity: Box-and-whisker plots of the evaluated vibratory sensations (answers to Q1) are shown in Fig. 9a. The evaluated movement sensation (answers to Q2) is also shown in Fig. 9b. If the data value v satisfies the following conditions, the data are treated as an outlier: v ≤ v 25 − 1.5 × IQR, v ≥ v 75 + 1.5 × IQR,(7) where v 25 and v 75 are the 25-percentile value and 75percentile value, respectively, and IQR is the interquartile range. Outliers were plotted as white dots in the graphs. As seven participants could not perceive the LM-M stimulus with A = 2 mm, their answers were excluded. In total, 13 data of the LM-M with A = 2 mm were excluded from each graph. The results showed that the highest median value of the vibratory sensation score was 7, and the stimulus condition was LM-S at 25 Hz with A = 4, 5, 6 mm. The lowest median value was 1, and the condition was LM-M at 5 Hz with A = 2 mm. The highest median value of the movement sensation score was 6.5, and the stimulus condition was LM-S at 5 Hz with A = 5 mm. The lowest median value was 1, and the condition was LM-M at 5 Hz with A = 2 mm. We conducted the Wilcoxon signed-rank test with Bonferroni correction to compare the results between the stimulus conditions (LM-M, LM-S at 5 Hz, and LM-S at 25 Hz) for each stimulus radius A. The results of the LM-M stimulus with A = 2 mm were excluded from the analysis. The test results showed that at all values of A, the perceived vibratory sensation of the LM-S at 25 Hz was significantly higher than that of the other LM stimuli (p < 0.005). At A = 3 mm, the vibratory sensation of the LM-S at 5 Hz was significantly higher than that of the LM-M (p < 0.05). The results also showed that at A = 3, 4, 5, 6 mm, the perceived movement sensation of the LM-S at 25 Hz was significantly lower than that of the other LM stimuli (p < 0.05). At A = 3, 4, 5 mm, the movement sensation of the LM-M was significantly lower than that of the LM-S at 5 Hz (p < 0.05). Fig. 9 shows these pairs with significant differences as "*" and "**" for p < 0.05 and p < 0.005, respectively. Moreover, we conducted the Friedman test with Bonferroni correction using stimulus radius A and stimulus type (LM-M, LM-S at 5 Hz, and LM-S at 25 Hz) as factors. The test results showed that A and stimulus type had a significant effect on both vibration and movement sensation (p < 0.0005). 2) Surface Curvature: Box-and-whisker plots of the evaluated tactile shape (answers to Q3) with LM-M, LM-S at 5 Hz, and LM-S at 25 Hz are shown in Fig. 10a, Fig. 10b, Fig. 10c, respectively. 13 data of LM-M with A = 2 mm were excluded (Section VI-B). The highest median value for the flat score was 5.5, and the condition was LM-S at 25 Hz with A = 4, 5 mm. The lowest median was 3.5, and the condition was LM-S at 5Hz with A = 6 mm. The highest median for the convex score was 5, and the conditions were LM-M with A = 3, 4 mm and LM-S at 5 Hz with A = 2, 3, 4 mm. The lowest median was 2, and the condition was LM-S at 25 Hz with A = 5 mm. The highest median value of the concave score was 5, and the condition was LM-S at 5 Hz with A = 6 mm. The lowest median was 1.5, and the condition was LM-M with A = 3 mm. We conducted the Wilcoxon signed-rank test with Bonferroni correction to compare the score between the shapes (i.e., flat, convex, concave) at each stimulus condition. The test result showed that in the LM-M with A = 3, 4 mm, the flat and convex scores were significantly higher than the concave scores (p < 0.05). With A = 4 mm, the convex score was significantly higher than the flat score (p < 0.005). In the LM-S at 5 Hz with A = 2, 3 mm, the flat score was significantly higher than the concave score (p < 0.05). For A = 2, 3, 4 mm, the convex score was significantly higher than the concave score (p < 0.05). With A = 2 mm, the convex score was also significantly higher than the flat score (p < 0.05). In the LM-S at 25 Hz, all flat scores were significantly higher than the convex scores (p < 0.005). For A = 2, 3, 4, 5 mm, the flat score was significantly higher than the concave score (p < 0.05). VII. EXPERIMENT2: PERCEIVED SIZE In this experiment, we changed the radius of LM stimulus A and evaluated the perceived stimulus size. (1) (2) (3) (4) (5) Fig. 11. Presented picture to evaluate the perceived size of the presented LM stimulus. The five pictures with different radii (2, 3, 4, 5, and 6 mm) were presented simultaneously. Participants selected one of these images showing the circle whose size matches the perceived haptic size. A. Procedure Eight males (24-31 age) and (24 and 28 age) two females participated in this experiment. The experimental setup was the same as that used in Experiment 1 (Fig. 3). The tactile stimulus was always presented while the fingertip was touching the marker. The stimulus conditions were identical to those used in Experiment 1, which is explained in Section VI-A. 30 experimental trials were conducted (i.e., 3 different LM stimuli × 5 stimulus radii × 2 sets = 30 experimental trials). A real-time video of the participants' fingers was presented to them during the experiment. The screenshot of the presented video is shown in Fig. 11. In this video, a blue circular image corresponding to the trajectory of the LM stimulus is superimposed on the finger pad of the participant. Participants selected one of the videos showing a circle whose size matched the perceived haptic size to evaluate the perceived size of the presented stimulus. The center of the circular image was changed in real-time to match the center of the presented LM stimulus r cnt . The radii of the circular images were 2, 3, 4, 5, and 6 mm, which were the same as the radii of LM stimuli A used in this experiment. Five videos with different radii were simultaneously presented to the participant. This video was captured using an RGB camera built into the depth camera. Fig. 12 presents the confusion matrix for the stimulus size identification results. The highest accuracy was 0.6, and the stimulus condition was LM-S at 5 Hz with A = 5 mm. The lowest accuracy was 0.15, and the condition was LM-S at 25 Hz with A = 2, 4 mm. Chance rate in this experiment was 0.2, and accuracy exceeded the chance rate in all conditions, except for the lowest-accuracy condition. B. Results and Analysis We compared the perceived size across the stimulus condition (LM-M and LM-S at 5 Hz and 25 Hz, respectively). Fig. 13 shows box-and-whisker plots of the perceived stimulus sizes. The highest perceived stimulus radius was 5 mm, and the condition was LM-S at 5 and 25 Hz with A = 5 mm and all LM stimuli with A = 6 mm. The lowest radius was 2 mm, and the conditions were LM-M with A = 2, 3 mm. We applied the Wilcoxon signed-rank test with Bonferroni correction to the results of the perceived size. The test results showed that the perceived radii of the LM-S at 25 Hz were significantly higher than that of the LM-M with A = 3, 4, 5 (p < 0.05) and LM-S at 5 Hz with A = 2, 4, 5 mm (p < 0.05). The results also showed that the radius of the LM-S at 5 Hz was significantly larger than that of the LM-M with A = 3, 4 mm (p < 0.05). VIII. EXPERIMENT3: EQUIVALENT PHYSICAL STIMULUS This experiment investigated physically static force which is equivalent to the pressure sensation evoked by LM stimulus at a finger pad. Physical force was presented by pushing a force gauge against the finger pad. A. Setup and Stimulus Fig. 14 illustrated the experimental setup. In this experiment, we used a force gauge whose z-position was automatically controlled by a 3-axis stage (QT-AMM3 and ALS-7013-G1MR, CHUO PRECISION INDUSTRIAL Co., Ltd.) and the visual-haptic system (Fig. 3) used in the other experiments. This force gauge (IMADA ZTS-50N) can measure forces up to 50 N with a resolution of 0.01 N. The stimulus condition was the same as that used in Experiment 1, which is explained in Section VI-A. There were 30 experimental trials conducted (3 different LM stimuli × 5 stimulus radii A × 2 sets = 30 experimental trials). B. Procedure Eight males (23-28 age) and two females (24 and 28 age) participated in this experiment. Participants were instructed to place their index fingers of their right hands on the marker presented by the midair image display. Participants were also instructed to place their index fingers of their left hands such that the finger pad faced the tip of the force gauge. At this point, the force gauge did not touch the finger pad. The force gauge was fixed in midair in a horizontal orientation (Fig. 14). Participants grasped the aluminum handle and fixed their finger position by placing it in front of an acrylic auxiliary plate. A plastic cylinder with a radius of 1 cm was attached to the tip of the force gauge. The basal plane of the cylinder was beveled to 1 mm so that the participants did not perceive its edges. Participants wore headphones and listened to white noise during the experiment to avoid hearing the driving noise of the AUTD. A force gauge was pressed against the finger pad of the participant by moving along the z-axis. After the force gauge reached the specified position (the initial pushing depth was 4 mm), an LM stimulus was presented to the finger pad of the right hand. After 2 s, the LM stimulus was stopped, and the force gauge returned to its initial position. The force gauge immediately started pushing again, and the LM stimulus was presented again. This 2 s tactile stimulation was repeated automatically. In this experimental loop, participants compared the physical pushing force with the LM stimulus and orally reported the results. Based on the participants' answers, we changed the pushing depth of the force gauge such that the Repeat automatically Force gauge Auto 3-axis stage Fig. 14. Setup to evaluate the perceived force. A force gauge was pressed against the finger pad of the left hand, and the LM stimulus was presented to the right finger. The pushing depth was automatically controlled by the 3-axis stage. These stimuli were terminated after 2 s and automatically repeated. Participants compared the pushing force with the LM stimulus and orally reported the comparison results. perceived intensity of the two stimuli is the same. For example, pushing depth in the 2nd stimulus was shortened to weaken the pushing force if the participant answered that the pushing force was stronger than the LM stimulus in the 1st stimulus. The force gauge kept pushing the finger pad and recorded the pushing force for 2 s when the participants reported that the intensities of the two stimuli were the same. The median value of the pushing force time series data was finally adopted as the measured force. After the measurement, the stimulus conditions were changed, and the same procedure was repeated. The adjustment resolution of the pushing depth is 0.25 mm and the speed of the force gauge was 5 mm/s. The maximum number of pushing depth adjustments was 20, and all participants completed the experiment within 30 min. cm In the stimulus comparison, we instructed the participants to ignore the perception at the moment when the LM stimulus and the pushing force were presented to assess the steady-state perceived intensity of the LM stimulus. C. Results and Analysis In this experiment, the median value of the measured force time series data was adopted as the participant's answer. The maximum standard deviation (SD) of the time series data, median value, and minimum values were 0.522, 0.186, and 0, respectively. The maximum, median, and minimum values were answered by a different participant. Fig. 16 shows the times series data whose SD is the maximum (0.522) and data whose SD is the median (0.186). The median force of each time series data is shown in Fig. 16 as a red line. Fig. 17 shows the box-and-whisker plots of the pushing forces. Outliers were calculated using eq. 7, and are plotted Data with medium SD Median Data with maximum SD Fig. 16. Time series data of measured force. We calculated the standard deviation (SD) of each recorded time series data, and the data with the maximum SD (0.522) and with the median value of the SD (0.186) was plotted in this figure. We also plotted the median value of these plotted time series force. as white dots. One participant was unable to perceive the LM at 5 Hz with A = 2 mm; thus, this value was plotted as 0 N. The forces lower than 0.01 N, which is the lowest measurable force of the force gauge, were also plotted as 0 N. The results showed that the highest median value of the perceived force was 0.53 N, and the stimulus condition was LM-S at 25 Hz with A = 4 mm. The lowest median value was 0.16 N, and the condition was LM-M at 5 Hz with A = 2 mm. We also conducted the Wilcoxon signed-rank test with Bonferroni correction to compare the perceived force between the stimulus conditions (LM-M, LM-S at 5 Hz, and LM-S at 25 Hz) at each stimulus radius A. The test results showed that with A = 2, 3, 4, 6 mm, the perceived force of the LM-S stimulus at 25 Hz was significantly higher than that of the LM-M stimulus (p < 0.05). For A = 4, 6 mm, the perceived force of the LM-S stimulus at 25 Hz was significantly higher than that of the LM-S stimulus at 5 Hz (p < 0.05). For A = 2, 3 mm, the perceived force of the LM-S stimulus at 5 Hz was significantly higher than that of the LM-M stimulus (p < 0.05). IX. DISCUSSION A. Static Pressure Sensation at Finger Pad The results of Experiment 1 showed that LM at 5 Hz (including both LM-M and LM-S) can produce a non-vibratory pressure sensation on a finger pad. Moreover, with stimulus radii of A = 2, 3 mm, the movement sensations were barely perceivable, and the pressure sensation was well static. The vibration sensation of the LM stimulus at 5 Hz was 4 or less in all conditions except LM-M with A = 5 mm, which was significantly lower than that of the LM-S at 25 Hz. For A = 2, 3, the movement sensations of the LM-M were 2 or less. The results of Experiment 3 also showed that the perceived intensity of the pressure sensation on the finger pad was perceptually comparable to 0.16 N or more physical contact force on average. With the lowest vibration and movement sensation (LM-M with A = 3 mm), the perceived force was 0.24 N, which was 12 times the radiation pressure at the focus presented in the setup. However, in Experiment 3, extremely low and high forces were identified causing large variance. For the LM-M with A = 3 mm, the minimum and maximum values were 0 and 1.22 N, respectively. Note that the participant who answered 0 N could perceive the LM-M stimulus with A = 3 mm. Since the answered equivalent force is less than 0.01 N, which is the measurable minimum force of the force gauge, the force is recorded as 0 N. This large difference in perceived force could be attributed to the individual differences in the tactile receptor-adaptation speed to the pushing stimulus presented by the force gauge. The pushing force is static, and the perceived intensity of such stimulus gradually weakens with stimulus duration owing to SA-I (slowly-adaptive type I) tactile receptor adaptation [35]. In Experiment 3, the contact time with the force gauge was controlled for 2 s to prevent this adaptation effect, and participants were instructed to ignore the perception of the moment of the contact. However, if the adaptation speed greatly differs among participants, even under this control, there could be a large difference in the answered equivalent pushing force. For example, we considered that the adaptation speed of the participants answered an extremely high force was fast. When the adaptation speed is fast, the perceived intensity of the contact force rapidly weakens over the period of 2 s, resulting in a high pushing force as the equivalent force. Conversely, the adaptation speed of the participants answering an extremely low force could be slow. The evaluation of the individual differences in adaptation speed is important for future work. The experimental results also indicated that the perceived intensity of the LM-M stimulus with A = 2 mm was extremely weak. In Experiments 1 and 2, eight participants could not perceive the LM-M stimulus with A = 2 mm. We considered that the weakness is because the circumference with a 2 mm radius and the length of the curved line-shaped stimulus distribution used in LM-M (9 mm) were almost the same. As an exception, in Experiment 3, only one participant could not perceive the LM-M stimulus with A = 2 mm, and the average perceived force was 0.16 N. This difference could be attributed to the difference in the presentation time of the LM stimulus [35]. In Experiment 3, the stimulus duration was 2 s, but in Experiments 1 and 2, the participants continued to be presented with the LM stimulus without any time limit. Therefore, in most participants in Experiments 1 and 2, their SA-I tactile receptors completely adapted to the LM-M stimulus, and they could not perceive the stimulus. B. Perceived Curvature In Experiments 1 and 2, since eight participants could not perceive the LM-M stimulus with a radius of 2 mm, we excluded it from the following discussions. The results of Experiments 1 and 2 suggest that a circular LM stimulus with A = 2-4 mm can render a contact sensation with a convex surface with radii of 2-4 mm. As described in Section IX-A, particularly for the A = 2, 3 mm, the contact sensation was well static. In the LM at 5 Hz with A = 2-4 mm, the convex score was significantly higher than the concave score (p < 0.05). In LM-M with A = 4 mm and LM-S with A = 2, 4 mm, the convex score was significantly higher than the flat score (p < 0.05). The perceived radii for LM-M with A = 3 mm and LM-S with A = 4 mm were 2 and 4, respectively. The comments of participants also suggest that convex sensation was rendered. Four participants commented that they sometimes felt in contact with sharp or rounded objects. Based on the authors' subjective view, we felt the LM-M and LM-S stimuli at 5 Hz with A = 3 mm as a contact sensation with a rounded convex surface. However, in some cases, participants found it difficult to determine whether the perceived contact shape was convex or flat. Two of the participants commented that this determination was difficult. Moreover, no significant differences were observed between the convex scores for the LM-S at 5 Hz with A = 3 mm and LM-M with A = 3 mm. In the future, we will quantitatively evaluate the curvature of the perceived surface and explore a control method for the curvature. In the LM at 5 Hz with A = 2-4 mm, all concave scores were less than 2, and a concave sensation was not perceived. We considered that the periphery of the LM was hardly perceived in the radius range as three participants commented that they had high concave scores when they strongly perceived the perimeter of the stimulus. The characteristics of the timeaveraged radiation pressure distribution of the LM stimulus were also consistent with this consideration. Fig. 18 shows the simulated time-averaged pressure distribution. The simulation setup is the same that shown in Fig. 7. The results indicates that the periphery of the LM stimulus is the peak of the timeaveraged radiation pressure only above a radius of 5 mm, where the concave score is high. Finally, we compared the perceived curvature to the 5 Hzvibration intensity distribution produced by the LM stimulus at 5 Hz. Fig. 19 shows the simulated distribution of the 5 Hz vibration intensity (power spectrum of 5 Hz) produced by LM-M and LM-S at 5 Hz with A = 3 mm. The power spectrum distribution was obtained by simulating the time variation of the radiation pressure at each point in the stimulus area and Fourier transforming the time variation. The simulation setup is the same that shown in Fig. 7. The simulation results showed that the physical intensity of the 5 Hz vibration was the highest on the focal orbits and does not match the perceived stimulus shape (perceived curvature). With A = 3 mm, the LM-M and LM-S at 5 Hz were perceived as contact with a convex surface. However, even under these conditions, the peaks of vibration intensity formed a circle, which is a contact shape with concave. In the future, we will investigate the relationship between perceived curvature and vibration intensity distribu-tion by measuring or simulating skin displacement generated by the LM stimulus as in previous studies [36], [37]. C. Comparison of LM-M and LM-S at 5 Hz The results of Experiment 1 showed that the curved lineshaped pressure distribution, which consists of four ultrasound foci and is used in LM-M, can suppress the movement sensation of low-frequency LM stimuli. With A = 3, 4, 5, the movement sensation of the LM-M was significantly lower than that of the LM-S at 5 Hz. We considered that the reason for the suppression of motion perception was that the simultaneously stimulated area of LM-M was wider than that of LM-S. The LM-M stimulus was perceived to be smaller than the LM-S stimulus at 5 Hz. For A = 3, 4 mm, the perceived size of LM-M was significantly smaller than that of LM-S at 5 Hz (p < 0.05). The trend in perceived size is consistent with the difference in the size of the time-averaged radiation pressure distribution. The simulation results shown in Fig. 18 indicate that the time-averaged distributions of LM-M with A = 3, 4 mm were smaller than those of LM-S. In terms of vibratory sensation and perceived shape (curvature), there were no huge differences between LM-M and LM-S at 5 Hz. Except for A = 3 mm, there were no significant differences in vibration sensations. For A = 3, 4, 5 mm, the convex scores was higher than the flat and concave scores for both the M-M and LM-S at 5 Hz. D. Comparison of Movement Sense Between LM Frequencies The results of Experiment 1 showed that the movement sensation of the LM at 25 Hz was lower than that of the LM at 5 Hz. At A = 3, 4, 5, 6 mm, the movement sensation of the LM-S at 25 Hz was significantly lower than that of LM-M and LM-S at 5 Hz. We considered that this was because the focus speed at f LM = 25 Hz was too fast for the participants to perceive movement different from the vibration. This results consisted with the previous study [38]. They presented circular STM stimuli with a diameter of 4-7 cm on the palm and found that the focal movement can not be perceived when the movement speed of the focus was above 18 Hz. The results also showed that rendering a convex surface was difficult with the vibratory sensation produced by focused ultrasound. As the vibration score of LM-S at 25 Hz was 6 or higher, this stimulus evoked a vibratory sensation in the experiments. In the LM at 25 Hz, the flat score is the highest for all radii and was significantly greater than the convex score. One participant commented that the contact shape often felt flat when vibration was perceived. X. CONCLUSION In this study, we verified that ultrasound radiation pressure distribution, which spatiotemporally varies at 5 Hz, can provide a static pressure sensation on a finger pad. We also demonstrated that the pressure sensation on the finger pad was perceived as a static contact sensation with a convex surface. In the experiment, four ultrasound focal points were presented on the finger pads of the participant and they were simultaneously rotated in a circle at 5 Hz. When the radius of the focal trajectory was 3 mm, the perceived vibration and movement sensations were the lowest, 1.5 and 2 out of 7 on average, respectively. The perceived intensity of this evoked pressure sensation was equivalent to a 0.24 N physically constant force lasting for 2 s, which is 12 times the physically presented radiation force at the focus. Under the most static condition, the pressure sensation was perceived as a contact sensation on a convex surface with a radius of 2 mm. The average perceptual similarity was 5 out of 7. From these results, we conclude that focused ultrasound can render a static contact sensation at a finger pad with a small convex surface. This contact sensation rendering enables the noncontact tactile reproduction of a static-fine uneven surface. In the future, we will investigate curvature control of the rendered convex surface. Fig. 1 . 1One unit of airborne ultrasound tactile display (AUTD) used in this study. The one AUTD unit is equipped with 249 ultrasound transducers operating at 40 kHz. Fig. 2 . 2Schematic of LM-S (single focus) stimulus and LM-M (multi foci) stimulus. In the LM-S, a single focus is periodically moved in a circle on a finger pad. In the LM-M, multiple foci are simultaneously rotated. The foci are placed along with the circular trajectory. Fig. 3 . 3Experimental equipment used in all subject experiments in this study. This equipment presents a midair image marker. An ultrasound tactile stimulus (LM stimulus) is presented when the finger of a participant touches this marker. The image marker was used to indicate a finger positron to a participant. Fig. 5 . 5Setup for measuring radiation force. The tip of the force gauge to which a 1.5 cm diameter acrylic disk was attached was placed at the focal point. The force gauge was tilted 50 deg so that this disk opposed the propagation direction of the ultrasound wave. Fig. 6 .Fig. 7 . 67Simulated radiation pressure distribution of focus. The white circle with a diameter of 1.5 cm means the area for measuring the radiation force. AUTD setup for the simulation. Fig. 8 . 8Example of the presented picture to evaluate the perceived curvature (A = 2, 6 mm). The radius of the object was changed according to the radius of the presented LM stimulus A. For one stimulus condition, the image of flat, convex, and concave was sequentially presented in random order. Participants reported the perceptual similarity between the perceived curvature and the image. Fig. 9 . 9Evaluated perceptual stationarity of LM stimulus on a finger pad in experiment 1. Participants evaluated the perceived intensity of the vibratory sensation and the focal movement sensation of the LM stimulus with a seven-point Likert scale. ) LM-S at 25 Hz. Fig. 10 . 10Evaluated perceived curvature in experiment 1. The reference images with flat, convex, and concave was presented, and the participants answered perceptual similarity between the perceived tactile shape (curvature) and the image with a seven-point Likert scale. Fig. 12 . 12Confusion matrix of the stimulus size identification. The chance rate in this experiment was 0.2. Fig. 13 . 13Evaluation result of the perceived size of the circular LM stimulus. Fig. 15 . 15Plastic cylinder attached to the tip of the force gauge, used to push a finger pad. The radius was 1 cm, and the basal plane of the cylinder was beveled 1 mm. Fig. 17 . 17Physically static pushing force perceptually equal to the intensity of the LM stimulus. Fig. 18 .Fig. 19 . 1819Simulated time-averaged radiation pressure distribution. These values were normalized. Simulated 5 Hz-power spectrum distribution of time variation of radiation pressure produced by LM-M and LM-S at 5 Hz. The power spectrum distribution was obtained by simulating the time variation of the radiation pressure at each point in the stimulus area and Fourier transforming the time variation. These values were normalized. Tao Morisaki Tao Morisaki is a Ph.D. student with the Graduate School of Frontier Sciences, the University of Tokyo, since 2020. He received the M.S. degree from the Department of Complexity Science and Engineering from the University of Tokyo, Chiba, Japan, in 2020. His research interests include haptics, ultrasound midair haptics, and human-computer interaction. He is a member of VRSJ.Masahiro Fujiwara He is a project assistant professor in the Graduate School of Frontier Sciences, the University of Tokyo, Japan. He received the BS degree in Engineering, the MS degree and the PhD degree in Information Science and Technology from the University of Tokyo, in 2010, 2012, and 2015, respectively. His research interests include information physics, haptics, non-contact sensing and application systems related to them. He is a member of IEEE. Hiroyuki Shinoda Hiroyuki Shinoda is a Professor at the Graduate School of Frontier Sciences, the University of Tokyo. After receiving a Ph.D. in engineering from the University of Tokyo, he was an Associate Professor at Tokyo University of Agriculture and Technology from 1995 to 1999. He was a Visiting Scholar at UC Berkeley in 1999 and was an Associate Professor at the University of Tokyo from 2000 to 2012. 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[]
[ "Phonon-induced electron relaxation in weakly-confined single and coupled quantum dots", "Phonon-induced electron relaxation in weakly-confined single and coupled quantum dots" ]
[ "J I Climente \nCNR-INFM S3\nVia Campi 213/A41100ModenaItaly\n", "A Bertoni \nCNR-INFM S3\nVia Campi 213/A41100ModenaItaly\n", "G Goldoni \nCNR-INFM S3\nVia Campi 213/A41100ModenaItaly\n\nDipartamento di Fisica, Università degli Studi di Modena e Reggio Emilia\nVia Campi 213/A41100ModenaItaly\n", "E Molinari \nCNR-INFM S3\nVia Campi 213/A41100ModenaItaly\n\nDipartamento di Fisica, Università degli Studi di Modena e Reggio Emilia\nVia Campi 213/A41100ModenaItaly\n" ]
[ "CNR-INFM S3\nVia Campi 213/A41100ModenaItaly", "CNR-INFM S3\nVia Campi 213/A41100ModenaItaly", "CNR-INFM S3\nVia Campi 213/A41100ModenaItaly", "Dipartamento di Fisica, Università degli Studi di Modena e Reggio Emilia\nVia Campi 213/A41100ModenaItaly", "CNR-INFM S3\nVia Campi 213/A41100ModenaItaly", "Dipartamento di Fisica, Università degli Studi di Modena e Reggio Emilia\nVia Campi 213/A41100ModenaItaly" ]
[]
We investigate charge relaxation rates due to acoustic phonons in weakly-confined quantum dot systems, including both deformation potential and piezoelectric field interactions. Single-electron excited states lifetimes are calculated for single and coupled quantum dot structures, both in homonuclear and heteronuclear devices. Piezoelectric field scattering is shown to be the dominant relaxation mechanism in many experimentally relevant situations. On the other hand, we show that appropriate structure design allows to minimize separately deformation potential and piezolectric field interactions, and may bring electron lifetimes in the range of microseconds.
10.1103/physrevb.74.035313
[ "https://export.arxiv.org/pdf/cond-mat/0604655v1.pdf" ]
119,334,784
cond-mat/0604655
104b80f5f82b7730d9f1224ff8e2b4f8179c25b8
Phonon-induced electron relaxation in weakly-confined single and coupled quantum dots 28 Apr 2006 J I Climente CNR-INFM S3 Via Campi 213/A41100ModenaItaly A Bertoni CNR-INFM S3 Via Campi 213/A41100ModenaItaly G Goldoni CNR-INFM S3 Via Campi 213/A41100ModenaItaly Dipartamento di Fisica, Università degli Studi di Modena e Reggio Emilia Via Campi 213/A41100ModenaItaly E Molinari CNR-INFM S3 Via Campi 213/A41100ModenaItaly Dipartamento di Fisica, Università degli Studi di Modena e Reggio Emilia Via Campi 213/A41100ModenaItaly Phonon-induced electron relaxation in weakly-confined single and coupled quantum dots 28 Apr 2006(Dated: March 23, 2022)numbers: 7321La7361Ey7210Di We investigate charge relaxation rates due to acoustic phonons in weakly-confined quantum dot systems, including both deformation potential and piezoelectric field interactions. Single-electron excited states lifetimes are calculated for single and coupled quantum dot structures, both in homonuclear and heteronuclear devices. Piezoelectric field scattering is shown to be the dominant relaxation mechanism in many experimentally relevant situations. On the other hand, we show that appropriate structure design allows to minimize separately deformation potential and piezolectric field interactions, and may bring electron lifetimes in the range of microseconds. I. INTRODUCTION Single-electron excited states lifetimes in GaAs-based quantum dots (QDs) are limited to the order of nanoseconds. 1,2 This limit is set mainly by acoustic-phonon scattering, which has been shown to be the dominant scattering mechanism between discrete energy states with few-meV gap in both single 1,2 and vertically coupled 3,4 quantum dots (SQDs and CQDs). Prediction of, and control over the electron relaxation rates is desirable for many possible QD applications, ranging from decoherence-free implementation of quantum gates 5 to efficient QD lasers. 6,7 It is also desirable in order to achieve high-resolution optical spectroscopy, since this requires that (non-radiative) phonon scattering rates are smaller than (radiative) photon emission and absorption rates. Therefore, the knowledge of the physics of electron coupling to acoustic phonons in these nanostructures is of great interest. In a pioneering work, Bockelmann investigated theoretically the influence of lateral (spatial and magnetic) confinement on the single-electron relaxation rates in parabolic QDs. 8 To this end, he considered the deformation potential (DP) coupling between electrons and longitudinal acoustic phonons, while neglecting piezoelectric (PZ) coupling on the grounds of its weaker contribution in 2D GaAs/Ga x Al 1−x As structures, and the fact that a similar qualitative behavior could be expected in QDs. Using the same assumption, it has been recently proposed that orders-of-magnitude suppression of phonon-induced scattering rates can be achieved 5,9,10 in properly designed CQD structures, and the effect of electron-electron interaction in multi-electron QD structures has been investigated. 11 On the other hand, recent theoretical and experimental works suggest that electronacoustic phonon scattering due to PZ field interaction is indeed relevant for momentum and spin-relaxation processes in QDs, 2,12 and may even provide the leading contribution to charge decoherence in laterally coupled QDs. 13,14 Therefore, the use of a theoretical model which considers simultaneously both DP and PZ coupling of electrons to acoustic phonons in QDs appears necessary to evaluate quantitatively electron relaxation rates and assess to which extent previous predictions are affected by the PZ field. In this paper we study the phonon-induced single-electron relaxation rates in realistic models of weakly-confined single and vertically coupled QDs, taking into account both DP and PZ mechanisms. The contribution of each relaxation channel is singled out, and the regimes where each coupling mechanism prevails are established. The PZ coupling is shown to be the prevailing one in several experimental situations. Furthermore, we show that QD devices with maximum lifetimes, in the range of microseconds, can be realized not only in coupled QDs, as previously predicted, 5,9 but also in single QDs. II. THEORETICAL CONSIDERATIONS The theoretical model we use is essentially the same of our previous works, 9,10 but now including the scattering rate arising from the PZ field due to longitudinal-acoustic (LA) and transverse-acoustical (TA) phonons. We study GaAs/AlGaAs dots with disk shape and confinement energy in the range of few meV. 1,2, 15 The confinement potential has been modeled as a quantum well in the growth direction z, formed by the heterostructure bandoffset, while in the xy plane a 2D parabolic confinement is assumed, which gives rise to the Fock-Darwin level structure. 16 The three-dimensional single-particle electronic states are computed within the envelope function approximation. The electron states are labelled by the three quantum numbers (n, m, g), where n = 0, 1, 2 . . . denotes the n-th state with azimuthal angular momentum m = 0, ±1, ±2 . . . and parity g. Here g = 0 (1) stands for even (odd) parity with respect to the reflection about the z = 0 plane. We consider bulk phonons with linear dispersion ω σq = c σ |q|, where c σ is the sound velocity of LA (σ = LA) or TA (σ = TA) phonon modes in the QD material, and q is the momentum. 17 The electronphonon interaction Hamiltonian reads H e−p = νq M ν (q) b q e iqr + b † q e −i qr , where b q and b † q are the phonon annihilation and creation operators respectively, and M ν (q) is the scattering matrix element corresponding to the electron scattering mechanism ν (see below). For simplicity we assume zero temperature, so that phonon absorption and multi-phonon processes are neglegible. The relaxation rate between the initial (occupied) state |Ψ i and the final (unoccupied) state |Ψ f is determined by Fermi golden rule: τ −1 if = 2π h νq |M ν (q)| 2 | Ψ f | e −iqr |Ψ i | 2 δ(|E f − E i | − E q ),(1) where E f and E i stand for the final and initial electron states energy and E q =hω σq represents the phonon energy. It is clear from the above equation that the relaxation is mediated by phonons whose energy matches that of the transition between the initial and final electron states. The electron-phonon scattering is composed of the following contributions: 18,19 (i) The electron-LA phonon scattering due to the deformation potential (ν =LA-DP): |M LA-DP (q)| 2 =h D 2 2 d c LA Ω |q|,(2) (ii) the electron-LA phonon scattering due to the PZ field (ν =LA-PZ): |M LA-PZ (q)| 2 = 32π 2h e 2 h 2 14 ǫ 2 d c LA Ω (3 q x q y q z ) 2 |q| 7 ,(3) (iii) the electron-TA phonon scattering due to the PZ field (ν =TA-PZ): it is useful to realise that differences between the various scattering mechanisms in Eq. (1) arise from i) the diverse matrix elements M ν (q); ii) the different value of q associated with a given transition energy for TA and LA phonons which enters | Ψ f | e −iqr |Ψ i | 2 . The former factor introduces some qualitative differences in the behavior, whereas the latter mostly shifts the transition energy values at which LA-and TA-phonon scattering with the same phonon momentum occur. |M TA-PZ (q)| 2 = 32π 2h e 2 h 2 14 ǫ 2 d c TA Ω q 2 x q 2 y + q 2 y q 2 z + q 2 z q 2 x |q| 5 − (3 q x q y q z ) 2 |q| 7 .(4) In this work we consider mostly relaxation rates from the first excited to the ground electron state in SQDs and CQDs. This is often the most relevant transition, since many QD applications rely on the formation of two-level systems, and it can be monitored e.g. by means of pump-and-probe techniques. 1,2 Assuming Fock-Darwin level structure, for SQDs this corresponds to a transition from a p (m = 1) to a s (m = 0) level. For CQDs this corresponds to the isospin transition from an antisymmetric (g = 1) to a symmetric (g = 0) state. The behavior of transition rates from higher excited states is qualitatively similar, except for the presence of a larger number of decay channels which smears the features of direct scattering between two selected states. 8 We use GaAs/Al 0.3 Ga 0.7 As material parameters: 20 electron effective mass m * = 0.067, band-offset V c = 243 meV, d = 5310 kg/m 3 , D = 8.6 eV, ǫ = 12.9, and h 14 = 1.41 · 10 9 V/m. For the sound speed c σ , we take into account that in cylindrical QDs most of the scattering arises from phonon propagation close to the growth direction. 10 We then assume that the QDs are grown along the [1 0 0] direction and use the corresponding values c LA = 4.72 · 10 3 m/s and c TA = 3.34 · 10 3 m/s. 21 III. RELAXATION IN SINGLE QUANTUM DOTS In this section we study the (n, m, g) = (0, 1, 0) → (0, 0, 0) transition in SQDs as a function of the spatial and magnetic confinement. for weak confinements (hω 0 < 0.4 meV) PZ coupling prevails, so that the total scattering rate shows two maxima instead of one. This is because the different wave vector dependence in the DP and PZ matrix elements ( √ q for DP versus 1/ √ q for PZ interaction) yields maxima at different confinement energies (i.e., different energies of the emitted phonons). Moreover, we observe that the PZ contribution coming from TA-phonons is larger than that of LA-phonons. This holds for almost all calculations throughout this paper. Although the shape of PZ scattering rate is different from the DP one, the limiting behavior is similar: it tends to zero at very weak confinement potentials because of the decreasing phonon density of states; it also tends to zero in the strong confinement limit, due to the orthogonality of the electron states which makes the factor Ψ f | e −iqr |Ψ i vanish rapidly when the phonon wavelength λ q is shorter than the lateral (i.e., the largest) confinement length. 8,10 The lateral confinement and transition energy in a QD can be also modulated by a magnetic field (B) applied along the growth direction. In Figure 1(b) we show the B-dependence of the scattering rate for a SQD with L z = 10 nm andhω 0 = 2 meV. We note that the PZ rate is neglegible at zero magnetic field, but it rapidly increases with B and soon exceeds the DP rate. This is because the energy of the (0,1,0) and (0,0,0) states converge with increasing B, 16 so that the energy of the emitted phonon becomes ever smaller (see upper scale in the figure) and the PZ matrix elements are maximized. As a consequence, the maximum of the total scattering rate, including the PZ interaction, is over three times higher than the one calculated including only the DP mechanism; this difference is even larger for dots with weaker lateral confinement (not shown). Therefore, the inclusion of PZ interactions to properly describe the scattering rate in SQDs in the presence of magnetic fields is critical. This result also justifies recent findings reported by other authors, which claim that spin relaxation rate is mostly determined by TA-PZ scattering: 12 spin-orbit mixing is only significant when B drives the electron states involved close in energy 22 , and then TA-PZ is by far the most important source of phonon scattering. One may note in Figure 1(b) that the maxima of TA-PZ and LA-PZ scattering rates take place at different values of B, in spite of the fact that the matrix elements of both mechanisms depend on the phonon wave vector as 1/ √ q. This is because of the different sound velocities of TA and LA phonon modes, which associate the same phonon energy with different wave vectors, |q| = E q /(hc σ ). IV. CONTROLLING ELECTRON RELAXATION A. Single Quantum Dots Electron-acoustic phonon interaction in SQDs is to a large extent determined by the interplay of lateral confinement length, quantum well width and phonon wavelength. 8 However, in weakly-confined QDs vertical confinement is usually much stronger than lateral confinement. As a result, the energy of the transition (0, 1, 0) → (0, 0, 0) (and thus the wavelength of the emitted phonon) is exclusively determined by the lateral confinement. Therefore, if we fix the lateral confinement and change only the well width we could expect that the scattering rate exhibits periodic oscillations, with maxima when L z ≃ (j + 1/2)λ q (i.e., the electron wave function along the quantum well direction is in-phase with the phonon wave) and minima when L z ≃ jλ q (i.e., the electron wave function is in anti-phase with the phonon wave). We explore this possibility in Figure 2 This accounts for the fact that only one damping dip is observed in the total scattering rate, even though TA-PZ scattering shows two dips due to the smaller speed of TA phonons. From a practical point of view, it is clearly difficult to fabricate QDs with the exact aspect ratio (i.e., lateral vs vertical confinement) which meets the scattering rate minima plotted in Figure 2. Therefore, we next show that the damping of phonon-induced relaxation rate in SQDs can be also achieved by means of external magnetic fields. Let us consider a SQD of lateral confinementhω 0 and well width L z > L crit z , where L crit z is a well width value leading to a scattering rate minimum. A magnetic field applied along the growth direction decreases the energy spacing between the (0, 1, 0) and (0, 0, 0) states and, as a consequence, increases the wavelength of the emitted phonon. It is then possible to tune B in order to obtain the phonon wavelength which is in anti-phase with the electron wave function in the quantum well of width L z . This is illustrated in Figure 3(a) for a SQD withhω 0 = 3 meV and L z = 13 nm, and in Figure 3(b) for a SQD withhω 0 = 5 meV and L z = 15 nm. In both cases scattering rate dips similar to those observed in Figure 2 are forced by the magnetic field. Moreover, the total scattering rate suppressions are again well described in terms of DP coupling, since PZ coupling becomes significant at larger values of B only. This indicates that, as a general rule, for usual QD heights, phonon energies leading to scattering rate suppression derive from lateral (spatial or magnetic) confinements which are strong enough to disregard PZ interaction. B. Vertically Coupled Quantum Dots CQDs constitute an interesting system for phonon scattering modulation because they allow the electron charge to distribute among the various quantum wells along the vertical direction. As a consequence, the total height of the artificial molecule, as CQDs are often called, replaces the SQD height as the critical parameter leading to relaxation rate suppressions when it is in-phase with the phonon wavelength. It is then possible to minimize the scattering rate of the (0, 1, 0) → (0, 0, 0) (intradot) transition by proper structure design 5 or using magnetic fields 9,10 even when the lateral confinement is so weak that the suppression described above for SQDs cannot occur. In particular, it has been predicted that for periodic values of the interdot barrier thickness, L b , decoherence-free systems may be built. 5 This prediction was obtained considering only DP scattering, and one may wonder whether this picture holds when we include PZ interactions. To this end, in Figure 4 Recently we have shown that B-induced control of the DP scattering rate in CQDs is also possible. 9,10 Again, the question arises concerning the influence of PZ scattering in this picture. Figure 5 illustrates the relaxation rate for two CQD structures with L z = 12 nm, L b = 5 nm and different lateral confinement. The arrows point to the position of the expected minima according to the DP description. 9,10 One can see that forhω 0 = 1 meV the minimum of DP scattering coincides with the maximum of TA-PZ scattering, and therefore the expected suppression of the relaxation rate is greatly inhibited. Yet, this no longer occurs forhω 0 = 2 meV, where PZ interactions are less relevant at weak fields due to the stronger spatial lateral confinement. It is worth stressing that this suppression is found for a lateral confinement strength which is too weak for the SQD-based suppression to be efficient [seehω 0 = 2 meV curve in Figure 2(a)]. On the other hand, simultaneous control of DP and TA-PZ scattering rates, aimed at making their respective minima coincide, is an intrincate process which follows from the interplay among QD geometry, composition and growth direction. Therefore, we conclude that CQDs stand as a firm alternative to SQDs for control of intradot transitions phonon relaxation only when both scattering sources have significantly different weights. This can be achieved for instance using QDs with lateral confinement ∼ 2 meV, or building QDs with narrow-gap materials where the spacing between conduction band states is increased and hence PZ influence is smaller. Next, we analyze the transition rate between the lowest bonding and antibonding states of two CQDs, (n, m, g) = (0, 0, 1) → (0, 0, 0) (interdot or isospin transition). The interest of this transition is in part connected with the demand of coherent tunneling between the two dots for isospin-based implementations of quantum gates 23 , as well as with quantum dot cascade laser devices performance. 7 The wavelength of the emitted phonon is in this case controlled by the tunneling energy, which can be tuned by means of either the barrier thickness or an electric field applied along the z direction (E z ). 9 In Figure 6 we study the effect of both mechanisms for CQD devices with L z = 5 nm andhω 0 = 5 meV. Figure 6(a) represents the transition rate vs the barrier width. The total rate depends strongly on L b . However, only a few non-periodic oscillations are observed, as opposed to the intradot transition case (Figure 4). This is due to the changing tunneling energy (see inset), which makes the phonon wavelength vary exponentially with L b . On the other hand, we note that PZ interactions become dominant when the barrier thickness is large, owing to the small tunneling energy. These results can be related to recent experiments which reported different relative photoluminescence intensities for bonding and antibonding exciton states in a pair of CQDs as a function of the interdot distance. 4 Since the intensity of photoluminescence peaks is proportional to the lifetime of conduction states, and acoustic phonon coupling was identified as the main relaxation source in the experiment, these observations indicated a non-monotonic dependence of the antibonding state lifetime on the barrier thickness, in agreement with Figure 6(a). In Figure 6(b) we depict the E z -dependent scattering rate corresponding to CQDs with L b = 15 nm. As in previous works 9 , we find order-of-magnitude oscillations of the DP scattering rate, which also take place for PZ rates albeit with different period for the TA phonons. Interestingly, the PZ scattering rate decay is much faster than that of the DP, due to the different wave vector dependence of the matrix element. As a result, even if PZ prevails in the absence of external fields, the use of an electric field soon turns DP into the dominant relaxation mechanism, and one can think of electric-fieldinduced suppressions of the scattering rate in terms of DP interaction only. We would like to point out that the interdot transition rate minima observed in Figure 6 have no counterpart in laterally coupled dots. 14 This represents a fair advantage of vertically coupled structures with regard to quantum gate implementation schemes, since it allows one to reduce charge decoherence times dramatically. The underlying reason for this difference lies in the larger total vertical dimension of the vertically coupled double QD system, which makes it possible to reach in-phase relation between the electron wavefunction and the phonon plane wave along z for small tunneling energies (i.e., long phonon wavelengths). So far we have considered CQDs composed by two identical dots (in analogy with atomic molecules, we refer to these systems as 'homonuclear'). However, in practice this is hard to achieve and the size of the coupled dots is usually slightly different, leading to 'heteronuclear' systems. 24 (1/ns) L (nm) L (nm) 5 6 4 (1/ns) 24 24. Figure 1 ( 1a) illustrates the scattering rate of a SQD with well width L z = 10 nm as a function of the harmonic confinement energy,hω 0 . For most lateral confinements, DP coupling (the only source of relaxation considered in Ref. 8) gives the largest contribution. However, (a), where the effect of vertical confinement on the total scattering rates of SQDs with differenthω 0 is illustrated. Forhω 0 = 1 and hω 0 = 2 meV no scattering minima are observed, indicating that λ q is still too large for the range of L z shown (which are indeed typical well widths of realistic devices). However, for hω 0 = 3 meV we observe a striking dip at about L z = 10.5 nm, where the scattering rate is suppressed by orders of magnitude due to the anti-phase relation between the electron wave function and the phonon plane wave. Forhω 0 = 5 meV order-of-magnitude damping of the scattering rate at several well widths values is observed. A similar geometry-induced suppression of the scattering rate was previously proposed for CQDs 5 , but here we show that it is also feasible for SQDs with realistic dimensions if the confinement energy is in the proper range. This result may be of great significance for quantum coherence preservation in SQD devices. Figure 2 ( 2b) depicts separately the LA-DP, TA-PZ and LA-PZ scattering rates in the SQD withhω 0 = 3 meV. One can see that, in the absence of external fields, LA-DP scattering rate constitutes the dominant relaxation mechanism by at least one order of magnitude. we plot the scattering rate of two CQDs with L z = 5 nm andhω 0 = 1 meV, as a function of the barrier thickness. The total scattering rate oscillates periodically, with most of the contribution coming from DP interactions. Nevertheless, the TA-PZ contribution becomes predominant at the values of L b where complete suppression of LA-DP scattering is predicted (see insets around L b = 5 nm and L b = 25 nm inFigure 4); indeed, the DP and PZ mechanism have minima at different L b due to the difference in the sound velocity and, therefore, in the wavelength of the emitted phonons. This imposes a limit on the barrier-thickness-induced suppression of intradot transition rates reported in Ref. 5, which can be nonetheless minimized by growing CQDs with stronger lateral confinement, so that PZ interaction strength rapidly decreses (seeFigure 1). , 25 FIG. 1 :FIG. 2 :FIG. 3 : 25123Molecular levels are very sensitive to small changes in the size of the coupled dots,24,26 so that one may also expect significant effects on the transition rates. To analyze this case, inFigure 7we consider a CQD structure with L b = 10 nm. Both QDs have the same lateral confinementhω 0 = 5 meV, but the bottom well width is L z = 5 nm, while the upper well one is L z = 5.5 nm. The effect of an electric field on this heteronuclear system is to introduce order-of-magnitude oscillations which first lead to an enhancement of the scattering rate and then to a reduction in a clearly symmetric fashion. These results are interpreted as follows. In the absence of external fields, bonding and antibonding states (solid and dotted lines in the insets, respectively) localize in opposite wells. As a consequence, the factor Ψ f | e −iqr |Ψ i in Eq. (1) is reduced as compared to the homonuclear case, where the charge density of bonding and antibonding states are equally distributed between the dots.For a given electric field we can compensate for the different well width of the two QDs, and force an identical charge density distribution of bonding and antibonding states. This maximizes Ψ f | e −iqr |Ψ i and therefore the relaxation rate. Finally, for stronger fields the original localization of states is reversed. It is also worth noting that the total scattering rate inFigure 7is essentially given by the DP interaction, except in the vicinity of the 'homonuclearity' point, where the energy of bonding and antibonding levels are very close and TA-PZ relaxation prevails. For practical purposes, one is often interested in obtaining coherent delocalization of the particles between the two dots, i.e. reaching a homonuclear system with long electron lifetimes.23 Figure 7shows that a relatively small departure from the electric field which gives the homonuclear charge distribution, from E z = 570 to E z = 485 kV/m, increases the electron lifetime by three orders of magnitude, while partially preserving electron delocalization.V. SUMMARYSingle-electron relaxation rates in GaAs SQDs and CQDs due to coupling with acousticphonons have been investigated. PZ interactions due to TA phonons, often disregarded in the literature, constitute the major source of scattering for small (< 0.5 meV) electron transition energies. For larger gaps, DP interactions due to LA phonons prevail. Indeed, we identify many situations where PZ effects become critical, such as SQDs subjected to an axial magnetic field or the isospin transition in CQDs with large interdot distances. Nevertheless, we have shown that proper structure design makes it possible to control the scattering rates of both SQDs and CQDs by orders of magnitude. Suppression of the scattering rate in SQDs can be attained for moderately large (few meV) lateral confinement, while CQD structures permit to extend it down to fairly weak lateral confinements. Similar results can be achieved using external fields even when the structures do not have the optimum geometry. Unlike laterally CQDs, vertically CQDs can be used to obtain enhanced electron lifetimes (in the range of microseconds) in excited 'molecular' states, which has important implications for quantum computation devices. These results are found to be robust for size-mismatched CQDs. tures, owing to the similar lattice constants of the low-dimensional structure and embedding matrix materials. However, if the materials were elastically dissimilar, confined acoustic phonon modes should be considered, see e.g. A.A. Balandin, J. Nanosci. Nanotech. 5, 1015 (2005), and references therein. 18 V.F. Gantmakher, and Y.B. Levinson, Carrier Scattering in Metals and Semiconductors, (Mod-(Color online) Acoustic phonon scattering rate as a function of (a) lateral confinement, and (b) magnetic field, for the electron transition (n, m, g) = (0, 1, 0) → (0, 0, 0) in a GaAs/Al 0.3 Ga 0.7 As QD with L z = 10 nm. In panel (b)hω 0 = 2 meV, and the upper scale shows the emitted phonon energy E q . (Color online) (a) Acoustic phonon total scattering rate as a function of the quantum well width L z for the electron transition (n, m, g) = (0, 1, 0) → (0, 0, 0) in a SQD, at selected values of lateral confinementhω 0 . (b) Individual contributions from each scattering mechanism in a QD withhω 0 = 3 meV. (Color online) Acoustic phonon scattering rate as a function of the magnetic field for the electron transition (n, m, g) = (0, 1, 0) → (0, 0, 0) in selected SQDs. FIG. 4 : 1 FIG. 7 : 417(Color online) Acoustic phonon scattering rate as a function of the barrier width L b for the electron transition (n, m, g) = (0, 1, 0) → (0, 0, 0) in a CQD structure withhω 0 = 1 meV and L z = 5 nm, . The insets zoom in the regions around the DP scattering minima. Scattering mechanisms are represented as in the legend of Figure two CQD structures with L z = 12 nm, L b = 5 nm and different lateral confinement. The arrows point to the position of the relaxation rate minima expected including the DP interaction only (Refs. online) Acoustic phonon scattering rate as a function of (a) the barrier width L b , and (b) the electric field, for the electron transition (n, m, g) = (0, 0, 1) → (0, 0, 0) in a CQD structure withhω 0 = 5 meV and L z = 5 nm. Inset of panel (a): energy of the lowest bonding (solid line) and antibonding (dashed line) states vs barrier width. In panel (b) L b = 15 nm. Scattering mechanisms are represented as in the legend of Figure 1. (Color online) Acoustic phonon scattering rate as a function of the electric field for the electron transition (n, m, g) = (0, 0, 1) → (0, 0, 0) in a heteronuclear CQD structure, with L b = 10 nm,hω 0 = 5 mev, L z = 5 nm (bottom QD) and L z = 5.5 nm (upper QD). Insets: electron wavefunction (arbitrary units) in the double quantum well for E z = 0 kV/cm (left), E z = 568 kV/m (center) and E z = 1200 kV/m (right), with solid (dotted) lines representing the lowest bonding (antibonding) state. Scattering mechanisms are represented as in the legend of Figure 1. account for the two transverse phonon modes (both identical under Ref. 19 approximation for zinc-blende crystals). For the analysis of the results presented in the following sections,In the above expressions D, d and Ω stand for the crystal acoustic deformation potential constant, density and volume, respectively. e is the electron charge, h 14 the PZ constant and ǫ the static dielectric constant. Note that the sum in Eq. 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Elsevier Science Publishers19The use of bulk phonons is a reasonable approximation for GaAs/Al 1−x Ga x As heterostruc- ern Problems in Condensed Matter Sciences, vol. 19, Elsevier Science Publishers, 1987). . J D Zook, Phys. Rev. 136869J.D. Zook, Phys. Rev. 136, A869 (1964). Physics of Hot Electron Transport in Semiconductors. C.S. TingWorld ScientificC.S. Ting (ed.), Physics of Hot Electron Transport in Semiconductors, (World Scientific, 1992). Landolt-Börnstein, Numerical Data and Functional Relationships in. Group IV Elements and III-V Compounds. O. MadelungSpringer-Verlag17O. Madelung (Ed.), Landolt-Börnstein, Numerical Data and Functional Relationships in Science and Technology, Vol. 17 Semiconductors, Group IV Elements and III-V Compounds, (Springer- Verlag, 1982). . M Florescu, S Dickman, M Ciorga, A Sachrajda, P Hawrylak, Physica E (Amsterdam). 22414M. Florescu, S. Dickman, M. Ciorga, A. Sachrajda, and P. Hawrylak, Physica E (Amsterdam) 22, 414 (2004); . M Florescu, P Hawrylak, Phys. Rev. B. 7345304M. Florescu, and P. Hawrylak, Phys. Rev. B 73, 045304 (2006). . M Bayer, P Hawrylak, K Hinzer, S Fafard, M Korkusinski, Z R Wasilewski, O Stern, A Forchel, Science. 291451M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z.R. Wasilewski, O. Stern, and A. Forchel, Science 291, 451 (2001). . M Pi, A Emperador, M Barranco, F Garcias, K Muraki, S Tarucha, D G Austing, Phys. Rev. Lett. 8766801M. Pi, A. Emperador, M. Barranco, F. Garcias, K. Muraki, S. Tarucha, and D. G. Austing, Phys. Rev. Lett. 87, 066801 (2001). . N N Ledentsov, V A Shchukin, M Grundmann, N Kirstaedter, J Böhrer, O Schmidt, D Bimberg, V M Ustinov, A Yu, A E Egorov, P S Zhukov, S V Kop&apos;ev, N Zaitsev, Yu, Gordeev, . I Zh, A I Alferov, A O Borovkov, S S Kosogov, P Ruvimov, U Werner, J Gösele, Heydenreich, Phys. Rev. B. 548743N.N. Ledentsov, V.A. Shchukin, M. Grundmann, N. Kirstaedter, J. Böhrer, O. Schmidt, D. Bim- berg, V.M. Ustinov, A. Yu. Egorov, A.E. Zhukov, P.S. Kop'ev, S.V. Zaitsev, N.Yu. Gordeev, Zh.I. Alferov, A.I. Borovkov, A.O. Kosogov, S.S. Ruvimov, P. Werner, U. Gösele, and J. Hey- denreich, Phys. Rev. B 54, 8743 (1996). . L R C Fonseca, J L Jimenez, J P Leburton, Phys. Rev. B. 589955L.R.C. Fonseca, J.L. Jimenez, and J.P. Leburton, Phys. Rev. B 58, 9955 (1998).
[]
[ "Resolving the trans-Planckian problem along the lines of a finite geometry", "Resolving the trans-Planckian problem along the lines of a finite geometry" ]
[ "Arkady Bolotin \nBen-Gurion University of the Negev\nBeershebaIsrael\n" ]
[ "Ben-Gurion University of the Negev\nBeershebaIsrael" ]
[]
In black hole physics, inflationary cosmology, and quantum field theories, it is conjectured that the physical laws are subject to radical changes below the Planck length. Such changes are due to effects of quantum gravity believed to become significant at the Planck length. However, a complete and consistent quantum theory of gravity is still missing, and candidate models of quantum gravity have not yet overcome major formal and conceptual difficulties. Another problem is how to determine a geometry of physical space that features a minimal length scale such as the Planck length. In the present paper it is demonstrated that the said geometry can be any geometric system omitting continuity, i.e., a geometry that possesses only a finite number of points.
null
[ "https://export.arxiv.org/pdf/2304.10396v1.pdf" ]
258,236,486
2304.10396
72840e17738cdf0daa53f84ddf1b9d1dcdc17884
Resolving the trans-Planckian problem along the lines of a finite geometry 15 Apr 2023 April 21, 2023 Arkady Bolotin Ben-Gurion University of the Negev BeershebaIsrael Resolving the trans-Planckian problem along the lines of a finite geometry 15 Apr 2023 April 21, 2023minimal lengthgeneralized uncertainty principleholographic principlefinite plane geometryEuclidean distance In black hole physics, inflationary cosmology, and quantum field theories, it is conjectured that the physical laws are subject to radical changes below the Planck length. Such changes are due to effects of quantum gravity believed to become significant at the Planck length. However, a complete and consistent quantum theory of gravity is still missing, and candidate models of quantum gravity have not yet overcome major formal and conceptual difficulties. Another problem is how to determine a geometry of physical space that features a minimal length scale such as the Planck length. In the present paper it is demonstrated that the said geometry can be any geometric system omitting continuity, i.e., a geometry that possesses only a finite number of points. Introduction Does nature feature a minimal length scale ℓ min ? Is it true to say that ℓ min coincides with the Planck length ℓ P ? On account of the generalized uncertainty principle (GUP) [1,2,3] and the holographic principle (HP) [4,5,6], one may incline to think that both those questions have a positive answer. Consider for example a thought experiment proposed in [7] that concerns the registration of photons reflected off a mirror at some distance D from a non-relativistic detector. Let the variance of the position of the detector ℓ ≥ 0 be Var(ℓ). Then, in accordance with the Heisenberg uncertainty principle, the variance of the detector's velocity must be Var(l) ≥ 2 /4Var(ℓ)M 2 , where M is the detectors's mass. Observe that the time needed for a photon to travel to the mirror and come back is T = 2D/c. Let the detector have moved bylT during that time. On the assumption that ℓ andlT are uncorrelated random variables, the variance of their sum must be equal to the sum of their variances, or, expressed symbolically: Var(ℓ +lT ) = Var(ℓ) + Var(l)T 2 . Provided that the registration of photons reflected off the mirror will not be connected to the rest of the world if the distance from the detector to the mirror D is closer than or equal to the Schwarzschild radius 2GM/c 2 , one finds: Var ℓ +lT ≥ Var(ℓ) + 4ℓ 4 P Var(ℓ) ,(1) where ℓ 4 P = 2 G 2 /c 6 . The above is a generalization of the uncertainty principle that accounts for gravitational effects in the measure of positions. The variance Var(ℓ +lT ) gets the minimum when Var(ℓ) = min(Var(ℓ)) = 2ℓ 2 P . As min(Var(ℓ)) ∼ ℓ 2 min , this entails the lower bound on the variable ℓ, namely, ℓ min = √ 2ℓ P ≈ ℓ P . Regarding HP, it states that physics inside a bounded region is fully captured by physics at the boundary of the region [8]. As a consequence, the vacuum enclosed inside a region with a boundary of area A is fully described (up to a factor of log 2) by no more than A/ℓ 2 P degrees of freedom. Stipulating A min /ℓ 2 P = 1 and A min ∼ ℓ 2 min , this implies ℓ min ∼ ℓ P . On the other hand, neither GUP nor HP has implications for a geometry of physical space. In more detail, since GUP is expressed in terms of variances, there is a nonzero chance of ℓ being within the interval ℓ < ℓ P . Additionally, HP relates to discretization, i.e., the process of transferring continuous geometrical magnitudes such as areas into sets of primitive objects, e.g., Planck areas ℓ 2 P . As such, it prompts to discretization error that satisfies the equation: A < A ℓ 2 P ℓ 2 P + ℓ 2 P .(2) Denoting the floor function ⌊A/ℓ 2 P ⌋ by m ∈ N and supposing A = mℓ 2 P + ℓ 2 indicates that there are distances ℓ < ℓ P . Hence, despite GUP and HP, a geometry of physical space continues to be without limits or bounds. A case in point is a gravitational singularity where all spatial dimensions become of size zero. The appearance of distances beyond the Planck length presents a problem (known as the trans-Planckian problem [9,10]) because one expects the physical laws to undergo fundamental changes beyond ℓ P . To address the problem, one may propose that gravitational singularities do not exist. The idea can be stated in the form that due to effects of quantum gravity, there is a significant deviation from 1/ℓ 2 law of Newtonian gravity beyond the minimum distance ℓ min = ℓ P . Consequently, distances ℓ < ℓ P are unreachable by way of the gravitational force. Be that as it may, a complete and consistent quantum theory of gravity is still missing, and candidate models of quantum gravity still need to overcome major formal and conceptual problems [11]. Giving that, alternative ways of preventing distances ℓ < ℓ P from appearing deserve to be considered and analyzed. One of those ways is to assume that a geometry of physical space excludes continuum, that is, the number of points that make up physical space is finite. This approach will be discussed in the present paper. A finite plane geometry A geometry can be defined as a system of axioms which identify what things are that constitute fundamental objects such as points and lines [12]. In terms of this definition, a finite geometry is any of axiomatic systems which permit only a finite number of points. Suppose that M 2 is a plane (i.e., a flat, two-dimensional space) that can be called a finite geometry and A is a region in M 2 whose area is A. Let the cardinality of A (i.e., the number of points that constitute A) be denoted by P (A) and let it conform to the area A expressed in units of the Planck area: card(A) ≡ P (A) ∼ A ℓ 2 P .(3) Consider C(A), the system of coordinates on A, that is, the set of numbers that specify the position of each point in A. Since M 2 is finite, the region A is finite and so is the set C(A). Moreover, to be qualified as a system of coordinates, the set C(A) must be such that any binary operation on its elements is defined. More precisely, a mapping f: C(A) × C(A) → C(A)(4) is required to exist (meaning that binary operations are supposed to be closed on C(A)). This implies that the set C(A) must be a finite field F q whose size is a prime power q = p n with a prime number p and a positive integer n [13]. Therefore, the cardinality of C(A) can be determined as card (C(A)) = card(F q ) = p mP (A) .(5) where m ∈ {1, 2, . . . }. This system of coordinates can store log 2 card (C(A)) = m P (A) log 2 p(6) bits of information (m log 2 p bits per each point in A). Picking out p and m to be 2 and 1 respectively, i.e., choosing the size of A to be card (C(A)) = 2 P (A) ,(7) allows one to construe each point in A as a bit of information. To make our discussion more tangible, let us assume that A consists of just 4 points, i.e., P (A) = 4. Then, card (C(A)) = 2 4 or card (C(A)) = card(F 4 ) 2 ,(8) where card(F 4 ) 2 is the cardinality of a vector space of dimension 2 over the finite field F 4 . The field F 4 consists of four elements called O, I, α and β [14]. Therewithal, O plays the role of the additive identity element, 0, at the same time as I fulfils the role of the multiplicative identity element, 1. Furthermore, α 2 = β, β 2 = α, and ∀x ∈ {O, I, α, β}: x + x = O .(9) The expression (8) Fig.1 has no ordinary lines (i.e., ones that contain exactly two of the set of points) but possesses "parallel" lines (i.e., ones that have no common points); for example, the line 11 is parallel to the line 16. The configuration 16 5 20 4 is known as Sylvester-Galai configuration (SGC) [15]. It cannot be realized by points and lines of the Euclidean plane. This suggests that a geometry based on SGC is not metric, to be specific, such a geometry cannot be equipped with the Euclidean distance satisfying all the metric axioms. The last can be demonstrated explicitly. The Euclidean distance on an affine geometry Given that subtraction is identical to addition, as is the case for every filed F m 2 with m ∈ {1, 2, . . . }, one can introduce the distance d(x 1 , x 2 ) between distinct elements x 1 and x 2 in F 4 by the formula ∀x 1 , x 2 ∈ F 4 : d(x 1 , x 2 ) = x 1 + x 2 .(11) Such a distance is metric because it satisfies all the metric axioms, which are: (M1) Identity of indiscernibles: ∀x 1 , x 2 ∈ F 4 x 1 = x 2 → d(x 1 , x 2 ) = O , (M2) Positiveness: ∀x 1 , x 2 ∈ F 4 x 1 = x 2 → d(x 1 , x 2 ) ∈ F 4 \ O , (M3) Symmetry: ∀x 1 , x 2 ∈ F 4 d(x 1 , x 2 ) = d(x 2 , x 1 ) , (M4) Triangle inequality: ∀x 1 , x 2 , x 3 ∈ F 4 d(x 1 , x 3 ) = d(x 1 , x 2 ) + d(x 2 , x 3 ) . Certainly, those axioms can be easily verified by replacing the distance between two elements in F 4 for their total. For example, since x 2 + x 2 = O, one finds x 1 + x 3 = (x 1 + x 2 ) + (x 2 + x 3 ) = x 1 + O + x 3 .(12) The above means to imply that the ordered pair (F 4 , x 1 + x 2 ) is a metric space over the field F 4 . Following the treatment of [16], consider the direct product of two metric spaces over F 4 (F 4 , x 1 + x 2 ) × (F 4 , y 1 + y 2 ) = F 4 × F 4 , (x 1 + x 2 ) × (y 1 + y 2 ) ,(13) where the distance between two points on the plane F 16 = F 4 × F 4 with coordinates (x 1 , y 1 ) and (x 2 , y 2 ) is the Euclidean distance d ((x 1 , y 1 ), (x 2 , y 2 )) = (x 1 + x 2 ) × (y 1 + y 2 ) = (x 1 + x 2 ) 2 + (y 1 + y 2 ) 2 . In view of (13), the set C(A) is the coordinate system that specifies every point in A by a pair of the elements (x, y) in F 16 which are the distances x + O = x and y + O = y to the point from two fixed coordinate lines. Therefore, (x, y) are Cartesian coordinates of points in A and, correspondingly, C(A) is the Cartesian coordinate system for A. By virtue of the identity ∀x 1 , x 2 ∈ F 4 : (x 1 + x 2 ) 2 = x 2 1 + x 2 2(15) the Euclidean distance on the plane F 16 takes the form of sum d ((x 1 , y 1 ), (x 2 , y 2 )) = (x 1 + x 2 ) 2 + (y 1 + y 2 ) 2 √ a 2 +b 2 = √ (a+b) 2 =a+b = x 1 + x 2 + y 1 + y 2 .(16) If this sum is O + I + α + β or x + x, where x ∈ F 4 , then the Euclidean distance will be O. This means that the Euclidean distance on F 16 does not satisfy the metric axiom M2: ∃(x 1 , y 1 ), (x 2 , y 2 ) ∈ F 16 (x 1 , y 1 ) = (x 2 , y 2 ) → d ((x 1 , y 1 ), (x 2 , y 2 )) = O .(17) As an illustration, the Euclidean distance between any two distinct points on the line 11 or on the line 16 pictured in Fig. 1 is equivalent to zero. E.g., d(P 6 , P 11 ) = I + α + I + α = O and d(P 10 , P 4 ) = I + β + α + O = O. Consequently, the Euclidean distance on the plane F 16 is not metric. Provided the metric Euclidean distance is the natural way of measuring physical length of a line segment between two arbitrary points, it can be concluded that there is no notion of physical distance which can be defined everywhere on the plane F 4 × F 4 . Let F (P ) be a real function defined in each point P = (x, y) on the plane F 4 × F 4 . The difference between two points is known as their delta, ∆P , while the function difference, ∆F (P ), divided by the point difference ∆P is known as "difference quotient" [17]: ∆F (P ) ∆P = F (P + ∆P ) − F (P ) ∆P .(18) On condition that the difference between two distinct points (x 1 , y 1 ) and (x 2 , y 2 ) on F 4 × F 4 is the Euclidean distance d ((x 1 , y 1 ), (x 2 , y 2 )), the difference quotient is given by the formula: ∆F (P ) ∆P = F (x 1 , y 1 ) − F (x 2 , y 2 ) x 1 + x 2 + y 1 + y 2 .(19) Based thereon, ∆F (P )/∆P is the slope of the secant line passing through the points with coordinates (x 1 , y 1 ), F ((x 1 , y 1 )) and (x 2 , y 2 ), F ((x 2 , y 2 )) . Since for all x in F 4 , O · x = O, the division by O must remain undefined. According to (17), this implies that for some pairs of points on F 4 × F 4 , the slope ∆F (P )/∆P is undefined: F (x 1 , y 1 ) = F (x 2 , y 2 ) → F (x 1 , y 1 ) − F (x 2 , y 2 ) O .(20) Due to that, the difference quotient ∆F (P )/∆P cannot be considered as the mean value of the derivative of F over the interval [(x 1 , y 1 ), (x 2 , y 2 )]. One can infer then that the function F does not have a derivative at the points of this interval because the function is not continuous there. In formal terms, the above implies that one cannot introduce a globally defined structure that makes possible differential calculus on the plane F 4 × F 4 . As a result, differential equations of motion cannot be applicable in the case of such a plane. Emergent metricity of a plane geometry The number P (A) of points that constitute the region A can be presented as the number of elements in a finite set S, namely, P (A) = card(S) .(21) In this way, using (7) one gets card (C(A)) = 2 P (A) = 2 card(S) . At the same time, since S is finite, there is a bijection from S to the set of those natural numbers that are less than some specific natural number n = card(S), namely, f : S → {1, . . . , n} .(23) For the region A of a macroscopic scale ℓ ∈ [1, 10 24 ] m, or ℓ/ℓ P ∈ [10 35 , 10 59 ], this number, i.e., n = P (A) ∼ ℓ 2 ℓ 2 P ,(24) is greater than 10 70 . In that instance, it can be believed that n = ∞ and so S can be considered to be the set of all natural numbers N. Symbolically, card (C(A)) −→ ℓ ≫ ℓ P 2 card(N) .(25) By Cantor-Bernstein-Schroeder theorem [18], 2 card(N) = card(R). Thus, when ℓ ≫ ℓ P , the cardinality of the Cartesian coordinate system for A becomes the cardinality of the continuum: card (C(A)) −→ ℓ ≫ ℓ P card(R) .(26) Now, recall that in a field F q of size q = p k (with a prime number p and a positive integer k), adding p copies of any element in F q results in zero, specifically, ∀x ∈ F p k : x + x + · · · + x p copies = 0 .(27) Accordingly, one can say that the characteristic of the field F q is p [19]. If the above sum never reaches 0, the field F q is said to have characteristic zero. For example, the field R consisting of all real numbers has characteristic 0. In keeping with (26), a field having the characteristic 2 at ℓ ∼ ℓ P becomes a field of characteristics 0 when ℓ ≫ ℓ P . Given that the metric axiom M2 may hold true in fields of characteristics zero but breaks down in fields of characteristics 2, one can infer that a geometry of a plane seen as a metric at macroscopic scales ℓ ≫ ℓ P ceases to be such in a scale ℓ ∼ ℓ P whereat the Euclidean distance stops being metric. For that reason, the Planck length ℓ P can be thought about as a length which is smaller than all possible physical distances. Conclusion At this point, the solution of the trans-Planckian problem along the lines of a finite geometry can be formulated in the following way. Seeing that there is no notion of physical distance defined across a finite field F q of characteristics 2, the Einstein field equations -the set of nonlinear partial differential equations -cannot be given meaning at scales ℓ ∼ ℓ P , let alone ℓ < ℓ P . The same applies to differential equations of any quantum field theory at ℓ ℓ P . Thus, a junction between general relativity and quantum mechanics is unable to render gravitational singularities: Neither of the theories has meaning in a scale ℓ ℓ P . This conclusion may serve as corroboration of the conjecture about radical modifications of the physical laws below the Planck length. implies that the elements of C(A) are the points of the affine plane F 16 = F 4 ×F 4 . These points are identified with ordered pairs (x, y) of numbers modulo 4 and can be connected with lines satisfying a linear equation ax + by = c (mod 4) . Figure 1 : 1The affine plane F 4 × F 4 .In this way, C(A) can be presented as the configuration in the plane of 16 points and 20 (straight) lines. TheFig.1shows this configuration (the lines 9, 10, 12-15 are drawn as curves and the lines 17-20 are pictured as circles). In the language of configuration, the one shown in theFig.1has the notation 16 5 20 4 (meaning that there are 16 points, 5 lines per point, 20 lines, and 4 points per line). The affine plane displayed in the Observable Consequences of Fundamental-Length Hypotheses. C , Alden Mead, Phys. Rev. 143C. Alden Mead. Observable Consequences of Fundamental-Length Hypotheses. Phys. Rev., 143:990-1005, 1966. On gravity and the uncertainty principle. J Ronald, David I Adler, Santiago, Modern Physics Letters A. 1420Ronald J. Adler and David I. Santiago. On gravity and the uncertainty principle. Modern Physics Letters A, 14(20):1371-1381, 1999. Minimal Length Scale Scenarios for Quantum Gravity. Sabine Hossenfelder, Living Reviews in Relativity. 162Sabine Hossenfelder. Minimal Length Scale Scenarios for Quantum Gravity. Living Reviews in Relativity, 16(2):1-90, 2013. Black holes and the dimensionality of space-time. Gerard &apos;t Hooft, Proceedings of the Symposium. Oskar Klein and Ulf Lindströmthe SymposiumStockholm, SwedenWorld ScientificGerard 't Hooft. Black holes and the dimensionality of space-time. In Oskar Klein and Ulf Lindström, editors, Proceedings of the Symposium "The Oskar Klein Centenary", 19-21 Sept. 1994, Stockholm, Sweden, pages 122-137. World Scientific, 1995. The World as a Hologram. Leonard Susskind, J. Math. Phys. 36Leonard Susskind. The World as a Hologram. J. Math. Phys., 36:6377-6396, 1995. TASI lectures on the Holographic Principle. Daniela Bigatti, Leonard Susskind, Daniela Bigatti and Leonard Susskind. TASI lectures on the Holographic Principle. https://arxiv.org/abs/hep-th/0002044, Feb. 2000. Quantum limitations of the measurement of space-time distances. H Salecker, E P Wigner, Phys. Rev. 109H. Salecker and E. P. Wigner. Quantum limitations of the measurement of space-time dis- tances. Phys. Rev., 109:571-577, 1958. The holographic principle. Raphael Bousso, Reviews of Modern Physics. 74Raphael Bousso. The holographic principle. Reviews of Modern Physics, 74:825-874, 2002. Black-hole evaporation and ultrashort distances. Theodore Jacobson, Phys. Rev. D. 446Theodore Jacobson. Black-hole evaporation and ultrashort distances. Phys. Rev. D, 44(6):1731-1739, 1991. Hawking radiation without trans-Planckian frequencies. R Brout, S Massar, R Parentanit, Ph Spindel, Phys. Rev. D. 528R. Brout, S. Massar, R. Parentanit, and Ph. Spindel. Hawking radiation without trans- Planckian frequencies. Phys. Rev. D, 52(8):4559-4568, 1995. Claus Kiefer, Conceptual Problems in Quantum Gravity and Quantum Cosmology. ISRN Mathematical Physics. 509316Claus Kiefer. Conceptual Problems in Quantum Gravity and Quantum Cosmology. ISRN Mathematical Physics, 509316:1-18, 2013. Classical Topics in Discrete Geometry. Károly Bezdek, SpringerNew York, NYKároly Bezdek. Classical Topics in Discrete Geometry. Springer, New York, NY, 2010. Handbook of Finite Fields. L Gary, Daniel Mullen, Panario, Chapman and Hall/CRCNew YorkGary L. Mullen and Daniel Panario. Handbook of Finite Fields. Chapman and Hall/CRC, New York, 2013. Finite Fields and Applications. L Gary, Carl Mullen, Mummert, American Mathematical SocietyGary L. Mullen and Carl Mummert. Finite Fields and Applications. American Mathematical Society, 2007. Endre Boros, Zoltan Füiredi, L M Kelly, On Representing Sylvester-Gallai Designs. Discrete and Computational Geometry. 4Endre Boros, Zoltan Füiredi, and L. M. Kelly. On Representing Sylvester-Gallai Designs. Discrete and Computational Geometry, 4:345-348, 1989. A Course in Metric Geometry. Dmitri Burago, Yuri Burago, Sergei Ivanov, American Mathematical SocietyDmitri Burago, Yuri Burago, and Sergei Ivanov. A Course in Metric Geometry. American Mathematical Society, 2001. Numerical Treatment of Partial Differential Equations. Christian Grossmann, Martin Hans-Görg Roos, Stynes, Springer-VerlagBerlin HeidelbergChristian Grossmann, Hans-Görg Roos, and Martin Stynes. Numerical Treatment of Partial Differential Equations. Springer-Verlag, Berlin Heidelberg, 2007. Schröder-Bernstein Theorem. From MathWorld-A Wolfram Web Resource. Eric W Weisstein, Eric W. Weisstein. Schröder-Bernstein Theorem. From MathWorld-A Wolfram Web Resource. https://mathworld.wolfram.com/Schroeder-BernsteinTheorem.html, 2023. First Course in Abstract Algebra. B John, Neal Fraleigh, Brand, PearsonJohn B Fraleigh and Neal Brand. First Course in Abstract Algebra, A. Pearson, 2021.
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[ "Projective Manifold Gradient Layer for Deep Rotation Regression", "Projective Manifold Gradient Layer for Deep Rotation Regression" ]
[ "Jiayi Chen \nCFCS\nPeking University\n\n\nBeijing Institute for General AI\n\n", "Yingda Yin \nCFCS\nPeking University\n\n", "Tolga Birdal \nStanford University\n\n\nImperial College London\n\n", "Baoquan Chen \nCFCS\nPeking University\n\n", "Leonidas J Guibas \nStanford University\n\n", "He Wang \nCFCS\nPeking University\n\n" ]
[ "CFCS\nPeking University\n", "Beijing Institute for General AI\n", "CFCS\nPeking University\n", "Stanford University\n", "Imperial College London\n", "CFCS\nPeking University\n", "Stanford University\n", "CFCS\nPeking University\n" ]
[]
Regressing rotations on SO(3) manifold using deep neural networks is an important yet unsolved problem. The gap between the Euclidean network output space and the non-Euclidean SO(3) manifold imposes a severe challenge for neural network learning in both forward and backward passes. While several works have proposed different regression-friendly rotation representations, very few works have been devoted to improving the gradient backpropagating in the backward pass. In this paper, we propose a manifold-aware gradient that directly backpropagates into deep network weights. Leveraging Riemannian optimization to construct a novel projective gradient, our proposed regularized projective manifold gradient (RPMG) method helps networks achieve new state-of-the-art performance in a variety of rotation estimation tasks. Our proposed gradient layer can also be applied to other smooth manifolds such as the unit sphere. Our project page is at https://jychen18.github.io/RPMG.
10.1109/cvpr52688.2022.00653
[ "https://arxiv.org/pdf/2110.11657v3.pdf" ]
239,616,243
2110.11657
7684a7889bd8d2ef63e025b175d016663b176616
Projective Manifold Gradient Layer for Deep Rotation Regression Jiayi Chen CFCS Peking University Beijing Institute for General AI Yingda Yin CFCS Peking University Tolga Birdal Stanford University Imperial College London Baoquan Chen CFCS Peking University Leonidas J Guibas Stanford University He Wang CFCS Peking University Projective Manifold Gradient Layer for Deep Rotation Regression Regressing rotations on SO(3) manifold using deep neural networks is an important yet unsolved problem. The gap between the Euclidean network output space and the non-Euclidean SO(3) manifold imposes a severe challenge for neural network learning in both forward and backward passes. While several works have proposed different regression-friendly rotation representations, very few works have been devoted to improving the gradient backpropagating in the backward pass. In this paper, we propose a manifold-aware gradient that directly backpropagates into deep network weights. Leveraging Riemannian optimization to construct a novel projective gradient, our proposed regularized projective manifold gradient (RPMG) method helps networks achieve new state-of-the-art performance in a variety of rotation estimation tasks. Our proposed gradient layer can also be applied to other smooth manifolds such as the unit sphere. Our project page is at https://jychen18.github.io/RPMG. Introduction Estimating rotations is a crucial problem in visual perception that has broad applications, e.g., in object pose estimation, robot control, camera relocalization, 3D reconstruction and visual odometry [8,13,16,22,37]. Recently, with the proliferation of deep neural networks, learning to accurately regress rotations is attracting more and more attention. However, the non-Euclidean characteristics of rotation space make accurately regressing rotation very challenging. As we know, rotations reside in a non-Euclidean manifold, SO(3) group, whereas the unconstrained outputs of neural networks usually live in Euclidean spaces. This gap between the neural network output space and SO(3) manifold becomes a major challenge for deep rotation regression, thus tackling this gap becomes an important research topic. One popular research direction is to design learningfriendly rotation representations, e.g., 6D continuous rep- † : He Wang is the corresponding author ([email protected]). resentation from [46] and 10D symmetric matrix representation from [27]. Recently, Levinson et al. [25] adopted the vanilla 9D matrix representation discovering that simply replacing the Gram-Schmidt process in the 6D representation [46] with symmetric SVD-based orthogonalization can make this representation superior to the others. Despite the progress on discovering better rotation representations, the gap between a Euclidean network output space and the non-Euclidean SO(3) manifold hasn't been completely filled. One important yet long-neglected problem lies in optimization on non-Euclidean manifolds [1]: to optimize on SO(3) manifold, the optimization variable is a rotation matrix, which contains nine matrix elements; if we naively use Euclidean gradient, which simply computes the partial derivatives with respect to each of the nine matrix elements, to update the variable, this optimization step will usually lead to a new matrix off SO(3) manifold. Unfortunately, we observe that all the existing works on rotation regression simply rely upon vanilla auto-differentiation for backpropagation, exactly computing Euclidean gradient and performing such off-manifold updates to predicted rotations. We argue that, for training deep rotation regression networks, the off-manifold components will lead to noise in the gradient of neural network weights, hindering network training and convergence. To tackle this issue, we draw inspiration from differential geometry, where people leverage Riemannian optimization to optimize on the non-Euclidean manifold, which finds the direction of the steepest geodesic path on the manifold and take an on-manifold step. We thus propose to leverage Riemannian optimization and delve deep into the study of the backward pass. Note that this is a fundamental yet currently under-explored avenue, given that most of the existing works focus on a holistic design of rotation regression that is agnostic to forward/backward pass. However, incorporating Riemannian optimization into network training is highly non-trivial and challenging. Although methods of Riemannian optimization allow for optimization on SO(3) [5,31], matrix manifolds [1] or general Riemannian manifolds [34,44], they are not directly applicable to update the weights of the neural networks that are Euclidean. Also, approaches like [17] incorporate a Riemannian distance as well as its gradient into network training, however, they do not deal with the representation issue. In this work, we want to propose a better manifoldaware gradient in the backward pass of rotation regression that directly updates the neural network weights. We begin by taking a Riemannian optimization step and computing the difference between the rotation prediction and the updated rotation, which is closer to the ground truth. Backpropagating this "error", we encounter the mapping function (or orthogonalization function) that transforms the raw network output to a valid rotation. This projection, which can be the Gram-Schmidt process or SVD orthogonalization [25], is typically a many-to-one mapping. This nonbijectivity provides us with a new design space for our gradient: if we were to use a gradient to update the raw output rotation, many gradients would result in the same update in the final output rotation despite being completely different for backpropagating into the neural network weights. Now the problem becomes: which gradient is the best for backpropagation when many of them correspond to the same update to the output? We observe that this problem is somewhat similar to some problems with ambiguities or multi-ground-truth issues. One example would be the symmetry issue in pose estimation: a symmetric object, e.g. a textureless cube, appears the same under many different poses, which needs to be considered when supervising the pose predictions. For supervising the learning in such a problem, Wang et. al. [39] proposed to use min-of-N loss [14], which only penalizes the smallest error between the prediction and all the possible ground truths. We therefore propose to find the gradient with the smallest norm that can update the final output rotation to the goal rotation. This back-projection process involves finding an element closest to the network output in the inverse image of the goal rotation and projecting the network output to this inverse image space. We therefore coin our gradient projective manifold gradient. One thing to note is that this projective gradient tends to shorten the network output, causing the norms of network output to vanish. To fix this problem, we further incorporate a simple regularization into the gradient, leading to our full solution regularized projective manifold gradient (RPMG). Note that our proposed gradient layer operates on the raw network output and can be directly backpropagated into the network weights. Our method is very general and is not tied to a specific rotation representation. It can be coupled with different non-Euclidean rotation representations, including quaternion, 6D representation [46], and 9D rotation matrix representation [25], and can even be used for regressing other non-manifold variables. We evaluate our devised projective manifold gradient layers on a diverse set of problems involving rota-tion regression: 3D object pose estimation from 3D point clouds/images, rotation estimation problems without using ground truth rotation supervisions, and please see Appendix E.3 for more experiments on camera relocalization. Our method demonstrates significant and consistent improvements on all these tasks and all different rotation representations tested. Going beyond rotation estimation, we also demonstrate performance improvements on regressing unit vectors (lie on a unit sphere) as an example of an extension to other non-Euclidean manifolds. We summarize our contribution as below: • We propose a novel manifold-aware gradient layer, namely RPMG, for the backward pass of rotation regression, which can be applied to different rotation representations and losses and used as a "plug-in" at no actual cost. • Our extensive experiments over different tasks and rotation representations demonstrate the significant improvements from using RPMG. • Our method can also benefit regression tasks on other manifolds, e.g. S 2 . Related Work Both rotation parameterization and optimization on SO(3) are well-studied topics. Early deep learning models leverage various rotation representations for pose estimation, e.g., direction cosine matrix (DCM) [19,43], axisangle [12,15,35], quaternion [11,21,23,42,45] and Eulerangle [24,30,33]. Recently, [46] points out that Euler-angle, axis-angle, and quaternion are not continuous rotation representations, since their representation spaces are not homeomorphic to SO(3). As better representations for rotation regression, 6D [46], 9D [25], 10D [27] representations are proposed to resolve the discontinuity issue and improve the regression accuracy. A concurrent work [7] examines different manifold mappings theoretically and experimentally, finding out that SVD orthogonalization performs the best when regressing arbitrary rotations. Originating from general Riemannian optimization, [31] presents an easy approach for minimization on the SO(3) group by constructing a local axis-angle parameterization, which is also the tangent space of SO(3) manifold. They backpropagate gradient to the tangent space and use the exponential map to update the current rotation matrix. Most recently, [32] constructs a PyTorch library that supports tangent space gradient backpropagation for 3D transformation groups, (e.g., SO(3), SE(3), Sim (3)). This proposed library can be used to implement the Riemannian gradient in our layer. Preliminaries Riemannian Geometry Following [3,4], we define an m-dimensional Riemannian manifold embedded in an ambient Euclidean space X = R d and endowed with a Riemannian metric G (G x ) x∈M to be a smooth curved space (M, G). A vector v ∈ X is said to be tangent to M at x iff there exists a smooth curve γ : [0, 1] → M s.t. γ(0) = x andγ(0) = v. The velocities of all such curves through x form the tangent space T x M = {γ(0) | γ : R → M is smooth around 0 and γ(0) = x}. Definition 1 (Riemannian gradient). For a smooth function f : M → R and ∀(x, v) ∈ T M, we define the Riemannian gradient of f as the unique vector field gradf satisfying [6]: Df (x)[v] = v, gradf (x) x (1) where Df (x)[v] is the derivation of f by v. It can further be shown (see Appendix B.1) that an expression for gradf can be obtained through the projection of the Euclidean gradient orthogonally onto the tangent space gradf (x) = ∇f (x) = Π x ∇f (x) .(2) where Π x : X → T x M ⊆ X is an orthogonal projector with respect to ·, · x . Definition 2 (Riemannian optimization). We consider gradient descent to solve the problems of min x∈M f (x). For a local minimizer or a stationary point x of f , the Riemannian gradient vanishes gradf (x ) = 0 enabling a simple algorithm, Riemannian gradient descent (RGD): x k+1 = R x k (−τ k gradf (x k ))(3) where τ k is the step size at iteration k and R x k is the retraction usually chosen related to the exponential map. Rotation Representations There are many ways of representing a rotation: classic rotation representations, e.g. Euler angles, axis-angle, and quaternion; and recently introduced regression-friendly rotation representations such as e.g. 5D [46], 6D [46], 9D [25] and 10D [27] representations. A majority of deep neural networks can output an unconstrained, arbitrary ndimensional vector x in a Euclidean space X = R n . For Euler angle and axis-angle representations which use a vector from R 3 to represent a rotation, a neural network can simply output a 3D vector; however, for quaternions, 6D, 9D or 10D representations that lies on non-Euclidean manifolds, manifold mapping function π : R n → M is generally needed for normalization or orthogonalization purposes to convert network outputs to valid elements belonging to the representation manifold. This network Euclidean output space X is where the representation manifolds reside and therefore are also called ambient space. Definition 3 (Rotation representation). One rotation representation, which lies on a representation manifold M, defines a surjective rotation mapping φ :x ∈ M → φ(x) ∈ SO(3) and a representation mapping function ψ : R ∈ SO(3) → ψ(R) ∈ M, such that φ(ψ) = R ∈ SO(3). Definition 4 (Manifold mapping function). From an ambient space X to the representation manifold M, we can define a manifold mapping function π : x ∈ X → π(x) ∈ M, which projects a point x in the ambient, Euclidean space to a valid elementx = π(x) on the manifold M. We summarize the manifold mappings, the rotation mappings and representation mappings for several non-Euclidean rotation representations below. Unit quaternion. Unit quaternions represent a rotation using a 4D unit vector q ∈ S 3 double covering the non-Euclidean 3-sphere i.e. q and −q identify the same rotation. A network with a final linear activation can only predict x ∈ R 4 . The corresponding manifold mapping function is usually chosen to be a normalization step, which reads π q (x) = x/ x . For rotation and representation mapping, we leverage the standard mappings between rotation and quaternion (see Appendix G). 6D representation and Gram-Schmidt orthogonalization. 6D rotation representation [46], lying on Stiefel manifold V 2 (R 3 ), uses two orthogonal unit 3D vectors (ĉ 1 ,ĉ 2 ) to represent a rotation, which are essentially the first two columns of a rotation matrix. Its manifold mapping π 6D is done through Gram-Schmidt orthogonalization. Its rotation mapping φ 6D is done by adding the third column c 3 =ĉ 1 ×ĉ 2 . Its representation mapping ψ 6D is simply getting rid of the third columnĉ 3 from a rotation matrix. 9D representation and SVD orthogonalization. To map a raw 9D network output M to a rotation matrix, [25] use SVD orthogonalization as the manifold mapping function π 9D , as follows: π 9D first decomposes M into its left and right singular vectors {U, V } and singular values Σ, M = UΣV ; then it replaces Σ with Σ = diag(1, 1, det(UV )) and finally, computes R = UΣ V to get the corresponding rotation matrix R ∈ SO(3). As this representation manifold is SO(3), both the rotation and representation mapping functions are simply identity. 10D representation. [27] propose a novel 10D representation for rotation matrix. The manifold mapping function π 10D maps θ ∈ R 10 to q ∈ S 3 by computing the eigenvector corresponding to the smallest eigenvalue of A(θ), expressed as π 10D (x) = min q∈S 3 q A(x)q, in which A(θ) =     θ 1 θ 2 θ 3 θ 4 θ 2 θ 5 θ 6 θ 7 θ 3 θ 6 θ 8 θ 9 θ 4 θ 7 θ 9 θ 10     .(4) Since the representation manifold is S 3 , the rotation and representation mapping are the same as unit quaternion. Deep Rotation Regression We conclude this section by describing the ordinary forward and backward passes of a neural network based rota- In the forward pass, the network predicts a raw output x, which is then transformed into a valid rotation R = φ(π(x)). We leave this forward pass unchanged and only modify the backward pass. In the backward pass, we first use Riemannian optimization to get a goal rotation Rg and map it back toxg on the representation manifold M. After that we find the element xgp which is closest to the raw output in the inverse image ofxg, and finally get the gradient g RPM we want. tion regression, as used in [25,46]. Forward and backward passes. Assume, for a rotation representation, the network predicts x ∈ X , then the manifold mapping π will map x tox = π(x) ∈ M, followed by a rotation mapping φ that finally yields the output rotation R = φ(x) = φ(π(x)). Our work only tackles the backward pass and keeps the forward pass unchanged, as shown in the top part of Figure 1. The gradient in the backwardpass is simply computed using Pytorch autograd method, that is g = f (R)φ (x)π (x). Loss function. The most common choice for supervising rotation matrix is L2 loss, R − R gt 2 F , as used by [25,46]. This loss is equal to 4−4 cos(< R, R gt >), where < R, R gt > represents the angle between R and R gt . Method Overview. In this work, we propose a projective manifold gradient layer, without changing the forward pass of a given rotation regressing network, as shown in Figure 1. Our focus is to find a better gradient g of the loss function L with respect to the network raw output x for backpropagation into the network weights. Let's start with examining the gradient of network output x in a general case -regression in Euclidean space. Given a ground truth x gt and the L2 loss x − x gt 2 that maximizes the likelihood in the presence of Gaussian noise in x, the gradient would be g = 2(x − x gt ). In the case of rotation regression, we therefore propose to find a proper x * ∈ X for a given ground truth R gt or a computed goal rotation R g when the ground truth rotation is not available, and then simply use x − x * as our gradient to backpropagate into the network. Note that finding such an x * can be challenging. Assuming we know R gt , finding an x * involves inverting φ and π since the network output R = φ(π(x)). Furthermore, we may not know R gt under indirect rotation supervision (e.g., flow loss as used in PoseCNN [42]) and self-supervised rotation estimation cases (e.g., 2D mask loss as used in [38]). In this work, we introduce the following techniques to mitigate these problems: (i) we first take a Riemannian gradient to compute a goal rotation R g ∈ SO(3), which does not rely on knowing R gt , as explained in Section 4.1; (ii) we then find the set of all possible x g s that can be mapped to R g , or in other words, the inverse image of R g under π and φ; (iii) we find x gp which is the element in this set closest to x in the Euclidean metric and set it as "x * ". We will construct our gradient using this x * , as explained in 4.2. (iv) we add a regularization term to this gradient forming g RP M G as explained in 4.3. The whole backward pass leveraging our proposed regularized projective manifold gradient is shown in the lower half of Figure 1 Riemannian Gradient and Goal Rotation To handle rotation estimation with/without direct rotation supervision, we first propose to compute the Riemannian gradient of the loss function L with respect to the output rotation R and find a goal rotation R g that is presumably closer to the ground truth rotation than R. Assume the loss function is in the following form L(f (R)), where R = π(φ(x)) is the output rotation and f constructs a loss function that compares R to the ground truth rotation R gt directly or indirectly. Given R(x) and L(f (R(x))), we can perform one step of Riemannian optimization yielding our goal rotation R g ← R R (−τ grad L(f (R))), where τ is the step size of Riemannian gradient and can be set to a constant as a hyperparameter or varying during the training. For L2 loss R − R gt 2 F , the Riemannian gradient is always along the geodesic path between R and R gt on SO(3) [20]. In this case, R g can generally be seen as an intermediate goal between R and R gt dependent on τ . Gradually increasing τ from 0 will first make R g approach R gt starting with R g = R, and then reach R gt where we denote τ = τ gt , and finally going beyond R gt . Although, when R gt is available, one can simply set R g = R gt , we argue that this is just a special case under τ = τ gt . For scenarios where R gt is unavailable, e.g., in self-supervised learning cases (see in Section 5.3), we don't know R gt and τ gt , thus we need to compute R g using Riemannian optimization. In the sequel, we only use R g for explaining our methods without loss of generality. See Section 4.3 for how to choose τ . Projective Manifold Gradient Given R g , we can use the representation mapping ψ to find the correspondingx g = ψ(R g ) on the representation manifold M. However, further inverting π and finding the corresponding x g ∈ X is a non-trivial problem, due to the projective nature of π. In fact, there are many x g s that satisfy π(x g ) =x g . It seems that we can construct a gradient g = (x − x g ) using any x g that satisfies π(x g ) =x g . No matter which x g we choose, if this gradient were to update x, it will result in the same R g . But, when backpropagating into the network, those gradients will update the network weights differently, potentially resulting in different learning efficiency and network performance. Formally, we formulate this problem as a multi-groundtruth problem for x: we need to find the best x * to supervise from the inverse image ofx g under the mapping π. We note that similar problems have been seen in pose supervision dealing with symmetry as in [39], where one needs to find one pose to supervise when there are many poses under which the object appears the same. [39] proposed to use a min-of-N strategy introduced by [14]: from all possible poses, taking the pose that is closest to the network prediction as ground truth. A similar strategy is also seen in supervising quaternion regression, as q and −q stand for the same rotation. One common choice of the loss function is therefore min{L(q, q gt ), L(q, −q gt )} [27], which penalizes the distance to the closest ground truth quaternion. Inspired by these works, we propose to choose our gradient among all the possible gradients with the lowest level of redundancy, i.e., we require x * to be the one closest to x, or in other words, the gradient to have the smallest norm, meaning that we need to find the projection point x gp of x to all the valid x g : x gp = argmin π(xg)=xg x − x g 2(5) We then can construct our projective manifold gradient (PMG) as g P M = x − x gp . We will denote the naive gradient g M = x −x g as manifold gradient (MG). Here we provide another perspective on why a network may prefer PMG. In the case where a deep network is trained using stochastic gradient descent (SGD), the final gradient used to update the network weights is averaged across the gradients of all the batch instances. If gradients Figure 2. Illustration for regularized projective manifold gradient. Left: In the forward pass, we simply project x tox by π. In the backward pass, first we compute a Riemannian gradient, which is shown as the green arrow. After getting a next goalxg ∈ M by Riemannian optimization, we find the inverse projection xgp of xg, which leads to our projective manifold gradient, shown as the blue arrow. With a regularization term, we can get our final regularized projective manifold gradient, as the purple arrow. Right: Projection pointxgp in the case of quaternion. from different batch instances contain different levels of redundancy, then the averaged gradient may be biased or not even appropriate. This argument is generally applicable to all stochastic optimizers (e.g., Adam [2]) Inverting π. There are many ways to solve this projection problem for different manifold mapping functions π. For example, we can formulate this as a constrained optimization problem. For the manifold mapping functions we consider, we propose the following approach: we first solve for the inverse image π −1 (x g ) ofx g in the ambient space X analytically, which reads π −1 (x g ) = {x g ∈ X | π(x g ) = x g }; we then project x onto this inverse image space. Note that, sometimes only a superset of this inverse image can be found analytically, requiring certain constraints on x gp to be enforced. Here we list the inverse image π −1 (x g ) and the projection point x gp for different rotation representations and their corresponding manifold mapping π. Please refer to Appendix B.2 for detailed derivations. Quaternion. With π q (x) = x/ x , x ∈ R 4 , andx g ∈ S 3 : π −1 q (x g ) = {x | x = kx g , k ∈ R and k > 0} , which is a ray in the direction ofx g starting from the origin. Without considering the constraint of k > 0 , an analytical solution to this projection point x gp of x onto this line can be derived: x gp = (x ·x g )x g . 6D representation. With π 6D as Gram-Schmidt process, x = [u, v] ∈ R 6 , andx g ∈ V 2 (R 3 ): π −1 6D (x g ) = {[k 1ûg , k 2ûg + k 3vg ] | k 1 , k 2 , k 3 ∈ R and k 1 , k 3 > 0} (the former is a ray whereas the latter spans a half plane). Without considering the constraint of k 1 , k 3 > 0, the projection point x gp can be analytically represented as (3), the analytical expression for π −1 9D is available when we ignore the positive singular value constraints, which gives π −1 x gp = [(u ·û g )û g , (v ·û g )û g + (v ·v g )v g ] 9D representation. With π 9D (x) as SVD orthogonaliza- tion, x ∈ R 3×3 , andx g ∈ SO9D (x g ) = {Sx g | S = S }. We can further solve the projection point x gp with an elegant representation x gp = xx T g +xgx T 2 . 10D representation. Please refer to Appendix B.2 for the derivation and expression of x qp . Regularized Projective Manifold Gradient Issues in naive projective manifold gradient. In the right plot of Figure 2, we illustrate this projection process for several occasions where x takes different positions relative to x g . We demonstrate that there are two issues in this process. First, no matter where x is in, the projection operation will shorten the length of our prediction because x gp < x is always true for all of 4D/6D/9D/10D representation. This will cause the length norm of the prediction of the network to become very small as the training progresses (see Figure 3). The shrinking network output will keep increasing the effective learning rate, preventing the network from convergence and leading to great harm to the network performance (see Table 2 and Figure 3 for ablation study). Second, when the angle between x andx g becomes larger than π/2 (in the case of x = x 3 ), the naive projection x gp will be in the opposite direction ofx g and can not be mapped back tox g under π q , resulting in a wrong gradient. The same set of issues also happens to 6D, 9D and 10D representations. The formal reason is that the analytical solution of the inverse image assumes certain constraints are satisfied, which is usually true only when eitherx g is not far from x or the network is about to converge. Regularized projective manifold gradient. To solve the first issue, we propose to add a regularization term x gp −x g to the projective manifold gradient, which can avoid the length vanishing problem. The regularized projective manifold gradient then reads: g RP M = x − x gp + λ(x gp −x g ),(6) where λ is a regularization coefficient. See the left plot of Figure 2 for an illustration. Discussion on the hyperparameters λ and τ . Our method apparently introduces two additional hyperparameters, λ and τ , however, we argue that this doesn't increase the searching space of hyperparameters for our method. For λ, the only requirement is that λ is small (we simply set to 0.01), because: (1) we want the projective manifold gradient (x − x gp ) to be the major component of our gradient; (2) since this regularization is roughly proportional to the difference in prediction length and a constant, a small lambda is enough to prevent the length from vanishing and, in the end, the prediction length will stay roughly constant at the equilibrium under projection and regularization. In the ablation study of Section 5.1, we show that the performance is robust to the change of λ. Note that, on the other extreme, when λ = 1, g RP M becomes g M . For τ , we propose a ramping up schedule which is wellmotivated. To tackle the second problem of reversed gradient, we need a small τ init to keep R g close to R at the beginning of training. But when the network is about to converge, we will prefer a τ converge which can keep R g close to R gt for better convergence. We cannot directly set τ converge to τ gt , which is introduced in 4.1, because τ gt is not a constant and cannot be used in Riemannian Optimization. However, if we want to tackle the problem of reversed gradient, we must need Riemannian Optimization and τ init . Thus we need a constant approximation of τ gt when the angle between R and R gt converges to 0. Note that τ converge can be derived analytically when the loss function is the most widely used L2 loss or geodesic loss(please refer to Appendix B.1 for details), and therefore doesn't need to be tuned. Therefore we propose to increase τ from a small value τ init , leading to a slow warm-up and, as the training progresses, we gradually increase it to the final τ = τ converge by ten uniform steps. This strategy further improves our performance. Experiments We investigate popular rotation representations and find our methods greatly improve the performance in different kinds of tasks. For our regularized projective manifold gradient (RPMG), we apply it in the backpropagation process of Quaternion, 6D, 9D and 10D, without changing the forward pass, leading to three new methods RPMG-Quat, RPMG-6D, RPMG-9D and RPMG-10D. We compare the following seven baselines: Euler angle, axisangle, Quaternion, 6D [46], 9D [25], 9D-Inf [25] and 10D [27]. We adopt three evaluation metrics: mean, median, and 5 • accuracy of (geodesic) errors between predicted rotation and ground truth rotation. For most of our experiments, we set the regularization term λ = 0.01 and increase τ from τ init = 0.05 to τ converge = 0.25 by ten uniform steps. We further show and discuss the influence of different choices of these two hyperparameters in our ablation studies. 3D Object Pose Estimation from Point Clouds Experimental setting. As in [9], we use the complete point clouds generated from the models in ModelNet-40 [40]. We use the same train/test split as in [9] and report the results of airplane, chair, sofa, toilet and bed those five categories because they exhibit less rotational symmetries. Given one shape point clouds of a specific category, the network learns to predict the 3D rotation of the input point clouds from the predefined canonical view of this category [39]. We replace the point clouds alignment task used in [25,46] (which has almost been solved) by this experiment since it is more challenging and closer to real-world applications (no canonical point clouds is given to the network). We use a PointNet++ [28] network as our backbone, supervised by L2 loss between the predicted rotation matrix Table 1. We see a great improvement of our methods in all three rotation representations. In this experiment, one may find 9D-Inf also leads to a good performance, which is actually a special case of our method with λ = 1, or in other words, it is MG with τ = τ gt . Nonetheless, in Table 3, we can observe a larger gap. Also, this simple loss may lead to bad performance when R gt is unavailable in Section 5.3. Ablation study on λ. As mentioned in Section 4.3, naively using PMG without any regularization, corresponding to setting λ = 0, will lead to length vanishing; To maintain the length of prediction roughly constant, we only need to add a small λ. In Figure 3, We show the length vanishing problem without regularization and stabilized length with a small regularization. In Table 1, we show that the network performs much better when we have a small λ (RPMG) than λ = 0 (PMG) or λ = 1 (MG), which deviates too far away from the desired projective manifold gradient. As for the exact value of λ, our experiments show that our method is robust to the choice of λ as long as it is small. Table 2 also shows that λ = 0.01, 0.005, 0.05 all lead to similar performance, thus freeing us from tuning the parameter λ. Ablation study on τ . For the choices of τ , Table 2 shows that our proposed strategy, which ramps up τ from a small τ init to τ converge , works the best. The reason is that: a big τ , when training begins, may cause the problem of reversed gradient discussed in Section 4.3. On the other side, a small τ at the end of training will slow down the training process and can do harm to convergence. Note that, the performance is not very sensitive to the exact value, which means we don't require a parameter tuning for τ even in general cases. We are good even with simply setting τ = τ gt . Methods Mean Table 2. Ablation study of pose estimation from airplane point clouds. Here MG stands for manifold gradient x−xg, corresponding to set λ = 1; PMG stands for projective manifold gradient x − xgp, corresponding to set λ = 0. ( • )↓ Med ( • )↓ 5 • Acc (%)↑ L2 6D - - 5. 3D Rotation Estimation from ModelNet Images In this experiment, we follow the setting in [25] to estimate poses from 2D images. Images are rendered from ModelNet-10 [40] objects from arbitrary viewpoints [26]. A MobileNet [18] is used to extract image features and three MLPs to regress rotations. We use the same categories as in Experiment 5.1 except airplane, since ModelNet-10 doesn't have this category. We didn't quote the numbers from [25] since we conduct all the experiments using the same set of hyperparameters to ensure a fair comparison. Please see Appendix F.2 for more details. The results are shown in Table 3. Our RPMG layer boosts the performance of all three representations significantly. See the curves with the same color for comparison. Table 3. Pose estimation from ModelNet10 images. Left: a comparison of methods by mean( • ), median( • ), and 5 • accuracy(%) of (geodesic) errors after 600k training steps. Mn, Md and Acc are abbreviations of mean, median and 5 • accuracy. Right: median test error of chair in different iterations during training. Rotation Estimation without Supervision Self-supervised instance-level rotation estimation from point clouds. For one complete chair instance Z, given a complete observation X, we estimate its pose R. We then use Chamfer distance between Z and R −1 X as a selfsupervised loss. The network structure and training settings are all the same as Experiment 5.1, except here we use τ = 2. See Appendix E.2 for how to find a suitable τ . The interesting thing here is that vanilla 9D-Inf fails while our methods still perform very well. We think that this is because the Chamfer distance loss will greatly enlarge the effect of the noisy part (which is introduced by λ) in gradient, leading to a very bad performance. Table 4. Self-supervised Instance-Level Rotation Estimation from Point Clouds. We report mean, median and 3 • accuracy of (geodesic) errors after 30K iterations. Regression on Other Non-Euclidean Manifolds In addition to SO(3), our method can also be applied for regression on other non-Euclidean manifolds as long as the target manifold meets some conditions: 1) the manifold should support Riemannian optimization. 2) the inverse projection π −1 should be calculable, although it doesn't need to be mathematically complete. Here we show the experiment of Sphere manifold S 2 . Unit vector regression. For rotational symmetric categories (e.g., bottle), the pose of an object is ambiguous. We'd rather regress a unit vector for each object indicating the up direction of it. We use the ModelNet-40 [40] bottle point cloud dataset. The network architecture is the same as in Experiment 5.1 except the dimension of output is 3. L2-loss-w/-norm computes L2 loss between the normalized predictions and the ground truth. L2-loss-w/o-norm computes L2 loss between the raw predictions and the ground truth, similar to λ = 1 and τ = τ gt . For MG-3D, PMG-3D and RPMG-3D, We increase τ from 0.1 to 0.5 since here τ converge = 0.5 (please see Appendix C for the derivation). The results are shown in Table 5. MG-3D performs on par with L2-loss-w/o-norm, and PMG-3D leads to a large error since the length vanishing problem similar to Figure 3. RPMG-3D outperforms all the baselines and variants. Table 5. Unit vector estimation from ModelNet bottle point clouds. We report mean, median, and 1 • accuracy of (geodesic) errors after 30K iterations. Conclusion and Future Work Our work tackles the problem of designing a gradient layer to facilitate the learning of rotation regression. Our extensive experiments have demonstrated the effectiveness of our method coupled with different rotation representations in diverse tasks dealing with rotation estimation. The limitation of our methods mainly lies in two fronts: 1) we introduce two new hyperparameters, i.e., τ and λ, though our performance is not sensitive to them, as long as they are in a reasonable range; 2) as discussed in Sec 5.4, our method can only be applied to manifolds with certain constraints. We leave how to relax those to future works. A. More on Riemannian Geometry In this part, we supplement the definitions in Section 3 to allow for a slightly more rigorous specification of the exponential map for interested readers. We denote the union of all tangent spaces as the tangent bundle: T M = ∪ x∈M T x M. Riemannian metric G x induces a norm u x , ∀u ∈ T x M locally defining the geometry of the manifold and allows for computing the length of any curve γ : [0, 1] → M, with γ(0) = x and γ(1) = y as the integral of its speed: (γ) = 1 0 γ(t) γ(t) dt. The notion of length leads to a natural notion of distance by taking the infimum over all lengths of such curves, giving the Riemannian distance on M, d(x, y) = inf γ (γ). The constant speed length minimizing curve γ is called a geodesic on M. By the celebrated Picard Lindelöf theorem [10], given any (x, v) ∈ T M, there exists a unique maximal geodesic γ v such that γ v (0) = x andγ v (0) = v. Hence, we can define a unique diffeomorphism or exponential map, sending x to the endpoint of the geodesic: exp x (v) = γ v (1). We will refer to the well-defined, smooth inverse of this map as the logaritmic map: log x y exp −1 x (v). Note that the geodesic is not the only way to move away from x in the direction of v on M. In fact, any continuously differentiable, smooth map R x : T x M → M whose directional derivative along v is identity, i.e. DR x (0)[v] = v and R x (0) = x allows for moving on the manifold in a given direction v. Such R x , called retraction, constitutes the basic building block of any on-manifold optimizer as we use in the main paper. In addition to those we also speak of a manifold projector π : X → M is available for the manifolds we consider in this paper. Note that, most of these definitions directly generalize to matrix manifolds such as Stiefel or Grassmann [1]. B. Projective Manifold Gradient on SO(3) B.1. Details of Riemannian Optimization on SO(3) Riemannian gradient on SO(3). Since we mainly focus on the SO(3) manifold in this paper, we will further show the specific expression of some related concepts of SO(3) below. Firstly, SO(3) is defined as a matrix subgroup of the general linear group GL(3): SO(3) = {R ∈ R 3×3 : R R = I, det(R) = 1}. (7) The tangent space of a rotation matrix in SO(3) is isomorphic to R 3 making SO(3) an embedded submanifold of the ambient Eucldiean space X . Hence, SO(3) inherits the metric or the inner product of its embedding space, X . maximal refers to the fact that the curve is as long as possible. Since SO (3) is also a Lie group, elements of the tangent space φ ∧ ∈ T I M can be uniquely mapped to the manifold M through the exponential map: (8) where I ∈ SO(3) is the identity matrix and ∧ is a skewsymmetric operator ∧ : exp I (φ ∧ ) = I + φ ∧ + 1 2! (φ ∧ ) 2 + 1 3! (φ ∧ ) 3 + ... ,R 3 → T I M as φ ∧ =   0 −φ z φ y φ z 0 −φ x −φ y φ x 0   (9) Due to the nature of the Lie group, we can expand the formula in Eq. (8) from the tangent space of the identity, T I M, to T R M by simply multiplying by an R: exp R (φ ∧ ) = R ∞ n=0 ( 1 n! (φ ∧ ) n )(10) If the vector φ is rewritten in terms of a unit vector ω and a magnitude θ, the exponential map can further be simplified as Exp R (φ) = R(I + sin θ ω ∧ + (1 − cos θ)(ω ∧ ) 2 ) (11) which is well known as the Rodrigues formula [29]. Following [31], we have ∂ ∂φx Exp R (φ) φ=0 = R cos θ ∂θ ∂φx ω ∧ φ=0 = Rx ∧(12) where x = (1, 0, 0) ∈ R 3 . For φ y and φ z , there are the similar expressions of the gradient. Finally we can have grad Lf (R) = ∂f (R) ∂R ∂ ∂φ Exp R (φ) φ=0 ∧(13) Riemannian gradient descent on SO(3). We are now ready to state the Riemannian optimization in the main paper in terms of the exponential map: R k+1 = Exp R k (−τ k ∇φ).(14) Note that if we consider the most commonly used L2 loss f (R) = R − R gt 2 F , where R =   a1 b1 c1 a2 b2 c2 a3 b3 c3   ∈ SO(3), Rgt =   x1 y1 z1 x2 y2 z2 x3 y3 z3   ∈ SO(3), we can get an analytical expression of ∇φ = (∇φ x , ∇φ y , ∇φ z ) as follows: ∇φ x = ∂f (R) ∂R * Rx ∧ = 2   a1 − x1 b1 − y1 c1 − z1 a2 − x2 b2 − y2 c2 − z2 a3 − x3 b3 − y3 c3 − z3     0 c1 −b1 0 c2 −b2 0 c3 −b3   1 = 2 * 3 i=1 (bi * zi − ci * yi)(15) Similarly, we have ∇φ y = 2 * 3 i=1 (c i * x i − a i * z i ) and ∇φ z = 2 * 3 i=1 (a i * y i − b i * x i ). τ converge in ablation study. We have mentioned in Section 4.3 that τ should be small at the beginning of training and be large when converging. This is because a small τ can yield R g closer to R and greatly alleviate the reverse problem at the beginning stage of training discussed in Section 4.3. Later in training, a large τ can help us converge better. The initial τ will not influence the final results too much, and we just need to choose a reasonable value. But the final τ matters. Right before convergence, our ideal choice for the final τ would be τ gt . Given that the value of τ gt will change according to the geodesic distance between R and R gt , we instead choose to find a suitable constant value to act like τ gt when converging, which we denotes as τ converge . Lemma 1. The final value of τ converge satisfies: Rgt = lim <R,R gt >→0 R R (−τconverge grad L(f (R)))(16) where < R, R gt > represents the angle between R and R gt . Proof. Considering the symmetry, without loss of generality, we assume that R = I, which will simplify the derivation. Based upon the conclusion in Eq. (15), when we use L2 loss, we have ∇φ = (2 * (z 2 − y 3 ), 2 * (x 3 − z 1 ), 2 * (y 1 − x 2 )) and grad Lf (R) = (∇φ) ∧ = 2(R gt − R gt ). Taking the manifold logarithm of both sides, we get: log R (Rgt) = lim <R,R gt >→0 −τconverge grad Lf (R)(17) The solution for τ converge can then be derived as follows: τconverge = lim <R,R gt >→0 − log R (Rgt) grad Lf (R) = lim θ→0 − (φ gt ) ∧ 2(R gt − Rgt) = lim θ→0 − (φ gt ) ∧ 2 sin θ(((ωgt) ∧ ) − (ωgt) ∧ ) = lim θ→0 θ 4 sin θ = 1 4(18) where (φ gt ) ∧ = log I (Rgt) = θ(ωgt) ∧ , θ =< I, Rgt > Note that though τ converge = 1 4 is only true for the L2 loss, we can solve τ converge for other frequently used loss formats, e.g., geodesic loss [27]. If we use geodesic loss θ 2 , it can be computed that τ converge = 1 2 . We leave the detailed derivation to the interested readers. B.2. Derivations of Inverse Projection For different rotation representations, we follow the same process to find its inverse projection: we first find the inverse image space π −1 (x g ), then project x to this space resulting in x gp , and finally get our (regularized) projective manifold gradient. Quaternion We need to solve x gp = argmin xg∈π −1 q (xg) x g − x 2 2 ,(19) where x is the raw output of our network in ambient space R 4 ,x g is the next goal in representation manifold S 3 , and x g is the variable to optimize in ambient space R 4 . Recall π −1 q (x g ) = {x | x = kx g , k ∈ R and k > 0}, and we can have x − x g 2 2 = x 2 − 2kx ·x g + k 2x2 g(20) Without considering the condition of k > 0, We can see when k = x·xĝ x 2 g = x ·x g the target formula reaches minimum. Note that when using a small τ , the angle between x g and x is always very small, which means the condition of k = x ·x g > 0 can be satisfied naturally. For the sake of simplicity and consistency of gradient, we ignore the limitation of k no matter what value τ takes. Therefore, the inverse projection is x gp = (x ·x g )x g . 6D representation We need to solve [ugp, vgp] = argmin [ug ,vg ]∈π −1 6D ([ûg ,vg ]) ( ug − u 2 2 + vg − v 2 2 )(21) where [u, v] is the raw output of network in ambient space R 6 , [û g ,v g ] is the next goal in representation manifold V 2 (R 3 ) and [u g , v g ] is the variable to optimize in ambient space R 6 . Recall π −1 6D ([û g ,v g ]) = {[k 1ûg , k 2ûg + k 3vg ] | k 1 , k 2 , k 3 ∈ R and k 1 , k 3 > 0}. We can see that u g and v g are independent, and u g is similar to the situation of quaternion. So we only need to consider the part of v g as below: v−v g 2 2 = v 2 +k 2 2û 2 g +k 2 3v 2 g −2k 2 v·û g −2k 3 v·v g (22) For the similar reason as quaternion, we ignore the condition of k 3 > 0 and we can see when k 2 = v ·û g and k 3 = v ·v g , the target formula reaches minimum. Therefore, the inverse projection is [u gp , v gp ] = [(u ·û g )û g , (v · u g )û g + (v ·v g )v g ] 9D representation For this representation, obtaining the inverse image π −1 9D is not so obvious. Recall π 9D (x) = UΣ V , where U and V are left and right singular vectors of x decomposed by SVD expressed as x = UΣV , and Σ = diag(1, 1, det(UV )). Lemma 2. The inverse image π −1 9D (R g ) = {SR g | S = S } satisfies that {x g | π 9D (x g ) = R g } ⊂ π −1 9D (R g ). Proof. To find a suitable π −1 9D , the most straightforward way is to only change the singular values Σ g = diag(λ 0 , λ 1 , λ 2 ), where λ 0 , λ 1 , λ 2 can be arbitrary scalars, and recompose the x g = UΣ g V . However, we argue that this simple method will fail to capture the entire set of {x g | π 9D (x g ) = R g }, because different U and V can yield the same rotation R g . In fact, U g can be arbitrary if x g = U g Σ g V g and U g Σ g V g = R g . Assuming R g is known, we can replace V g by R g and express x g in a different way: x g = U g Σ g 1 Σ g U −1 g R g . Notice that U g Σ g 1 Σ g U −1 g must be a symmetry matrix since U g is an orthogonal matrix. Therefore, {x g | π 9D (x g ) = R g } ⊆ π −1 9D (R g ) = {SR g | S = S }. Note that such x g ∈ π −1 9D (R g ) can't ensure π 9D (x g ) = R g , because in the implementation of SVD, the order and the sign of three singular values are constrained, which is not taken into consideration. Therefore, {x g | π 9D (x g ) = R g } = π −1 9D (R g ). Then we need to solve x gp = argmin xg∈π −1 9D (Rg) x g − x 2 2(23) where x is the raw output of our network in ambient space R 3×3 ,x g is the next goal in representation manifold SO (3), and x g is the variable to optimize in ambient space R 3×3 . We can further transform the objective function as below: x g − x 2 2 = SR g − x 2 2 = S − xR g 2 2(24) Now we can easily find when S equals to the symmetry part of xR g , the target formula reaches minimum. Therefore, the inverse projection admits a simple form x gp = xR g +Rgx 2 R g . 10D representation Recall the manifold mapping π 10D : R 10 → S 3 , π 10D (x) = min q∈S 3 q A(x)q, in which A(θ) =     θ 1 θ 2 θ 3 θ 4 θ 2 θ 5 θ 6 θ 7 θ 3 θ 6 θ 8 θ 9 θ 4 θ 7 θ 9 θ 10     .(25) We need to solve x gp = arg min A(xg)qg=λqg x g − x 2 2 ,(26) where x is the raw output of our network in ambient space R 10 , q g is the next goal in representation manifold S 3 , and x g is the variable to optimize in ambient space R 10 . Note that λ is also a variable to optimize. For the similar reason as before, for the sake of simplicity and consistency of analytical solution, here we also need to relax the constraint that λ should be the smallest eigenvalue of A(x g ). To solve Eq. 25, we start from rewriting A(x g )q g = λq g as M∆x = λq g − A(x)q g ,(27) where ∆x = x g − x and M =    q1 q2 q3 q4 0 0 0 0 0 0 0 q1 0 0 q2 q3 q4 0 0 0 0 0 q1 0 0 q2 0 q3 q4 0 0 0 0 q1 0 0 q2 0 q3 q4   (28) where q g = (q 1 , q 2 , q 3 , q 4 ) . For simplicity, we denote λq g − A(x)q g as b. Once we have finished the above steps for preparation, we solve λ and ∆x for the minimal problem by two steps as below. First, we assume λ is known and the problem becomes that given M and b, we need to find the best ∆x to minimize ∆x 2 2 with the constraint M∆x = b. This is a typical quadratic optimization problem with linear equality constraints, and the analytical solution satisfies I M M 0 ∆x v = 0 b (29) where v is a set of Lagrange multipliers which come out of the solution alongside ∆x, and I M M 0 is called KKT matrix. Since this matrix has full rank almost everywhere, we can multiple the inverse of this KKT matrix in both sides of Eq. 29 and lead to the solution of ∆x as below: ∆x v = I M M 0 −1 0 b(30) Recall that b = λq g − A(x)q g , therefore so far we have had the solution of ∆x respect to each λ: ∆x = ∆x v 0:10 = K(λq g − A(x)q g ) = λS − T (31) in which K isλ = (S T+T S) 2S S x gp = x + λS − T(32) Another thing worth mentioning here is that in this special case, the representation manifold S 3 is no longer a subspace of the abmient space R 10 , which means that we can't directly compute our regularization term x gp − q g because x gp ∈ R 10 while q g ∈ S 3 . However, the length vanishing problem still exists as shown in Figure 3. Therefore, to compute the regularization term, we need a simple mapping to convert q g to an element on R 10 with stable length norm. We use the mapping g : S 3 → R 10 , g(q) = A −1 (I−qq ), which is proposed in [27]. They also proved that π(g(q)) = q is always true, which makes g(q) better than simply normalizing x gp because the latter one will suffer from the problem of opposite gradient discussed in Section 4.3. C. Projective Manifold Gradient on S 2 C.1. Riemannian Optimization on S 2 Our methods can also be applied for the regression of other manifolds. Taking S 2 as an example, which is included in Experiment 5.4, we will show the detail of how our projective manifold gradient layer works in other manifolds. During forward, The network predicts a raw output x ∈ R 3 , which is then mapped tox ∈ S 2 through a manifold mapping π(x) = x/ x . Here we don't define the rotation mapping and representation mapping, and we directly compute the loss function on representation manifold S 2 . During backward, to apply a Riemannian optimization, we first need to know some basic concepts of S 2 . The tangent space of an arbitrary elementx ∈ S 2 is TxM, which is a plane. And we can map a geodesic path v ∈ TxM to an element on the manifold S 2 through expx(v) = cos( v )x + sin( v ) v v , where . means the ordinal Frobenius norm. For the definition of the mapping ∧ , which connects Euclidean space R 2 and the tangent space TxM, we need to first define two orthogonal axesĉ 1 ,ĉ 2 in the tangent plane. Note that the choice ofĉ 1 andĉ 2 won't influence the final result, which will be shown soon after. To simplify the derivation, we can assume ground truth unit vectorx gt is known and chooseĉ 1 = Logx(xgt) Logx(xgt) =x gt−(xgt·x) x gt−(xgt·x) and c 2 =x ×ĉ 1 . Then we can say φ ∧ = φ 1ĉ1 + φ 2ĉ2 , where φ = (φ 1 , φ 2 ) ∈ R 2 . The gradient of exponential mapping with respect to φ is ∂ ∂φ1 Expx(φ) φ=0 = ∂ ∂φ1 (cos( φ1ĉ1 )x + sin( φ1ĉ1 ) φ1ĉ1 φ1ĉ1 ) φ=0 =ĉ1(33) Similarly, we have ∂ ∂φ2 Expx(φ) φ=0 =ĉ 2 . When using L2 loss, we can have grad Lf (x) = (∇f (x)) ∧ = (∇φ) ∧ = ∂f (x) ∂x ∂ ∂φ Expx(φ) φ=0 ∧ = ((2(x −xgt)ĉ1, 2(x −xgt)ĉ2)) ∧ = 2((x ·xgt)x −xgt)(34) Note that this expression doesn't depend on the choice of c 1 andĉ 2 . Similar to Eq 18, we can also solve a τ converge τconverge = lim <x,x gt >→0 − Logx(xgt) grad Lf (x) = lim θ→0 θĉ1 2 sin θĉ1 = 1 2(35) where θ =<x,x gt >. Note that in Experiment 5.4, we change the schedule of τ according to this conclusion. We increase τ from 0.1 to 0.5 by uniform steps. C.2. Inverse Projection Similar to quaternion, we can have x gp = (x ·x g )x g . For the detail of derivation, see Section B.2. D. Computational Cost Our method does not alter the forward pass and thus incurs no cost at test time. For backward pass at training, we observe that, before and after inserting RMPG layers, the backward time for quaternion / 6D / 9D / 10D representations, averaged among 1K iterations on a GeForce RTX 3090, changes from 4.39 / 4.48 / 4.48 / 4.53 to 4.45 / 4.43 / 4.49 / 4.63 (unit: 10 −2 s), and the memory cost changes from 11449 / 11415 / 10781 / 11363 to 11457 / 11459 / 11545 / 11447 (unit: MiB). Note that the runtime is almost keep the same, as Riemannian optimization only performs an additional projection and we derive and always use analytical solutions in representation mapping, inversion and projection steps. RPMG also has a very marginal cost on the memory, as it does not introduce any weights but only a few intermediate variables. E. More Experiments E.1. Pascal3D+ Pascal3D+ [41] is a standard benchmark for object pose estimation from real images. We follow the same setting as in [25] to estimate object poses from single images. For training we discard occluded or truncated objects and augment with rendered images from [30]. In the Table 6 and Table 7, we report our results on sofa and bicycle categories. We use the same batch size as in [25]. As for the learning rate, we use the same strategy as in Experiment 5.2. See the discussion in Section F.2. It can be seen that our method leads to consistent improvements to quaternion, 6D, 9D and 10D representations on both sofa and bicycle classes. One may be curious about Table 7. Pose estimation from PASCAL3D+ bicycle images. We report the same metrics as Table 6; see the caption there. why our method can only outperform 9D-inf for a margin. We think that this is because this dataset is quite challenging. The number of annotated real image for training is only around 200 for each category. Though there are a lot of synthetic images generated from [30] for training, these images suffer from sim-to-real domain gap. Therefore, we argue that the bottleneck here is not in optimization, which makes the gains from less noise in gradient smaller(Note that 9Dinf is just a special case of our methods with λ = 1 and τ = τ gt ). But compared to vanilla 4D/6D/9D/10D representation, our methods can still bring a great improvement. E.2. Using Flow Loss for Rotation Estimation from Point Clouds. Apart from the most widely used L2 loss, our method can also be applied to the loss of other forms, e.g. flow loss. We mainly follow the setting in Experiment 5.1 with airplane point clouds dataset and the only difference is that we use flow loss RX − R gt X 2 F here, where X is the complete point clouds. Since the format of loss is changed, the previous schedule of τ is not suitable anymore, and we have to change the value of τ accordingly. Our selection skill is to first choose a τ as we like and visualize the mean geodesic distance between predicted R and R g during training. Then we can roughly adjust τ to make the geodesic distance looked reasonable. For this experiment, we use τ = 50 and λ = 0.01. In Table 8, we show our methods again outperform vanilla methods as well as 9D-inf. Methods Mean ( E.3. Camera Relocalization The task of camera relocalization is to estimate a 6 Degree-of-Freedom camera pose (rotation and translation) from visual observations, which is a fundamental component of many computer vision and robotic applications. In this experiment, we use all the settings (data, network, training strategy, hyperparameters, etc.) of PoseLSTM [36] except that we modify the rotation representations and the Table 9. Camera relocalization on Cambridge Landscape dataset. We report the median error of translation and rotation of the best checkpoint, which is chosen by minimizing the median of rotation. We only care about the rotation error here. gradient layers. We report the results on the outdoor Cambridge Landscape dataset [23] in Table 9. Notice that our RPMG layer performs the best on the rotation regression task, but not on the translation regression. We believe this results from a loss imbalance. We does not change the weights of the rotation loss and translation loss, otherwise it leads to an unfair comparison with existing results. We only care about the rotation error here. Data We generate the data from ModelNet dataset [40] by sampling 1024 points on the mesh surface, following the same generation method as in [46]. We uniformly sample M rotations for each data point and set them as the ground truth. We apply the sampled rotations on the canonical point clouds to obtain the input data. Network Architecture We use a PointNet++ MSG [28] backbone as our feature extractor. Our network takes input a point cloud with a resolution of 1024. It them performs three set abstractions to lower the resolution to 512, 128, and finally 1, resulting in a global feature of dimensionality 1024. The feature is finally pushed through a three-layer MLP [1024, 512, N ] to regress rotation, where N is the dimension of the rotation representation. Training details The learning rate is set to 1e-3 and decayed by 0.7 every 3k iterations. The batch size is 20. For each experiment, we train the network on one NVIDIA TI-TAN Xp GPU for 30k iterations. F.2. Experiment 5.2 Most of the training settings and strategies are all the same as [25] except learning rate. We find setting initial learning rate lr = 1e − 3 and decaying to 1e − 5 can perform much better than using lr = 1e − 5 as in [25], which accounts for the inconsistency of the results of those baseline methods compared to [25]. We believe that the methods should be compared under hyperparameters as optimal as possible. Thus, we stick to our lr schedule. G. Addition on Rotation Representations Standard mapping between rotation matrix and unit quaternion The rotation mapping φ : q → R algebraically manipulates a unit quaternion q into a rotation matrix: φ(q) =   2(q 2 0 + q 2 1 ) − 1 2(q1q2 − q0q3) 2(q1q3 + q0q2) 2(q1q2 + q0q3) 2(q 2 0 + q 2 2 ) − 1 2(q2q3 − q0q1) 2(q1q3 − q0q2) 2(q2q3 + q0q1) 2(q 2 0 + q 2 3 ) − 1   (36) where q = (q 0 , q 1 , q 2 , q 3 ) ∈ S 3 . In the reverse direction, the representation mapping ψ(R) can be expressed as:        q0 = √ 1 + R00 + R11 + R22/2 q1 = (R21 − R12)/(4 * q0) q2 = (R02 − R20)/(4 * q0) q3 = (R10 − R01)/(4 * q0) (37) Note that q = (q 0 , q 1 , q 2 , q 3 ) and −q = (−q 0 , −q 1 , −q 2 , −q 3 ) both are the valid quaternions parameterizing the same R. Figure 1 . 1Projective Manifold Gradient Layer. . Figure 3 . 3Average L2 norm of the network raw output x during training. Left: PMG-4D/6D/9D/10D (w/o reg. λ = 0). Right: RPMG-4D/6D/9D/10D (w/ reg. λ = 0.01) R and the ground truth rotation matrix R gt . To facilitate a fair comparison between multiple methods, we use the same set of hyperparameters in all the experiments. Please see Appendix F for more details. Analysis of results. The results are shown in the upper right part of the inverse of, S = Kq g and T = KA(x)q g .Next, we need to optimize λ to minimize our objective function ∆x 2 2 . In fact, using the results of Eq. 31, ∆x 2 2 becomes a quadratic functions on λ, thus we can simply get the final analytical solution of λ and x gp : Table 1 . 1Pose estimation from ModelNet40 point clouds. Left: a comparison of methods by mean, median, and 5 • accuracy of (geodesic) errors after 30k training steps. Mn, Md and Acc are abbreviations of mean, median and 5 • accuracy. Right: median test error of airplane in different iterations during training.Methods Airplane Chair Sofa Toilet Bed Mn↓ Md↓ Acc↑ Mn↓ Md↓ Acc↑ Mn↓ Md↓ Acc↑ Mn↓ Md↓ Acc↑ Mn↓ Md↓ Acc↑ Euler 125 131 0 13.6 9.0 17 120 125 0 127 133 0 113 122 0 Axis-Angle 10.8 8.2 22 16.4 10.9 9 24.1 14.6 6 21.9 13.0 9 25.5 11.0 16 Quaternion 9.7 7.6 27 16.7 11.4 12 20.4 12.7 10 16.0 9.3 17 27.8 11.3 14 6D 5.5 4.7 54 9.8 6.4 35 14.6 9.5 15 9.3 6.8 33 24.7 9.6 17 9D 4.7 3.9 67 7.9 5.4 44 15.7 10.0 14 10.3 6.9 30 22.3 8.5 20 9D-Inf (MG-9D) 3.1 2.5 90 5.3 3.7 69 7.8 5.0 50 4.2 3.3 75 12.9 4.6 55 10D 5.3 4.2 61 8.9 6.0 38 15.1 10.3 13 10.7 6.5 35 23.1 8.7 19 RPMG-Quat 3.2 2.4 88 6.3 3.7 67 8.1 4.5 57 4.9 3.5 74 13.3 3.6 70 RPMG-6D 2.6 2.1 94 5.0 3.1 74 6.6 3.6 70 3.8 2.9 83 13.5 2.7 81 RPMG-9D 2.5 2.0 94 5.1 3.1 76 6.1 3.1 77 4.3 2.7 83 10.9 2.5 86 RPMG-10D 2.8 2.2 93 5.1 3.2 75 6.5 3.2 72 4.9 2.8 82 13.5 2.7 82 5k 10k 15k 20k 25k 30k Iteration 0 5 10 15 20 25 30 Median error (°) Axis-Angle Quaternion 6D 9D 9D-Inf 10D RPMG-Quat RPMG-6D RPMG-9D RPMG-10D Table 6 . 6Pose estimation from PASCAL3D+ sofa images. Left: a comparison of methods by 10 • / 15 • / 20 • accuracy of (geodesic) errors and median errors after 60k training steps. Middle: median test error at different iterations during training. Right: test error percentiles after training completes. The legend on the right applies to both plots.Methods Accuracy(%) ↑ Med( • ) ↓ 10 • 15 • 20 • Err Euler 28.2 48.1 62.7 15.7 Axis-Angle 5.3 8.1 10.1 79.7 Quaternion 20.8 38.8 54.6 18.7 6D 21.8 39.0 55.3 18.1 9D 20.6 37.6 56.9 18.0 9D-Inf 38.0 53.3 69.9 13.4 10D 23.9 42.3 56.7 17.9 RPMG-Quat 32.3 50.0 65.6 15.0 RPMG-6D 35.4 57.2 70.6 13.5 RPMG-9D 36.8 57.4 71.8 12.5 RPMG-10D 40.0 57.7 71.3 12.9 10k 20k 30k 40k 50k 60k Iteration 10 15 20 25 30 35 40 Median error (°) 10% 30% 50% 70% 90% Percentile 0.1°1°5°1 0°4 5°1 80°E uler Axis-Angle Quaternion 6D 9D 9D-Inf 10D RPMG-Quat RPMG-6D RPMG-9D RPMG-10D • ) Med ( • ) 5 • Acc (%)Table 8. Flow Loss for Rotation Estimation from Point Clouds. All models are trained for 30K iterations.Euler 12.14 6.91 33.6 Axis-Angle 35.49 20.80 4.7 Quaternion 11.54 7.67 29.8 6D 14.13 9.41 23.4 9D 11.44 8.01 23.8 9D-Inf 4.07 3.28 76.7 10D 9.28 7.05 32.6 RPMG-Quat 4.86 3.25 75.8 RPMG-6D 2.71 2.04 92.1 RPMG-9D 3.75 2.10 91.1 RPMG-10D 3.30 2.70 86.8 Table 10 . 10Test error percentiles for Experiment 5.1 & 5.2 Left: test error percentiles of airplane for Experiment 5.1 after training completes. Right: test error percentiles of chair for Experiment 5.2 after training completes. Optimization algorithms on matrix manifolds. P-A Absil, Robert Mahony, Rodolphe Sepulchre, Princeton University Press111P-A Absil, Robert Mahony, and Rodolphe Sepulchre. Opti- mization algorithms on matrix manifolds. Princeton Univer- sity Press, 2009. 1, 11 A fractal dimension for measures via persistent homology. arXiv: Dynamical Systems. Henry Adams, M Aminian, Elin Farnell, M Kirby, C Peterson, Joshua Mirth, R Neville, P Shipman, C Shonkwiler, Henry Adams, M. 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[]
[ "On the h -adaptive PUM and the hp -adaptive FEM approaches applied to PDEs in quantum mechanics", "On the h -adaptive PUM and the hp -adaptive FEM approaches applied to PDEs in quantum mechanics" ]
[ "Denis Davydov es:[email protected] \nChair of Applied Mechanics\nUniversity of Erlangen-Nuremberg\nEgerlandstr. 591058ErlangenGermany\n", "Tymofiy Gerasimov [email protected] \nInstitute of Applied Mechanics\nTechnische Universität Braunschweig\nBienroder Weg 8738106BraunschweigGermany\n", "Jean-Paul Pelteret [email protected] \nChair of Applied Mechanics\nUniversity of Erlangen-Nuremberg\nEgerlandstr. 591058ErlangenGermany\n", "Paul Steinmann [email protected] \nChair of Applied Mechanics\nUniversity of Erlangen-Nuremberg\nEgerlandstr. 591058ErlangenGermany\n" ]
[ "Chair of Applied Mechanics\nUniversity of Erlangen-Nuremberg\nEgerlandstr. 591058ErlangenGermany", "Institute of Applied Mechanics\nTechnische Universität Braunschweig\nBienroder Weg 8738106BraunschweigGermany", "Chair of Applied Mechanics\nUniversity of Erlangen-Nuremberg\nEgerlandstr. 591058ErlangenGermany", "Chair of Applied Mechanics\nUniversity of Erlangen-Nuremberg\nEgerlandstr. 591058ErlangenGermany" ]
[]
In this paper the h -adaptive partition-of-unity method and the h -and hp -adaptive finite element method are applied to partial differential equations arising in quantum mechanics, namely, the Schrödinger equation with Coulomb and harmonic potentials, and the Poisson problem. Implementational details of the partition-of-unity method related to enforcing continuity with hanging nodes and the degeneracy of the basis are discussed. The partition-of-unity method is equipped with an a posteriori error estimator, thus enabling implementation of error-controlled adaptive mesh refinement strategies. To that end, local interpolation error estimates are derived for the partition-of-unity method enriched with a class of exponential functions. The results are the same as for the finite element method and thereby admit the usage of standard residual error indicators. The efficiency of the hadaptive partition-of-unity method is compared to the h -and hp -adaptive finite element method. The latter is implemented by adopting the analyticity estimate from Legendre coefficients. An extension of this approach to multiple solution vectors is proposed. Numerical results confirm the remarkable accuracy of the h -adaptive partition-of-unity approach. In case of the Hydrogen atom, the h -adaptive linear partition-of-unity method was found to be comparable to the hp -adaptive finite element method for the target eigenvalue accuracy of 10 −3 .
null
[ "https://arxiv.org/pdf/1612.02305v1.pdf" ]
119,022,718
1612.02305
59e86a94299cacc12084d96a31948ab613a52a72
On the h -adaptive PUM and the hp -adaptive FEM approaches applied to PDEs in quantum mechanics 7 Dec 2016 Denis Davydov es:[email protected] Chair of Applied Mechanics University of Erlangen-Nuremberg Egerlandstr. 591058ErlangenGermany Tymofiy Gerasimov [email protected] Institute of Applied Mechanics Technische Universität Braunschweig Bienroder Weg 8738106BraunschweigGermany Jean-Paul Pelteret [email protected] Chair of Applied Mechanics University of Erlangen-Nuremberg Egerlandstr. 591058ErlangenGermany Paul Steinmann [email protected] Chair of Applied Mechanics University of Erlangen-Nuremberg Egerlandstr. 591058ErlangenGermany On the h -adaptive PUM and the hp -adaptive FEM approaches applied to PDEs in quantum mechanics 7 Dec 2016Preprint submitted to Journal of Computational Physics August 31, 2018(Jean-Paul Pelteret),adaptive finite element methodpartition-of-unity methoderror estimatorsSchrödinger equationlocal interpolation error estimates * Corresponding author In this paper the h -adaptive partition-of-unity method and the h -and hp -adaptive finite element method are applied to partial differential equations arising in quantum mechanics, namely, the Schrödinger equation with Coulomb and harmonic potentials, and the Poisson problem. Implementational details of the partition-of-unity method related to enforcing continuity with hanging nodes and the degeneracy of the basis are discussed. The partition-of-unity method is equipped with an a posteriori error estimator, thus enabling implementation of error-controlled adaptive mesh refinement strategies. To that end, local interpolation error estimates are derived for the partition-of-unity method enriched with a class of exponential functions. The results are the same as for the finite element method and thereby admit the usage of standard residual error indicators. The efficiency of the hadaptive partition-of-unity method is compared to the h -and hp -adaptive finite element method. The latter is implemented by adopting the analyticity estimate from Legendre coefficients. An extension of this approach to multiple solution vectors is proposed. Numerical results confirm the remarkable accuracy of the h -adaptive partition-of-unity approach. In case of the Hydrogen atom, the h -adaptive linear partition-of-unity method was found to be comparable to the hp -adaptive finite element method for the target eigenvalue accuracy of 10 −3 . Introduction Recently there has been an increase of interest in applying Finite Element (FE) methods to partial differential equations (PDEs) in quantum mechanics [7,10,13,15,19,21,39,40,47,50,53,58,60], namely to the coupled eigenvalue and Poisson problems. In order to improve the accuracy of the solution, the basis set can be adaptively expanded through either refinement of the mesh (h -adaptivity) or the basis functions can be augmented by the introduction of higher polynomial degree basis functions (p -adaptivity). Since the solution is not smooth and contains cusp singularities, the application of the h -adaptive FEM may require very fine meshes and could be computationally inefficient. There are several approaches to circumvent this problem. From the physical point of view, for ab initio calculation of molecules often core electrons (as opposed to valence electrons) behave in a similar way to single atom solutions. Thus one possesses an a priori knowledge of a part of the solution vectors to the eigenvalue problem. One of the approaches used to introduce this into a FE formulation is the Partition-of-Unity Method (PUM) [2,43], which is a generalization of the classical FE method. In PUM the enrichment functions are introduced into a basis as products with standard FE shape functions, thereby enlarging the standard FE space. As the standard FE functions satisfy the partition-of-unity property (that is, they sum to one in the whole domain), the resulting basis can reproduce enrichment functions exactly. In the continuum mechanics community this method is known as XFEM [8,11,16,22,25,37,51,59], originally popularized by Belytschko and Black [8]. For an overview on this topic we refer the reader to [9,23,52]. An alternative approach to the above is to combine h -and p -adaptivity resulting in what is termed as hp -adaptive FEM. For an overview of hp -adaptive refinement strategies we refer the reader to [46]. The general idea is that when the exact solution is smooth on the given element, p-adaptive refinement is more efficient and leads to a faster convergence; whereas if the solution is non-smooth (singular), h -adaptive refinement is performed. Thus in addition to a reliable error estimate and the choice of the marking strategy of elements for refinement, hp -adaptive methods need to decide which type of refinement to perform on a given element. In this work we use methods based on smoothness estimation [5,18,20,28,34,42]. As those methods are normally employed for problems with a single solution vector, we propose an extension to multiple solution vectors as is required for the here considered eigenvalue problems. Herein, our main focus is application of h -adaptive PUM and hp -adaptive FEM to PDEs in quantum mechanics, namely to the Schrödinger equation and the Poisson problem, and comparison of efficiency of these approaches. Application of the PUM to the above problems holds a significant promise to improve on accuracy of a standard (nonenriched) FE approximation. The corresponding numerical evidence can be found in [49,53], where convergence studies for PUM solutions obtained on uniformly refined meshes are performed. In our paper, the PUM will be equipped with an a posteriori error estimator, thus enabling implementation of error-controlled adaptive mesh refinement strategies. As for the model problems, we limit ourselves to uncoupled eigenvalue and Poisson problems as analytic solutions are available for that case. All findings are expected to apply to more complicated cases when the two equations are coupled, such as those arising from the Density Functional Theory [33,36]. The outline of this paper is as follows: In section 2 the considered PDEs and their solution are introduced. The PUM and its implementational details are given in Section 3. Section 4 is devoted to the strategy to decide between h -and p -adaptive refinement. Results of numerical studies of the chosen systems are presented in section 5, followed by some conclusions in Section 6. Finally, in the Appendix we rigorously derive the local interpolation error estimates for enrichment with a class of exponential functions. Problem formulation In order to motivate the use of the PUM, it is necessary to understand some of the difficulties arising from the classes of problems that we will evaluate in this work. In this manuscript we consider the following three-dimensional problems that have analytical solutions: Eigenvalue problem The eigenvalue problem that we will consider is the Schrödinger equation, for which we seek lowest eigenpairs (λ α , ψ α ) of − 1 2 ∇ 2 + V(x) ψ α (x) = λ α ψ α (x) on Ω , ψ α (x) = 0 on ∂Ω,(1)with two different (spherical) potentials V(x) = V(|x|) 1 . The first case is the the Coulomb potential V(x) = −1/ |x|, which corresponds to the Hydrogen atom. The eigenvalues of this problem are degenerate. In R 3 , on each energy level n there are n 2 eigenvalues λ n = λ 1 /n 2 , where λ 1 = −1/2 [27]. The eigenvector corresponding to the lowest eigenvalue reads ψ 1 (x) = 1 √ π exp (− |x|) .(2) The radial component of the eigenfunctions at the next energy level are R 2,0 = [1 − |x| /2] exp(− |x| /2) and R 2,1 = |x| /2 exp(− |x| /2). The second potential we will consider is a harmonic potential V(x) = |x| 2 /2 that leads to a harmonic oscillator problem. The eigenvalues for this problem are also degenerate; in R 3 they are given by λ n = n + 1/2 for n-th energy level. The lowest two have a degeneracy of 1 and 3, respectively. The (unnormalized) eigenvector corresponding to the lowest eigenvalue is ψ 1 (x) = exp − |x| 2 /2 .(3) The radial component of the next eigenfunction is R 0,1 (x) = |x| exp − |x| 2 /2 . Figure 1 shows radial components of eigenfunctions for the Coulomb and harmonic potential. It is clear that in order to have a low interpolation error for a standard Lagrange FE basis, a very fine mesh will be required near the origin. For such non-smooth solutions we will see that by introducing enrichment functions the interpolation error of the resulting FE basis will be greatly reduced. Poisson problem The associated Poisson equation relates the electron density field and the electrostatic potential. In atomic units it reads −∇ 2 φ(x) = 4πρ(x) on Ω .(4) The density function on the right-hand-side is composed of the squares of eigenvectors, possibly with the addition of other terms. In case of the Hydrogen atom, the total charge density is composed of the electron density less the singular nucleus density ρ(x) ≡ ψ 2 1 − δ(x) ,(5) where ψ 1 is the electron wave-function given in (2). Note that ρ(x) is not in H −1 and thus the solution φ(x) is not in H 1 . The corresponding electrostatic potential produced reads φ = − exp (−2 |x|) 1 + 1 |x| .(6) For the numerical analysis below we will consider a regularized counterpart where the delta function is substituted by a Gaussian distribution ρ(x) ≡ ψ 2 1 − 1 π 3 /2 σ 3 exp − |x| 2 σ 2 .(7) This corresponds to a split of the nuclei Coulomb potential into an (almost) local short range part and smooth long range part [15]. The electrostatic potential produced in this case reads Figure 2 shows radial components of density and potential fields for different values of σ. It is clear that by varying σ, the character of the solution is changed from smooth to more singular. The limit σ → 0 corresponds to the singular solution in Equation 6. φ = − exp (−2 |x|) 1 + 1 |x| + 1 |x| − erf(|x| /σ) |x| .(8) partition-of-unity method Enriched FE space The classical FEM may fail when the solution is not smooth or is highly oscillatory. In either case, in order to obtain an accurate solution using piecewise polynomial spaces one has to employ a very fine mesh that increases the computational cost of solving the problem. The PUM proposed by Melenk and Babuska in [2,43] can address this issue. The main features of the PUM are (i) the inclusion of an a priori knowledge about the solution into the FE space, and (ii) the construction of an FE space of any desired regularity. It is the former attribute which is important in the context of this work. The PUM enriches the vector space spanned by standard FE basis functions N i (x) (e.g. polynomials) by products of these functions with functions f j (x) that contain a-priori knowledge about the solution u(x) = i∈I N i (x)         u i + j∈S f j (x) u i j         .(9) Here u i are standard degrees-of-freedom (DoFs) and u i j are additional DoFs associated with the shape functions N i (x) and the enrichment functions f j (x); I is a set of all nodes and S is the set of enrichment functions. Since (possibly global) enrichment functions f j (x) are multiplied with N i (x) which has local support, the product also has local support and therefore matrices arising from the weak form remain sparse. Also, since the standard shape functions satisfy the partition of unity property i N i (x) ≡ 1 on Ω, the resulting vector space can reproduce enrichment functions f j (x) exactly. Implementational details An enriched finite element class has been implemented for the general purpose objectoriented C++ finite element library deal.II [6]. The implementation is based on the FESystem class, which is used to build finite elements for vector valued problems from a list of base (scalar) elements. What differs from that class is that the developed FE implementation is scalar, but built from a collection of base elements and enrichment functions 2 u(x) = i∈I N i (x)u i + k∈S f k (x)           j∈I pum k N jk (x) u jk           ,(10) where I is the set of all DoFs with standard shape functions (see Figure 3(a)), I pum k is the set of all DoFs corresponding to shape functions enriched with f k (x) (see Figure 3(b)) and S is the set of enrichment functions. As distribution of DoFs in deal.II is element based, we always enrich all DoFs on the element. To restore C 0 continuity between enriched and non-enriched elements, additional algebraic constraints are added to force DoFs u jk associated with N jk f k on the face between the enriched and non-enriched elements to be zero. This is equivalent to enriching only those shape functions whose support is contained within the enriched elements. The h -refinement in deal.II is implemented using hanging nodes. In this case, extra algebraic constraints have to be added to make the resulting field conforming. We build these constraints separately for the non-enriched FE shape functions and enriched shape functions; that is, the following spaces are separately made conforming: {N i (x)}, {N j0 (x)}, {N j1 (x)}, etc. To illustrate this idea consider two separate FE spaces shown in Figure 3. We assume that functions in the first space are non-zero everywhere in the domain, whereas functions in the second space are non-zero only in the left part, marked by the blue shading. Therefore we do not have to introduce any DoFs in the right part, the underlying elements are denoted by Q zero . The standard procedure implemented in deal.II [5] will enforce continuity of the vector field by introducing algebraic constraints for DoFs associated with hanging nodes 3 (3,5,17,19), plus constraints for DoFs 14, 16, 22 to make functions in the second FE space zero at the interface between Q 1 and Q zero . We can observe now that if we take the constrained scalar field from the first FE space and add a scalar field from the second FE space multiplied by the enrichment functions f (x) (continuous in space), the resulting scalar FE field will also be continuous. Thus we arrive at a conforming h -adaptive PUM space where only some elements are enriched. With reference to Figure 3, the resulting PUM field will have enrichment associated with DoFs 23, 24, 20, 21, 17, 18, 15 whereas DoFs 22, 19, 16, 14, 17 will be constrained. In this procedure the algebraic constraints do not depend on the enrichment functions and are equivalent to those one would have for the vector-value bases build upon the same list of scalar FEs. Therefore, no extension of the existing functionality to build algebraic constraints was necessary. This allows us to reuse the code written for the FESystem class. Figure 4 depicts an example of enriched and non-enriched shape functions for the case of h -adaptive refinement with hanging nodes in two dimensions. The choice of enrichment Most of the time the a-priori knowledge of the solution is limited. In DFT calculation of molecules often core electrons (as opposed to valence electrons) behave in a similar way to single atom solutions. Thus the corresponding solution of single atom problems is used as enrichment functions. To mimic this in the here considered test eigenproblems, we will only use the lowest eigenvector as an enrichment. Therefore, for the eigenvalue problems we will employ exponential enrichment. To lower computational costs we enrich only a subset of elements, chosen based on the input mesh according to vicinity of the element's center to the origin. There is another, more important reason why one should limit the enrichment radius. There exist combinations of local approximation spaces and partitions of unity that lead to linear dependent local basis functions that, consequently, do not form a basis of the PUM space [2]. The authors in [2] give an example of piecewise linear hat functions, which form the partition-of-unity, enriched with polynomial local approximation spaces. In principle these shape functions can still be used but the resulting matrices become positive semidefinite (as opposed to positive definite). As a further example, consider a one dimensional mesh with two linear FEs where the first one is enriched with an exponential function, as is shown in Figure 5(a). Determinants of the mass and Laplace matrices quickly tend to zero as the singularity point x 0 moves away from the enriched element. Similar behavior can be expected in three dimensions. From the practical perspective we notice that when the enrichment radius is too big, the variational convergence of the eigenvalues is lost; that is, the eigenvalues do not necessarily converge from the above to the exact values. To avoid such behavior the radius of enrichment has to be limited. The exact radius is contingent upon the decay of the enrichment function and the initial mesh. Note that in [53] enrichment for the harmonic oscillator problem is also localized to a predefined maximum distance. The authors, however, do not discuss the rationale for their choice of cut-off radius. Numerical integration One of the particular features of PUM that needs careful treatment is numerical integration. Integrands in the weak form become less smooth and attaining a higher accuracy of the integration is therefore a more difficult task. There are several approaches to address this. One is adaptive integration schemes (for example [48]), when the element over which the integration is performed is subdivided into child elements iteratively until the convergence of the integral is attained. Implementation of this procedure in the deal.II library is, unfortunately, not straightforward. An alternative is to perform coordinate transformation, such as the cubic transformation proposed in [54]. However the generalization of this approach to 3D appears to require rectangular hexahedron elements, which would be a major constraint in generating input meshes. As a result, similar to [51] we have opted to utilize higher order Gaussian quadrature rules. For the numerical results presented below this approach was found to produce sufficiently accurate results while not becoming a bottleneck in calculations. Error estimator A posteriori error estimation analysis for FE approximations of (second-order) eigenvalue problems has been a topic of intensive study within the last several decades, both from theoretical and implementational standpoints. We refer the interested reader to [14,17,24,31,38,41,56], where both residual-and averaging-based error estimators are presented. Let {ψ h , λ h } denote the set of eigenpairs computed on a finite element mesh P h . In general, a discretization error in approximated eigenfunctions, ψ − ψ h , measured in a suitable norm (e.g. L 2 and energy norm), as well as in approximated eigenvalues, |λ − λ h |, can be estimated from above. That is, ψ − ψ h ≤ C 1 η,(11) and |λ − λ h | ≤ C 2 η 2 ,(12) where C 1 , C 2 are stability constants that are independent of the mesh size and η is the explicitly computable error upper-bound, see e.g. [14,38] for details. These equations are typically termed (global) error estimators. The bound η reads as η :=         K∈P h η 2 K         1 2 , where summation is performed over all elements in P h and η K is the (local) error indicator, a quantity showing a discretization error of {ψ h , λ h } element-wise, that is, on every fixed K. With multiple solutions available (in this case, eigenpairs {ψ h α , λ h α }), η K will be a sum of discretization errors of the corresponding eigenpairs on a given element K, that is η K :=        α η 2 K,α        1 2 . For a standard (non-enriched) Q 1 -based finite element solution of (1), a local indicator η K,α of so-called residual type reads as follows (see [14,24,31,38] for details): η 2 K,α := h 2 K K − 1 2 ∇ 2 + V(x) ψ h α − λ h α ψ h α 2 dx + h K e⊂∂K e − 1 2 ∇ψ h α · n 2 e da, (13) where [[− 1 2 ∇ψ h α · n]] e := − 1 2 ∇ψ h α | K + 1 2 ∇ψ h α | K ′ · n e represents the jump of the gradient across interface e between two adjacent elements K and K ′ , n e is the outward unit normal vector to e and h K := diam(K). One of the findings of our work is the proof that indicator (13) can also be used in the PUM with the exponential enrichment function f (x) = exp (−µ |x| p ). In the appendix, we derive and prove the related local interpolation error estimates required for the derivation of the error estimator in this case. hp-adaptive solution There have been numerous works devoted to hp -adaptive refinement [18,28,32,34,35,42,44] including a comparison of different methods [46]. The main difficulty that a posteriori hp -adaptive methods aim to address is the following: Once an error is estimated and a certain subset of elements is marked for refinement, one has to choose between h -or p -refinement for each element. It is a general knowledge that it is better to increase polynomial degree (p -refinement) of those elements where the solution is smooth, whereas it is better to refine the element (h -refinement) near the singularities of the solution. In this work we adopt a strategy based on the estimate of the analyticity of the solution 4 on the reference element via expansion into a Legendre basis [18,28,34,42]. In particular, we perform a least squares fit of Legendre coefficients a i for each element |a i | ∼ C exp(−σi).(14) The minimum decay coefficient σ in each direction is used to estimate analyticity as exp(−σ). This corresponds to an estimation of smoothness in the direction where the solution is roughest. As there is no anisotropic elements in deal.II that can be used with hp -refinement, distinguishing between different directions is not needed. When this value is below exp(−1), the solution is considered to be smooth and thus p -refinement is performed, otherwise h -refinement is executed. For linear FEs p -refinement is always performed. Finally, in order to avoid numerical issues with the evaluation of log a j , for the least squares fit we only consider coefficients that are two orders of magnitude greater than the minimal representable positive floating value. In order to extend this hp -refinement strategy to the eigenvalue problem, that is when there are multiple vectors represented using the same FE basis, we propose the following approach: For each element we find an eigenvector which contributes the most to the total element's error. The smoothness of this vector is the basis on which we decide to perform h -refinement or p -refinement. The rationale behind this approach is that we aim at minimizing the error the most during a single refinement step while being conservative and avoiding performing both h -and p -refinement on the same element. In our opinion the proposed strategy is a better choice than, for example, choosing minimum smoothness among all vectors for a given element. That may be considered to be a more robust approach but could also lead to a slower global convergence. Finally, for the error indicator we adopt the following expression [26] η 2 K,α := h 2 K p 2 K K − 1 2 ∇ 2 + V(x) ψ h α − λ h α ψ h α 2 dx + e⊂∂K h e 2p e e − 1 2 ∇ψ h α · n 2 e da ,(15) where h e is the face's diameter, p K is the element's polynomial degree and p e is the maximum polynomial degree over two elements K and K ′ adjacent to the face e. Results and Discussion If not explicitly stated otherwise, the results below are obtained for the following configuration: (i) the initial polynomial degree for non-enriched DoFs is one for hp -adaptive FEM; (ii) linear shape functions are used for the PUM; (iii) a Gaussian quadrature rule with 20 3 points is used for enriched elements in the eigenvalue problem; (iv) a Gaussian quadrature rule with [7 + p] 3 points is used for standard elements in the eigenvalue problem, where p is the polynomial degree of the basis; (v) the Dörfler marking strategy with θ = 0.6 is used to mark elements for refinement; (vi) integration of the jump of fields over faces in error estimators is performed with [1 + p] 2 Gaussian quadrature points, where p is the polynomial degree of the basis; (vii) we assume a Q1 mapping for elements; (viii) Gauss-Legendre-Lobatto supports points are used for the hp -adaptive FEM basis to improve the condition number; (ix) a standard residual-type error estimator similar to (13) [3] and parallel solvers for eigenvalue problems in the Scalable Library for Eigenvalue Problem Computations (SLEPc) [29] are used for the eigenvalue problem; (xii) Trillinos [30] vectors, matrices, solvers and preconditioners are used to solve the extended Poisson problem. In case of hp -adaptive refinement the highest polynomial degree is limited to 4. The rationale for that choice is as follows: In order to preserve variational convergence when solving a coupled eigenvalue and Poisson problem in DFT, the polynomial degree of the Poisson FE basis should be twice of that used for the eigenvalue problem [15]. Thus quartic FEs in the eigenvalue problem would require polynomials up to 8-th order in the Poisson problem. From our experience (not reported here) this is already challenging both from the number of DoFs as well as the condition number of the Laplace matrix in hp -adaptive refinement. Eigenvalue problem The initial mesh used to solve the Schrödinger equation is obtained from 3 global mesh refinements of the single element in Ω = [−20; 20] 3 for the Coulomb potential and Ω = [−10; 10] 3 for the harmonic potential. For the PUM only 8 elements adjacent to the singularity that is located at the origin are marked for enrichment. First, we examine the convergence in case when a single eigenpair is required in the Schrödinger equation with two different potentials. Figure 6 compares the h -adaptive FEM, hp -adaptive FEM and h -adaptive PUM, whereas Figure 7 shows the cross-sections of meshes for the last refinement step. For both combination of potentials and enrichment functions, the h -adaptive PUM is superior to h -adaptive FEM. In particular, for the last refinement step the PUM solution is about 2 orders more accurate than the h -adaptive FEM with the same number of DoFs in case of the Coulomb potential. For the harmonic potential this value is smaller. The asymptotic convergence rate of the h -adaptive PUM with the default enrichment radius is very similar to that of the h -adaptive FEM for both problems (compare green and red lines in Figure 6). The advantage of h -adaptive PUM also depends on the enrichment radius with respect to the underlying exact solution. To examine this effect we employ an initial mesh obtained only by 2 global refinements of a single element and mark the 8 elements adjacent to the origin for enrichment. With this approach we effectively consider a larger enrichment domain [−5; 5] 3 instead of [−2.5; 2.5] 3 . Importantly, the numerically non-zero part of the underlying analytical solution will be almost fully contained in those 8 elements (see Figure 1(b)). From the numerical results we observe that for the most refined stage the hadaptive PUM displays an error which is about 6 orders of magnitude less than the same method with the smaller enrichment domain (compare purple and green lines in Figure 6(b) ). Interestingly, the hp -adaptive FEM does not display a big advantage over the hadaptive quadratic FEM for the Hydrogen atom and the smoothness estimator considered here (compare blue and purple lines in Figure 6(a)). Now let us turn our attention to a more realistic scenario where one seeks multiple eigenpairs whereas an a priori knowledge is available only for the first eigenvector. Figure 8 plots convergences of the first 5 / 4 eigenvalues for the Schrödinger equation with Coulomb / harmonic potential solved with the different methods. For both problems the h adaptive PUM again has remarkable convergence properties, superior to h -adaptive FEM. It is important to note that even though in the PUM the enrichment function corresponds to the first eigenvector only, others eigenpairs in the case of the harmonic potential tend to converge faster than the standard h -adaptive FEM case, as can be observed in Figure 8(b). The same applies to the spherical orbital at the second energy level of the Hydrogen atom; see Figure 8(a) where the corresponding eigenvalue in the PUM case displays a faster convergence rate than the others on the same energy level. For the Hydrogen atom, in the case of the hp -adaptive refinement one observes a superior convergence rate of the first eigenvalue, whereas eigenvalues from the next energy level have higher errors at some stages when compared to h -adaptive linear FEM. This indicates that the suggested strategy of deciding between h -or p -refinement for multiple degenerate eigenvectors is not ideal. A possible issue could be related to smoothness estimation on elements with hanging nodes. In particular it is observed [4] that the smooth- ness is overestimated when using similar methods, albeit based on Fourier coefficients. This leads to unnecessarily high order polynomial degrees in these areas. In DFT calculations, the requested tolerance of eigenvalues is often at the order of 10 −3 . In this case, it is clear from Figure 8(b) that for the Hydrogen atom the linear PUM achieves this tolerance for all eigenvalues at a number of DoFs comparable to the hpadaptive FEM. Thus, depending on the required accuracy, the h -adaptive PUM can be as efficient as the hp -adaptive FEM. Poisson problem In this subsection we turn our attention to the solution of the Poisson problem with the physical interpretation here being the electrostatic potential produced by the charge density. We will consider the solution obtained for two different values of the regularization parameter σ, namely 1.0 and 0.1 (their influence is shown in Figure 2). As was mentioned in the introduction, a similar case was considered in [53] (albeit for periodic boundary conditions with global refinement only), however the authors constrained all enriched DoFs to be of the same value. The resulting space is, obviously, smaller than the unconstrained PUM space and thus the Galerkin projection will certainly lead to higher errors. Figure 9(b) compares the energy error norm for the standard FEM, and constrained and unconstrained PUM in the course of global refinement for the case σ = 1.0. It is seen that by constraining PUM DoFs to have the same value, the accuracy is reduced by half for the finest mesh. Remarkably for the case σ = 1.0 the PUM is only slightly more accurate than the standard FEM. The same observation can be made for the h -adaptive refinement, shown in Figure 9(a). By comparison, for the case σ = 0.1 the PUM is significantly more accurate than the standard FEM, both for the case of global and h -adaptive refinement (see Figure 10). This indicates that, not surprisingly, the efficiency of the PUM as compared to FEM is very much contingent upon the underlying exact analytical solution. Finally, we observe that the convergence rates in the case of h -adaptive refinement are roughly the same for both the standard FEM and PUM. This agrees with our observation for the eigenvalue problem. Moreover, the standard residual error indicator used for the Poisson problem with PUM shows similar convergence rate for both values of σ and therefore can be considered as a reliable error indicator for the here considered problem. Summary In this contribution we have applied the h -and hp -adaptive FEM, and the h -adaptive PUM to the relevant PDEs in quantum mechanics, namely the Schrödinger equation and the Poisson equation. The main findings are summarized below. • The PUM renders several orders of magnitude more accurate eigenvalues than the standard FEM when solving the Schrödinger equation for the lowest eigenpair with Coulomb and harmonic potential. For the case when more eigenpairs are sought but only the lowest eigenvector is introduced as an enrichment, the PUM is still more accurate, especially for the lowest eigenvalue. Remarkably other eigenvalues also exhibit a faster convergence. • For the here considered smoothness and residual error estimators, an application of the hp -adaptive FEM to the Hydrogen atom displays an exponential-like convergence rate for the first eigenvalue, whereas other eigenvalues tend to stagnate. This illustrates the challenge of applying the hp -adaptive FEM to eigenvalue problems, namely that there are multiple solution fields represented on the same FE space that are likely to have distinct smooth and non-smooth (singular) regions. • Constraining PUM DoFs to have the same value when solving the Poisson equation could decrease the accuracy of the solution by a factor of two. • The efficiency of the PUM problem is very much dependent on the underlying solution. On the one hand when applied to the Poisson problem with the here studied density field, which is composed of the Gaussian charge and the charge of the electron in the Hydrogen atom, the PUM is only slightly more accurate for the case of σ = 1.0. On the other hand, for σ = 0.1 the PUM is about two orders of magnitude more accurate than the standard FEM. • The residual error estimator used for the Poisson problem with PUM shows a similar convergence rate to the energy error and, therefore, can be considered as a reliable error indicator for here considered problem. • An element view to the implementation of PUM in FEM codes based on hexahedra is proposed. As a result, continuity of the enriched field along the edges with hanging nodes is enforced by treating FE spaces produced by each function in the local approximation space separately. The resulting algebraic constraints are independent on the enrichment functions. This allows one to directly reuse algorithms written for enforcing continuity of vector-valued FE spaces constructed from a list of scalar-valued FEs. • Local interpolation error estimates are derived for the PUM enriched with the class of exponential functions. In this case the results are the same as for the standard FEM and thereby admit the usage of the error indicator (13). Appendix: Local interpolation error estimates In this appendix, the local interpolation error estimates required for the derivation of the error indicator (13) in the case of PUM are obtained for linear finite element approximations enriched with f (x) = exp (−µ |x| p ), where 0 < µ ∈ R and 1 ≤ p ∈ N. These are v − q h v L 2 (K) ≤ c K h K |v| H 1 (ω K ) ,(16)v − q h v L 2 (e) ≤ c e h 1 2 K |v| H 1 (ω K ) ,(17) where, as usual, v : Ω → R is a scalar-valued function, which is assumed to be at least in H 1 (Ω), q h is a quasi-interpolation operator (of the averaging type), K is an element of the discretization P h of Ω, e ⊂ ∂K is an edge of K. Also, h K measures the size of K, ω K is the patch of elements neighboring K including K itself. Finally, c K , c e ∈ R are the interpolation constants independent of the mesh size. We fix the notations to be used throughout the appendix and make assumptions that are conventional for this kind of analysis. For the sake of simplicity and without loss of generality, we elaborate here for the two-dimensional setting. The obtained results are valid in three dimensions as well. First, we assume that the partition P h of Ω ⊂ R 2 consisting of open and convex quadrilaterals K is shape-regular (or non-degenerate), as well as locally quasi-uniform in the sense of [12,45]. For every K and its edge e we define h K := diam(K) and h e := |e| is the length of e. For every node i in P h we denote by ω i the union of quadrilaterals connected to node i and set h ω i := diam(ω i ). Furthermore, for every K, ω K represents the patch containing K and the first row of its neighbors; it is then set h ω K := diam(ω K ). Also, in what follows, by the notation a b we imply the existence of a positive constant C independent of a and b such that a ≤ Cb. Then a ∼ b means that a b and a b hold simultaneously. The symbol | · | will be used to denote either the H 1 -seminorm (as e.g. in (16) and (17)) or the length of a linear segment in R 2 or the area of a plane domain in R 2 . With these notations at hand, one can show that |K| 1 2 ∼ h K , |ω i | 1 2 ∼ h ω i and |ω K | 1 2 ∼ h ω K . Furthermore, the shape regularity of the mesh P h ensures that h e ∼ h K , whereas its local quasi-uniformity implies that h K ∼ h ω i ∼ h ω K . Finally, we also recall useful inequalities, which are • the Poincaré-type inequality (see e.g. [55]): v − 1 |ω| ω v dx L 2 (ω) ≤ h ω π |v| H 1 (ω) , ∀v ∈ H 1 (ω),(18) where ω ⊂ R n (n = 2, 3) is a Lipschitz domain and h ω := diam(ω); • the scaled trace inequality (e.g. in [57], Lemma 3.2): v L 2 (e) h − 1 2 e v L 2 (K) + h 1 2 e |v| H 1 (K) , ∀v ∈ H 1 (K).(19) Quasi-interpolation operator Herein, we construct an interpolation operator for obtaining the local error estimates (16) and (17). Let V := H 1 (Ω) be an admissible space and V h be its (enriched) finite element counterpart V h :=        v h ∈ C(Ω) : v h (x) := i∈I ⋆ a i N i (x) + f (x) i∈I ⋆ b i N i (x) + i∈I std. c i N i (x), a i , b i , c i ∈ R        ⊂ V,(20) where I ⋆ is the set of all enriched nodes of P h and I std. is the set of standard, i.e. nonenriched nodes of P h ; I ⋆ ∩ I strd. = ∅. Recall also that N i in our case is the Q 1 -shape function associated with node i and supported on ω i . Explicit construction of the operator q h : V → V h implies the explicit pattern of assignments of a i , b i , c i ∈ R through a function v ∈ V. In the case of the enriched FE approximation (20), the major challenge in deriving q h is imposition of the constant-preserving property on q h , which should be fulfilled on every element K ∈ P h regardless the element type (see Figure 11). The operator q h : V → V h with the desired property reads as follows: q h v(x) := i∈I ⋆ 1 2|ω i | ω i v(y)dy N i (x) + f (x) i∈I ⋆ 1 2 f (x i )|ω i | ω i v(y)dy N i (x) + i∈I strd. 1 |ω i | ω i v(y)dy N i (x),(21) with all notations as in (20) and where x i , entering the second term, denotes the coordinate of a node i. Below, for the proposed quasi-interpolation operator of the averaging type q h we establish that q h c| K = c on a standard element (note this is a classical result for a non-enriched FEM) and, more importantly, that q h c| K = c + O(h p K ) on a fully-enriched and a blended element. Estimates 7.2.1. Preliminaries. The three estimates that we start with are basic for the following local interpolation error analysis. On every K ∈ P k and its node i it holds that N i (x) L 2 (K) h K , N i (x) L 2 (e) h 1 2 K ,(22)1 |ω i | ω i v(y)dy h −1 K v L 2 (ω K ) + |v| H 1 (ω K ) ,(23) and f (x) f (x i ) = 1 + O(h p K ).(24) Results (22) rigorously follow from the isoparametric concept and related properties, see e.g. [1] for details. We note that they may be also derived in a less rigorous manner owing to a boundedness of the basis function N i on K along with |K| The inequality (23) is obtained as follows: 1 |ω i | ω i v(y)dy ≤ |ω i | −1 ω i |v(y)| dy ≤ |ω i | − 1 2 v L 2 (ω i ) h −1 K v L 2 (ω K ) + |v| H 1 (ω K ) . Here we used the Cauchy-Schwarz inequality, |ω i | 1 2 ∼ h ω i ∼ h K and also the extension- related result v L 2 (ω i ) ≤ v L 2 (ω K ) . Finally, to show (24) we explicitly use the properties of f (x). For any fixed K, x ∈ K and x i ∈ K being one of its nodes, we have the following upper bound estimate: f (x) f (x i ) = exp(µ|x i | p ) exp(µ|x| p ) ≤ exp µ max x∈K |x| p exp µ min x∈K |x| p ≤ exp µ min x∈K |x| + h K p exp µ min x∈K |x| p = 1 + µp min x∈K |x| p−1 h K + h.o.t. in min x∈K |x|, h K = 1 + O       min x∈K |x| p−1 h K       .(25) Notice that due to boundedness of min x∈K |x| for a given fixed K, there always exists ǫ > 0 such that min x∈K |x| = ǫh K . Using this in (25), we obtain min x∈K |x| p−1 h K = ǫ p−1 h p K , yielding, as a result, f (x) f (x i ) ≤ 1 + O(h p K ) . The lower bound estimate can be found similarly: f (x) f (x i ) = exp(µ|x i | p ) exp(µ|x| p ) ≥ exp µ min x∈K |x| p exp µ max x∈K |x| p ≥ exp µ min x∈K |x| p exp µ min x∈K |x| + h K p = 1 − µp min x∈K |x| p−1 h K + h.o.t. in min x∈K |x|, h K = 1 + O       min x∈K |x| p−1 h K       , and, eventually, f (x) f (x i ) ≥ 1 + O(h p K ) . The result (24) then follows. Stability of q h in L 2 -norm. The next step towards (16) and (17) implies obtaining the so-called stability result for the constructed q h . Using (22)-(24) one straightforwardly shows that q h v L 2 (K) v L 2 (ω K ) + h K |v| H 1 (ω K ) ,(26) and q h v L 2 (e) h − 1 2 K v L 2 (ω K ) + h 1 2 K |v| H 1 (ω K ) .(27) These estimates indeed hold for every K regardless of its type (standard, blended, enriched). Note that for a standard non-enriched FEM and the resulting interpolation operators, the estimates (26), (27) are classical. We have obtained and proved them for our specific operator q h adopted for the current enriched FEM setting. Constant-preserving property of q h . The final ingredient required for obtaining (16) and (17) is the determination of how "well" the constructed q h reproduces the constant on an element K, depending on its type. This constant-preserving property of the operator is of major importance particularly in the case of enriched FEM. The required result on a standard (non-enriched) element K follows immediately. Indeed, in this case q h v(x)| K = 4 i=1 1 |ω i | ω i v(y)dy N i (x), and the partition of unity 4 i=1 N i (x) = 1 on K yields q h c| K = c, c = const. The situation on a fully-enriched and partly-enriched (blended) element is more delicate. In the case of a fully enriched element we have q h v(x)| K = 4 i=1 1 2|ω i | ω i v(y)dy N i (x) + f (x) 4 i=1 1 2 f (x i )|ω i | ω i v(y)dy N i (x), that, owing to (24), results in q h c| K = 1 2 c + 1 2 c 4 i=1 f (x) f (x i ) N i (x) = 1 2 c + 1 2 c 1 + O(h p K ) 4 i=1 N i (x) = c + O(h p K ). Now, let K be a blended element, implying the representation: q h v(x)| K = ℓ i=1 1 2|ω i | ω i v(y)dy N i (x) + f (x) ℓ i=1 1 2 f (x i )|ω i | ω i v(y)dy N i (x) + 4 i=ℓ+1 1 |ω i | ω i v(y)dy N i (x), where ℓ ∈ {1, 2, 3} is the number of enriched nodes of K. Adding and subtracting the first sum in the above expression, enables us to rewrite it as follows: q h v(x)| K = − ℓ i=1 1 2|ω i | ω i v(y)dy N i (x) + f (x) ℓ i=1 1 2 f (x i )|ω i | ω i v(y)dy N i (x) + 4 i=1 1 |ω i | ω i v(y)dy N i (x). Note that the last term contains the summation over all four nodes and is the standard (non-enriched) FE contribution which will automatically reproduce a constant. We then need to estimate, in this context, the remaining part constituting of the first and the second sums. We obtain, q h c| K = 1 2 c ℓ i=1 f (x) f (x i ) − 1 N i (x) + c ≤ 1 2 c ℓ i=1 f (x) f (x i ) − 1 N i (x) + c ≤ 1 2 c 4 i=1 f (x) f (x i ) − 1 N i (x) + c = 1 2 c O(h p K ) 4 i=1 N i (x) + c = c + O(h p K ), where (24) was also used. (16), (17) The derivation of the estimates for v − q h v L 2 (K) and v − q h v L 2 (e) is based on a combined use of the above stability results for q h , the Poincaré and the scaled trace inequalities (18) and (19), respectively, as well as the constant-preserving property results. First, due to linearity of q h , we have Proof of local error estimates v − q h v L 2 (σ) = v − c − q h (v − c) + c − q h c L 2 (σ) ≤ v − c L 2 (σ) + q h (v − c) L 2 (σ) + c − q h c L 2 (σ) ,(28) where c = const and where, for the sake of brevity, we set σ = {K, e}. We are now in a position to dissect every term in (28) in either case of σ. When σ = K in (28): By the Poincaré inequality (18), it holds that v − c L 2 (K) ≤ v − c L 2 (ω K ) h K |v| H 1 (ω K ) ,(29) where one can choose c = |ω K | −1 ω K vdx and use h ω K ∼ h K . By the stability estimate (26) and the Poincaré inequality, it holds similarly to the above that q h (v − c) L 2 (K) v − c L 2 (ω K ) + h K |v − c| H 1 (ω K ) h K |v| H 1 (ω K ) .(30) Furthermore, using the results of Section 7.2.3 we obtain c − q h c L 2 (K) ≡ 0, if K is standard,(31) and c − q h c L 2 (K) = O(h p+1 K ), if K is fully enriched or blended. In the former case we also use that 1 L 2 (K) = |K| 1 2 ∼ h K . Using (29)- (32) in (28), the resulting local interpolation error of type (16) follows. Note that in the case of fully enriched and blended elements the term O(h p+1 K ) that appears in the corresponding upper bound can be neglected, being the higher order term with respect to the leading one h K |v| H 1 (ω K ) . When σ = e in (28): By the scaled trace inequality (19), it holds K |v| H 1 (ω K ) ,(33) where we also use h e ∼ h K along with result in (29). By the stability estimate (27) and the Poincaré inequality (18), we obtain the result that q h (v − c) L 2 (e) h − 1 2 K v − c L 2 (ω K ) + h 1 2 K |v − c| H 1 (ω K ) h 1 2 K |v| H 1 (ω K ) .(34) Finally, using the results of section 7.2.3 we derive c − q h c L 2 (e) ≡ 0, if K is standard, and c − q h c L 2 (K) = O(h p+ 1 2 K ), if K is fully enriched or blended.(36) In the former case we also use the fact that 1 L 2 (e) = |e| Using (33)- (36) in (28), the resulting local interpolation error estimate of type (17) follows as well. Again, in the case of fully enriched and blended elements the term O(h 3 2 K ) that appears in the corresponding upper bound can be neglected, being the higher order term with respect to the leading one h 1 2 K |v| H 1 (ω K ) . Figure 1 : 1Radial components of eigenfunctions for different potentials V(x). The dotted vertical line indicates the smallest initial mesh size which will be used in our numerical calculations. Figure 2 : 2Charge density and electrostatic potential for different values of σ. The dotted vertical line indicates the smallest initial mesh size which will be used in our numerical calculations. Figure 3 : 3Treatment of hanging nodes for the h -adaptive PUM. Q 1 denotes (bi)linear FE, whereas Q zero denotes elements on which functions in the FE space associated with the enrichment function f k (x) are zero and thus no DoFs need to be introduced. Figure 4 : 4h -adaptive mesh refinement and shape functions associated with the central node on the domain [0, 1] 2 for the standard and enriched element. Figure 5 : 5Enrichment with exponential functions. x 0 is the position of the singularity. Figure 6 : 6Solving an eigenproblem for a single eigenpair. Figure 7 : 7(a) h -adaptive FEM (linear) (b) h -adaptive FEM (quadratic) (c) hp -adaptive FEM (d)h -adaptive PUM (linear) Cross-sections of the final meshes for the Coulomb potential when solving for a single eigenpair. potential (4 out of 5 eigenvalues are degenerate). potential (3 out of 4 eigenvalues are degenerate). Figure 8 : 8Convergence of eigenvalues from the first two energy levels for the Schrödinger equation in the course of adaptive refinement. Red lines denote the lowest eigenvalue, whereas blue lines correspond to degenerate eigenvalues on the next energy level. Figure 9 : 9Convergence of the errors for the Poisson problem with σ = 1.0. Figure 10 : 10Convergence of the errors for the Poisson problem with σ = 0.1. Figure 11 : 11Types of elements in mesh P h with respect to the imposed enrichment. is used for the Poisson problem in h -adaptive FEM and PUM calculafor the FEM and PUM when applied to the Poisson problem; (xi) Parallel vectors, matrices and solvers for linear algebra problems in the Portable, Extensible Toolkit for Scientific Computation (PETSc)tions η 2 K := h 2 K K ∇ 2 φ + 4πρ 2 dx + h K e⊂∂K e [[∇φ · n]] 2 e da; (x) linear shape functions are used For spherically symmetric potentials one can separate eigenfunctions into radial R n,l (r) and angular Y m,l (θ, φ) parts, where the latter are spherical harmonics[27]. Here {n, l, m} are three quantum numbers. If we can find a FE space N l which contains N i and N jk , then the vector space of (10) is contained in one, built using(9) with N l . In practice one could use linear shape functions for enriched DoFs and possibly higher order shape functions for non-enriched DoFs. 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[ "A Declarative Metamorphic Testing Framework for Autonomous Driving", "A Declarative Metamorphic Testing Framework for Autonomous Driving" ]
[ "Journal Of L A T E X Class ", "Files " ]
[]
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Autonomous driving has gained much attention from both industry and academia. Currently, Deep Neural Networks (DNNs) are widely used for perception and control in autonomous driving. However, several fatal accidents caused by autonomous vehicles have raised serious safety concerns about autonomous driving models. Some recent studies have successfully used the metamorphic testing technique to detect thousands of potential issues in some popularly used autonomous driving models. However, prior study is limited to a small set of metamorphic relations, which do not reflect rich, real-world traffic scenarios and are also not customizable. This paper presents a novel declarative rule-based metamorphic testing framework called RMT. RMT provides a rule template with natural language syntax, allowing users to flexibly specify an enriched set of testing scenarios based on real-world traffic rules and domain knowledge. RMT automatically parses human-written rules to metamorphic relations using an NLP-based rule parser referring to an ontology list and generates test cases with a variety of image transformation engines. We evaluated RMT on three autonomous driving models. With an enriched set of metamorphic relations, RMT detected a significant number of abnormal model predictions that were not detected by prior work. Through a large-scale human study on Amazon Mechanical Turk, we further confirmed the authenticity of test cases generated by RMT and the validity of detected abnormal model predictions.
10.1109/tse.2022.3206427
[ "https://export.arxiv.org/pdf/2012.10672v4.pdf" ]
252,111,232
2012.10672
0b7ba6cc7a9973706f307a4846a4d12b1065e910
A Declarative Metamorphic Testing Framework for Autonomous Driving AUGUST 2015 1 Journal Of L A T E X Class Files A Declarative Metamorphic Testing Framework for Autonomous Driving 148AUGUST 2015 1Index Terms-Metamorphic testingAutonomous drivingtesting ! Autonomous driving has gained much attention from both industry and academia. Currently, Deep Neural Networks (DNNs) are widely used for perception and control in autonomous driving. However, several fatal accidents caused by autonomous vehicles have raised serious safety concerns about autonomous driving models. Some recent studies have successfully used the metamorphic testing technique to detect thousands of potential issues in some popularly used autonomous driving models. However, prior study is limited to a small set of metamorphic relations, which do not reflect rich, real-world traffic scenarios and are also not customizable. This paper presents a novel declarative rule-based metamorphic testing framework called RMT. RMT provides a rule template with natural language syntax, allowing users to flexibly specify an enriched set of testing scenarios based on real-world traffic rules and domain knowledge. RMT automatically parses human-written rules to metamorphic relations using an NLP-based rule parser referring to an ontology list and generates test cases with a variety of image transformation engines. We evaluated RMT on three autonomous driving models. With an enriched set of metamorphic relations, RMT detected a significant number of abnormal model predictions that were not detected by prior work. Through a large-scale human study on Amazon Mechanical Turk, we further confirmed the authenticity of test cases generated by RMT and the validity of detected abnormal model predictions. INTRODUCTION A UTONOMOUS driving has been through rapid development and has attracted increasing attention and investment from industry in recent years. In 2018, Waymo launched the first autonomous car service in the Phoenix metropolitan area, making one of the first steps towards commercializing autonomous vehicles [1]. Deep Neural Networks (DNNs) are widely used to solve perception and control problems in autonomous driving [2], [3]. However, recent traffic accidents caused by incorrect predictions of driving models have raised significant safety concerns. For instance, a crash caused by Tesla autopilot system led to the death of the driver, where the driving model failed to recognize the white truck against the bright sky [4]. Thus, it is crucial to detect erroneous behavior of driving models on various traffic scenarios to improve safety and robustness of autonomous driving. A common practice to test autonomous vehicles in industry is through road test [5], [6]. However, road test is quite expensive. It is difficult to cover various weather conditions, road conditions, and driving scenes. Simulationbased testing [7], [8], [9], [10], [11], [12], [13] is widely adopted to complement road test by mimicking driving scenarios in a simulated environment. However, existing stud- Deng ies have questioned the fidelity of simulation and whether simulated driving scenarios can faithfully reflect real-world scenarios [14], [15], [16], [17]. Recent work has applied metamorphic testing (MT) [18], [19] to automatically synthesize new road images for testing driving models [20], [21]. These techniques rely on predefined metamorphic relations (MRs) between model predictions of an original image and a new image transformed from it. An MR example is, the model prediction of steering angle should not change significantly after adding raindrops to a road image. • Y. However, prior works [20], [21] are limited to hardcoded MRs for generating testing cases based on a small set of affine transformations. The limited MRs are not enough to guide a comprehensive generation of test cases and the evaluation of autonomous driving systems because they cannot cover complicated and diverse driving scenarios. To our best knowledge, there is no existing work that supports generating diverse MRs based on customized rules for constructing new test cases of autonomous driving systems. Recently, the generation of MRs is researched in other application domains. For example, Blasi et al. [22] proposed a tool called MeMo to automatically generate MRs based on Javadoc comments, which contains rich information, such as the summary of implemented functions. Rahman et al. [23] also leveraged knowledge in Javadoc comments to create a dataset and then trained a text classification model to predict MRs. These work show that leveraging domain knowledge is a promising approach to reduce the effort of generating MRs. Along the same research direction, we propose to generate MRs for autonomous driving system testing based on domain knowledge from traffic rules or domain experts. For example, Table 1 shows three traffic rules from driver arXiv:2012.10672v4 [cs.SE] 23 Sep 2022 handbooks. These traffic rules describe the correct behavior of a driver when the driving environment changes, which could be converted to MRs. In addition, domain experts can describe interesting scenarios to test or modify traffic rules according to actual situations in different regions. In this way, we can create diverse MRs based on different rules to generate test cases for different driving scenarios. Therefore, we design and develop a novel declarative Rulebased Metamorphic Testing framework (RMT). RMT allows testers to define and create testing rules in natural language. Testers can create their testing rules by referring traffic rules or other domain knowledge. Then RMT leverages a NLP-based semantic parser to extract grammar dependency predicates, identify elements to change in a driving scene by matching extracted predicates with a predefined ontology list, identify transformation to be applied using predicate translation, and create corresponding MR of the testing rule. RMT then generates new road images based on the ontology elements and transformations and then validates the correctness of model predictions based on the MR. Since driving scenarios based on testing rules require sophisticated image transformations such as adding or removing objects in an image, RMT makes use of several advanced computer vision techniques such as image semantic manipulation [24] and image-to-image translation [25] to support these transformations. RMT is also extensible to import new ontologies and image generation techniques to create more testing scenarios. We evaluated RMT on three autonomous driving models that predict steering angle or driving speed. We experimented with seven testing rules to assess the usefulness of RMT. Given a dataset of 942 road images, RMT generated 3846 new images by applying the transformations induced by seven rules. Based on the metamorphic relations induced by the seven rules, RMT detected 2184, 1314, and 941 abnormal predictions of these three driving models respectively. While these many abnormal predictions are detected, one may question whether they are indeed traffic rule violations. Prior work has never answered this question but only reported the number of detected abnormal predictions. To answer this question, for the first time, we conducted a large-scale human study with 64 drivers to assess the validity of detected abnormal model predictions on Amazon Mechanical Turk [26]. The experiment shows the majority of detected abnormal model predictions are considered meaningful by human drivers. We also compared RMT with prior works [20], [21] and found that RMT is more effective to detect abnormal model predictions. In summary, this work makes the following contributions: • We, for the first time in its kind, proposed a declarative autonomous driving testing framework that allows users to flexibly specify metamorphic testing rules based on domain knowledge in natural language. An ontology was specifically defined to extract critical information from these rules. • We implemented the testing framework using ontology-based semantic parsing, image segmentation, and image translation techniques. Given the input testing rule written in natural language, RMT On the other hand, some works applied adversarial attacks [28] to generate test images that look similar to original driving images but can cause driving models make wrong predictions. In [29], DeepXplore was proposed to generate driving images that maximize the neuron coverage of the driving model under test using a optimization-based adversarial attack method. In [30], Wicker et al. proposed an adversarial attack method to add perturbations on the most vulnerable pixels in a image. Changes of such pixels would affect predictions of a CNN model for traffic sign detection. In [31], Deepbillboard was proposed to replace normal billboards with adversarial ones in driving images to test the robustness of driving models. Though adversarial attackbased methods can easily generate new driving images to expose misbehaviors of driving models, such images may not be realistic because adversarial attacks need to either directly modify images collected from cameras or modify objects such as traffic signs on road. In this work, we mainly focus on the generation of driving images that can occur in real world and may cause faults of driving models. Simulation-based Testing Besides generating driving images, several recent works create driving scenarios in simulation environments to test autonomous driving systems. In [32], Gambi et al. proposed to use NLP techniques to extract information from police reports and then reconstruct driving scenarios based on extracted information. In this study, we propose an ontology to describe traffic scenes, use NLP techniques to extract ontology elements from human defined rules, create metamorphic relations, and generate test driving images. Several works [33], [34], [35], [36] proposed to use search algorithms to generate critical driving scenarios that cause collision or deviation of the autonomous vehicle in the simulation environment. In their work, they adopted domain knowledge to design objective functions such as time to collision (TTC) to help find critical driving scenarios. In our work, we leverage domain knowledge to design MRs and new driving scenes. The MRs can also be applied in simulation environment to create driving scenarios. The main difference is to use a simulator to render driving scenarios instead of applying image generation techniques on real-world driving images. We leave this as a future work. In [36], Riccio and Tonella applied human study to assess whether generated images are recognizable from human's view. In our work, we applied human study to evaluate the authenticity of generated images and the validity of detected abnormal model predictions. Testing and Debugging of Deep Learning Models Recently, many techniques have been proposed for testing and debugging deep learning models [29], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48]. For example, Pei et al. [29] developed a white-box testing framework called DeepXplore, which optimizes neuron coverage to generate test inputs for activating previously uncovered neurons in a DL model. DeepGauge [37] extended neuron coverage and proposed multi-granularity coverage for DL models. Different from the white-box optimization-based methods, our proposed method is a black-box method to generate meaningful testing images based on traffic rules, without the knowledge such as neuron values of driving models. DeepConcolic [38] leverages concolic testing to generate adversarial examples for DL models. Lee et al. [44] proposed an adaptive neuron-selection strategy to select most vulnerable neurons in DL models to improve the testing coverage and efficiency. MODE [45] facilitates DL debugging by leveraging differential analysis to find faulty neurons in CNNs. TRADER [47] leverages trace divergence analysis and embedding regulation to debug RNNs. There is also a large body of verification techniques for deep learning models [49], [50], [51], [52], [53], [54], [55], [56], [57]. Pulina et al. [49] proposed an abstraction-refinement approach to examine the safety of a neural network. Reluplex [50] leveraged SMT solving to verify the robustness of DNNs with the ReLU activation function. Huang et al. [51] proposed a framework to find adversarial examples using SMT and exhaustive search. Metamorphic Testing Chen et al. first proposed metamorphic testing (MT) to address the oracle problem in software testing [58]. Given source test cases with corresponding outputs, MT transforms original test inputs to generate new test inputs and validates the program output on a new input based on metamorphic relations (MRs). MRs specify the relationships between the outputs of the original input and a new input. When test results violate MRs, the program has a bug. For example, suppose a program f implements the SINE function. We can define an MR, f (x) = f (x + 2π). Then we can generate a new test case x + 2π and test whether the output of the original test input x (i.e., f (x)) and the output of x + 2π (i.e., f (x + 2π)) are equal [59]. MRs are also able to reveal relationships between two outputs beyond equality. An example is for an accommodation booking system. The searching result of rooms with filtering conditions (e.g., the price of a room is in a certain range) should be a subset of that without filtering conditions. In this study, we proposed an MT testing framework that allows users to define their MRs in natural language to test autonomous driving systems. MT has been widely applied to many domains such as middleware [60], healthcare [61], and machine learning [62]. For example, Murphy et al. [63] proposed six MRs including additive, multiplicative, permutative, invertive, inclusive, and exclusive to transform input data for support vector machines. Based on this work, Xie et al. [62] further proposed MRs to test specific supervised ML models such as K-Nearest Neighbor (KNN) classifiers and Naive Bayes classifiers. In [64], Zhou et al. applied MT to test Lidar-based perception system in an autonomous driving system called Apollo. DeepTest [21] and DeepRoad [20] are representative approaches that apply MT to autonomous driving models. However, these two approaches only support equality MRs. In other words, the outputs of an original input and a new input should be the same or similar within a threshold. In addition, prior works (e.g., DeepTest [21] and Deep-Road [20]) were limited to often hardcoded rules. In this work, we propose a testing framework that allows users to specify testing rules, e.g., a car should slow down if a stop sign is added to the curbside. Then the framework automatically parses input testing rules using NLP techniques and uses the corresponding ontology to generate MRs beyond equality. Furthermore, our framework is integrated with more image transformations than DeepTest and DeepRoad to support other MRs. Generation of MRs The generation of MRs is the core task for MT. However, it is difficult and challenging to propose a general or universal MR generation method to automatically generate MRs. Prior work [20], [21], [64] proposed hardcoded MRs to guide the generation of testing cases and model evaluation. Recently, some works proposed MR generation methods for some application domains by using search-based techniques or using ML techniques to predict MRs. A tool called GAssertMRs [65] was proposed to automatically generate MRs for cyber-physical systems based on genetic programming. Blasi et al. [22] proposed MeMo to automatically generate equivalent MRs from Javadoc comments using NLP techniques. In addition, Rahman et al. [23] trained a text classification model to predict MRs based on Javadoc comments. In this study, we proposed the generation of MRs by parsing input testing rules originating from domain knowledge using NLP techniques, a proposed ontology, and predicate inference. The declarative approach of MR generation based on testing rules in natural language can support diverse and complicated testing scenarios in autonomous driving. THE RMT FRAMEWORK RMT allows users to interactively and flexibly define custom metamorphic driving scenarios for testing autonomous driving models. Figure 1 shows the architecture of RMT. It consists of three components: (1) an NLP-based semantic parser that induces metamorphic relations (MRs) from human-written test scenarios, (2) a test generator that generates new road images based on the induced MR, and (3) a validator that detects MR violations. The testing process is semi-automated. When the human-written test rules are provided, the tool can automate the process of test case generation driving model evaluation. First, a user describes a testing rule following the IFTTT (If-This-Then-That) paradigm [66] in natural language (Section 3.1). The testing rule specifies how to change a driving scenario as well as the expected change of the driving behavior. Then an NLP-based semantic parser leverages a driving scenario ontology to match and extract key information in the testing rule (Section 3.2). Then, an MR is established based on the extracted information. Based on this MR and a configuration file, the test generator invokes the corresponding image generation technique to generate new road images (Section 3.3). For each pair of the original and the newly generated images, the violation validation is processed to check whether the model predictions satisfy the MR (Section 3.4). If an MR is violated, an abnormal prediction is detected, indicating a violation of the human-written testing rule. Section 3.5 demonstrates the implemented prototype and describes how to flexibly configure and modify the driving scenario ontology and the test generator. The user of RMT needs to provide a testing rule, configurations of image transformation engines, and original test cases (i.e., road images) as input. Testing Rule Specification RMT accepts testing rules described in natural language based on the IFTTT paradigm. The IFTTT rule syntax contains one or more if-then statements. In the if clause, a user describes how to change a driving scenario. In the then clause, the user describes the expected change in driving behaviors caused by the scenario change. For example, we can specify a testing rule "If a pedestrian appears on the roadside, then the ego-vehicle should slow down at least 30%". The if clause specifies that a driving scenario can be changed by adding a pedestrian on the roadside. The then clause specifies the expected behavior change, which is the speed of the ego-vehicle should decrease at least 30%. We chose to use IFTTT paradigm because it is suitable to cover most driving scenarios in traffic rules that describes correct human behaviors when specific driving conditions are met. IFTTT paradigm may not be able to describe some complicated driving scenarios such as collisions involving multiple vehicles. However, such driving scenarios are not described in traffic rules and thus beyond the scope of this paper. NLP-based Semantic Rule Parsing RMT applies an NLP-based semantic parser to extract information in the IFTTT rule and create the corresponding MR. From the if clause, the semantic parser identifies information including the element to change in the driving scene, the transformation to be applied on the element, and how to apply the transformation (i.e., transformation parameters). From the then clause, the parser extracts the expected change of the driving behavior. Section 3.2.1 describes how RMT applies Part-of-Speech (POS) tagging and grammar dependency analysis to analyze the structure of a testing rule and build a grammar dependency graph. Section 3.2.2 describes how to identify the element to change by matching elements in the grammar dependency graph with a predefined ontology. Section 3.2.3 describes how to infer the metamorphic transformation and transformation parameters based on extracted key elements in the if clause. Section 3.2.4 describes how to infer the expected change based on the extracted key elements in the then clause. Section 3.2.5 describes how to handle dynamic scenarios with a sequence of rules. Dependency Analysis Given a testing rule, the parser first splits the rule into if and then clauses. Then, for each clause, the parser applies partof-speech (POS) tagging [67] to identify the part of speech of each word based its definition and context. The parser then uses grammar dependency analysis [68] to identify the grammatical relationships between words and generate a dependency graph. For example, given a sentence, "a pedestrian appears on the roadside", Figure 2 shows the resulting dependency graph and POS tags such as VERB and NOUN. The dependency nsubj means pedestrian is the nominal subject of appears. Table 2 shows common grammar dependencies in a sentence. Ontology-based Information Extraction Ontology [69] is a methodology to formally summarize categories and create a knowledge base to organize properties and relations in a specific domain. In this work, we define a driving scenario ontology to model elements in driving scenarios. Specifically, we define an ontology containing three categories of elements, as shown in Table 3. The first category contains elements for forming a road network such as lanes, lines, and crosswalk, and traffic infrastructure such as traffic signs, and traffic lights. The second category includes objects in a driving scenario such as pedestrians and vehicles. The third category is about weather and driving time. The ontology is inspired by the five-layer driving scene ontology proposed in [70]. We combine the first three layers in [70] to the category Road network and keep the rest two layers as two categories Object and Environment. We combine the first three layers in [70] because they are all related to road network elements including the topology and geometry of roads, traffic infrastructure on roads, and temporary manipulation of road elements. Then, we enrich each category with elements and properties. In [70], Bagschik et al. used their ontology to model hundreds of driving scenarios for German motorways, which has justified the ability of the ontology to describe diverse driving scenarios. An ontology element can contain specific properties. For example, a vehicle has type and color properties. For some ontology elements, we predefined some properties and their values as shown in Table 3. For these categories Road network, Object and Environment, ontology elements in the level-2 subcategory will be matched in the input testing rule. Given the POS tags from the previous step, the semantic parser identifies all nouns and matches them with elements in the ontology using WordNet Wu-Palmer (WUP) similarity [71]. WordNet is a lexical database of semantic relations including synonyms, hyponyms, and meronyms between words. WUP similarity is used to measure the semantic similarity between a pair of words based on WordNet database. The similarity score is between (0, 1] where 1 means two words have identical semantic meaning. Specifically, we use the WUP similarity API in a Python library Natural Language Toolkit (NLTK) [72] to calculate the similarity between each pair of extracted noun and ontology element. If the similarity score is above a threshold (0.75 in this paper), the noun is considered matched with the ontology element. The threshold is set as 0.75 based on the experiment result in a previous study [73] where the threshold of 0.75 helped achieve the optimal F1 score. In the previous example, the noun "pedestrian" is identified and matched with the ontology element pedestrian with the similarity of 1. If the extracted noun from the input rule is "person", it will also be matched with the ontology element pedestrian because the similarity between two words is 0.89, which is higher than the threshold. When an ontology element is matched, the parser further identifies its properties. The parser first checks the grammar dependency graph and extracts dependencies starting from the ontology element (i.e., the noun is the head node in the dependency graph) as predicates, where the dependency name is the predicate name. For example, for the description a black car, DET(a, car), AMOD(black, car) can be extracted but only the dependency AMOD(black, car) is kept because it shows "black" is an adjective to modify the ontology element "car". Finally, the parser matches extracted adjectives from predicates with predefined property values using WordNet similarity. In the example, "black" is matched with the property "color" of the ontology element "car". Inferring Metamorphic Transformation In this work, we define three transformations to change an ontology element in a driving scenario, including Add, Remove, and Replace. A new ontology element such as a car can be added to a traffic scene. An existing element such as a traffic sign can be removed from a driving scenario. Furthermore, an element can be replaced by another element. For example, a white car can be replaced with a black car, and the sunny day can be replaced with the rainy day. For different transformations, they have different parameters. The remove transformation only takes a target ontology element as an input parameter. The add transformation takes a target ontology element, a reference ontology element, and a position predicate, such as on and front, to describe the relative location with respect to the reference element. The replace transformation requires a target ontology element and a new element as parameters. To identify the transformation described in the if clause of a testing rule, the parser first extracts the root verb from the grammar dependency graph and matches it with the three predefined transformations using word2vec [74]. Word2vec is a method to learn word associations and convert words to vector representations. Words that are used in similar context or have similar meaning are close to each other in the vector representation space. Therefore, we can use the distance between two word vectors to measure the similarity between two words. In this work, we use this method to match extracted verb and pre-defined transformations. First, we calculate the distance between the verb and all pre-defined transformations using a pre-trained word2vec model in a Python library SpaCy [75], which can be downloaded in the link 4 . Then, we match the verb to the transformation within the shortest distance in the vector representation space. We use word2vec but not WordNet for transformation match because word2vec performs better to match verbs that implicitly present same meanings but are not synonyms in our pilot study. In the previous example, the verb "appears" is matched with Add. Thus, a proposition, TRANSFORMATION(''appears'', add), is generated accordingly. Then, the subject of this verb is extracted to produce another proposition in the form of NSUBJ(noun, verb), such as NSUBJ(''pedestrian'', ''appears'') in the previous example. The object of the verb is also extracted accordingly in the form of DOBJ(noun, verb). In some cases, a verb does not have a direct object but a prepositional phrase. In such cases, the parser extracts the transit dependency by traversing the dependency graph to find the noun in the prepositional phrase and uses the preposition as the predicate. In the previous example, ON(''roadside'', ''appears'') is extracted from the prepositional phrase "on the roadside." Table 4 describes the logic rules to infer a transformation from the generated propositions. Take the first infer rule of Add as an example. NSUBJ(n 1 , v) means the predicate NSUBJ should have been extracted from the testing rule. ONTOLOGY(n 1 ) means the noun extracted from the proposition NSUBJ(n 1 , v) should be matched as an ontology element. TRANSFORMATION(v, add) means the extracted verb should be matched as the add transformation. ON(n 2 , v) means the predicate ON should have been extracted and ONTOLOGY(n 2 ) means the noun in the proposition ON(n 2 , v) should be matched as an ontology element. When all of these conditions are met, ADD(n 1 , n 2 , on) is inferred, which means add n 1 on n 2 . More examples of testing rules that can be parsed by transformation inference rules can be found in Table 6 (e.g., "change" is inferred to the transformation Replace in Rule 7). Inferring Expected Change In this work, we consider three possible expected changes, including increase (left for steering angles), decrease (right for steering angles), and staying the same. Furthermore, a change can be described by a change modifier, such as at least and more than, or a change quantity, such as a number and a percentage. As the description of the then clause is often simple, the parser directly matches key information without performing 4. https://spacy.io/models/en Add NSUBJ(n 1 , v) ∧ ONTOLOGY(n 1 ) ∧ TRANSFORMATION(v, add) ∧ ON(n 2 , v) ∧ ONTOLOGY(n 2 ) → ADD(n 1 , n 2 , on) NSUBJ(n 1 , v) ∧ ONTOLOGY(n 1 ) ∧ TRANSFORMATION(v, add) ∧ FRONT(n 2 , v) ∧ ONTOLOGY(n 2 ) → ADD(n 1 , n 2 , front) NSUBJ(n 1 , v) ∧ ONTOLOGY(n 1 ) ∧ TRANSFORMATION(v, add) ∧ BEHIND(n 2 , v) ∧ ONTOLOGY(n 2 ) → ADD(n 1 , n 2 , behind) Remove NSUBJ(n, v) ∧ ONTOLOGY(n) ∧ TRANSFORMATION(v, remove) → REMOVE(n) Replace NSUBJ(n 1 , v) ∧ ONTOLOGY(n 1 ) ∧ TRANSFORMATION(v, replace) ∧ PREP(n 2 , v) ∧ ONTOLOGY(n 2 ) ∧ WEATHER (n 2 ) ∧ WEATHER(n 2 ) → REPLACE(n 1 , n 2 , weather) NSUBJ(n 1 , v) ∧ ONTOLOGY(n 1 ) ∧ TRANSFORMATION(v, replace) ∧ PREP(n 2 , v) ∧ ONTOLOGY(n 2 ) ∧ OBJECT (n 1 ) ∧ OBJECT(n 2 ) → REPLACE(n 1 , n 2 , object)CHANGE(decrease) → x1 > x2 CHANGE(increase) → x1 < x2 CHANGE(decrease) ∧ NEGATION(description) → x1 <= x2 CHANGE(increase) ∧ NEGATION(description) → x1 >= x2 CHANGE(decrease) ∧ (MODIFIER(at least) ∨ MODIFIER(more than)) ∧ QUANTITY (n, number) → x1 − x2 >= n CHANGE(decrease) ∧ (MODIFIER(at least) ∨ MODIFIER(more than)) ∧ QUANTITY (n, number) ∧ NEGATION(description) → x1 − x2 <= n CHANGE(increase) ∧ (MODIFIER(at least) ∨ MODIFIER(more than)) ∧ QUANTITY (n, number) → x2 − x1 >= n CHANGE(increase) ∧ (MODIFIER(at least) ∨ MODIFIER(more than)) ∧ QUANTITY (n, number) ∧ NEGATION(description) → x2 − x1 <= n CHANGE(decrease) ∧ (MODIFIER(at least) ∨ MODIFIER(more than)) ∧ QUANTITY (n, percentage) → (x1 − x2)/x1 >= n% CHANGE(decrease) ∧ (MODIFIER(at least) ∨ MODIFIER(more than)) ∧ QUANTITY (n, percentage) ∧ NEGATION(description) → (x1 − x2)/x1 <= n% CHANGE(decrease) ∧ (MODIFIER(less than) ∧ QUANTITY (n, number) → x1 − x2 <= n ∧ x1 > x2 CHANGE(decrease) ∧ (MODIFIER(less than) ∧ QUANTITY (n, number) ∧ NEGATION(description) → x1 − x2 >= n CHANGE(decrease) ∧ MODIFIER(less than) ∧ QUANTITY (n, percentage) → (x1 − x2)/x1 <= n% ∧ x1 > x2 CHANGE(decrease) ∧ MODIFIER(less than) ∧ QUANTITY (n, percentage) ∧ NEGATION(description) → (x1 − x2)/x1 >= n% CHANGE(increase) ∧ (MODIFIER(less than) ∧ QUANTITY (n, number) → (x2 − x1) <= n ∧ x2 > x1 CHANGE(increase) ∧ (MODIFIER(less than) ∧ QUANTITY (n, number) ∧ NEGATION(description) → x2 − x1 >= n CHANGE(increase) ∧ MODIFIER(less than) ∧ QUANTITY (n, percentage) → (x2 − x1)/x1 <= n% ∧ x2 > x1 CHANGE(increase) ∧ MODIFIER(less than) ∧ QUANTITY (n, percentage) ∧ NEGATION(description) → (x2 − x1)/x1 >= n% CHANGE(same) → |x1 > x2| <= δ dependency analysis. The parser first applies POS tagging and Name Entity Recognition (NER) on the then clause. Then, it matches the expected change from verbs or nouns, the change modifier (if any) from adjectives or adverbs, and the change quantity (if any) from numerals. Specifically, we define a lexicon for expected changes for each type of behavior, such as "increase", "decrease", "same", "accelerate", "slow", "deviate", etc. Then the parser matches extracted verbs with expected change lexicons using WUP similarity. In the previous example of "the vehicle should slow down", "slow" is matched with the decrease behavior. Therefore, the parser infers the expected change as decrease and generates a proposition CHANGE(decrease). To identify the change modifier, the parser first identifies adjectives and adverbs in the then clause. Then the parser checks the adjective as well as its neighbor words together and matches the phrase with a predefined lexicon of change extent including at least, more than and less than. In the example of then the ego-vehicle should slow down at least 30%, 'least" is first identified and then "at least" is found and matched with the change extent at least. Finally, a proposition, MODIFIER(at least), is generated. The parser also considers the condition of negation words such as "no", "not". When these words occur in a sentence, the meaning of described change modifier should be reversed, such as from "more than" to "no more than". Therefore, the parser matches the occurrence of negation words. When a negation word is matched, a proposition NEGATION(description) is generated. The change quantity is extracted from numeral words. The parser applies Name Entity Recognition (NER) [76] to check whether there is a number or a percentage in the then clause. In the example of then the ego-vehicle should slow down at least 30%, "30%" is found and identified as a percentage. A proposition, QUANTITY(''30'', percentage), is then generated. When all propositions are generated, the parser uses them to infer an expected change function E. Table 5 describes the logic rules to infer an expected change function from the generated propositions, where x 1 and x 2 refer to model predictions on the original image and the generated image. Take the first rule as an example. If no change modifier or change quantity propositions are generated, the expected change formula is simply defined as x 1 > x 2 or x 1 < x 2 . If a change modifier proposition and a change quantity proposition are generated, the formula is defined with respect to these two values. For example, if there are two propositions, MODIFIER(''least'', at least) and QUANTITY(''30'', percentage), then a formula, (x 1 − x 2 )/x 1 >= 30%, is generated. If the expected change is same, the formula is defined as |x 1 − x 2 | <= δ, meaning the difference between two model predictions should be smaller than a threshold δ. If a negation word occurs, the comparison operator will be reversed accordingly. After the transformation function and the expected change formula are obtained, RMT uses them to create an MR. Let the original driving image be x o , the transformation function be T and its parameters be p, the test generator be G, the driving model under test be F , and the expected change formula be E. The expected change E could be 0 or within a small range (e.g., [−0.01, 0.01]) if the MR represents an equality relation. Then the new driving image is represented as G(x o , T, p), and model predictions on two driving images are F (x o ) and F (G(x o , T, p)). The generated MR is shown as Formula 1, which describes that the change of the input driving image causes the change of the model prediction. G(x o , T, p) → E(F (x o ), F (G(x o , T, p)))(1) Testing Dynamic Scenarios with a Sequence of Rules Our framework also supports combing two or more IFTTT blocks in one rule to test dynamic scenarios. For example, an autonomous vehicle should understand the risk of hitting a pedestrian with respect to its distance to the pedestrian. Therefore, a tester may want to check whether the egovehicle slows down more when the pedestrian is closer to it. In the case, we can write a testing rule, "If a pedestrian appears on the roadside, the speed should decrease. If he gets closer to the vehicle, the speed should decrease more." This rule contains two IFTTT blocks. The first block "If a pedestrian appears on the roadside, the speed should decrease." describes the basic test scenario and the expected change is the comparison of model predictions on the original image x o and generated image g 1 . The second block describes a subsequent scenario where another image g 2 is generated based on the original image. The added pedestrian in g 2 should be closer to the egovehicle. The expected change between model predictions on g 1 and g 2 is also defined in the second block. To support such rules indicating the comparison between model predictions of two driving scenarios, the parser first applies dependency analysis and POS tagging on both two IFTTT blocks to extract key information as describe in the previous sections. From the first IFTTT block, the transformation function and expected change function are created. For the second block, if the parser cannot find the same ontology elements described in the first block, the parser uses a pronoun resolution technique [77] to match pronouns with ontology elements. In the example "If a pedestrian appears on the roadside, the speed should decrease. If he gets closer to the vehicle, the speed should decrease more.", "he" is extracted and matched with the ontology element pedestrian in the first if clause. Furthermore, the parser checks comparative adjectives or adverbs occurred in the second IFTTT block and matches them with comparative adjectives and adverbs such as closer, more, faster, and slower. The comparative adjective or adverb in the second if clause is the additional parameter in the transformation proposition. In the example "If he gets closer to the vehicle", closer is identified as a comparative adjective and it is added to the transformation proposition obtained from the first block. The new transformation propostion of the second block is thus Add(pedestrian, sidewalk, on, closer). The comparative adjective or adverb in the second then clause indicates the comparison of model predictions. In the example "the speed should decrease more", decrease and more are extracted, which infers to the expected change inequity x 2 > x 3 where x 2 is the model prediction on the generated image from the first IFTTT block and x 3 is generated from the second IFTTT block. For the rule containing two IFTTT blocks, two MRs are created and both of them should be satisfied, as shown in Formula 2. It expresses the condition to compare model predictions on two generated images. g 1 = G(x o , T, p 1 ) → E(F (x o ), F (g 1 )) g 2 = G(x o , T, p 2 ) → E(F (g 1 ), F (g 2 ))(2) Metamorphic Test Generator Given the inferred metamorphic transformation identified in the previous step, the test generator creates new driving images using three different image generation techniquesimage manipulation, image synthesis, and image-to-image translation. To make the image generation process more extensible, we parameterize the test generator and allow users to supplement new image generation techniques through a configuration file (detailed in Section 3.5). Essentially, this configuration file specifies the command to invoke a transformation technique and a mapping between the transformation technique and transformation propositions identified in the previous step. Image Manipulation We develop an image manipulation technique to directly add ontology elements in a road image. This is done through semantic label maps. A semantic label map provides image classification results in pixel level. As Figure 3 shows, image (a) is the driving scene, and image (b) is the semantic label map associating each pixel in the image (a) with a class of different color (e.g., the blue area in the semantic label map represents the sky in the original image). Based on the semantic label map, we are able to extract an object (i.e., also known as a mask in computer vision) belonging to a class from the original image, as shown in image (c). We use OpenCV [78] to identify the exact position of different objects and extract object masks as templates. In this work, we use a driving dataset with annotated semantic label maps. Semantic label maps can be automatically generated using semantic segmentation models such as DeepLab [79]. We leave the integration of such techniques as future work. By navigating through the semantic label maps in the original driving dataset, we create a gallery of object images for each ontology element. To add an ontology element into driving images, RMT first matches the ontology element with its template mask in the gallery. Then RMT adds the mask into the driving images using OpenCV. In the previous example, the ontology element is a pedestrian and the transformation parameter is roadside. Therefore, the corresponding template mask is selected and added into test images. In this step, one challenge is that for different road images, the position of roadside is very likely to be different. We solve this challenge by checking the boundary of a road using the semantic label map. More specifically, we move the mask horizontally by changing the x-axis coordinates of the mask and then check whether at the current position of the mask (i.e., coordinates in the driving image where the mask is moved to), the semantic label of the left bottom pixel is changing from road to other class' semantic label. If so, we add the mask at that position. Mathematically, let a driving image with height h and width w; let the bottom left point of the image be the coordinate origin; let an ontology element mask with height h and width w , the coordinate of the mask's bottom left point be (x, y), y + h <= h and x + w <= w; and let the semantic map of the image is M and M (p x , p y ) is the semantic class label of the point p x , p y . We identify the position of the mask as x + x , y, which satisfy conditions:      M (x + x − 1, y) = road M (x + x , y) = road x + x + w <= w(3) To add a pedestrian closer to the vehicle, we just need to firstly move the mask from (x, y) to (x, y − y ) and use above rules to identify the add position. Image Synthesis Though image manipulation technique can implement Add transformation fast with low cost, it falls short in several cases. First, the added element sometimes may look unnatural since it is extracted from another image with a different driving environment. Second, it cannot support Remove and Replace transformations. Therefore, we develop an image synthesis technique based on a generative model called Pix2pixHD [24]. RMT applies the model to remove and replace ontology elements. Pix2pixHD is a Generative Adversarial Network (GAN) [80] for image generation based on semantic label maps. Given a semantic label map of a driving scene image, Pix2pixHD synthesizes the corresponding driving scene image. For example, if the ontology element is "dashed lane lines" and the transformation is "Remove", the test generator detects pixels in the semantic label map with the class as dashed lines and then change the label of these pixels to road. The transformation effect is shown as from Figure 8e to Figure 8j Similarly, if the user wants to replace the buildings in the driving images with trees, the test generator changes the pixels of buildings in the semantic label to trees. Then the test generator invokes Pix2pixHD to generate a new driving image based on the modified semantic label map (i.e., from Figure 8c to Figure 8h). Image-to-Image Translation Even though the generative model-based image synthesis can replace objects by manipulating semantic label maps, it cannot be applied to ontology elements without specific shapes such as weather and time. Therefore, we adopt another technique image-to-image translation to solve this problem. We use an image-to-image translation GAN called UNIT [25]. The main function of UNIT is to transform images from one domain to another (e.g., changing a tiger to a lion). We integrate pre-trained models of UNIT on BDD100K [81], which is an autonomous driving dataset containing a large amount of driving images in different weather conditions and driving time. Image Filtering When generating new driving images, not all testing images are qualified to be transformed. For example, if the user wants to add a vehicle in the front of the ego-vehicle but there is already a vehicle or other objects at the same position, such images should be filtered out. Therefore, we develop a test case filtering mechanism in RMT to automatically filter invalid images. For object addition, RMT checks whether another object also exists at the position to put the mask. For object removal or replacement, RMT checks whether the size of the object is too small. If an object is very small, removing or replacing the object is unlikely to make an influence on the model prediction. Though recent research found that adversarial examples that have slightly modification on original images would make driving models produce wrong prediction, this paper aims to generate meaningful new test images (test generation) not for robustness testing. New images generated by removing or replacing small objects are rather relevant to the robustness testing, and thus not considered in this study. For weather and time replacement, RMT filters generated driving images that are similar to original driving images within the bounds of the mean squared error (MSE), since in this condition the original driving images are most possibly already in the target weather or driving time. Metamorphic Relation Validator Given a metamorphic testing set with source test cases (i.e., original driving images) and follow-up test cases (i.e., generated new driving images), the metamorphic relation validator feeds each test case into the driving model under test and obtains model predictions. Then, the validator uses the created MR to validate each prediction pair (prediction x 1 for a source test and prediction x 2 for the corresponding follow-up test). If the prediction pair violates the MR, an abnormal model prediction is detected. If the testing rule contains two IFTTT blocks, a prediction tuple (x 1 , x 2 , x 3 ) is fed to the validator and two created MRs are evaluated. Both two MRs should be satisfied simultaneously, otherwise a violation is said to be detected. Implementation We implemented a prototype of RMT in Python and Py-Torch. Figure 4 shows the main GUI of the prototype. A model tester can specify the driving rule to apply in the text field and then select the driving model and the original test set. The prototype can automatically parse the rule, create the MR, generate source and follow-up test cases, and detect violations. A separate window ( Figure 5) will pop up to show the metamorphic testing result. It first shows the number of detected violations and the total number of generated test cases (e.g., 294 violations were found out of 532 test cases). A sample of detected violations is rendered in the middle of the popup window. In this example, a pedestrian is added in the new image, while the driving speed does not decrease, thus the violation. By comparing the predictions, the tester can verify whether the MR is indeed violated. Finally, a line of text at the bottom shows the storage path of images for all detected violations. RMT uses a YAML configuration file to manage and coordinate components in the RMT framework. Specifically, the configuration file stores pre-definend ontology elements and their properties, the transformation list, and the invocation command of each image generation technique. Figure 6 shows a subsection of a configuration file. In the example, we define ontology elements including lane, building, and tree, transformations including remove and replace. We then configure a image generation model Pix2pixHD to implement transformations. For the image generation EXPERIMENTAL SETTINGS We conducted two experiments to evaluate the effectiveness of RMT and answered the following research questions: For RQ1, we proposed seven rules as shown in Table 6 to test three driving models. We then measured how many violations were detected by MRs generated from these rules. To design such rules, we read traffic rule handbooks and pick up rules that contain supportable transformations and ontology elements by RMT prototype. Rules 1, 2, and 7 are directly derived from traffic rules from official driver handbooks in three different countries by rephrasing the rules in IFTTT syntax, as shown in Table 1. These rules check whether a driving model can handle potential hazards (e.g., a pedestrian may cross the road), recognize traffic signs, and recognize different driving scenes (e.g., daytime vs. nighttime). Rules 3-6 are custom rules. Rule 3 If: a pedestrian appears on the roadside, Then: the ego-vehicle should slow down. Rule 2 If: a speed limit sign appears on the roadside, Then: the ego-vehicle should slow down. Rule 3 If: a pedestrian appears on the roadside, Then: the ego-vehicle should slow down at least 30%. Rule 4 If: a pedestrian appears on the roadside, Then: the ego-vehicle should slow down. If: he gets closer to the ego-vehicle, Then: the speed should decrease more. is a customized from Rule 1 with a specific deceleration threshold 30%, which can be set flexibily based on user expectation. We further evaluated all rules with different thresholds from 0 to 50% for speed and steering angle. Rule 4 extends the traffic rule of NSW in Table 1 to express a more complicated scenario-if the ego-vehicle is getting closer to a pedestrian, it should decelerate more. Rules 5 and 6 are not derived from traffic rules, but they demonstrate how users can specify their own interesting driving scenes by adding, removing, or replacing objects in a road image. We further designed more complicated rules based on Rules 1-7 to compare the performances of simple rules and complicated rules. We used A2D2 dataset [82] to train autonomous driving models for speed and steering angle predictions. The dataset contains 41, 227 images collected from 15 different locations and times. These images are stored in 15 folders. We used images collected in 13 folders as the training set to train autonomous driving models for steering and speed predictions, one folder as the validation set, and the last one as the test set containing 942 images. We used A2D2 in our experiment because this dataset provides rich labels including semantic label maps, speed, and steering angles needed in transformation engines and training autonomous driving models. When training models for steering angle prediction, we converted the source label "steering wheel angle" in the dataset to "steering angle" based on the steering ratio 14.4:1, because the steering wheel angle from 0 to 360 degrees corresponds to the turn of vehicle wheels (i.e., "steering angle") from 0 to 25 degrees. As A2D2 dataset does not provide the parameter of steering ratio, we followed the same setting of the dataset released by Udacity [83]. Finally, the range of steering angles is [− 25,25], where negative degrees mean that the vehicle is turning left. For Rules 1, 2, 4, and 7, the threshold was set as 0 by default. For Rule 3, the threshold was set as 30% as described in the rule. For Rules 5 and 6, the threshold was set as 1.39. In [20], the tolerance setting started from 10 degrees. However, in the previous study [20], the range of steering wheel angle is [-180, 180] while in our paper the range of steering angle is [-25, 25]. We thus scaled the value of accordingly to a small value to reflect a similar standard (1.39 10/(180/25)). Note that our experiment results do not show much differences when becomes larger. We implemented and trained three CNN-based autonomous driving models for the evaluation: Epoch [84], Resnet101 [85], , and VGG16 [86]. Epoch is a top-performing self-driving model released by Udacity. The driving model architecture is adapted from Nvidia Dave-2 [3] and achieves better performance in Udacity Challenge [87]. We chose Epoch to represent driving models including Chauffeur [88] proposed in Udacity challenge because Epoch has the simplest architecture while achieves similar performances. ResNet is a state-of-the-art CNN architecture with residual blocks as components. VGG16 is a widely used architecture for general-purpose computer vision tasks. We adapted both ResNet and VGG16 into regression driving models by replacing the classifiers in two models as a three-layer feedforward network where the last layer contains 1 neuron to predict speeds or steering angles. For these driving models, we standardized their input image size to 320×160. We chose the input size as 320×160 which has the same scale as the input dataset and does not distort the resized driving scene. For the images from A2D2, we first cropped the center part from the original size of 1920 × 1208 and then resized the images to 320 × 160. Specifically, we removed the top 248 rows of an image to make the size as 1920 and then resized it to 320 × 160. We kept most part of the original image to ensure that the added objects such as pedestrians on the roadside would not be removed if we only keep the center crop of an image. For all autonomous driving models, we used Adam Optimizer with the default learning rate 0.0001. The error rates of these driving models on the A2D2 were measured by Mean Absolute Error (MAE), which is a common metric to measure the performance of regression models. The MAEs of three driving models for steering angle prediction with Epoch, VGG16, and Resnet101 are 2.68, 2.65, and 2.73 respectively. The MAEs of three driving models for speed prediction are 3.09, 3.30, and 3.02, respectively. We also used the test set to train the transformation engine Pix2pixHD, which ensures that Pix2pixHD can generate authentic follow-up test sets that look similar to the original test set. Since we used the default network architecture and hyper-parameter setting of Pix2pixHD, it will generate transformed images with size 1024 × 512. To fit the input requirement for three driving models under test, the generated images were cropped and resized to 320 × 160. As introduced in Section 3.3, each transformation engine has a built-in filter to select images that are applicable for transformation in the original test set. For seven proposed rules, transformation engines generated seven sets of metamorphic group of test cases with 532, 425, 532, 335, 476, 604, and 942 pairs (source test cases, follow-up test cases) respectively. For RQ2 and RQ3, we published an evaluation task on a crowdsourcing platform, Amazon Mechanical Turk (mTurk) [26]. An example test in a evaluation task is shown in Figure 7. We sampled test image pairs-a source test image and a corresponding transformed test image-as well as the corresponding model predictions of VGG16. We selected VGG16 as the target model in this experiment, since RMT detected abnormal model predictions using all seven rules. We asked the workers to view the test image pairs and rate (1) whether the transformed image looks real and (2) whether the model prediction on the transformed image is reasonable based on workers' own driving experience on a 7-point Likert Scale (1 for pretty unreal/pretty unreasonable and 7 for pretty real/pretty reasonable). The choice Normal means the rater maintains neutral attitude for the authenticity of a generated image and the reasonableness of the model prediction. In RQ3, we used both MR nonviolation cases (the model prediction change is deemed as rule compliance) and MR violation cases (the model prediction change is deemed rule violation) to investigate rater's general opinion towards MRs used. We didn't disclose such information (non-violation or violation cases) to raters to get unbiased rating information. To ensure a 95% confidence level with a 5% confidence interval, we randomly sampled 345 test case pairs from generated metamorphic test sets based on proposed rules except Rule 4. We did not use the test set of Rule 4 because test cases in Rule 4 are generated by the same transformation engine as in Rule 1 but with different MRs. We split 345 test cases into 15 groups, each containing 23 test cases. We created a human intelligence task (HIT) for each group of test cases. Each task takes about 15 minutes to finish. There are 15 HITs and for each HIT, we wanted to assign 8 unique workers. All together there shall be 120 workers. Since some workers finished multiple HITs, in the end there were 64 unique workers in total for the 15 HITs. Since different drivers have different driving expertise and preference, we computed the inter-rater agreement using Krippendorff's alpha. To ensure the quality of the human evaluation, we only allow workers whose task approval rate is greater than 95% to participate. We required workers to have at least one year of driving experience. We paid each worker $0.5 US dollars after they finished each task. For RQ4, we compared RMT with prior work DeepTest and DeepRoad in two settings. In the first setting, we reproduced the common MR "If the weather changes to a rainy day, the steering angle of ego-vehicle should keep the same" supported by DeepTest and DeepRoad. We also integrated another image-to-image translation model UGATIT [89] in RMT to implement the MR. Then we compared the detected violations of three methods. For DeepTest and DeepRoad, we did not find their source code or pre-trained models to generate the raining effect. Instead, we used the public code that applies OpenCV to add rain to reproduce DeepTest. For DeepRoad, we used the same network architecture as the prior work and trained the model on the A2D2 dataset. For UGATIT, we used the same dataset to train the model. In the second setting, as RMT supports to generate more complicated scenarios by composing multiple transformations, we evaluated whether RMT can detect more violations by extending test scenarios generated by DeepTest and Deep-Road. Therefore, we first applied DeepTest and DeepRoad to generate new test scenarios in rainy weather and further applied RMT to add a pedestrian on the roadside as more complicated scenarios. For RQ5, we quantified the efficiency of RMT on generating new testing images and evaluating MR violations for different rules. For each rule, we calculated the time cost to generate the testing set and the mean cost of evaluations on three driving models. In addition, we also evaluated and compared the efficiency of RMT for generating test images from the same MR supported by DeepTest and DeepRoad. EVALUATION RESULTS Section 5.1 demonstrates how effective RMT is based on seven proposed rules. Section 5.2 presents how authentic generated test cases are and how valid are detected violations in the generated test cases from the view of human raters. A sample of Rule 5 test images is shown in Figure 9. To better present these images in the paper, we clipped the size of these images from 320 × 160 to 224 × 224. Table 7 presents experiment results of violation detected on test cases generated by seven rules. For Rules 1-4 and 7, Epoch performs worst among all five rules, with violation ratios in the range from 55.26% to 97.93%. The performances It is notable that the violation ratios increase drastically on Rule 3 compared with Rule 1. In Rule 3, we set a more strict condition "the deceleration should be at least 30%" rather than simply "the speed should decrease". Such customized rules are useful because they enable domain experts to test more strict requirements for driving models. For Rule 4 that combines two scenarios, it provides further evaluation for driving models. In Rule 1, when a driving model passes a test case, we cannot ensure whether it predicts a lower speed value because it observed a pedestrian is at the roadside. More violations are detected in Rule 4, which means some hidden erroneous behaviors that cannot be disclosed by simple rules are detected. Therefore, such rules combining multiple transformations (specified in the If statements) and MRs (specified in the Then statements) could be powerful tools to comprehensively evaluate driving models. RQ1: The capability of RMT for violation detection Evaluation on proposed rules Rules 5 and 6 aim to evaluate driving models for steering angle predictions. Epoch and Resnet101 driving models perform well. No violation is detected for Resnet101 on Rules 5 and 6 and only 1.49% violations are detected for Epoch on Rule 6. The removal of lane lines (in Rule 5) and the change of buildings (in Rule 6) do not affect the decision-making of Epoch and Resnet101. However, for VGG16, the violation ratios on Rules 5 and 6 are 15.97% and 42.38% respectively, which means VGG16 leverages features of lane lines and buildings to make predictions and is more sensitive to the change of driving environment when predicting steering angles. When considering the results of different driving models on different rules more comprehensively, more interesting insights could be obtained. When driving models are trained, from the original test set, we can only evaluate their performances by single metric MAE or MSE. When the values of MAE or MSE are similar in three models, it is intuitive to think their performances are similar. However, when applying different MTs based on different rules, we can find that their performance varies in different testing scenarios and we are able to find the optimal driving model capable of meeting specific requirements (e.g., the model should perform well in the nighttime.). In addition, by analyzing the difference, we could find out characteristics of driving models with different architectures. In our experiment, Epoch model performs worse than other two models on rules for evaluating speed prediction (expected model predictions shall change). However, Epoch model performs better on rules for evaluating steering angle prediction (expected model predictions shall keep the same). This result means Epoch tends to make predictions conservatively, which may be caused by its simple network architecture. For Resnet101, it performs well on both rules for speed and steering angle predictions. It implies that this model is complicated enough to handle both tasks by learning important features for speed and steering angle predictions. The support of setting different thresholds makes RMT effective to meet different testing requirements (e.g., different countries may have different standards for the same traffic rule) and reveal prediction patterns of driving models. For Rules 1 and 2, when the threshold is changed from 0 to 10%, the number of detected violations on three models drastically increases. The result means that the decreases of speed predictions on most of testing images generated by Rules 1 and 2 are less than 10%. With the more strict setting of the threshold, such predictions are detected as violations by RMT. For Rule 7, the result shows the same pattern that with the increase of the threshold, the number of detected violations also increase. For Rules 5 and 6, the detected number of violations on Epoch and Resnet101 keeps close to 0. The result means models' predictions of Epoch and Resnet101 on most of testing images from Rules 5 and 6 are close to predictions on original images. Evaluation on complicated rules To further investigate the effectiveness of complicated rules (e.g., Rule 4) over the simple rules (e.g. Rule 1), we propose corresponding complicated rules for Rules 2, 5, 6, and 7. Specifically, for Rule 2, we generate another new image that adds a sign closer to the autonomous vehicle and check whether the speed of the driving model decrease more (called CR 1). For Rule 5 and 6, we combine these two rules to generate testing images that remove lane lines and replace buildings with trees simultaneously and check whether the steering angle of the driving model changes (called CR 2). For Rule 7, we combine it with Rule 1 to generate testing images that contain a pedestrian on the roadside in night driving time and check whether the speed of the driving model decrease more (called CR 3). Table 9 shows detected violations of complicated rules on three driving models. Comparing CR 1 and Rule 2, the violation ratios increase significantly on three driving models, from 57.41% to 78.46% on Epoch, from 9.18% to 56.27% on VGG16, and from 14.82% to 59.48% on Resnet101 respectively. The results disclose erroneous behaviors of driving models that they cannot decelerate properly when meeting traffic signs. For CR 2, detected violations on three driving models are similar as Rules 5 and 6, which means driving models have similar behaviors for predicting steering angles when the driving environment is become more complicated. For CR 3, violations on Epoch increase about 4% and keep similar on VGG16 and Resnet101, compared with Rule 7. Overall, complicated rules that combine multiple transformations and MRs help detect more abnormal behaviors and disclose hidden problems of driving models. Result 1: RMT can detect a large amount of abnormal behaviors of driving models using customized metamorphic relations. The customized setting of threshold and the combination of multiple test scenarios are able to cover and detect hidden problems of driving models. Fig. 10: Human assessment of the image authenticity in a 7-Point Likert Scale 64 workers were recruited in mTurk to participate in the human evaluation. Among 64 workers, 1 of them did not have driving experience; 2 had less than one year of driving experience; 8 had less than three years of driving experience, and 53 have more than three years of driving experience. We filtered out workers that did not have driving experience or had less than one year of driving experience to ensure that all results are from experienced drivers. Figure 10 demonstrates the box plot of scores on image authenticity of six transformations (Rules 1 & 3 use the same transformation) in a 7-point Likert Scale. For each box, the green triangle is the mean, and the yellow line is the median. It can be seen for transformations of Rules 1-4, majority of ratings are in the range [6,7], and the mean rating is about 6. For transformations of Rules 5 and 7, the mean ratings are above 5. For the transformation of Rule 6, the mean rating is about 4.7. This relatively lower rating for Rule 6 is largely contributed by the limited capabilities of the transformation engine for this specific rule scenarios. The results show that workers think most of the transformed images are at the level between slightly real and real, which means the quality of generated test cases is accepted. RQ2: The authenticity of transformed images generated by RMT Result 2: Human raters consistently considered that transformed images are authentic. RQ3: the validity of abnormal model predictions detected by RMT olds Figure 11 shows results on the human evaluation of the reasonability of model prediction on the transformed images. In the human evaluation, we randomly sampled test cases that violated MRs and did not violate MRs. We thus report results based on the categorization of non-violation cases and violation cases. For the non-violation cases, if a rater gave rating 7, it implies that the rater was in agreement with RMT that the model prediction on the transformed image is reasonable and model does not exhibit abnormal behavior (as the underlying MR also holds), and vice versa. For the violation cases, if a rater gave rating 7, the detected violation is invalid because it implies that the rater was not in agreement with RMT that the model prediction on the transformed image exhibits abnormal behavior (as the underlying MR is violated), and vice versa. For the non-violation cases, except for Rules 5 and 6 (with median value 5), in all other rules (with median value 6), raters in general agreed with RMT that when the For the violation cases, we have an interesting observation. Except for Rules 6 and 7 (with average value 4 and 5 respectively and the third quartile of ratings in both rules are much lower than those on non-violation cases), in all other rules (with median value 6), raters seemed in disagreement with RMT that when the underlying MR violates, the model exhibits abnormal behavior. But after taking a close look at how we randomly sampled the violation and non-violation cases, we observed that for Rules 6 and 7, the violation cases have the ratios of 31.25% and 35.29% respectively in the total samples for each rule. And for other rules, the violation cases have much lower ratios in the rule samples for the human rater's evaluation (Rules 1 and 3 have only 5.36% and Rule 5 has 6.90%). Along with the last group of box plots for total samples show that the mean rating of reasonability on non-violation cases is less than that on violation cases. All these results mean that in general raters believe violations detected by RMT are likely true erroneous behaviors in the underlying autonomous driving models. To verify how consistent the ratings of different workers, we used Krippendorff's alpha to measure the agreement of rating amount multiple raters. Krippendorff's alpha is a metric generalized from other inter-rater measurements such as Fleiss' kappa [90], and it is more suitable to the small size of samples and ordinal rating than Fleiss' kappa. A score less than 0 means disagreement, and equal to 1 means perfect agreement. Therefore, we considered that the Krippendorff's alpha score should be greater than 0. We thus calculated Krippendorff's alpha score for each HIT. The agreements of authenticity (on transformation) and reasonability (on MRs) are highest at HIT 14, 0.75, and 0.60, respectively. The average scores are 0.48 and 0.37, which means the ratings of workers are fairly consistent. The result then can support the findings that test cases generated by RMT are authentic and detected violations are more likely actual erroneous behaviors of driving models. Result 3: Human raters consistently considered that MR violations as potential erroneous behaviors. Table 10 shows the experiment result under the first setting above. For the same MR, RMT can detect more abnormal behaviors of driving models using test images generated by the more advanced image generator. Table 11 shows the experiment result under the second setting. With the more complicated testing scenarios generated by RMT, more abnormal behaviors of three driving models are detected. Compared with DeepTest, the improvement on VGG16 achieves about 40%, from 18.61% to 56.58%. Compared with DeepRoad, the biggest improvement is also on VGG16 driving model, from 15.03% to 60.34%. The experiment result means that RMT can detect more abnormal behaviors of driving models by more complicated testing scenarios extended beyond prior work. RQ4: Comparison with prior work Result 4: RMT can detect more abnormal behaviors of driving models than prior work using more complicated test scenarios constructed by more advanced image generation techniques. Table 12 shows the efficiency of RMT on generating new testing images and evaluating MR violations of model predictions. For Rules 1-3 and Rule 7 that apply image manipulation and image-to-image translation, the cost of RMT on generating and filtering new test images on the testing set is about 130 seconds. For Rule 4 that needs to generate two images, the generation cost is about 280 seconds. For Rules 5 and 6 that apply a large-scale GAN model to generate new images, the image generation cost is about 195 seconds. The bottleneck is that the image generation and filtering process are applied individually on images, because the position to add objects or pixels to remove and replace are different for each image. Overall, RMT can finish the test image generation and model evaluation in less than 300 seconds in the worst case of the proposed rules. When the rule is simple and low-cost image generation method is used, the time cost of RMT is less than 150 seconds. RQ5: Efficiency of RMT For generating test images from the same MR supported by DeepTest, DeepRoad and RMT, the time costs are 70.41 seconds, 118.35 seconds, and 130.90 seconds, respectively. DeepTest requires the shortest time because it does not use any image generation model to generate images. RMT requires about 10% more time than DeepRoad because it uses a more advanced image generation model with higher resource consumption. In summary, RMT has marginally longer time in image generation, but it is still in the same order of magnitude as that of DeepTest and DeepRoad. Considering RMT's higher effectiveness in detecting abnormal behaviors, such marginally longer time is negligible. Result 5: RMT is efficient to generate test images and evaluate driving models. THREAT TO VALIDITY Internal Validity In this work, the main threats to internal validity are the completeness of proposed methods to describe driving scenes, the image generation methods and the experiment dataset. We propose to use IFTTT paradigm to describe test rules. Though the syntax of IFTTT is simple, it is suitable to cover driving scenarios in traffic rules. We propose a traffic scene ontology (Table 3), which is extended from a prior work evaluated on Germany highway driving scenes. The ontology is comprehensive to describe driving scenes derived from traffic rules. We define transformation and expected change inference rules (Tables 4 and 5) to identify image transformations and MRs. Though the transformation inference rules may not cover all possible conditions, they are powerful enough to support most of change of traffic scenes based on driving images and the relations between driving model predictions. The expected change inference rules covered all possible MRs for steering and speed predictions. In this work, we aim to generate high-quality driving images and apply different image generation techniques. The authenticity of images is important for driving models to make reasonable predictions. To guarantee the meaningfulness of generated images, we implement an image filtering to remove unreasonable images (e.g., a pedestrian stands on a wall). We also apply GAN-based image generation, which has been used in prior works including DeepRoad [20] to generate authentic driving images. A problem of GAN is that it is difficult to train and thus generates low quality images. To mitigate this problem, we adopt the state-of-theart GAN architectures and augment training datasets. We use A2D2 dataset to evaluate the proposed method. Though the size of the test set (942 images) is not large, it contains traffic scenes under various conditions, including high way, rural area, and urban area. All proposed testing rules can be evaluated on the dataset. Therefore, the dataset is sufficient to evaluate abnormal behaviors that may imply faults in driving models. External Validity The threats to external validity are regarding to generalization of the proposed framework RMT on testing autonomous driving systems. This work focuses on the testing of image-based driving models. We select a top-performance driving model Epoch from Udacity Challenge contest and adapt two well-known and complicated classification networks VGG16 and ResNet101 to regression driving models. Thanks to the extensiveness of RMT, it is straightforward to add more test generation techniques, including those based on driving videos or simulations. It is thus possible to generate more complicated driving scenarios for the testing of large-scale driving systems such as Apollo [91], which is our future work. Construct Validity The threats to construct validity are related to the evaluation of driving models and the comparison with prior works. In the present work, an abnormal behavior of a driving model is considered to be detected when an MR is violated. However, in the context of autonomous driving, the abnormal behavior only implies the potential fault in a driving model. We applied qualitative user study to mitigate this issue. If the driving model's behavior is not agreed by experienced human drivers, the driving model possibly contains faults and prone to take dangerous actions. RMT supports a diverse set of MRs based on traffic rules and is not restrictive for equality-based MRs, while existing techniques such as DeepTest and DeepRoad only support equality relations. Hence, in Section 5.4, we are not arguing that this comparison is a head-to-head, fair comparison because the declarative language in RMT can express a superset of what can be expressed by DeepTest and DeepRoad. We are simply assessing how much additional benefit can be provided by this improvement in MR expressiveness and extensibility. RMT also implements the affine transformations in DeepTest and the GAN-based transformations in DeepRoad. Therefore, RMT should be considered as a more general framework compared to prior work. CONCLUSION In this paper, we proposed RMT to test the robustness of autonomous driving models. RMT allows users to define testing rules in natural language leveraging domain knowledge. RMT then implements an NLP-based rule parser for understanding testing rules, generates MRs, creates new testing images, and follow-up test sets to detect erroneous behaviors of autonomous driving models. The declarative way of defining testing rules makes RMT more flexible to generate different MRs and support more driving scenarios for testing. The complete testing process is semi-automated, with the only manual work being the user's provision of testing rules. We built a prototype of RMT and evaluated it with seven rules. The evaluation results showed that RMT effectively detected a large number of potential erroneous behaviors in three DNN-based autonomous driving models. Our qualitative study on Amazon Mechanical Turk involved experienced drivers confirmed the authenticity of generated driving scenarios and the truthfulness of detected erroneous behaviors of driving models. In addition, by comparing results of the driving model on different rule-based MT, we further evaluated performances of the model under different scenarios, which cannot be done by common testing metrics for deep learning model such as MSE. We also analyzed learned patterns of the driving model to interpret how it makes predictions. The first prototype version of RMT is only used to test DNN-based driving models for predicting speed and steering angle on static road images. Currently, RMT does not support expressing MRs in complex driving scenarios such as vehicle overtaking, lane merging, and parallel parking. Furthermore, industry-scale autonomous driving systems, such as Baidu Apollo, consist of multiple DNNbased modules that take multi-modal input data collected from multiple kinds of sensors such as camera, Radar, and Lidar. In future work, we will extend RMT to test more complex, multi-model driving systems with multi-modality sensor data. In our random sampling process for RQ3, we did not implement sample balance between violation and non-violation cases leading to much less violation cases in our study. In future work, we will also implement sample balance to explore further the correlation between MRs violations and models faults. In the future, we will extend RMT to support more dynamic driving scenarios and multiple modality sensor data to evaluate more complicated autonomous driving systems such as Baidu Apollo and Autoware. We will also explore the possibility to use generated test cases for improving the training of driving models. Fig. 1 : 1An overview of RMT Fig. 2 : 2An example of dependency parsing Fig. 3 : 3An example of image segmentation model, the attribution entry describes how to call the model and the attribution support transformations defines what transformation with its specific ontology parameters can be implemented by the generation model. Such setting allows flexible extension of RMT. For instance, we can enrich the ontology list by adding new elements and their parameters into the configuration file. If we implement a new image generation model, we can add it into the configuration file as well and specify its supportable transformations and parameters. Fig. 4 : 4The main interface for specifying traffic rules in RMTFig. 5: A sample of detected traffic rule violations for user inspection. A pedestrian is added on the roadside while the ego-vehicle does not slow down. Rule 5 5If: lane lines are removed from the road, Then: the steering angle of ego-vehicle should keep the same. Rule 6 If: the buildings are replaced with trees, Then: the steering angle of ego-vehicle should keep the same. Rule 7 If: the driving time changes into night, Then: the ego-vehicle should slow down. Fig. 7 : 7An example test in a task on Amazon Mechanical Turk Fig. 8 :Fig. 9 : 89Examples of source images and transformed images for Rules 1Examples of source images and transformed images for Rule 4 of VGG16 and Resnet101 are close. The violation ratios of VGG16 are lowest on Rules 1 and 2 and Resnet101 achieves the lowest violation ratios on Rules 4 and 7. Violation ratios of three driving models on Rule 3 are similar and all above 95%. Epoch driving model performs not well for scenarios where there are pedestrians or traffic signs, or the driving scene is in the nighttime. On the other hand, VGG16 and Resnet101 perform better. Fig. 11 : 11Human assessment of the reasonability of new prediction in a 7-Point Likert Scale. , X. Zheng, G. Lou are with the Department of Computing, Macquarie University, Sydney, NSW.E-mail: [email protected], [email protected], [email protected] • T. Zhang is with the Department of Computer Science, Purdue University, West Lafayette, IN.E-mail: [email protected] • H. Liu, T.Y. Chen are with the School of Software and Electrical Engineering, Swinburne University of Technology, Melbourne, VIC.E-mail: [hliu, tychen]@swin.edu.au • M. Kim is with the Department of Computer Science, University of California, Los Angeles, Los Angeles, CA. E-mail: [email protected] received April 19, 2005; revised August 26, 2015. TABLE 1 : 1Traffic rule examplesCountry/District Traffic rules NSW, Australia 1 When you see potential hazards, slow down and prepare to stop, for example when pedestrians are close to the road or when other vehicles may turn in front of you. California, USA 2 A 3-sided red YIELD sign indicates that you must slow down and be ready to stop, if necessary, to let any vehicle, bicyclist, or pedestrian pass before you proceed. Germany 3 Drive more slowly at night because you cannot see as far ahead and you will have less time to stop for a hazard. maximizes the automation for the testing and eval- uation of autonomous driving systems, including the rule parsing, MR creation, and follow-up test case generation. The link of RMT prototype is https: //github.com/ITSEG-MQ/RMT-TSE. • We evaluated our framework on three autonomous driving models and demonstrated that our frame- work is capable of detecting a significant number of abnormal model predictions. We also conducted the first large-scale human study with 64 workers on Amazon Mechanical Turk to evaluate the validity of detected abnormal model predictions. 2 RELATED WORK 2.1 Testing Autonomous Driving Systems 2.1.1 Image-based Testing Many works focused on generating test cases based on real-world driving images for testing autonomous driving models. DeepTest [21] applied affine transformations such as rotation and blurring to generate test images. Deep- Road [20] applied a generative adversarial network (GAN) to create driving images in snowy or rainy weathers. In [27], Dreossi et al. proposed to insert vehicles with different sizes into driving images to test CNN models for detect- ing and classifying vehicles. In this work, we propose to use three kinds of image generation techniques including image manipulation, image synnthesis, and image-to-image translation to construct new test images for different driving scenes that are derived from traffic rules. TABLE 2 : 2Common grammar dependencies in a sentenceGrammar Dependency Description NSUBJ Nominal subject DOBJ Direct object NMOD Nominal modifier AMOD Adjectival modifier ADVMOD Adverb modifier NPADMOD Noun phrase as adverbial modifier DET Determiner TABLE 3 : 3The Ontology of Traffic ScenesCategory Level-1 Subcategory Level-2 Subcategory Properties Road network Road part Lane direction: [forward, reverse] orientation: [vertical, horizontal] position: [left, right] Line Type: [solid, dash] Color: [white, yellow] Crosswalk Orientation: [vertical, horizontal] Sidewalk - Traffic infrastructure Traffic sign Type: [stop, speed limit, turn] Shape: [circle, square] Traffic light Color: [red, yellow, green] Stop and yield line - Object Static object Tree - Building - Dynamic object Pedestrian - Vehicle Type: [car, truck, van, school bus] Color: [white, black, blue, ...] Bicyclist - Environment Weather rainy, cloudy, snowy Level: [light, normal, heavy] Time day, night - NSUBJ PREP appears (VERB) DET pedestrian (NOUN) a (DET) POBJ on (ADP) DET roadside (NOUN) the (DET) TABLE 4 : 4Transformation Inference Rules TABLE 5 : 5Expected Change Inference Rules TABLE 6 : 6Proposed rulesRule 1 TABLE 7 : 7# violations (ratio %) detected by seven rulesEpoch VGG16 Resnet101 Rule 1 294 (55.26%) 46 (8.65%) 68 (12.78%) Rule 2 244 (57.41%) 39 (9.18%) 63 (14.82%) Rule 3 521 (97.93%) 509 (95.68%) 510 (95.86%) Rule 4 239 (71.34%) 115 (34.33%) 74 (22.09%) Rule 5 0 (0%) 76 (15.97%) 0 (0%) Rule 6 9 (1.49%) 256 (42.38%) 0 (0%) Rule 7 877 (93.10%) 273 (28.98%) 226 (23.99%) TABLE 8 : 8# violations (ratio %) detected by rules with different thresholds Threshold Rule Model 0 10% 20% 30% 40% 50% Epoch 55.26% 96.80% 97.74% 97.93% 98.50% 98.68% VGG16 86.47% 88.72% 93.42% 95.68% 96.80% 97.18% Rule 1 Resnet101 12.78% 88.91% 94.92% 95.86% 96.62% 96.80% Epoch 57.41% 97.65% 99.06% 99.29% 99.53% 99.76% VGG16 9.18% 89.18% 94.12% 96.47% 98.11% 98.59% Rule 2 Resnet101 14.82% 94.82% 96.47% 96.94% 97.18% 97.65% Epoch 0 0 0 0 0 0 VGG16 51.89% 15.97% 4.83% 1.68% 0.42% 0.21% Rule 5 Resnet101 0.21% 0 0 0 0 0 Epoch 4.30% 1.49% 0.50% 0 0 0 VGG16 67.88% 42.38% 25.83% 14.90% 9.11% 5.46% Rule 6 Resnet101 2.81% 0 0 0 0 0 Epoch 93.10% 99.15% 99.36% 99.36% 99.47% 99.58% VGG16 28.98% 68.79% 86.84% 92.46% 94.28% 95.86% Rule 7 Resnet101 23.99% 26.75% 43.52% 62.42% 73.14% 78.56% 5.1.2 Evaluation on rules with different thresholds Table 8 8shows the experiment results of applying different thresholds on rules 1, 2, 5, 6, and 7. The setting of thresholds is similar as Rule 4. For Rules 1, 2, and 7, the threshold 0 means that the driving model should slow down and the thresholds 10% to 50% mean that the driving model should slow down at least 10% to 50% respectively. For Rules 5 and 6, the threshold 0 means that the driving model should keep the same steering angle and other thresholds mean that the driving model should deviate at least 10% to 50% respectively. TABLE 9 : 9# violations (ratio %) detected by complicated rules Epoch VGG16 Resnet101 CR 1 244 (78.46%) 175 (56.27%) 185 (59.48%) CR 2 0 (0%) 112 (42.91%) 0 (0%) CR 3 516 (96.99%) 143 (26.88%) 132 (24.81%) TABLE 10 : 10Comparison with DeepTest and DeepRoad for detected # violations (ratio %) under setting I underlying MR holds, the model does not exhibit abnormal behavior. For Rules 5 and 6, the average ratings are 4.97 and 4.80 respectively. Since both rules are steering angle basoldsed, raters might be quite conservative in determining the reasonability of model's prediction.DeepTest DeepRoad RMT Epoch 2 (0.21%) 11 (1.17%) 20 (2.12%) VGG16 501 (53.18%) 541 (57.43%) 546 (57.96%) Resnet101 0 (0%) 1 (0.11%) 2 (0.21%) TABLE 11 : 11Comparison with DeepTest and DeepRoad for detected # violations (ratio %) under setting IIRain added (DeepTest) Rain (DeepTest) and person added Rain added (DeepRoad) Rain (DeepRoad) and person added Epoch 325 (61.09%) 429 (80.64%) 355 (66.73%) 452 (84.96%) VGG16 99 (18.61%) 301 (56.58%) 80 (15.03%) 321 (60.34%) Resnet101 199 (37.41%) 337 (63.35%) 284 (53.38%) 366 (68.80%) TABLE 12 : 12Efficiency of RMT for rulesImage generation cost (s) Evaluation cost (s) Total cost (s) Rules 1 & 3 137.50 5.58 143.08 Rule 2 131.30 4.80 137.10 Rule 4 279.17 4.80 283.97 Rule 5 196.67 4.78 201.45 Rule 6 195.07 6.08 201.15 Rule 7 126.74 4.37 131.11 ACKNOWLEDGMENTThe authors would like to thank the anonymous reviewers for the valuable comments to improve this work. This work is in part supported by an Australian Research Council Project (DP210102447), and an ARC Linkage Project (LP190100676).Tianyi Zhang is a Tenure-Track Assistant Professor in Computer Science at Purdue University. Prior to that, he was a Postdoctoral Fellow at Harvard University. He worked to build interactive systems that help domain experts explore and make sense of large collections of complex data, e.g., health records and code corpora. He received his Ph.D. from UCLA in 2019. His research is in the intersection of Human-Computer Interaction, Software Engineering, and AI. Huai Liu received the BEng degree in physioelectronic technology, the MEng degree in communications and information systems, both from Nankai University, China, and the PhD degree in software engineering from the Swinburne University of Technology, Australia. He is a senior lecturer in the Department of Computing Technologies, Swinburne University of Technology, Melbourne, Australia. 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K He, X Zhang, S Ren, J Sun, proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionK. He, X. Zhang, S. Ren, and J. Sun, "Deep residual learning for image recognition," in proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2016, pp. 770-778. Very deep convolutional networks for large-scale image recognition. K Simonyan, A Zisserman, proceedings of International Conference on Learning Representations. International Conference on Learning RepresentationsK. Simonyan and A. Zisserman, "Very deep convolutional net- works for large-scale image recognition," proceedings of Interna- tional Conference on Learning Representations, 2015. Udacity challenge 2: Steering angle prediction. Udacity, Udacity, "Udacity challenge 2: Steering angle prediction," https: //bit.ly/2E3vWyo, 2017. chauffeur. mjshiggins, "chauffeur," https://shorturl.at/gjqvM, 2017. U-gat-it: Unsupervised generative attentional networks with adaptive layer-instance normalization for image-to-image translation. J Kim, M Kim, H Kang, K Lee, arXiv:1907.10830arXiv preprintJ. Kim, M. Kim, H. Kang, and K. Lee, "U-gat-it: Unsupervised generative attentional networks with adaptive layer-instance normalization for image-to-image translation," arXiv preprint arXiv:1907.10830, 2019. The kappa statistic in reliability studies: use, interpretation, and sample size requirements. J Sim, C C Wright, Physical Therapy. 853J. Sim and C. C. Wright, "The kappa statistic in reliability studies: use, interpretation, and sample size requirements," Physical Ther- apy, vol. 85, no. 3, pp. 257-268, 2005. . &quot; Apolloauto, Apollo, 2021ApolloAuto, "Apollo," https://bit.ly/2E3vWyo, 2021.
[ "https://github.com/explosion/spaCy," ]
[ "Closed-book Question Generation via Contrastive Learning", "Closed-book Question Generation via Contrastive Learning" ]
[ "Xiangjue Dong \nTexas A&M University\n\n", "Jiaying Lu [email protected] \nEmory University\n\n", "Jianling Wang \nTexas A&M University\n\n", "James Caverlee [email protected] \nTexas A&M University\n\n" ]
[ "Texas A&M University\n", "Emory University\n", "Texas A&M University\n", "Texas A&M University\n" ]
[ "Proceedings of the 17th Conference of the European Chapter of the Association for Computational Linguistics" ]
Question Generation (QG) is a fundamental NLP task for many downstream applications.Recent studies on open-book QG, where supportive answer-context pairs are provided to models, have achieved promising progress. However, generating natural questions under a more practical closed-book setting that lacks these supporting documents still remains a challenge. In this work, we propose a new QG model for this closed-book setting that is designed to better understand the semantics of long-form abstractive answers and store more information in its parameters through contrastive learning and an answer reconstruction module. Through experiments, we validate the proposed QG model on both public datasets and a new WikiCQA dataset. Empirical results show that the proposed QG model outperforms baselines in both automatic evaluation and human evaluation. In addition, we show how to leverage the proposed model to improve existing question-answering systems. These results further indicate the effectiveness of our QG model for enhancing closed-book questionanswering tasks.
10.48550/arxiv.2210.06781
[ "https://www.aclanthology.org/2023.eacl-main.230.pdf" ]
252,873,463
2210.06781
89f96a9d98417c618b27a478ddf675e23567349f
Closed-book Question Generation via Contrastive Learning May 2-6, 2023 Xiangjue Dong Texas A&M University Jiaying Lu [email protected] Emory University Jianling Wang Texas A&M University James Caverlee [email protected] Texas A&M University Closed-book Question Generation via Contrastive Learning Proceedings of the 17th Conference of the European Chapter of the Association for Computational Linguistics the 17th Conference of the European Chapter of the Association for Computational LinguisticsMay 2-6, 2023 Question Generation (QG) is a fundamental NLP task for many downstream applications.Recent studies on open-book QG, where supportive answer-context pairs are provided to models, have achieved promising progress. However, generating natural questions under a more practical closed-book setting that lacks these supporting documents still remains a challenge. In this work, we propose a new QG model for this closed-book setting that is designed to better understand the semantics of long-form abstractive answers and store more information in its parameters through contrastive learning and an answer reconstruction module. Through experiments, we validate the proposed QG model on both public datasets and a new WikiCQA dataset. Empirical results show that the proposed QG model outperforms baselines in both automatic evaluation and human evaluation. In addition, we show how to leverage the proposed model to improve existing question-answering systems. These results further indicate the effectiveness of our QG model for enhancing closed-book questionanswering tasks. Introduction Question Generation (QG) has a wide range of applications, such as generating questions for exams (Jia et al., 2021;Lelkes et al., 2021;Dugan et al., 2022) or children's story books (Zhao et al., 2022;Yao et al., 2022), recommending questions for users in a dialogue system (Shukla et al., 2019;Laban et al., 2020), improving visual (Li et al., 2018;Lu et al., 2022) or textual question-answering tasks (Duan et al., 2017;Lewis et al., 2019a;Zhang and Bansal, 2019;Sultan et al., 2020;Lyu et al., 2021), asking clarification questions (Rao and Daumé III, 2019;Ren et al., 2021), and generating queries for SQL or multimodal documents (Kim et al., 2021). * Equal Contribution Previous works on QG are mainly under the openbook setting, which aims to generate questions based on factoid or human-generated short answers under the assumption that there is access to external knowledge like retrieved documents or passages (Du et al., 2017;Zhao et al., 2018;Kim et al., 2019;Fei et al., 2021). After demonstrated that feeding a large pre-trained model input questions alone without any external knowledge can lead to competitive results with retrieval-based methods on open-domain question-answering benchmarks, there is an increasing interest in the closed-book setting. This closed-book setting is appealing in practice and can be widely applied, e.g., in question suggestion (Laban et al., 2020;, query recommendation (Kim et al., 2021), and other practical settings where extensive external knowledge is unavailable. However, generating questions without access to such external knowledge is challenging for two key reasons. First, without access to retrieved documents (or passages), simple open-domain strategies like basing the answers on these documents (or passages) are not possible under the closed-book setting. Instead, models must rely on the answers alone. Second, the data used by most of the closedbook works (Lewis et al., 2021; are variants of existing open-domain datasets, e.g., SQuAD (Rajpurkar et al., 2018), TriviaQA (Joshi et al., 2017), WebQuestions (Berant et al., 2013) that ignore the answer-related passages. These answers in open-book works are usually short, e.g., entities, and easier to be remembered by the language model and stored in the parameters of the model than long-form answers. Thus, this leads to our motivating research question -How can we empower a QG model to better understand the semantics of long-form abstractive answers and store more information in its parameters? To tackle the aforementioned challenges existing in the closed-book setting, this paper proposes a new QG model with two unique characteristics: (i) a contrastive learning loss designed to better understand the semantics of the answers and the semantic relationship between answers and ground-truth questions at a contextual-level; and (ii) an answer reconstruction loss designed to measure the answerability of the generated question. Contrastive learning has shown promising results in many NLP tasks, e.g., (Giorgi et al., 2021;Gao et al., 2021; and aligns positive pairs better with available supervised signals (Gao et al., 2021); here we show how to learn question representations by distinguishing features of correct question-answer pairs from features of incorrectly linked question-answer pairs. Further, to ensure the generated questions are of good quality and can be answered by the answer that is used for question generation, we frame the model as a generationreconstruction process (Cao et al., 2019;Zhu et al., 2020), by predicting the original answers given the generated questions by a pre-trained seq2seq model. In addition, we introduce a new closedbook dataset with long-form abstractive answers -WikiCQA -to complement existing datasets like GooAQ (Khashabi et al., 2021) and ELI5 (Fan et al., 2019) and show how to leverage our model to generate synthetic data to improve closed-book question-answering tasks. Through experiments, we find that the proposed QG model shows improvement through both automatic and human evaluation metrics on WikiCQA and two public datasets. Compared to the baseline, the proposed QG framework shows an improvement of up to 2.0%, 2.7%, and 1.8% on the ROUGE-L score on WikiCQA, GooAQ-S, and ELI5, respectively, and 1.3% and 2.6% in terms of relevance and correctness. Furthermore, we leverage the QG framework to generate synthetic QA data from WikiHow summary data and pre-train a closed-book QA model on it in both an unsupervised and semi-supervised setting. The performance is evaluated on both seen (WikiCQA) and unseen (GooAQ-S, ELI5) datasets. We find consistent improvements across these datasets, indicating the QG model's effectiveness in enhancing closedbook question-answering tasks. In conclusion, our contributions can be summarized as follows: • We propose a contrastive QG model, which to our knowledge is the first work to explore con-trastive learning for QG under a closed-book setting. • The proposed model outperforms baselines on three datasets. The human evaluation also indicates that the questions generated by our model are more informative compared to other baselines. • We leverage the QG model as a data augmentation strategy to generate large-scale QA pairs. Consistent improvements shown on both seen datasets and unseen datasets indicate the QG model's effectiveness in enhancing closed-book question-answering tasks. Related Work Many previous works on QG are under the openbook setting, which takes factoid short answers (Rajpurkar et al., 2016) or human-generated short answers (Kočiský et al., 2018) with the corresponding passages to generate questions . Early approaches for question generation rely on rule-based methods (Labutov et al., 2015;Khullar et al., 2018). To bypass handcrafted rules and sophisticated pipelines in QG, Du et al. (2017) introduce a vanilla RNN-based sequence-to-sequence approach with an attention mechanism. The recently proposed pre-trained transformer-based frameworks (Lewis et al., 2020; also improve the performance of QG. In addition, Sultan et al. (2020) shows that the lexical and factual diversity of QG provides better QA training. However, their success can not directly adapt to the closed-book setting, where the model is supposed to generate questions solely relying on answers. In this work, we explore the widely applicable closed-book QG setting, which is still under-explored. Contrastive Learning aims to pull semantically similar neighbors close and push non-neighbors apart. It has achieved great success under both supervised and unsupervised settings. In pioneer works, the contrastive loss function (Hadsell et al., 2006;Chopra et al., 2005) has been proposed as a training objective in deep metric learning considering both similar and dissimilar pairs. Recently, proposes the SimCLR framework to learn useful visual representations. Viewing contrastive learning as dictionary look-up, He et al. (2020) present Momentum Contrast (MoCo) to build dynamic dictionaries for contrastive learning. Some works apply contrastive learning into the NLP domain to learn better sentence representations (Giorgi et al., 2021;Gao et al., 2021). In addition, contrastive learning has been applied in multilingual neural machine translation (Pan et al., 2021), abstractive summarization , and multi-document question generation (Cho et al., 2021). The recent most relevant work is , where they design two contrastive losses for paraphrase generation. In this work, we adopt contrastive learning for improving representation learning in question generation under a closed-book setting. Proposed Approach To answer our research question -How can we empower a QG model to better understand the semantics of long-form abstractive answers and store more information in its parameters? -we propose a closed-book QG model, which generates questions directly without access to external knowledge. Formally, given an answer sentence x, a closed-book QG engine generates a natural question y. Figure 1 illustrates an overview of the proposed QG framework, which consists of three parts: question generation, contrastive learning, and answer reconstruction. The framework is optimized with the joint losses from these three parts simultaneously. Question Generation We first focus on question generation through a sequence-to-sequence architecture which consists of an encoder and a decoder (Sutskever et al., 2014;Vaswani et al., 2017). The encoder takes an input sequence of source words x = (x 1 , x 2 , . . . , x n ) and maps it to a sequence of continuous representations z = (z 1 , z 2 , . . . , z n ). Then, the decoder takes z and generates a sequence of target words y = (y 1 , y 2 , . . . , y m ) at a time. The closed-book QG task is defined as findingŷ: y = arg max y P (y|x),(1) where P (y|x) is the conditional likelihood of the predicted question sequence y given answer x. P (y|x) = T i=1 p(y t |y <t , x),(2) Given the answer-question pairs, the training objective of the generation part in the proposed framework is to minimize the Negative Log-Likelihood (NLL) of the training data, L qg = − N i=1 log p(q i |A),(3) where q i is the i-th token in the generated question and A is the answer. A naive question generation model will generate questions based on answers but lacks a rich model of the semantics of answers nor can it guarantee the generated questions have a semantic relationship with the answers. Intuitively, an encoded answer should be similar to its question and dissimilar to others. In addition, the generated question should be able to be answered by the answers. Hence, this motivates the following contrastive learning and answer reconstruction modules. Contrastive Learning Contrastive learning aims to pull positive pairs and push apart negative pairs to learn effective representations. Further, the supervised signals can produce better sentence embeddings by improving alignment between positive pairs . An effective QG model should be able to understand the semantics of the answers and the semantic relationship with the ground-truth questions. Especially, the encoded answer should have semantic similarity with its ground-truth question and dissimilarity with other questions. Thus, aiming to learn a similarity function that pulls the distance between the answer sequence representation and its ground-truth question sequence representation closer, we design a contrastive loss in the representation space. Specifically, given a positive pair S = {(x i , y i )} n i=1 , where x i and y i are semantically related inputs, the other 2(n − 1) examples within a mini-batch are treated as negative examples. The training objective for (x i , y i ) is: L cl = − log exp(sim(z x i , z y i )/τ cl ) 2n i=1 exp(sim(z x i , z y i )/τ cl ) ,(4) where z x i and z y i is the representation of input x i and y i , sim(z i , z j ) = z ⊤ i z j / ∥z i ∥ ∥z j ∥ denotes cosine similarity, and τ cl is a temperature parameter. In this work, aiming to learn better answer representations and force the encoder to drive representations of correct question-answer pairs closer than representations of incorrect question-answer pairs, we take the ground-truth question as the positive instance of an answer and fine-tune the model parameters based on the contrastive loss function (Eq. 4), where z is the embedding of the special token [CLS] from the transformer encoder, representing the meaning of the entire sentence. Answer Reconstruction The questions that are generated by the model should be of good quality and should also be able to be answered by the answer that is used for question generation. To measure the answerability of the generated question, we design an answer reconstruction module, which uses a pre-trained seq2seq model to predict the original answer given the generated question. The loss is calculated by a negative log-likelihood loss function: L ar = − N i=1 log p(a i |Q)(5) where a i is the i-th token in the answer and Q is the generated question. A major challenge is that the generated questions from Section 3.1 are not differentiable. Gradients cannot be back-propagated directly. To solve this challenge, we employ the Straight-Through (ST) Gumbel-Softmax for gradient computation (Jang et al., 2017). The ST Gumbel-Softmax is a discrete version of the Gumbel-Softmax and takes different forward and backward paths. In the forward pass, the embedding is discretized by using the argmax, whereas in the backward pass the gradients are computed by the gumbel-softmax (Qader et al., 2019;Lu et al., 2021). y i = exp((log(p i ) + g i )/τ gs ) |V | j=1 exp((log(p j ) + g j )/τ gs ) ,(6) where τ gs is a temperature parameter and g i is the Gumbel noise drawn from a uniform distribution (0, 1). In this work, the one-hot embedding from ST Gumbel-Softmax is multiplied with the vocabulary embedding and then fed into the encoder of the pre-trained seq2seq model as the representation of the generated question. Overall Loss Function As a result, all three losses are summed together to provide the overall loss function L as follows: L = λ 1 L qg + λ 2 L cl + λ 3 L ar ,(7) The weights λ 1 ,λ 2 , and λ 3 are tuneable hyperparameters to balance losses and the final objective is to minimize the overall loss. The overall structure of the proposed QG framework is presented in Algorithm 1. Algorithm 1: QG framework. Input: Pre-trained language model p(q|a), answer reconstruction model p(a|q) and answer-question pairs Output: Question generator p(q|a) 1 for i ← 1 to Epoch do 2q = p(q|a) 3 Compute L qg via Eq. 3 /* contrastive learning */ 4 a i = Encoder([CLS] ⊕ a)[:, 0 :] 5 a + i = Encoder([CLS] ⊕ q)[:, 0 :] 6 Get L cl to (a i , a + i ) via Eq. 4 /* answer reconstruction */ 7â = p(a|q) 8 Compute L ar via Eq. 5 9 Calculate total loss L via Eq. 7 10 Update generator p(q|a) with L 11 end 12 return question generator p(q|a) Dataset We conduct the experiments on two public datasets -GooAQ-S (Khashabi et al., 2021) and ELI5 (Fan et al., 2019) -and a new dataset we curate called WikiCQA. GooAQ-S is a sub-sampled dataset from GooAQ which contains three different sub-tasks: short, snippet (i.e., multi-sentence description), and collection response questions (Khashabi et al., 2021). Under the long-form closed-book setting, we adopt the snippet part (i.e., questions with snippet answers) for our experiments as in the original paper. Furthermore, from the paper, the model performance on the snippet task does not vary much when supervised with 200K and 2M training instances. Thus, to improve the experimental efficiency, we take 200k instances from the snippet set and split them based on their original scripts. 1 ELI5 (Fan et al., 2019) is a widely-used large-scale corpus for long-form question-answering with supporting web documents. In this work, we use the question-answer pairs from the dataset and ignore the supporting context to fit the closed-book setting like (Khashabi et al., 2021). We follow the data split from huggingface. 2 WikiCQA is a new closed-book long-form QA dataset (Section 4.2) introduced here. It contains 20,202 question-answer pairs and is collected from a wiki-style website to complement existing datasets. We shuffle the data and split it to train, dev, and test sets by the ratio of 80%/10%/10%. Table 1 shows detailed data statistics and splits of these three datasets. WikiCQA WikiCQA contains real user QA pairs collected from WikiHow 3 . WikiHow is a wiki-style website featuring over 200K how-to articles. Different from the existing dataset, ELI5, containing questions/answers from the Reddit forum and leveraging evidence queried from the web to help answer the question (Fan et al., 2019) and GooAQ, containing questions from search auto-complete and answers from answer boxes (Khashabi et al., 2021), WikiCQA involves long-form question-answering grounded on WikiHow articles. Dataset Construction We construct the new dataset by collecting the question-answer pairs from the Q&A section of articles on WikiHow. The questions and answers are related to the specific articles, asked by WikiHow users, and answered by a knowledgeable reader or editor 4 . The questions can not be answered directly by the content of the article. After removing duplicates and questionanswer pairs with meaningless answers, 23,037 QA pairs remain. To keep the same format as other existing QA datasets (e.g., ELI5, GooAQ), we discard questions not starting with a question-type word (e.g., what, how). After the dataset processing steps, we arrive at 20,202 question-answer pairs. More details can be found in Appendix A. In Table 2, we show an example article from Wikihow, which includes the title and summary of an article and question-answer pairs from the corresponding Q&A section. From the example, we can see that answers are abstractive and long-form, written by real users, and not contained within the context passage. These question-answer pairs from the Q&A section are collected in the newly constructed dataset. Thus, this dataset is different from reading comprehension datasets, where the answers are a short text span in context. More comparisons with ELI5 and GooAQ are shown in Appendix B. Title: How to Prepare a Healthy Meal for Your Pet Dog. Summary: To prepare a healthy meal for your dog, choose lean meat with the bones and fat removed, like chicken or beef . . . Q: What can I feed my dog if I have run out of dog food? A: In the short term, any bland human food such as chicken or white fish with rice or pasta is just fine . . . Q: How much homemade dog food do you feed your dog? A: Great question because it highlights one of the problems of feeding home prepared foods . . . Q: What should I not feed my dog? A: There are many human foods that are toxic to dogs. Top of the list of foods NOT to give are . . . Table 2: Question-answer pairs from WikiHow Q&A section. Q: question; A: answer. Models & Hyper-parameters We use BART-base (Lewis et al., 2020), a widely used sequence-to-sequence framework with 12 layers and a hidden size of 1024, as the backbone model. Following previous works (Fan et al., 2019;Khashabi et al., 2021), we finetune it using answers as inputs and questions as outputs as the baselines. To get the answer reconstruction loss (Section 3.3), we use a BART-base model which is fine-tuned on the target QA datasets. Based on the dataset analysis in Appendix B, we set the maximum sequence length to 128 for the question and 256 for the answer sentence to improve calculation efficiency. We train the models for five epochs with a learning rate of 5 × 10 −5 and evaluate the checkpoint for each epoch. We select the checkpoint with the highest ROUGE-L score on the validation set and report its corresponding score on the test set. We run each model three times and record the average scores. After performing manual hyper-parameter searching, we set loss parameters λ 1 , λ 2 , and λ 3 in Equation 7 to 1.0, 0.1, 0.1, respectively, which give the best ROUGE-L score on validation set. The temperature of contrastive loss τ cl is set to 0.3. The experiments are run on 1 NVIDIA Tesla V100 GPU. The training time takes about 48 hours. Evaluation Metrics To evaluate the performance of the models, we use ROUGE (Lin, 2004) scores, which evaluate the ngrams recall of generated sentences with reference sentences, as automatic evaluation metrics (Fan et al., 2019;Khashabi et al., 2021). We report the F1 for ROUGE-1, ROUGE-2, ROUGE-L, and ROUGE-Lsum. ROUGE-1, 2, L measures the unigram, bigram, and longest common subsequence between the pair of sentences, respectively. The difference between ROUGE-L and ROUGE-Lsum is that ROUGE-Lsum splits text using " \n". The higher ROUGE score indicates higher similarity between generated questions and references. We further perform human evaluations for the quality of generated questions in terms of Fluency: whether the questions are grammatically correct and fluent; Relevance: whether the questions are related to the answers; Correctness: whether the questions can be answered by the answers. Experiment Results In this section, we answer the four experimental research questions in turn. Generation Performance (RQ1) First, does the proposed QG framework have good performance? Table 3 shows the performance of the proposed closed-book QG model (denoted as QG ours ) on the new dataset and two public datasets compared with the baseline model trained only with question generation loss (denoted as QG b ). We observe that the proposed framework outperforms the baseline on three datasets for all ROUGE scores. For example, the performance increases by up to 2.0%, 2.7%, and 1.8% on the ROUGE-L score, respectively, which means the proposed framework can generate questions having a longer longest common subsequence (LCS) with ground-truth questions. We attribute these improvements to how contrastive learning pulls the answer representations closer to the ground-truth questions and thus, the generated questions have a higher chance to have similar words, phrases, or sentences with the ground-truth questions. The results demonstrate the effectiveness of our proposed QG framework. Quality of Generated Questions (RQ2) Next, are the generated questions of good quality? To answer this question, we perform human evaluation to measure the quality of generated questions in terms of fluency, relevance, and correctness. We randomly sample 100 answer-question pairs from the WikiCQA dataset, which contains answers, ground-truth questions, and generated questions from the baseline model and our best model. Then, we ask three annotators to rate the generated question pairs, comparing them with the ground-truth questions. The generated questions are rated on a 1-5 scale (5 for the best) from the aspects of the three aspects above. Further, we calculate the percentage of questions that have higher ratings in each group. Table 4: Results of human evaluation for baseline and our best model. score: is the average score from raters; %: represents the proportion of generated questions from one model that were rated higher than those from its counterpart. Synthetic Data Generation (RQ3) Further, can this QG framework be leveraged to generate effective synthetic data that can improve the closed-book QA task? Data augmentation is one of the main directions that question generation has been used for previously, with several studies finding improvements on the QA task (Lewis et al., 2019b;Alberti et al., 2019). Here, we show how to leverage the proposed QG framework to improve closed-book QA tasks on seen data (Wi-kiCQA) and unseen data (GooAQ and ELI5). Since freely available summary data is a good resource to generate synthetic data (Lyu et al., 2021), we use WikiSum (Cohen et al., 2021), which contains 39,775 coherent-paragraph summaries written by the article's authors on the WikiHow website. We take each sentence from the article summary as an answer and pass it into the best QG model, described in Section 3.1, to generate a question. Then, we train an unsupervised QA model based on the synthetic QA pairs and optimize it by the following negative log-likelihood loss function: L qa = − N i=1 log p(a i |Q),(8) where a i is the i-th token in the generated answer and Q is the question. In this work, we pre-train a BART-base model on the 200K synthetic QA pairs that are generated through the best QG model (denoted as QA s ) and evaluate it on the test set of the seen dataset (Wi-kiCQA) and unseen datasets (GooAQ-S and ELI5). This approach is unsupervised since the model is trained on no labeled question-answer pairs. The results are summarized in Table 5a, showing 22.4%, 12.5%, and 11.4% improvement than the BARTbase model without any synthetic pre-training (denoted as QA b ), on WikiCQA, GooAQ-S, and ELI5, respectively. This shows that pre-training on the generated question-answer pairs derived from our QG model leads to significant improvements, echoing findings in previous works that find synthetic data can be helpful (Khashabi et al., 2021;Lewis et al., 2021;Ding et al., 2021). After further fine-tuning on the target training set (denoted as QA s+f ), from Table 5b we can see that QA s+f also achieves better results than fine-tuned baselines model QA b+f by 3.6%, 0.4%, and 4.3% on the same three datasets. Ablation Study (RQ4) Finally, we investigate the contribution of each component in the proposed closed-book QG framework. Table 6 shows the results on the WikiCQA dataset. As described in Section 3.2, the contrastive learning module takes ground-truth questions as positive pairs to the input answers (denoted as CL t ) We also explore its variant -in which we feed the same input (answers) to the encoder with different dropouts to obtain the positive pairs (Gao et al., 2021) (denoted as CL s ). We find that the choice of CL t is slightly better than CL s in most evaluation metrics. In addition, by adding the answer reconstruction (AR) module, the model performance can be further improved. Thus, we can observe the effectiveness of the proposed contrastive learning module and the overall design of the framework. Table 7a, we can see our model generates "how many" for the statistics-type answer, which is closer to the reference "what percentage" than the baseline model, generating a "yes/no" question. For the second one, although the reference contains the detailed information "textured surface" which is not in the answer, our model captures the information "without nails", which is more informative than the baseline model. Thus, the proposed model can produce better questions that are more relevant to the answers and contain more detailed information than questions generated from baseline models. There are also some cases where the proposed model fails to generate good questions shown in Table 7b. Compared with the reference question "what food can be included on a clear liquid diet", our model generates "what can I eat to lose weight", which fails in automatic evaluation metrics and is hard for annotators to justify in terms of relevance. For the second one, although the "broken door" generated from our model is relevant to the answer, it's still far from the reference "door sticking to the weather strip". The question generated from the baseline model related to "bedroom" is even less relevant. These examples encourage us to explore how to capture more semantics in the answers and generate questions that have more detailed information in the future. Conclusion In this work, aiming to empower a QG model to better understand the semantics of long-form abstractive answers and store more information in its parameters, we propose a closed-book QG model empowered by a contrastive learning module and an answer reconstruction module. We present a new closed-book long-form QA dataset -WikiCQA involving more than 20K real user QA pairs and show that WikiCQA is a valuable training resource, complementing public datasets. Through the experiments, the proposed QG model shows better performance than baselines through automatic and human evaluations. Moreover, we show how to leverage the proposed model as a data augmentation strategy to improve existing closed-book QA systems. The closed-book QA model, pre-trained on our generated synthetic QA pairs, achieves better performance on the seen dataset (WikiCQA). In addition, it shows strong generalization on unseen datasets (GooAQ and ELI5), which further demonstrates the effectiveness of our QG framework for enhancing closed-book QA performance. Limitations While our model shows promising results on English datasets, its efficiency and performance in other languages require further investigation. Moreover, our evaluation is limited to datasets with oneto-one QA pairs, without considering situations where multiple answers correspond to a single question. Before feeding the input into the model, we ignore the tokens that exceed the maximum length we set, which may sometimes bring in an information loss for the input corpus. Additionally, our model is built on transformer architecture and therefore requires huge computation resources, especially for large training data sizes. Our focus in this paper is solely on the closed-book setting and our experiments center around four research questions to evaluate the effectiveness of the proposed framework, which we believe is an essential and emerging area deserving of specialized research investigation. In future work, we aim to explore the feasibility of adapting our framework for other QG scenarios. in ELI5 have a broader range of lengths. By measuring the proportion of the leading unigram that starts a question (Figure 2c), the dominant pattern is "how" questions in WikiCQA, while ELI5 has many "why" questions. We think WikiCQA has the potential to benefit the community in three ways. Firstly, its source is unique compared to the current two closed-book data sources, as it is collected from the Q&A section of articles on WikiHow. The questions and answers are related to the specific articles, asked by WikiHow users who read the articles and answered by a knowledgeable reader or editor. Secondly, compared to large-scale QA datasets, WikiCQA has longer answers and more open-ended questions with a wider range of question types. Additionally, it has a different data distribution compared to the other two datasets, which is crucial for testing the generalization ability of question-answering models. Finally, the question-answer pairs within relevant articles in WikiCQA can also be used as a valuable resource for conversational questionanswering tasks. Our code and data can be found at https://github.com/dongxiangjue/Closed-book-Question-Generation-via-Contrastive-Learning. Figure 1 : 1An overview of the proposed closed-book QG framework, which consists of three parts: contrastive learning, question generation, and answer construction. A i : represents answer i; Q i represents question i. Figure 2 : 2Comparison of the distribution of QA length and question types among three datasets. Table 1 : 1Dataset statistics. Table 3 : 3QG results on three datasets. QG b : baseline model; QG ours : our proposed QG framework. Table 4 4shows the average scores and the percent- ages of preferred questions on the three criteria. Both models are equally good at generating fluent questions. In terms of relevance and correctness, our approach shows 1.3% and 2.6% higher scores than the baseline method. These are consistent with what we expect: the contrastive learning part pulls the answer representation closer to the ground-truth question representation and generates more rele- vant questions; the reconstruction part can ensure that generated questions are of good quality and an- swerable. In addition, among 100 answer-question pairs, 24.7% and 23.3% of generated questions from our best model are rated higher while 56.3% and 57.4% of them have the same rating as those from baselines, which indicates that there is still substantial room to improve. To get a sense of the stability of the human evaluation results, we mea- sure the inter-agreement among annotators using the Correlation Coefficient, which is 0.998, 0.944, and 0.976 in terms of these three aspects, showing excellent reliability. Model fluency relevance correctness score % score % score % QG b 4.83 8.3 3.84 19.0 3.49 19.3 QGours 4.83 7.3 3.89 24.7 3.58 23.3 Table 5 : 5Evaluation on QA tasks. QA s/b : QA model w/o pre-training on synthetic data; QA * +f : QA * fine- tuned on target dataset. Table 6 : 6Ablation study of our proposed QG framework on WikiCQA. CL * : contrastive learning module; AR: answer reconstruction module.6 Case Study Finally, we showcase some examples of the gen- erated questions on the WikiCQA dataset. For the first example in QB Are small businesses audited by the irs? QG How many small businesses are audited by the irs? QR What percentage of small businesses are audited? A Try using poster putty to secure the images in your collage. if that doesn't work, you may have to use nails. QB How do i put pictures in a collage? QG How do i make a collage without nails? QR How can i make a collage on a wall with a textured surface? (a) Good examples. A Foods such as popsicles, hard candy, and gelatin can be eaten on a clear liquid diet. QB what can you eat on a clear liquid diet? QG What can I eat to lose weight? QR What food can be included on a clear liquid diet? A You'll have to remove the door and sand, prime, and repaint it. QB What do i have to do to make my bedroom look nice? QG How do i fix a broken door? QR What do i do if my door is sticking to the weather strip? (b) Bad examples.A The most recent statistics show that around 2.5% of small businesses are audited by the irs. Table 7 : 7Generated question examples. Q B : generated questions from baseline model; Q G : generated questions from proposed QG model; Q R : reference questions. 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Experimental SetupTo evaluate the effectiveness of the proposed QG framework, we aim to answer the following research questions (RQ) via experiments: RQ1: Can this proposed QG framework improve the performance of the closed-book QG tasks? RQ2: Are the generated questions of good quality? That is, are they fluent and relevant to the answer? RQ3: Can this QG framework be leveraged as a good resource to generate synthetic data for the QA task? RQ4: How much does each component in the framework contribute? https://github.com/allenai/gooaq/blob/main/ experiments/create_splits.py 2 https://huggingface.co/datasets/eli5 3 https://www.wikihow.com, under an Attribution-Noncommercial-Share Alike 3.0 Creative Commons License. https://www.wikihow.com/Use-wikiHow# Reading-and-Learning-from-wikiHow AcknowledgementsWe thank Zhuoer Wang, Yun He, Ziwei Zhu, anonymous reviewers, and the action editor for providing valuable feedback.Appendix A Data FilteringFirst, we choose the question-answer pairs where the questions start with a question-word in the list['how', 'what', 'can', 'is', 'do', 'why', 'are', 'does', 'where', 'when', 'should', 'will', 'did', 'which', 'who', 'would', 'if', 'about', 'for', 'as', 'could', 'in', 'after', 'at', 'while', 'to', 'am', 'has', 'any'] and end with a question mark. Then, we filter out the pairs where the length of answers is less than 8 tokens or the answers are meaningless. For the "meaningless" answers we mean, answers are "please refer to certain website http://xxx" or "please refer to the article xxx".Appendix B Dataset AnalysisTo better understand the content of WikiCQA in comparison with existing QA datasets,Figure 2shows the distributions of question length, answer length, and common question word. The questions and answers are tokenized by BART(Lewis et al., 2020). FromFigure 2aand 2b we could see more than 80% of the questions in WikiCQA and GooAQ-S have less than 15 tokens and more than 90% of answers of them lie in the range of 16-127 tokens while both questions and answers Synthetic QA corpora generation with roundtrip consistency. 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[ "https://github.com/dongxiangjue/Closed-book-Question-Generation-via-Contrastive-Learning.", "https://github.com/allenai/gooaq/blob/main/" ]
[ "SODAPOP: Open-Ended Discovery of Social Biases in Social Commonsense Reasoning Models", "SODAPOP: Open-Ended Discovery of Social Biases in Social Commonsense Reasoning Models" ]
[ "Haozhe An [email protected] \nUniversity of Maryland\nCollege Park\n", "Zongxia Li \nUniversity of Maryland\nCollege Park\n", "Jieyu Zhao [email protected] \nUniversity of Maryland\nCollege Park\n", "Rachel Rudinger [email protected] \nUniversity of Maryland\nCollege Park\n" ]
[ "University of Maryland\nCollege Park", "University of Maryland\nCollege Park", "University of Maryland\nCollege Park", "University of Maryland\nCollege Park" ]
[ "Proceedings of the 17th Conference of the European Chapter of the Association for Computational Linguistics" ]
A common limitation of diagnostic tests for detecting social biases in NLP models is that they may only detect stereotypic associations that are pre-specified by the designer of the test. Since enumerating all possible problematic associations is infeasible, it is likely these tests fail to detect biases that are present in a model but not pre-specified by the designer. To address this limitation, we propose SODAPOP 1 (SOcial bias Discovery from Answers about PeOPle), an approach for automatic social bias discovery in social commonsense question-answering. The SODAPOP pipeline generates modified instances from the Social IQa dataset (Sap et al., 2019b) by(1)substituting names associated with different demographic groups, and (2) generating many distractor answers from a masked language model. By using a social commonsense model to score the generated distractors, we are able to uncover the model's stereotypic associations between demographic groups and an open set of words. We also test SODAPOP on debiased models and show the limitations of multiple state-of-the-art debiasing algorithms. 2 We adopt the definition of race/ethnicity from the US census survey. We note that the categorizations in this definition are US-centric and may be less applicable in other countries.3 https://www.ssa.gov/oact/babynames/1566EA female
10.48550/arxiv.2210.07269
[ "https://www.aclanthology.org/2023.eacl-main.116.pdf" ]
252,907,310
2210.07269
0ea77398e28c5d32ac7b9cd42153c02266371b00
SODAPOP: Open-Ended Discovery of Social Biases in Social Commonsense Reasoning Models May 2-6, 2023 Haozhe An [email protected] University of Maryland College Park Zongxia Li University of Maryland College Park Jieyu Zhao [email protected] University of Maryland College Park Rachel Rudinger [email protected] University of Maryland College Park SODAPOP: Open-Ended Discovery of Social Biases in Social Commonsense Reasoning Models Proceedings of the 17th Conference of the European Chapter of the Association for Computational Linguistics the 17th Conference of the European Chapter of the Association for Computational LinguisticsMay 2-6, 2023 A common limitation of diagnostic tests for detecting social biases in NLP models is that they may only detect stereotypic associations that are pre-specified by the designer of the test. Since enumerating all possible problematic associations is infeasible, it is likely these tests fail to detect biases that are present in a model but not pre-specified by the designer. To address this limitation, we propose SODAPOP 1 (SOcial bias Discovery from Answers about PeOPle), an approach for automatic social bias discovery in social commonsense question-answering. The SODAPOP pipeline generates modified instances from the Social IQa dataset (Sap et al., 2019b) by(1)substituting names associated with different demographic groups, and (2) generating many distractor answers from a masked language model. By using a social commonsense model to score the generated distractors, we are able to uncover the model's stereotypic associations between demographic groups and an open set of words. We also test SODAPOP on debiased models and show the limitations of multiple state-of-the-art debiasing algorithms. 2 We adopt the definition of race/ethnicity from the US census survey. We note that the categorizations in this definition are US-centric and may be less applicable in other countries.3 https://www.ssa.gov/oact/babynames/1566EA female Introduction Researchers are increasingly aware of how NLP systems, especially widely used pre-trained language models like BERT (Devlin et al., 2019), capture social biases. Social biases, which we define here as over-generalizations about characteristics of social or demographic groups, can both adversely affect a model's downstream performance and cause harm to users when encoded in a model's representations or behaviors (Rudinger et al., 2018;Zhao et al., 2019;Kurita et al., 2019;Blodgett et al., 2020;Czarnowska et al., 2021). In this paper, we propose an approach to uncovering social biases in social commonsense reasoning models. It is particularly important to examine social commonsense 1 Code is available at https://github.com/haozhe-an/ SODAPOP. In an open-ended fashion, we generate distractors (Answer A and B) that contain words uncovering model social biases when names associated with different demographic groups are inserted into the context and question. In this example, Answer A, with the presence of "ruthless", is a more successful distractor for African American female names, whereas Answer B is a more successful distractor for European American female names due to the word "funny". Answer C is the correct answer choice from the Social IQa dataset. reasoning models because they are designed to reason about people and social interactions, and hence susceptible to stereotyped inferences. Biased inferences based on social group identities mentioned or alluded to in the input may cause representational harms to those group members. There have been consistent efforts to diagnose multiple types of social biases in NLP systems. Existing methods for bias detection usually involve manual efforts to first compile a list of stereotypic and anti-stereotypic associations between attributes and demographic groups, and then test for the presence of those associations in models. Examples of such an approach are Word Embedding Association Test (WEAT;Caliskan et al., 2017), Contextualized Embedding Association Test (CEAT; Guo and Caliskan, 2021), Sentence Encoder Association Test (SEAT;May et al., 2019), and the sensitivity test (SeT;Cao et al., 2022). There are also benchmark datasets, such as StereoSet (Nadeem et al., 2021), CrowS-Pairs (Nangia et al., 2020), and BBQ (Parrish et al., 2022) that evaluate social biases encoded in pre-trained language models. Although effective, these tests have a shortcoming: they may only be able to detect stereotyped attributes that the designers are aware of, as a result of searching pre-specified stereotypic model behavior within a defined scope. These approaches will not uncover any extant harmful associations that have not been specified in advance. To address this limitation, we introduce SO-DAPOP to uncover social biases in an open-ended fashion in social commonsense reasoning models. SODAPOP stands for SOcial bias Discovery from Answers about PeOPle. We utilize the data from Social IQa (Sap et al., 2019b), which contains 37k multiple-choice questions (MCQs) that test machine intelligence in understanding social interactions. As shown in Fig. 1, each MCQ contains a context, a question, and three choices. A model is trained to distinguish the correct choice from the remaining two distractors to answer the question. SODAPOP uses modified Social IQa examples to discover group-attribute associations in models. The Social IQa examples are systematically modified via (1) name substitution (to represent different social groups), and (2) open-ended distractor generation (representing different attributes). While SO-DAPOP requires the target social group identities to be pre-specified (e.g., female, African American), associated attributes are automatically discovered rather than pre-specified. Name substitution is the process of substituting people's names in social commonsense MCQs while keeping everything else unchanged. A fair model should not make radically different predictions given this change. If a model systematically makes disparate predictions after name substitution, we hypothesize these differences arise from demographic associations (e.g., gender, race/ethnicity) reflected by the names. While contexts may exist in which models could reasonably treat different names differently ("The name is Christine/Kristine with a C/K."), we believe this is generally not true of Social IQa contexts. Open-ended distractor generation produces new distractor answers by replacing a few tokens in the original answer using a masked language model. The resulting distractors draw from a large vocabulary, reflecting an openended set of possible attributes. To reveal a model's biased associations between a social group and an open set of words, we construct new MCQs with the generated distractors and analyze the model behavior when names are substituted. Fig. 1 illus-trates an example of a newly constructed MCQ. We use SODAPOP to uncover biased groupattribute associations in a finetuned BERT MCQ model for social commonsense reasoning. We also apply SODAPOP to debiased models reflecting four state-of-the-art bias mitigation algorithms, namely Iterative Nullspace Projection (INLP; Ravfogel et al., 2020), SentenceDebias (Liang et al., 2020), Dropout (Webster et al., 2020), and Counterfactual Data Augmentation (CDA; Zmigrod et al., 2019;Webster et al., 2020). SODAPOP reveals that these models persist in treating names differently based on demographic associations, despite their nominal purpose of mitigating such biases. To summarize, our contributions are: (1) We propose SODAPOP, a bias detection pipeline for social commonsense reasoning models via name substitution and open-ended distractor generation, without the need to pre-specify the potentially biased attributes we are looking for ( § 3). (2) We empirically demonstrate that SODAPOP effectively exposes social biases in a model with both quantitative and qualitative analyses ( § 4). (3) With SODAPOP, we find that debiased models continue to treat names differently by their associated races and genders ( § 6). Motivating Observations We obtain preliminary observations that suggest BERT produces different internal representations for names associated with different demographic groups. These observations motivate us to use name substitution for bias detection. Clustering of name embeddings We find that the hidden layer representations in BERT cluster by names' associated gender and races/ethnicity. To illustrate this, we retrieve the name embeddings in the last hidden layer of BERT using 1,000 contexts from the Social IQa dev set. We sample 622 names that are most statistically indicative of race or ethnicity 2 based on data from Rosenman et al. (2022). Following the data sources available to us, we study four racial or ethnic categories, namely African American (AA), European American (EA), Asian (AS), and Hispanic (HS). We obtain the gender statistics of names by referencing the SSA dataset. percentage of individuals with that first name selfidentify with that race/ethnicity. We set the percentage threshold to be 0.9 for EA and AA, 0.8 for HS and 0.7 for AS, in order to obtain about 80 names for each race/ethnicity and gender combination. From the dataset, we include only names with a frequency of 200 or greater. 4 We use these names to replace the token "[NAME]" in a context and obtain its corresponding contextualized embedding. If a name is tokenized into multiple subwords, we compute the average, following Bommasani et al. (2020); Wolfe and Caliskan (2021). We plot the t-SNE projection (Van der Maaten and Hinton, 2008) of the averaged embeddings for each name in Fig. 2. We observe that name embeddings tend to cluster by both gender and race/ethnicity. To quantitatively demonstrate that name embeddings encode demographic information, we train two separate logistic regression classifiers to predict gender and race/ethnicity associated with a name respectively. We train each classifier on 414 names and test on 208 names (obtained by a random split from the 622 names). We report the prediction accuracy for both settings in Table 1. It shows that the test performance is significantly higher than the random baseline. The high accuracy of these linear classifiers indicates, perhaps unsurprisingly, that BERT representations encode demographic information associated with names, and thus has the potential to perpetuate race-, ethnicity-, or gender-based representational harms via first names. The observations that name embeddings reveal demographic traits motivate us to use name sub-4 Rarer names should be studied in future work, but we omit them here as we anticipate they may elicit different model behavior. stitution in Social IQa samples to uncover model social biases towards different groups of people. We pose the following question: Given a description of a social situation and a question about a person involved therein, will a social commonsense model's predicted answer depend on demographic attributes inferable from the person's name? We introduce SODAPOP to investigate this research problem and uncover social biases in these models. Fig. 3 shows an overview of our proposed framework. SODAPOP composes two steps. The SODAPOP Pipeline Step 1 takes the context, question, and the correct answer choice from a Social IQa sample as the input. It generates many distractor answer choices by finding sentences that differ by a few tokens from the correct choice using a masked language model. Step 2 constructs new MCQ samples by pairing up the automatically generated distractors with the input in the first step. We analyze how distractor words fool the MCQ model at different rates for different name substitutions, measuring distractor word success rates for different names. Open-Ended Distractor Generation Following Zhang et al. (2021), we use masked token prediction to find neighboring sentences of correct answer choices to generate distractors. Alg. 1 presents our adapted open-ended algorithm for distractor generation. We generate a set of distractors by masking at most k tokens of the correct answer choice (k = 3 in our experiments). We adopt a recursive approach to replace one token at a time. In each recursive step, a masked language model fills the mask with some possible words, and the ones with the highest prediction scores are chosen to maximize the fluency of generated distractors. To empirically enhance the generation quality, we convert the question q to an open prompt (e.g., "How would you describe [NAME]?" becomes "[NAME] is "). We gather all unique distractors generated by Figure 3: Overview of our SODAPOP pipeline. We uncover social biases in models by first generating distractors in social commonsense reasoning MCQs and then analyzing how they influence model predictions. Algorithm 1 Open-ended distractor generation Input: Correct answer choice x 0 , masked language model LM , max distance k ≥ 1, context c, question q Output: X distract , a set of generated distractors 1: function GEN(x 0 , k) 2: if k ≥ 2 then 3: return x∈Gen(x 0 ,1) Gen(x, k − 1) 4: end if 5: X distract ← ∅ 6: ▷ ⊕ denotes string concatenation 7: x concat ← c ⊕ q ⊕ x 0 8: for i ∈ [0, len(x)) do 9: i ← i + len(c) + len(q) 10: x ← x concat ▷ Create a copy 11: x (i) ← '[MASK]' 12: T ok, Scores ← LM.f illmask(x) 13: ▷ Get predictions with topM scores 14: T m , S m ← topM(T ok, Scores) 15: return X distract 19: end function the algorithm as the final set of distractors for bias detection. Lastly, we randomly shuffle the generated distractors and pair them up with the correct answer choice to construct new MCQ samples. An example is shown in Fig. 1. X new ← x|x (i) ← t, t ∈ T m 16: X distract ← X distract ∪ X Seed names We obtain several lists of names that represent demographic groups (genders and races/ethnicities) as our seed names for distractor generation. Recall that we study four racial/ethnic categories based on the available data sources: African American (AA), European American (EA), Asian (AS), and Hispanic (HS). We borrow AA and EA names from WEAT (Caliskan et al., 2017). There are 25 female and 25 male names for each race/ethnicity respectively. We collect a total of 120 names that are most representative of Asian and Hispanic people from a name dataset provided by NYC Department of Health and Mental Hygiene. 5 There are about 30 names per gender for AS and HS each. More details are in appendix A. In Step 1 of SODAPOP, we insert each name into a Social IQa MCQ as we run Alg. 1. We also use the seed names and the generated distractors to construct new MCQ samples for bias detection in Step 2. Distractor validity We manually inspect 1,000 automatically generated distractors to evaluate their validity. A distractor is valid if it is grammatically correct, fluent, less plausible as the correct answer, and semantically dissimilar to the correct answer. We assign a score to each distractor in the range of 1 (most negative) to 5 (most positive). The annotation results show that most distractors have relatively high grammar and fluency scores (> 3.8) but low plausibility and semantic similarity scores (< 1.6). This shows the distractors are generally valid. More detailed results are in appendix B. Quantifying Group-Attribute Associations With many instances of modified Social IQa examples produced through name substitution and distractor generation, we can now quantify how a BERT-based Social IQa model associates groups with different attributes based on what kind of distractor answers it is most likely to select for a particular name. Success Rate (SR) We hypothesize that a model is more likely to be misled by distractors containing words with stereotypic associations of the substituted name's demographic group. Hence, we study the success rate (SR) of a word w for some name n by finding the probability of a distractor τ success-fully misleading the model, given that the word w is in the distractor and the name n is in the context. Thus, the success rate is SR(w, n) = τ ∈Tsuc,n 1 [w ∈ tokenize(τ )] τ ∈T all,n 1 [w ∈ tokenize(τ )](1) where 1 is the indicator function, and T suc,n , T all,n are respectively the set of successful distractors and all distractors appearing with name n. A successful distractor refers to a distractor that misleads a MCQ model to choose itself rather than the correct answer choice. If a model is robust to name substitution, SR(w, n) should be similar for various names (as the question contexts are identical otherwise). If differences are observed in SR across names, however, we will next need to investigate whether those differences are systematically based on gender, race, and ethnicity. Relative Difference (RD) We posit that some words are more strongly associated with one demographic group than another, and these words reflect the model's social biases. We find such words by computing the relative difference of SR. Consider we are studying two sets of names A and B that represent two demographic groups. We compute the difference of average SR of w for each group d(w, A, B) = 1 |A| |A| i=1 SR(w, A i ) − 1 |B| |B| i=1 SR(w, B i ) (2) and also the mean of SR for the two groups m(w, A, B) = 1 2 · G∈{A,B}   1 |G| |G| i=1 SR(w, G i )   . (3) Then, we compute relative difference (RD) of success rates of word w for two demographic groups A, B by RD(w, A, B) = d(w, A, B) m(w, A, B) .(4) The sign of RD indicates to which group the word w is more strongly associated with. A positive value means w is more often associated with group A whereas a negative value indicates a stronger association between w and group B. Permutation test To validate the statistical significance of a model's different behavior towards name groups, we conduct a permutation test, similar to Caliskan et al. (2017). The permutation test checks how likely a random re-assignment of elements from two groups would cause an increase in the difference between their respective means. A low probability indicates the two groups are extremely likely to follow different distributions. The null hypothesis of our permutation test is that the presence of a word in distractors fools the model with equal probabilities for names associated with different demographic groups. We compute the two-sided p-value by with a multiple choice classification head. We concatenate the context and question with each choice in a MCQ sample, and then obtain a logit for each concatenation. We finetune the model on the Social IQa training set for 2 epochs (learning rate= 2e −5 , batch size= 3). The finetuned model achieves 60.51% accuracy on the original development set. P r d(w, A † , B † ) > d(w, A, B)(5) Success Rate in Multiple Contexts Using 220 seed names (balanced in both gender and racial/ethnic categories as described in § 3), we follow Alg. 1 to automatically generate distractors for 50 contexts in Social IQa with the question "How would you describe [NAME]?" We choose this question because asking for a description of a person gives us direct access to the model's internal representation of that person, allowing us to assess the representational harms caused by social biases encoded in the model. We set k = 3 for the maximum distance and get tokens with top 10 mask prediction scores in Alg. The ideal accuracy is 0.5 (random binary classification). Full results are available in Table 9 in the appendix. choice and two generated distractors. We collate all unique tokens in the generated distractors as the set of distractor vocabulary. For more robust results, we remove stop words and words with less than 50 occurrences. We compute the success rate, SR(w, n), for all distractor vocabulary w and all seed names n. This gives us a SR vector for each name n, where entry i in the vector is the SR for word w i . The final dimension of SR vectors is 443. Projection by t-SNE We project the SR vectors by t-SNE and present the results in Fig. 4. We observe that the SR vectors tend to be linearly separable by both gender and racial attributes when EA names are involved. The clustering of SR vectors in Fig.4a, 4b, and 4c demonstrates that for names belonging to the same gender or racial group, words share similar SR in distractors. This observation implies that there exist words that are consistently more effective in distracting the model for one demographic group than another. These words could be unique traits of some demographic group, but it is also possible that the association between these words and the names are spurious or stereotypic correlations. However, for AS and AA names in Fig. 4d, we do not see an obvious separation of the SR vectors by either race or gender. To quantify the separation of clusters, we conduct a binary classification of the SR vectors for each pair of name groups that are associated with different demographic attributes. This evaluation is similarly used by Gonen and Goldberg (2019); An et al. (2022). For two name groups representing different social groups (e.g., AA female names and EA female names), we use the classical KMeans algorithm (K = 2) to cluster the SR vectors and make a binary prediction that indicates the membership of either cluster. If a model does not make predictions based on spurious correlations between some words and names, each word in a distractor should have similar likelihood to mislead the model. It follows that the ideal classification accuracy should be 0.5. We report the classification accuracy in Table 2. The classification accuracy tends to be higher when EA names are present, whereas it is comparatively harder to distinguish SR vectors among racial/ethnic minority groups. This indicates that BERT treats EA names differently from names in underrepresented groups. Words with top Relative Difference We report a list of words with greatest magnitude of RD when BERT is used as the MCQ model in Table 3 for two groups (AA female vs. EA female). More results are in Table 10 (appendix). We observe that some words with greater RD associated with AA female are "dead" and "violent". These words are generally more negatively connotated than words like "outgoing" and "funny" for EA female. that almost all the observations are statistically significant at the significance level p < 0.01. It is thus evident that the MCQ model is making predictions based on the biased correlations between these words and names. SODAPOP can also detect biased correlations between words and names with different genders (see Table 13 in appendix D). Success Rate in a Single Context We generate distractors and collate the words with greatest magnitude of RD in a single context only, as an individual context may reveal specific stereotypic traits. Setup We use the sample in Fig. 3 as the input to SODAPOP. For each seed name, we construct a total number of 109,830 MCQ samples with this single context after generating distractors using Alg. 1. Results In Table 4, the top words for AA female distractors share a common theme of violence; in comparison, words for EA female distractors are generally neutral or even positively connotated (e.g., "educated"). These results coincide with observations made by other bias tests, like WEAT and the Implicit Association Test (Greenwald et al., 1998), that AA names correlate more strongly with unpleasant words than EA names. Table 21 in appendix D.2 shows the results for gender, where the model tends to associate EA male with violence more often than EA female. Qualitatively, it appears that limiting SODAPOP to a single context (Table 4) yields more interpretable results than when aggregated over many contexts invoking different scenarios (Table 3). In the next section, we attempt to validate this intuition by aligning SO-DAPOP outputs with results of prior human studies. Validation with Human Stereotypes Since NLP models have been repeatedly shown to reflect human biases, one way to validate SO-DAPOP would be to show that a subset of its discovered biases align with known human social stereotypes. To attempt this, we adopt the Agency-Belief- Table 5. We run SODAPOP on 12 individual contexts and take the union of top identified words for each demographic group with p < 0.05. Working independently, three authors manually mapped each word to zero, one, or more ABC-model traits, without awareness of the group-word associations. E.g., all annotators mapped the word "brutal" to the ABC trait threatening. For each group A and trait t, a raw count C(A, t) represents the number of times any annotator aligned a word SODAPOP associated with group A to trait t. A word could be aligned with a trait t (powerful) or its opposite ¬t (powerless). For a trait t and groups A and B, we then say SODAPOP supports the ordering (2022). Legend: "N/A" -no words from SODAPOP are mapped to the trait; ‡ -the absolute SODAPOP score difference is at least 5; † -the absolute difference between human scores for the two groups is at least 20; * -same as † but absolute difference is at least 10; shaded cells -SODAPOP yields orderings that are consistent with human annotators. A ≺ t B if and only if the SODAPOP score difference [C(A, t) − C(A, ¬t)] − [C(B, t) − C(B, ¬t)] < 0.≺ † high status dominated-dominant ≻ ‡ dominant ≺ † dominant ≺ * dominant ≺ † dominant ≺ † dominant ≺ † dominant poor-wealthy ≺ wealthy ≺ † wealthy ≺ † wealthy ≻ wealthy ≺ † wealthy ≺ wealthy unconfident-confident ≺ ‡ confident ≺ † confident ≻ confident ≺ confident ≺ † confident ≺ † confident unassertive-competitive ≺ competitive ≺ † competitive ≺ competitive ≺ competitive ≺ † competitive ≺ * competitive traditional-modern ≺ ‡ modern ≺ † modern ≺ † modern ≺ modern ≻ modern ≻ * modern religious-science oriented ≺ science oriented ≺ † science oriented ≺ † science oriented ≻ science oriented ≺ science oriented ≻ science oriented conventional-alternative ≻ alternative ≺ alternative ≻ alternative ≺ alternative ≺ alternative ≻ * alternative conservative-liberal N/A ≻ * liberal ≺ liberal N/A ≻ * liberal ≻ * liberal untrustworthy-trustworthy ≺ trustworthy ≺ † trustworthy ≺ * trustworthy ≺ trustworthy ≻ trustworthy ≺ trustworthy dishonest-sincere ≺ sincere ≺ * sincere ≺ * sincere ≻ † sincere ≻ sincere ≻ sincere cold-warm ≺ ‡ warm ≺ warm ≻ warm ≺ warm ≻ warm ≻ warm threatening-benevolent ≺ ‡ benevolent ≺ † benevolent ≻ * benevolent ≻ benevolent ≻ † benevolent ≻ † benevolent repellent-likable ≺ ‡ likable ≺ * likable ≻ * likable ≺ likable ≺ likable ≻ likable egoistic-altruistic ≺ ‡ altruistic ≺ altruistic ≺ † altruistic ≻ altruistic ≻ altruistic ≻ altruistic (indicated with †). Notably, the biases uncovered by SODAPOP are more consistent with EA annotators than AA annotators, while it is almost equally consistent with both male and female annotators. In a few cases, SODAPOP-derived orderings deviate from human results (e.g., powerless-powerful for AA female vs. EA female), perhaps owing to intersectional differences. Overall, SODAPOP appears capable of uncovering human-aligned stereotypes without pre-specifying attributes. This makes it a promising method to uncover other kinds of social overgeneralizations present in models -possibly those present in humans but less well studied, or possibly ones entirely peculiar to machines -in either case, carrying the potential for harm. Conclusion To the best of our knowledge, SODAPOP is the first open-ended pipeline for bias detection in social commonsense reasoning models. Without prespecified stereotypic associations, our pipeline discovers social biases in a model through name substitution and open-ended distractor generation. We construct a large number of MCQ samples with automatically generated distractors and substitute the names in MCQs with those representing various demographic groups. Analyzing the success rate of words in distractors reveals a model's learned social biases. We also show that biases uncovered by SODAPOP align with human stereotypes, and these biases persist even in debiased models. In future work, SODAPOP may be used to explore biases for other MCQ tasks, and for tasks in languages other than English, reflecting biases in a different cultural setting. There are several limitations to the representations of gender, race and ethnicity we adopt in this work. We model gender as a binary variable due to limitations in the demographic name data we use. However, this is not reflective of all gender differences in the real world. Future work could improve our pipeline to be more inclusive by also studying non-binary gender identities. Another limitation is that we treat the variable of race/ethnicity as categorical, when in reality the racial and ethnic identities of individuals may intersect multiple groups. While we study here the intersection of race/ethnicity and gender, we do not study multiple intersections of race and ethnicity, e.g., Black Hispanic. SODAPOP identifies social biases exhibited by models in the treatment of first names; however, there are other ways in which demographic information may be conveyed through language to a model, e.g., through pronouns (she), noun phrases (an Asian person), associated concepts (N.A.A.C.P.), dialect, etc. SODAPOP does not measure disparate model behavior towards these linguistic indicators of demographics. Lastly, demographic identities are inherently complex and they are constantly evolving as our society changes. Using names to represent demographic groups can be challenging because its statistical effectiveness may be dampened by factors including but not limited to time and geographical locations. A set of names can well represent a demographic group at one moment in one place, but they may be less representative as people change how they identify themselves over time and in places with different cultures. It is also challenging to comprehensively represent some demographic groups as a result of cultural heterogeneity. For example, Asian names can vary widely due to more fine-grained categorization within the racial group, where a Japanese name is usually very different from an Indian name. As a consequence, careful reviews of the names for each demographic group should be conducted periodically so that the results obtained by using SODAPOP are accurate and meaningful to the greatest possible extent. Ethics Statement Gender, race, ethnicity, and other demographic attributes are more complex in reality than simple categorical labels. Although many names demonstrate a strong association with a particular demographic group through census data, these correlations are seldom absolute. Therefore, SODAPOP is a method that works over aggregate statistics, though conclusions may be harder to draw from individual instances. The purpose of SODAPOP is to further research into the manifestation of social biases in social commonsense reasoning models. Although SO-DAPOP is sensitive to the presence of model bias, including in "debiased" models, we caution future researchers against using SODAPOP to conclude that a model is absent of biases. Furthermore, the results produced by SODAPOP should not be exploited to incite hatred towards any demographic groups or individuals. A Names A.1 Asian and Hispanic Name Collection for SODAPOP We collect Asian and Hispanic names from Popular Baby Names 7 provided by Department of Health and Mental Hygiene (DOHMH), in addition to African American and European American names from WEAT (Caliskan et al., 2017). Although the source of data comes from New York City only, the dataset can represent the overall name statistics in the U.S. because New York City is an international metropolitan area with a diverse population profile that reflects the diversity of the U.S. population (Gaddis, 2017). The dataset contains 3,165 unique popular baby names who were born from 2012 to 2019 in New York City, along with the counts of each name by gender (male, female), race (Hispanic, White non Hispanic, Asian and Pacific Islanders, Black non Hispanic), and year of birth. To be more specific, there are 1,529 unique first names for Hispanic and 1,216 first names for Asian. Fig. 6 shows the name distribution in terms of genders and the two races. Hispanic Female 32% Hispanic Male 31% Asian Female 19% Asian Male 19% Figure 6: Distribution of Hispanic and Asian names in the dataset, with European American and African American names excluded. Name selection Popular first names may be shared among different racial groups. Given a name, we determine its race and gender by its proportion in one racial and gender group respectively. The formula to determine the proportion of a first name n in a race r is as follows: P roportion r = count(n, r) r j ∈R count(n, r j )(6) where R is the set of all races and count(n, r) is the total counts of name n in race r. Similarly, we determine the proportion of a first name n for gender g by P roportion g = count(n, g) g j ∈G count(n, g j ) where G is the set consisting of male and female. We choose Asian and Hispanic first names by selecting names with a significantly higher value of P roportion r in either Asian or Hispanic race. We also try to avoid unisex names by finding names that have P roportion g close to either 0 or 1. Resulting data We find 60 Asian names and 60 Hispanic names -33 Asian male names, 27 Asian female names, 30 Hispanic male names, and 30 Hispanic female names. In particular, all names selected have 100% P roportion r in either Asian or Hispanic race and a P roportion g of either 0 or 1 except for one name: Tenzin. Tenzin is a 100% Asian name with gender ratio 0.4871. A.2 Lists of All Names We list all the names used in our experiments to generate distractor choices and analyze spurious correlations. To reiterate, the four races we study in this paper are European American (EA), Afirican American (AA), Asian (AS), and Hispanic (HS). to happen given a social interaction context. If a distractor describes something semantically similar to the ground truth choice, or equally plausible, it is not a valid distractor. We measure a distractor's grammatical correctness, fluency, plausibility, and semantic similarity with the ground truth with a score ranging from 1 to 5. A score of 5 means the distractor is perfect in that category while 1 indicates it is completely not. Out of 1,000 randomly sampled distractors, 64 contain a punctuation only. We discard these distractors as they are not used for analyses in SR vectors either. The results of the other 936 distractors are shown in Table 7 and Fig. 7. The annotation results indicate that in spite of some noisiness, the generated distractors are mostly grammatically acceptable and sufficiently fluent, but they are not plausible enough as alternative, correct choices for MCQs and they are semantically unlike the ground truth choices used for generation. For illustration, we include some distractors generated using Alg. 1 in Table 8. C Additional Analysis on Success Rate Vectors in Debiased Models We present our additional visualization of SR vectors for all debiasing algorithms that we have experimented. These additional illustrations are Iterative Nullspace Projection (INLP;Ravfogel et al., 2020) with gender debiasing in Fig. 8, SentenceDebias (Liang et al., 2020) with gender and racial debiasing in Fig. 9 and 10, Dropout (Webster et al., 2020) with general debiasing in Fig. 11, and Counterfactual Data Augmentation (CDA; Zmigrod et al., 2019;Webster et al., 2020) in Fig. 12 and 13. Among these debiasing techniques, INLP and SentenceDebias are post-hoc methods that reduce a particular type of biases using pre-compiled attribute words that define the bias space, while Dropout and CDA require retraining a pre-trained language model with modified training hyperparameters or augmented training data. We apply these debiasing algorithms to BERT. We implement them using the Bias Bench repository (Meade et al., 2022). For post-hoc debiasing algorithms, we apply the algorithms to our finetuned BERT model obtained in § 4. 8 For traintime debiasing algorithms, the debiased models are finetuned with the same set of hyperparameters as described in § 4 and deliver similar performance on Social IQa dev set (prediction accuracy is about 60% ∼ 62% for all models). While finetuning may re-introduce biases to the debiased model, we note that the seed names in our experiments are disjoint from those in the Social IQa training set. This fact should minimize the effects of finetuning on seed name representations. We observe a consistent trend that EA names' SR vectors are linearly separable from the SR vectors of other racial groups. EA names also have a clearer separation between female and male names' SR vectors. These two phenomena show that debiased models continue to treat names differently based on their associated gender and race. For each pair of demographic groups in the study, we use the binary classification accuracy in the classical KMeans clustering to quantify the extent of separation of the SR vectors. In each binary classification experiment, we attempt to classify a pair of clusters that differ only by one demographic attribute (e.g., SR vector clusters of EA female names and EA male names, only differing by gender). The results are available in Table 9. The trend is that debiased models, regardless of being racially debiased or gender debiased, still treat EA names significantly differently from other racial groups' For each undebiased and debiased model in our study, we collate a list of words with the greatest magnitude of RD values as we compare the SR of distractor vocabulary towards different racial and gender groups. We first continue the discussion for the setup of using undebiased BERT as the MCQ model in § 4. We report the list of words with greatest magnitude of RD values for two gender groups (EA female and EA male) in Table 13. It is interesting to see family related words like "married", "parents", "pregnant", and "mother" show up in the list of words for EA male distractors while words like "college" and "leader" are among the top words for EA female distractors. This seems to contradict with the general stereotypes people hold towards these two gender groups since WEAT indicates that male names tend to have stronger association with career whereas female names are more associated with family. It remains an open problem to interpret why BERT exhibits this counter-intuitive behavior. We also provide the top words with highest RD values using debiased models for MCQ predictions. Results for INLP model with gender bias mitigated are shown in Table 14. Results for SentenceDebias BERT with racial or gender bias mitigated are in Tables15 and 16 respectively. Dropout reduces general biases and its results are in Table 17 and 18. Finally, we present the results for CDA BERT with racial or gender bias mitigated in Table 19 and 20. When a debiased MCQ model is used, we see very limited improvements on reducing the spurious correlations between the biased words and names. A considerable number of words still have very small p-values. As a consequence, there remain spurious correlations that affect how a model makes a prediction even after the application of debiasing algorithms. D.2 Relative Difference in a Single Context We analyze words' RD in the same single context as studied in § 4.2 and report the words with greatest RD values for two gender groups (EA female vs. EA male). Table 21 presents the results. Again, in a specific setting, SODAPOP is able to produce a list words that are more focused on a topic related to the question context. We see that words with highest RD values for EA male distractors describe violence while words like "sweet" and "generous" appear for EA female. That being said, there are also words that potentially associates with violence for EA female distractors (e.g., "rebellious"). Figure 1 : 1An example of a modified Social IQa MCQ sample. Figure 2 23 A name is indicative of a race/ethnicity if a : t-SNE projections of name embeddings in BERT. Name embeddings cluster by the associated demographic traits (race/ethnicity and gender). Figure 7 : 7Distribution of annotated scores for 936 randomly sampled distractors. Grammar Fluency Plausibility Semantic sim Mean ± std 3.85 ± 1.2 3.83 ± 1.22 1.81 ± 1.15 1.57 ± 1.03 Table 2 : 2KMeans classification accuracy of SR vectors. Table 3 : 3Top 5 words with greatest magnitude of RD for two racial groups and their permutation test p-values. Table 4 : 4Top 5 words with greatest magnitude of RD in the specific context ( § 4.2) for two racial groups and their p-values. More results are in Table 11 (appendix). We coarsen this to an ordering of groups along traits, e.g., women ≺ powerful men. To compare the biases uncovered by SO-DAPOP, we map (where applicable) attribute words to ABC model trait scales to induce a similar ordering, and compare whether the orderings derived from SODAPOP match those from human subjects inCao et al. (2022), reporting results inCommunion (ABC) stereotype model (Koch et al., 2016) and cross-reference SODAPOP results with the the findings of Cao et al. (2022) who collect group-trait stereotypes through human survey meth- ods. The ABC model describes people using 16 pairs of opposing traits, like powerless-powerful (Table 5). Cao et al. (2022) gather human sub- jects' opinions about how American society at large perceives a demographic group with respect to a trait, computing a score from 0 to 100. E.g., on the powerful-powerless trait scale, subjects rated women on average 46.8 (less powerful) and men 81.4 (more powerful). (See Tables A14 to A17 from Cao et al. (2022).) Table 5 5compares the orderings derived from SODAPOP to those derived from human subjects inCao et al. (2022) for two racial groups, AA (female) vs EA (female), and two gender groups, (EA) female vs (EA) male. We note that group alignment betweenSODAPOP and Cao et al. (2022) is imperfect, as the former is intersectional. Nonetheless, we observe that the orderings for most group-trait pairs produced by SODAPOP are consistent with orderings produced by human annotators, particularly in cases where human results are strongest 1571 AA Female vs. EA Female EA Female vs. EA MaleTraits SODAPOP EA Annotators AA Annotators SODAPOP Male Annotators Female Annotators powerless-powerful ≻ ‡ powerful ≺ † powerful ≺ * powerful ≺ † powerful ≺ † powerful ≺ † powerful low status-high status ≺ high status ≺ † high status ≺ † high status ≺ high status ≺ † high status Table 5 : 5Comparison of SODAPOP to human stereotypes as measured in Cao et al. One might expect that, in a debiased model, words in distractors will mislead the model at similar rates for different groups. In this section, however, we demonstrate that biases uncovered by SODAPOP persist in debiased models. We apply the INLP algorithm to our finetuned BERT model in § 4 to reduce biases along the racial dimension. Our implementation uses Bias Bench(Meade et al., 2022). This racially debiased INLP model (INLP-race) is used as the new MCQ model. 6 6 Besides INLP-race, we present similar results with other debiasing algorithms in appendix C and appendix D.6 Debiased Models Continue to Treat Names Differently AA Female Distractors EA Female Distractors Word RD p-value Word RD p-value innocent 0.062 1.6E-05 sticking -0.042 2.7E-02 dead 0.046 1.9E-03 outgoing -0.040 1.9E-03 violent 0.041 0.0E+00 loud -0.037 3.0E-06 cousin 0.040 6.3E-05 funny -0.035 9.5E-05 ally 0.038 0.0E+00 cook -0.032 3.1E-01 Table 6 : 6Top 5 words with greatest magnitude of RD for two racial groups with their p-values in permutation tests. Here, INLP-race is used for MCQ predictions.Success Rate VectorsFig. 5visualizes the SR vector for various racial groups using INLP-race.Fig. 5a, 5b, and 5c show that the SR vectors for each demographic group remain in separable clusters even if the model is debiased along the racial dimension. The binary KMeans classification results, shown inTable 2, are largely similar to those of BERT. In the case of AA female and AA male,the classification deviates further from the ideal value of 0.5 to 0.86, compared to 0.58 for BERT. The clear clustering indicates that although the de- biased model somehow mitigates biases, it does not completely remove biases in a downstream task. Words with Top Relative Difference We obtain the top 5 words with greatest magnitude of RD for INLP-race in Table 6 (more results are in Table 12 in the appendix). INLP reduces racial bias to some extent because a subset of negatively connotated words for AA female distractors no longer show up ciation for Computational Linguistics, pages 5454-5476, Online. Association for Computational Linguistics. Rishi Bommasani, Kelly Davis, and Claire Cardie. 2020. Paula Czarnowska, Yogarshi Vyas, and Kashif Shah. 2021. 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Table 8 : 8Examples of generated distractors using Alg. 1 with their respective contexts and ground truth choices as the input. All samples share the same question "How would you describe [NAME]?" names. Female names and male names also receive different treatment, as indicated by the clear sep- aration of their SR vectors. Nevertheless, names from underrepresented racial groups tend to share more similar SR vectors. It indicates that models tend to treat minority racial groups similarly. D Additional Analysis on Words with Top Relative Difference D.1 Relative Difference in Multiple Contexts BERT INLP-race INLP-gender SentenceDebias-race SentenceDebias-gender Dropout CDA-race CDA-genderGender EA female and EA male 0.98 0.98 0.98 0.98 0.98 1.00 0.98 0.88 AA female and AA male 0.58 0.86 0.90 0.58 0.58 0.68 0.54 0.62 HS female and HS male 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 AS female and AS male 0.52 0.54 0.52 0.52 0.52 0.52 0.52 0.52 Race EA female and AA female 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.88 EA male and AA male 0.84 0.84 0.84 0.84 0.84 0.88 0.90 0.88 EA female and AS female 0.76 0.76 0.76 0.76 0.76 0.80 0.78 0.76 EA male and AS male 0.80 0.80 0.74 0.80 0.80 0.82 0.76 0.80 EA female and HS female 0.80 0.80 0.80 0.80 0.80 0.80 0.84 0.82 EA male and HS male 0.64 0.64 0.64 0.64 0.64 0.66 0.66 0.82 AA female and AS female 0.64 0.64 0.64 0.64 0.64 0.72 0.62 0.58 AA male and AS male 0.60 0.60 0.60 0.60 0.60 0.58 0.60 0.56 AA female and HS female 0.58 0.58 0.58 0.58 0.58 0.58 0.68 0.58 AA male and HS male 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.72 HS female and AS female 0.54 0.54 0.54 0.54 0.54 0.54 0.58 0.56 HS male and AS male 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 Table 9 : 9KMeans classification accuracy of SR vectors over all debiased models we have experimented. The ideal accuracy is 0.5 (random binary classification).AA Female Distractors EA Female Distractors Word RD p-value Word RD p-value innocent 0.060 2.1E-05 sticking -0.042 2.7E-02 cousin 0.042 2.7E-05 outgoing -0.040 1.8E-03 dead 0.042 5.4E-03 loud -0.036 6.0E-06 ally 0.040 0.0E+00 funny -0.035 1.0E-04 violent 0.039 1.0E-05 cook -0.032 2.9E-01 watches 0.034 5.4E-04 told -0.025 1.3E-05 associate 0.034 0.0E+00 sporting -0.025 0.0E+00 acquaintance 0.033 0.0E+00 front -0.022 1.0E-06 ofof 0.029 2.2E-05 talkative -0.019 4.8E-05 beyond 0.027 2.0E-06 vegetarian -0.018 3.5E-01 wary 0.026 2.0E-06 cool -0.018 1.2E-05 next 0.025 4.8E-05 Happy -0.017 5.0E-01 simple 0.022 5.0E-06 convinced -0.016 1.0E-03 even 0.022 2.8E-04 old -0.016 1.9E-01 college 0.021 4.0E-02 friends -0.016 8.2E-02 Table 10 : 10Top 15 words with greatest magnitude of RD for two racial groups and their permutation test p-values. This extends the content inTable.3.Word RD p-value Word RD p-value vicious 0.073 0.0E+00 educated -0.074 8.8E-05 brutal 0.071 0.0E+00 caring -0.063 6.9E-04 stubborn 0.066 1.7E-01 aroused -0.063 0.0E+00 possessive 0.065 1.5E-03 sweet -0.060 7.5E-05 arrogant 0.065 1.9E-04 interesting -0.058 7.2E-04 ruthless 0.064 0.0E+00 sophisticated -0.052 0.0E+00 nasty 0.057 3.1E-03 sound -0.050 2.3E-03 violent 0.055 0.0E+00 charming -0.050 5.0E-06 fierce 0.052 2.8E-05 sounds -0.049 1.6E-03 cruel 0.050 0.0E+00 confident -0.048 7.6E-04 gentle 0.043 7.5E-01 soft -0.048 3.9E-03 hostile 0.039 0.0E+00 demanding -0.048 2.1E-03 man 0.033 7.5E-05 loving -0.047 1.5E-01 rebellious 0.029 2.0E-03 serious -0.045 2.9E-05 personality 0.029 1.8E-04 young -0.045 2.0E-06 Table 11 : 11Top 15 words with greatest magnitude of RD in the specific context ( § 4.2) for two racial groups with their permutation test p-values. This extends the content inTable.4.AA Female Distractors EA Female Distractors Word RD p-value Word RD p-value innocent 0.062 1.6E-05 sticking -0.042 2.7E-02 dead 0.046 1.9E-03 outgoing -0.040 1.9E-03 violent 0.041 0.0E+00 loud -0.037 3.0E-06 cousin 0.040 6.3E-05 funny -0.035 9.5E-05 ally 0.038 0.0E+00 cook -0.032 3.1E-01 watches 0.036 4.8E-04 vegetarian -0.024 2.1E-01 associate 0.032 0.0E+00 sporting -0.024 0.0E+00 acquaintance 0.030 0.0E+00 front -0.022 1.0E-06 beyond 0.027 2.0E-06 told -0.020 1.7E-04 ofof 0.027 3.1E-05 talkative -0.018 5.8E-05 next 0.024 6.9E-05 cool -0.017 1.1E-05 animal 0.024 3.8E-04 old -0.017 1.3E-01 simple 0.023 3.0E-06 friends -0.017 4.7E-02 angry 0.023 2.0E-03 Happy -0.017 5.0E-01 even 0.023 1.7E-04 dog -0.016 2.8E-01 Table 12 : 12Top 15 words with greatest magnitude of RD for two racial groups with their p-values in permutation tests. INLP-BERT with racial bias mitigated is used for social commonsense MCQ predictions. This extends the content inTable. 61582 Table 13 : 13Top 15 words with greatest magnitude of RD for two gender groups and their p-values. We use BERT as the MCQ model.EA Female Distractors EA Male Distractors Word RD p-value Word RD p-value cook 0.061 8.1E-02 brilliant -0.052 5.0E-02 vegetarian 0.060 2.1E-02 married -0.042 0.0E+00 college 0.056 2.0E-05 animal -0.034 2.9E-03 outgoing 0.037 9.3E-04 name -0.027 2.6E-03 old 0.032 2.5E-02 shocked -0.025 3.4E-04 camp 0.024 0.0E+00 pregnant -0.024 0.0E+00 play 0.022 0.0E+00 friendly -0.023 3.0E-05 summer 0.021 0.0E+00 beyond -0.023 2.0E-05 leader 0.021 4.5E-04 parents -0.022 0.0E+00 husband 0.021 6.0E-06 former -0.022 1.7E-02 sticking 0.019 3.0E-01 Happy -0.022 2.8E-01 boot 0.018 1.4E-02 wary -0.021 1.2E-05 vegan 0.018 3.9E-01 mother -0.020 0.0E+00 spoke 0.016 2.7E-01 blessed -0.019 0.0E+00 liked 0.015 0.0E+00 seen -0.019 1.8E-05 Table 14 : 14Top 15 words with greatest magnitude of RD for two gender groups and their p-values in permutation tests. Here we use INLP BERT with gender bias mitigated for MCQ predictions.Word RD p-value Word RD p-value innocent 0.060 2.1E-05 sticking -0.042 2.7E-02 cousin 0.042 3.2E-05 outgoing -0.040 1.8E-03 dead 0.042 5.4E-03 loud -0.036 5.0E-06 ally 0.040 0.0E+00 funny -0.034 1.6E-04 violent 0.039 9.0E-06 cook -0.032 2.9E-01 watches 0.034 5.4E-04 told -0.025 1.2E-05 associate 0.034 0.0E+00 sporting -0.025 0.0E+00 acquaintance 0.033 0.0E+00 front -0.022 2.0E-06 ofof 0.028 2.7E-05 talkative -0.019 4.6E-05 beyond 0.027 2.0E-06 vegetarian -0.018 3.5E-01 wary 0.026 2.0E-06 cool -0.018 1.1E-05 next 0.025 4.8E-05 old -0.017 1.6E-01 college 0.022 2.7E-02 Happy -0.017 5.0E-01 simple 0.022 4.0E-06 convinced -0.016 1.1E-03 even 0.022 2.8E-04 brilliant -0.016 7.4E-01 Table 15 : 15Top 15 words with greatest magnitude of RD in the specific context for two racial groups with their p-values in permutation tests. Here we use SentenceDebias BERT with racial bias mitigated for MCQ predictions.EA Female Distractors EA Male Distractors Word RD p-value Word RD p-value cook 0.061 5.0E-02 brilliant -0.053 4.6E-02 college 0.056 9.0E-06 married -0.041 0.0E+00 vegetarian 0.054 8.6E-02 animal -0.027 2.6E-03 outgoing 0.047 1.2E-04 parents -0.026 0.0E+00 innocent 0.030 2.0E-01 pregnant -0.024 0.0E+00 old 0.027 2.3E-02 friendly -0.024 2.3E-05 camp 0.024 0.0E+00 wary -0.023 8.0E-06 leader 0.022 7.8E-04 beyond -0.023 1.8E-05 play 0.022 0.0E+00 former -0.022 1.6E-02 husband 0.021 4.0E-06 shocked -0.022 4.3E-04 summer 0.020 0.0E+00 name -0.021 1.6E-02 boot 0.020 5.7E-03 mother -0.020 0.0E+00 vegan 0.018 3.9E-01 Angry -0.020 2.5E-01 fairly 0.017 2.0E-04 seen -0.019 2.3E-05 talkative 0.016 2.8E-05 excellent -0.017 3.1E-04 Table 16 : 16Top 15 words with greatest magnitude of RD in the specific context for two gender groups with their p-values in permutation tests. Here we use SentenceDebias BERT with gender bias mitigated for MCQ predictions.Word RD p-value Word RD p-value associate 0.054 0.0E+00 vegetarian -0.054 8.0E-03 ally 0.052 2.3E-05 rude -0.047 0.0E+00 acquaintance 0.051 4.0E-06 cat -0.045 4.0E-02 vegan 0.048 2.8E-02 excellent -0.044 1.0E-06 property 0.044 1.5E-01 brilliant -0.031 7.7E-03 relative 0.034 2.0E-05 innocent -0.029 2.3E-01 ofof 0.032 3.8E-03 nine -0.027 4.0E-06 pet 0.026 3.8E-01 college -0.027 3.0E-02 simple 0.026 2.8E-03 convinced -0.025 7.2E-05 mine 0.026 0.0E+00 cousin -0.022 9.0E-02 winter 0.022 2.0E-01 sticking -0.021 2.1E-03 animal 0.021 3.4E-01 loud -0.018 0.0E+00 complained 0.020 9.1E-05 Angry -0.018 7.9E-03 beyond 0.020 8.6E-05 bit -0.016 4.0E-04 father 0.019 1.4E-03 anyone -0.016 8.0E-06 Table 17 : 17Top 15 words with greatest magnitude of RD in the specific context for two racial groups with their p-values in permutation tests. Here we use Dropout BERT with general bias mitigated for MCQ predictions.EA Female Distractors EA Male Distractors Word RD p-value Word RD p-value college 0.043 2.3E-03 vegan -0.056 1.2E-02 ally 0.040 4.6E-02 winter -0.055 4.9E-05 innocent 0.029 2.3E-01 father -0.052 0.0E+00 used 0.028 0.0E+00 six -0.046 0.0E+00 camp 0.026 0.0E+00 married -0.042 0.0E+00 pet 0.020 4.9E-01 nine -0.041 0.0E+00 asked 0.018 5.2E-04 ten -0.039 0.0E+00 acquaintance 0.017 2.1E-01 parents -0.038 0.0E+00 prefers 0.017 1.7E-01 next -0.037 1.0E-06 cousin 0.016 1.8E-01 10 -0.037 0.0E+00 rude 0.016 5.9E-03 former -0.035 3.0E-03 summer 0.015 3.7E-05 blessed -0.032 0.0E+00 read 0.015 8.7E-04 cat -0.027 5.4E-02 today 0.014 8.5E-04 would -0.027 0.0E+00 glad 0.012 1.7E-02 pregnant -0.025 1.8E-02 Table 18 : 18Top 15 words with greatest magnitude of RD in the specific context for two gender groups with their p-values in permutation tests. Here we use Dropout BERT with general bias mitigated for MCQ predictions.Word RD p-value Word RD p-value cat 0.069 3.5E-02 outgoing -0.060 1.5E-04 innocent 0.054 7.8E-02 funny -0.048 5.0E-06 asked 0.052 6.0E-05 sporting -0.036 1.3E-04 pet 0.041 2.2E-02 quiet -0.035 0.0E+00 acquaintance 0.040 1.0E-04 nice -0.032 6.8E-05 next 0.040 4.4E-02 intelligent -0.019 1.0E-06 watches 0.038 5.2E-02 loud -0.018 1.0E-06 vegan 0.037 2.1E-02 hoping -0.018 1.0E-06 violent 0.036 0.0E+00 friendly -0.017 1.0E-06 promotes 0.034 1.0E-06 normally -0.017 1.0E-06 animal 0.033 2.4E-05 caring -0.017 3.0E-06 said 0.033 2.7E-03 nursing -0.016 1.1E-05 among 0.032 0.0E+00 convinced -0.016 7.9E-04 fo 0.032 9.6E-02 pretty -0.014 0.0E+00 personal 0.031 8.0E-06 talkative -0.014 0.0E+00 Table 19 : 19Top 15 words with greatest magnitude of RD in the specific context for two racial groups with their p-values in permutation tests. Here we use CDA BERT with racial bias mitigated for MCQ predictions.EA Female Distractors EA Male Distractors Word RD p-value Word RD p-value acquaintance 0.045 7.3E-03 sticking -0.052 8.3E-03 three 0.023 1.1E-03 brilliant -0.052 1.4E-02 outgoing 0.022 1.0E-02 father -0.039 0.0E+00 son 0.020 1.5E-04 Happy -0.037 3.3E-01 babies 0.019 2.1E-05 fo -0.032 1.2E-05 na 0.019 2.9E-01 excellent -0.031 3.4E-05 used 0.018 2.4E-04 former -0.029 1.5E-03 dead 0.018 1.6E-01 next -0.027 2.2E-02 husband 0.018 5.8E-03 boot -0.025 4.7E-03 vegetarian 0.017 1.8E-01 read -0.024 3.5E-05 dear 0.015 5.4E-02 stressed -0.022 0.0E+00 cat 0.014 4.9E-01 run -0.022 1.5E-02 kids 0.013 1.2E-05 beyond -0.021 3.1E-04 old 0.013 8.4E-02 wary -0.020 3.9E-03 watched 0.012 4.4E-01 movie -0.020 4.9E-01 Table 20 : 20Top 15 words with greatest magnitude of RD in the specific context for two gender groups with their p-values in permutation tests. Here we use CDA BERT with gender bias mitigated for MCQ predictions. https://data.cityofnewyork.us/Health/ Popular-Baby-Names/25th-nujf https://data.cityofnewyork.us/Health/ Popular-Baby-Names/25th-nujf In an alternative setup, we first apply a post-hoc debiasing algorithm to a BERT model and then finetune the debiased model on Social IQa. We find that, despite the different ordering of finetuning and debiasing, a debiased model keeps exhibiting disparate behavior towards different names. AcknowledgementsWe would like to thank the anonymous reviewers for their constructive feedback on this paper. We would also like to thank Hal Daumé III, Trista Cao, Shramay Palta, and Chenglei Si for their helpful comments.B Distractor ValidityWe manually inspect 1,000 random distractors to ensure their validity. A valid distractor for social commonsense reasoning MCQ should describe a consequence or reaction that is almost impossible Learning bias-reduced word embeddings using dictionary definitions. Haozhe An, Xiaojiang Liu, Donald Zhang, 10.18653/v1/2022.findings-acl.90Findings of the Association for Computational Linguistics: ACL 2022. Dublin, IrelandAssociation for Computational LinguisticsHaozhe An, Xiaojiang Liu, and Donald Zhang. 2022. Learning bias-reduced word embeddings using dic- tionary definitions. In Findings of the Association for Computational Linguistics: ACL 2022, pages 1139- 1152, Dublin, Ireland. Association for Computational Linguistics. Language (technology) is power: A critical survey of "bias" in NLP. Solon Su Lin Blodgett, Hal Barocas, Iii Daumé, Hanna Wallach, 10.18653/v1/2020.acl-main.485Proceedings of the 58th Annual Meeting of the Asso. the 58th Annual Meeting of the AssoSu Lin Blodgett, Solon Barocas, Hal Daumé III, and Hanna Wallach. 2020. Language (technology) is power: A critical survey of "bias" in NLP. In Pro- ceedings of the 58th Annual Meeting of the Asso-
[ "https://github.com/haozhe-an/" ]
[ "Forecasting of COVID-19 cases using Statistical Models and Ontology-based Semantic Modelling: A real time data analytics approach", "Forecasting of COVID-19 cases using Statistical Models and Ontology-based Semantic Modelling: A real time data analytics approach" ]
[ "Sadhana Tiwari \nIndian Institute of Information Technology\nAllahabadIndia\n", "Ritesh Chandra \nIndian Institute of Information Technology\nAllahabadIndia\n", "Sonali Agarwal \nIndian Institute of Information Technology\nAllahabadIndia\n" ]
[ "Indian Institute of Information Technology\nAllahabadIndia", "Indian Institute of Information Technology\nAllahabadIndia", "Indian Institute of Information Technology\nAllahabadIndia" ]
[]
SARS-COV-19 is the most prominent issue which many countries face today. The frequent changes in infections, recovered and deaths represents the dynamic nature of this pandemic. It is very crucial to predict the spreading rate of this virus for accurate decision making against fighting with the situation of getting infected through the virus, tracking and controlling the virus transmission in the community. We develop a prediction model using statistical time series models such as SARIMA and FBProphet to monitor the daily active, recovered and death cases of COVID-19 accurately. Then with the help of various details across each individual patient (like height, weight, gender etc.), we designed a set of rules using Semantic Web Rule Language and some mathematical models for dealing with COVID-19 infected cases on an individual basis. After combining all the models, a COVID-19 Ontology is developed and performs various queries using SPARQL query on designed Ontology which accumulate the risk factors, provide appropriate diagnosis, precautions and preventive suggestions for COVID Patients. After comparing the performance of SARIMA and FBProphet, it is observed that the SARIMA model performs better in forecasting of COVID cases. On individual basis COVID case prediction, approx. 497 individual samples have been tested and classified into five different levels of COVID classes such as Having COVID, No COVID, High Risk COVID case, Medium to High Risk case, and Control needed case.
10.48550/arxiv.2206.02795
[ "https://export.arxiv.org/pdf/2206.02795v2.pdf" ]
249,431,593
2206.02795
6e16a21fb216faf2cedb6cc79b980fdb7e23add7
Forecasting of COVID-19 cases using Statistical Models and Ontology-based Semantic Modelling: A real time data analytics approach Sadhana Tiwari Indian Institute of Information Technology AllahabadIndia Ritesh Chandra Indian Institute of Information Technology AllahabadIndia Sonali Agarwal Indian Institute of Information Technology AllahabadIndia Forecasting of COVID-19 cases using Statistical Models and Ontology-based Semantic Modelling: A real time data analytics approach [email protected] 1 , [email protected] 2 , [email protected] 3Semantic modellingOntology developmentSARIMAFBProphetSPARQL query, COVID-19 prediction SARS-COV-19 is the most prominent issue which many countries face today. The frequent changes in infections, recovered and deaths represents the dynamic nature of this pandemic. It is very crucial to predict the spreading rate of this virus for accurate decision making against fighting with the situation of getting infected through the virus, tracking and controlling the virus transmission in the community. We develop a prediction model using statistical time series models such as SARIMA and FBProphet to monitor the daily active, recovered and death cases of COVID-19 accurately. Then with the help of various details across each individual patient (like height, weight, gender etc.), we designed a set of rules using Semantic Web Rule Language and some mathematical models for dealing with COVID-19 infected cases on an individual basis. After combining all the models, a COVID-19 Ontology is developed and performs various queries using SPARQL query on designed Ontology which accumulate the risk factors, provide appropriate diagnosis, precautions and preventive suggestions for COVID Patients. After comparing the performance of SARIMA and FBProphet, it is observed that the SARIMA model performs better in forecasting of COVID cases. On individual basis COVID case prediction, approx. 497 individual samples have been tested and classified into five different levels of COVID classes such as Having COVID, No COVID, High Risk COVID case, Medium to High Risk case, and Control needed case. Introduction Coronavirus (COVID- 19) illness is basically a worldwide spreading pandemic, which affects most of the people's lives (in billions) across the world. It is generated by acute respiratory syndrome coronavirus 2 (SARS-CoV-2) virus in 2019. The new SARS-CoV-2 virus was initially introduced in December 2019, Wuhan, China [1]. Plenty of affected countries are taking important measures to set the limit on spread of coronavirus around the world and they have failed yet. On January 30, 2020, a catastrophic situation was announced by the World Health Organization (WHO) and on March 11, 2020 the pandemic became an international level issue [2]. This is the deadliest event in history. Time series data forecasting is necessary for analysing dynamically changing COVID data [2] [3]. The technologies related to big data analytics, real time data mining, machine learning (ML), Artificial Intelligence (AI), statistical data modelling and semantic modelling have played a very very essential part in identifying and tackling the situation of COVID-19 emergence [3] [4] [5]. There is an immense need to tackle the spread of SARS-CoV-2 Infectious Disease outbreak by applying data analytics techniques to strengthen the current economic, social and medical emergency situations [2]. The shortage of a consistent framework or prediction model creates a barrier which can be solved with the help of data analytics, machine learning, and semantic web modelling [3] [6]. The ontology is helpful to provide a formal definition and description of COVID-19 basics like age group, symptoms, infection rates, contact tracing, drug modelling, and provides a compelling solution to fill this gap [6][7] [8]. Many different data sources can be combined through Ontology-based solutions to provide better support for epidemiological situations, and identify pandemic hotspots to support argument, evidence, and knowledge-based recommendations for smart lock solutions [9] [10]. To handle the various issues created through SARS-CoV-2, the fusion of statistical time series modelling, semantic modelling through ontologies and big data analytics (BDA) provides an efficient solution for the improvement of the economy, human lives, businesses and occupations across the world [11] [12].The stock market is collapsing day by day as the global financial market is undergoing major changes. However, due to the current epidemic, the Indian economy is slowing down and it is very difficult to recover from it [13]. There are also many significant effects of this infectious disease on the human health systems and social facets of the society. The shortage of diagnostic kits / accessories is also an obstacle to the effective identification and treatment of infected people to control other infections. As a result, the need for sensitive diagnostic tools among healthcare professionals remains important to enable faster identification of potential COVID-19 cases [8] [13] . Researchers are exploring the potential computation models to contribute in this situation of crisis, as current traditional AI models pose challenges, while exploring ways to improve the accuracy of COVID-19 case prediction [12] [14]. In this sense, the performance of the traditional machine learning prediction models and big data analytics forms the basis for its use in the detection and prediction of Covid-19 cases [15]. This study investigated an ontology based decision support system integrated with the statistical models for time series data forecast and prediction using big data technology as a real-world model for COVID-19 detection and prediction. The prime contribution of this research is as follows: • Identify the nature of COVID data. • Convert the non-stationary COVID data into stationary data using some signification transformation like Log, Subtracting simple Rolling Averages and using the shift() method for accurate prediction. • To perform accurate forecasts use ARIMA and FBprophet models. • Compare the performance of both the models using MAE, MAPE and RMSE performance measures. • Prediction of COVID cases using these models. • Propose an Ontology based Decision Making System using SWRL rules. • Incorporation of the proposed mathematical model for precautionary suggestion of COVID patients. Background and Related work On March 11, 2020, WHO declared the novel coronavirus (COVID-19) outbreak a global pandemic [1]. With more than 5 million deaths around the world, the COVID-19wide spread pandemic is considered to be the most serious and critical incident after World War 2. In such unprecedented circumstances, it becomes essential to understand the growth and behaviour of the pandemic [1] [2]. Many prediction methods have been developed in the literature for the detection and prediction of active case, deaths and recovered cases from the COVID-19 dataset. These prediction models are also helpful in predicting the future cases in any particular country or region [3]. Time series data forecasting using Statistical models One of the most prominent methods used in time series prediction is the Auto Regressive Integrated Moving Average (ARIMA) model [4] [5]. This model allows the customization of time series data for prediction of future occurring data of the series. Seasonal ARIMA (SARIMA) is another model for time series data prediction including seasonality [16]. Statistical Time series models are helpful for modelling and predicting time-indexed data. ARIMA model can be broken into three main terms as AR, I, MA: where, AR (p) is used for an autoregressive model by passing parameter p which represents an integer and which certifies the number of lagged series used for future predictions. I (d) represents the differential part and parameter d informs about differencing orders which are being applied for constructing the series stationary. MA (q) is used for moving average where parameter q tells about the terms of total lagged forecast error in prediction of time indexed data. By carrying out a pragmatic analysis of the dataset, the identification of a suitable model is necessary, that could be readily used by researchers, medical experts, organizations, society, and governments to investigate the recent future outbreak of this event. The logic behind the selection of ARIMA and SARIMA time series model are as follows: i) These models provide very satisfactory results in predicting natural adversity compared to other predictive models like the Wavelet Neural Network (WNN) model and the Support Vector Machine (SVM) model. ii) The statistical models have been used before in similar crisis situations, such as during the SARS epidemic. This model is used to predict the number of beds occupied by the Tan Tock Bed Hospital in Singapore in real time [5]. Furthermore, Anwar et al. Developed a tool to predict malaria patterns in Afghanistan using the ARIMA model [16]. Time series data forecasting through Facebook Prophet (FB-Prophet) models Facebook Prophet is an additive model-based time series forecasting method that uses nonlinear trends to determine daily, weekly, monthly, and yearly inclination, and also focuses on holiday effects. The FB-Prophet model is resilient to various data pre-processing issues like missing value, repeated values and trend changes and generally handles outliers very well [18]. This model can be trained just like a curve fitting practice and ignores the time sensitive part of the data. Therefore, improper observation may be permitted in the dataset [19]. The various benefits of applying this model on time series data, are: Offers multi-period seasonality; Suitable for known and customized holidays; Offers flexibility with two popular options: 1. Linear model 2. Saturation growth model which can adapt changes very quickly. [20] This predictive modelling technique is beneficial to rename the Date and the output column to predict future cases. Furthermore, the Date should be converted to Date Time format. Combining Statistical data models with Semantic data modelling using Ontologies Statistical data models, machine learning and semantic data modelling using ontologies are the most commonly used technologies helping behaviour, trends and data prediction. Although the performance of these models is not limited to the data availability but also depends on the quality of input data [7] [11]. For efficient data processing structured information is needed for machine learning and training to achieve good prediction accuracy of Covid-19 dataset. So, transformation of unstructured data into meaningful information is one of the immense challenges. For this purpose computational ontologies are helping a lot for handling unstructured, completely insufficient and heterogeneous data. Earlier Machine learning, statistical data modelling and semantic modelling through ontologies are considered as different approaches, but nowadays, fusion of these technologies are becoming popular due to huge amounts of data and increasing complexity [7]. Table 1 presents the summarized literature review in the area of predictive modelling and Semantic web modelling with the help of ontologies and machine learning. After studying various research in COVID-19 prediction, it is identified that there is significant scope for work in forecasting and tracking the situation of COVID-19. Applying different machine learning models, statistical forecasting approaches [23][24] and semantic web modelling through ontology [9][13], seems a novel interdisciplinary idea for COVID-19 prediction, prevention and treatment. In such unprecedented circumstances, it becomes essential to understand the growth and behaviour of the pandemic. This research is broadly classified as follows: (1) Application of machine learning, big data analytics and statistical modelling to identify patterns, nature and trends of several time variant events related to various infectious diseases. (2) Latest research techniques such ontology based modelling is used, which strictly focuses on predicting COVID-19 outbreak-related statistics such as active cases, deaths, recovered cases, chances of getting infected on behalf of body symptoms and chances of getting recovered using statistical models and semantic models. Proposed Methodology Dataset Description In present research, the Statistical data published by Our World In Data (OWID) Organization [25] is obtained from kaggle [26]. The Data is available in CSV file format and includes information regarding new cases, total cases, and total deaths. Total number of columns in the dataset are 10 named as date, location, new_cases, new_deaths, total_cases, total_deaths, weekly_cases, weekly_deaths, biweekly_cases, biweekly_deaths and size of dataset is approx 4 MB. Structure of the proposed model The primary goal of the predictive analytics and decision support system is to detect severe instances of COVID-19. The proposed decision support has several components as shown in figure 1 and it is utilizing collective knowledge provided by applying various rule editing tools, data mining concepts, statistical data modelling, big data analytics, machine learning, AI and ontology based modelling. These methodologies are helping in both diagnosis and prognosis. It takes account of a variety of body vital parameters and symptoms related to COVID-19. The capacity of human biological responses to combat with the virus is a subject of study in this model. Patients with severe COVID-19 infections are diagnosed by general practitioners, specialists, nurses, and physicians with the use of clinical decisionsupport technologies. In this work, a four layer architecture has been proposed as shown in figure 1. The first layer is responsible for data collection, where data is gathered from various publicly available repositories and from existing biomedical literature. The second layer is responsible for prediction and Ontology development. The total no. of active, deaths recovered cases is predicted using SARIMA and FBProphet model. For ontology development and defining all the precautionary measures the medical literature available on COVID-19 is used. Then we add the prediction model in Ontology. The COVID-19 dataset is also being analysed using ontologies, followed by ontology-based queries processing. Data annotation and storage is the third layer of the architecture where we added the individual in the ontology in which the COVID cases are categorized into five different labels for individual basis COVID case prediction and the training should be properly annotated as per COVID use case. Store the data into Resource description Format (RDF) using COVID-19 Ontology. In the last layer, using SPARQL to query the ontology for different case situations on behalf of to identify COVID positive and negative cases on the basis of some essential symptoms, as well as other critical symptoms and visualize. In this paper, we perform structural analysis of SARS-CoV data through statistical data modelling and predicting the COVID cases through SARIMA and FBprophet models. The COVID-19 dataset is also being analysed using ontologies, followed by ontology-based queries processing to identify COVID positive and negative instances on the basis of some essential symptoms, as well as other critical symptoms. Predictive data analytics is basically a part of advanced data analytics through which future predictions can be made with the help of existing historical data and also identify the data patterns to figure out the future trends, scope, and risks. This stream data mining can be visualized as a combination of statistical data modelling, data analysis with data mining concepts and machine learning methods [12] [15]. Many data analytics tools, such as Hadoop, Storm, Spark, Mahout, Drill, and SCALATION, are available to support the analysis of large datasets. In the proposed model, statistical data model ARIMA and FB-Prophet and semantic modelling technology (ontology) are used for selecting, constructing, and explaining the prediction of daily active cases, new cases, and total death cases. Statistical models can be properly explained with the help of analytics ontology [7] [15]. Figure 2 shows the primary flow diagram of the proposed model. The stepwise processing the proposed predictive analytics model is as follows: Step I-There are two parts to this first step: loading COVID-19 data and making a process that can be used to manipulate data for simple data pre-processing. Step II-The data extraction step helps in the extraction of key information from the dataset using data and metadata. This step is very essential for the model type selection as well as the validation step. Step III-This step is useful for data refinement and preparation of the data for analysis purposes. This part includes the handling of missing values, handling of duplicate values and identification of auxiliary features of the dataset. Step IV-Exploratory data analysis step is essential for understanding the trends and patterns present in the COVID-19 dataset and discovering the anomalies with the help of graphical visualization and statistical summaries. Step V-The selection of an appropriate model type depends on the dataset characteristics. It is performed by applying domain description across each column of the dataset, and after that, applying predictive analysis using SARIMA and FB-Prophet models to identify the promising model for the respective dataset. Step VI-This step is needed for validation of the selected model and to move back if the model is not performing well as per the requirements and expectations, then the model needs to be refined. Step VII-The performance of the various models will be compared with respect to various performance metrics, and the best suitable model will be selected. Step VIII-In ontology based semantic modelling, some important features will be selected from the dataset and other factors will be associated, such as height, weight, age, gender, and prior illness from some other chronic disease for negative and positive COVID case prediction. Step IX-After comparison, when the best model is selected and validated, and the model is used for prediction of COVID-19 cases, the COVID-19 ontology generates inference rules using Semantic Web Rule Language (SWRL) to cover all queries related to COVID-19 case prediction, including precaution and chances of having a positive or negative RT-PCR. Model description The proposed model employs a statistical time series approach to forecast the coronavirus cases in India for the next 6 months during the first wave of COVID-19. Time series data analysis consists of a series of data that arrives (is indexed or graphed) in a timely manner. Thus, the data must be arranged with moderately deterministic timestamps and can be compared with random samples having some supplementary information. The objective of time series data analysis can be conveyed in two ways: the first way figure out a model which processes an observed sequence of real time data using stochastic mechanisms and the second one forecasts the upcoming values of that sequence according to the past records of the data [27], as shown in figure 3. Figure 3: Time series Analysis of daily new cases in India The analysis of time series data adds an explicit order dependence between observations. It adds a time dimension, which is both a constraint and a structure that provides a source of additional information. ARIMA and SARIMA Modelling for COVID-19 prediction The Seasonal Auto-Regressive Integrated Moving Average (SARIMA) model is a class of models that explain a given time series dataset based on its lags and the lag forecast errors. The initial steps taken in this work deal with performing exploratory data analysis on the COVID-19 dataset so that we can familiarize ourselves with the underlying trends and patterns present in the data [5]. During this process, the missing values present in the data were removed. Another important criterion that serves as the basis for all ARIMA models is that the data should be stationary. A series is said to be stationary if the marginal distribution of the output Y at time t is the same as at any other point in time. In other words, basic statistics such as mean and variance are time-invariant. One popular way of detecting stationarity in the data is by using the Augmented Dickey-Fuller (ADF) test. The ADF test is a statistical test that returns a set of parameters. If the p-value is less than 0.05, then the data is considered to be stationary [15] [16]. 6) Combination of all The proposed work uses three significant transformations, the Log, Subtracting simple Rolling Averages and using the shift() method for forecasting COVID-19 using statistical models as given in figure 5 [16] [17]. Figure 5: Final results of ADF Test The ARIMA model can be applied by using three main factors, the moving averages (q), the auto regressing lags (p), and the integrating factor (d). So, it is required to find the optimal values of p, d, q to find the best fitting ARIMA model. These values can either be determined using the ACF, PACF plots given in figure 6 or the auto ARIMA function. Figure 6: ACF and PACF Plots Algorithm 1 specifies the sequence of steps applied in predicting COVID-19 infected cases using ARIMA and SARIMA models. Algorithm -I 1) Get the list of source data files in the specified directory. 2) Reading the CSV file using the read_csv function. 3) Extracting India's Information: country1=df.loc[(df['location']=='India'),:] 4) Performing ADF Test to check for stationarity of data. 5) If data! = stationary: Making data stationary using above methods 6) Determine ACF and PACF plots for getting optimal values for p, d, q. 6) Divide data in train and test datasets for getting predictions. 7) The arima_model function is used to determine the appropriate values for p,q,d. 8) The predict() of function is used to forecast the cases for the next nine months FB-Prophet model for COVID-19 prediction Another model used in this work is FB-Prophet for prediction, first define and configure a Prophet () object, then call the fit () function to pass the data to fit the dataset. The argument is passed to the Prophet () object to configure the required model type, such as growth type, seasonality type, and so on. The input of the fit () function is a Data Frame of COVID-19 dataset, which requires a specific format. The first column of the dataset must contain the date and time. The next column must contain the observations. This means renaming the columns in the dataset [18] and [20]. Also, if the first column hasn't been converted to a date time object yet, it needs to be converted. Further analysis of COVID-19 data is obtained by considering the FB Prophet model in the Spark Environment. The COVID-19 dataset obtained from Kaggle [26] is analysed and the output is streamed using Spark Context [28] [29]. Semantic modelling through Analytics ontology for COVID-19 prediction Scientific communities are increasingly looking to ontologies to facilitate web-based management and interchange of scientific data, thanks to recent improvements in data modelling and increased usage of the Semantic Web. Ontologies can be used inside a domain to formally express ideas and links between concepts. The resulting logic-based representations provide a conceptual model that can help with data storage, management, and sharing among numerous research groups. In this paper developed COVID-19 ontology is developed, which improves the accuracy of prediction models and aids decision-making as shown in Figure 7. a) Ontology Design phase An ontology is built via a seven-step process: (1) Determine the domain and scope of the ontology. (2) Consider repurposing existing ontologies (3) A list of significant concepts in Ontology (core classes) (4) Classify the classes and their hierarchies. (5) Slots are used to define the properties of classes (6) The facets of the slots must be defined and (7) Instances are created. b) Covid-19 Core Classes and Property The Covid- 19 Ontology, an open source software for creating ontologies, is being developed with the Protégé editor [30]. Several concepts, attributes, and individuals are included in this Covid-19 Ontology, which is based on COVID-19 literature data, Daily medical conceptual data, prescription suggestions, and key metrics and parameters connected to Covid-19 statistical prediction. An ontology starts from the "Owl:Thing" class, which will be broken down into sub-concepts like "case type," "Dependent model prediction," "Disease Cases," "Gender," "Patient report," "Recovery," and so on. Figure 8 represents the class hierarchy in which numerous Object Properties are included such as "has_Symptom," "hasGender," "has_Symptom_Severity," "hasSuggestion", "has_joint_pain," "has_report" and so on. Further, many Data Properties are added such as "has_asthma", "has_depression", "has_age", "has_type1Diabeties", has_weak_immune", "has_type2Diabeties", "is Asymptomatic" and so on. The "OWL individual" declarations are another semantically ordered relation within the classes. The "Ideas" notation within an COVID-19 Ontology implies a range of OWL individual suggestions, such as SUG 1, SUG 2, SUG 3, SUG 4, SUG 5, and so on, that assist the patient in taking precautions in accordance with this proposal. Patient information with several "cases" concepts denote a variety of case types represented by OWL individuals, like CASE 01, CASE 02, CASE 03 etc. Technical specifics concerning semantic rules and inferencing will be discussed in the next section. The system that follows and uses those concepts to produce a prediction about the chances of getting affected by COVID-19, as well as individualised supportive action suggestions will be provided based on the criticality of the case. [31] is used to reintegrate this data into ontology. Through Cellfie we easily convert spreadsheet rows into instances of class and property. Each row has a specific patient, details of COVID, weekly death cases, weekly infected cases, total cases etc. Table 2 shows the transformation rules of how the dataset is converted into an ontology. Individuals: @B*(mm:hashEncode rdfs:label=("Patient", @B*)) Types: Patient Facts: 'diagnosed on' @B*(xsd:dateTime), 'Age' @C*(xsd:decimal), 'Has gender' @D*, 'State' @F*, 'City' @E*, 'Traveled from' @G*, 'Nationality' @I*, 'Weeklydeath'@W* 'Location'@l*, 'Toatlcases'@T*, 'Newdeaths'@D*, 'Biweeklydeaths@BW*, 'Status' @J*, 'Deaths'@Z*, 'has resulted in any other infections' @L*(xsd:boolean) c) The Reasoning Rules for COVID-19 Ontology The Semantic SWRL [32] is a programming language which is used for defining semantic rules to provide improved provable reasoning capacity. SWRLs are being used to build and validate semantic rules that provide the user with a mix of issue definition facts and knowledge base inference. For designing this ontology, the SWRL rules and OWL axioms use the ROWL plugin [33], available on the top portion of Protege's SWRL user interface. This interface gives the facility to supply input in the form of semantic rules. Any user can generate a rule by applying the SWRL syntax available in the ROWL tab. When any one uses the ROWL plugin to add a rule to the ontology and converts it to OWL axioms, also ROWL increases the amount of OWL axioms. For Covid -19 Ontology, 43 SWRL rules have been designed based on four main criteria: (1) Mathematical statements, (2) Considerate having COVID-19 through probability, (3) Precautionary guidelines and tailored supporting action recommendations, and (4) SWRL rules for predicting COVID-19 Cases using FBprophet and ARIMA model. Various parameters related to any person, such as weight, height, age, gender, temperature, and some newly observed symptoms (as per their severity levels, likewise, severe, moderate, and mild) are used as inputs in the majority of the rules. Furthermore, the construction of the rules includes various parameters of symptom categories from the mathematical formulation used. By using the patient data and symptom factors, a prediction score is calculated for the likelihood of the chances of having COVID-19. As per the severity level, the COVID infected patients have been divided into five categories: (1) Asymptomatic cases, (2) Severe COVID symptoms, (3) Moderate COVID symptoms, (4) Healed Covid symptoms, (5) and Mild Covid symptoms. The category to which the case belongs is selected based on the score, and a selection of specific supportive action options designated for that category is offered. Following the calculation of the chance of obtaining Covid-19 scores based on statistical models, the suggestion is given according to the score of case type classification, which is mentioned in figure 9. Each case has a different precaution and suggestion linked to it. A total of seven cases are included in our ontology. Figure 9. Hierarchy of the Case type d) SWRL rules for Mathematical statements The complex mathematical formulation can be generated with the help of SWRL rules applying SWRL mathematical expressions resident library [34]. A rule for computing Body Mass Index (BMI) using the SWRL's built-in function is given below: Where, "? p" represents patient information, "?h", "?w", "?BMI" represents height, weight, and BMI values, respectively. The "swrlm:eval" function determines the value of the "?BMI" variable. Here the symbol "^" is used as a "AND" in logical operators. Input is taken as height and weight for solving BMI of the patient. e) Probability Calculation Rules having COVID-19 The first rule is based on the patient attribute age, sex, loss of smell, loss of flavour, cough, severe fatigue, and missed meals. The second rule is based on the patient attribute age, sex, loss of smell, loss of taste, cough, severe fatigue, and skipped meals. In equation 1, if symptoms are yes then '1', if no means '0'. Gender represented '1' as male and '0' as female. As a result of solving equation 1, this value will be translated to the expected probability using the formula exp(x)/ (1+exp (x)). Estimated COVID-19 cases are assigned odds more than 0.5, whereas controls are assigned probabilities less than 0.5. All SWRL rules for equation 1 are correlated to each other which is shown in figure 10. f) Rules for Precautionary guidelines and tailored supporting action recommendations Some Rules are divided into five categories (1) Asymptomatic Case, (2) Severe COVID symptoms, (3) Moderate COVID symptoms, (4) Healed COVID symptoms, (5) and Mild COVID symptoms. Each category has different rules and parameters which define the case type (which is shown in Figure 12) and precautionary steps (https://www.who.int/emergencies/diseases/novel-coronavirus-2019/advice-for-public). In Table 3, some other case situation rules of COVID-19 like "Self Quarantine No Medication", "Intensive Case with Medication", etc. are listed. In Figure 11, we demonstrate how the Pellet Reasoner [37] works in Protege finds CASE 03, which comes under the Moderate COVID Symptoms case category. Patient number 101 is male and adult with mild symptoms. What precautionary steps have to be taken, also provided by inference rules. A total of 1250 patient details are included in the COVID-19 ontology and 43 SWRL rules. The pellet reasoner in protege is used to test this ontology. It takes inferences from SWRL rules and outputs the results based on the patient's information, as illustrated in Figure 11. Figure 11: Working of Protege tool Figure 11 depicts the protege tool's interface face structure, which demonstrates this reasoning. 1. Inference output gives the moderate COVID-19 case, which comes under the CASE 03 class. 2. Moderate inference rules retrieve the suggestions in accordance with the parameters, which is revealed in step 5 of the process. 3. SWRL regulations determine whether a patient is an adult or a minor. 4. The SWRL rules are required to verify the patient's gender. 5. Take a look at the patient's parameter information, which is depicted in figure 11 on the right side with a yellow outline. 6. Figure 12 illustrates which parameters should be checked in moderate cases, such as fever, cough, and body aches. Four explanations for each output: "genderReport" is "true", "has_moderate_COVID_19_symtoms" is "true" and "teenagerORadult" is "adult". Each explanation is linked with precautionary suggestions for COVID-19. This explanation is shown by the interface face framework of the protege tool which is shown in Figure 11. Fusion of FBprophet and ARIMA models in designing COVID-19 Ontology for predicting COVID-19 Cases We examined the ARIMA and FBprophet models in the preceding section. Now we will explain how we used time series data to create inference rules that made our ontology more perfect. We get the result with a time series which helps in taking precautionary measures on time. It also helps to find out the average death rate for every place using SWRL rules based on time, since weekly deaths, biweekly deaths, new deaths, and total deaths are all accessible in a triplestore format [36]. Figure 14 in yellow shows the inference results after running the pellet ARIMA and FB prophet models, as well as a description of the patient's location, date, and new cases. Figure. 14. ARIMA and FBprophet model applied on desperation of Patient details Prediction model 1, Prediction model 2, ARIMA, and FP_ prophet model are all correlated in COVID-19 ontology, making it more suited and excellent. It also employs 43 SWRL rules for mathematical reasoning and ontology restrictions. Most sorts of questions are addressed by our ontology, such as (1) death-related inquiries handled by the ARIMA and FB prophet models, such as average death per week, month, and year, as well as new cases. (2) Prediction Mode 1 and 2 cover specific patient inquiries such as the likelihood of having COVID-19 or not. It also notifies about the patient's chances of survival based on their history of sickness (like diabetes, having any lung diseases etc). It reassures all the preventive steps that should be taken based on the scenario. Figure 15 depicts how the four models are connected to one another. All models that work on patient details include details about time series, gender, age, and so on. It also explains case types, which were already covered in the preceding section. Querying on Covid-19 Ontology with SPARQL Query The SPARQL query engine allows us to query on ontologies [39]. SPARQL may be used to express searches across a range of data sources, whether the data is stored natively as RDF [36] or accessible as RDF via middleware. SPARQL may query both required and optional graph patterns, as well as their conjunctions and disjunctions. SPARQL also supports extensive value checking and query restrictions that are based on the underlying RDF graph. In addition to querying, SPARQL may delete, insert, and change data. Figure 16 shows the Protege tools output for the patient with an omicron using a SPARQL query. Table 4, two other query which have finding age who greater than 30, has COVID positive, patient who have diagnosis. Another query finding each day's total active case is given in Table 5. Same way query On COVID-19 Ontology about any scenario of patient related to COVID which helps in Decision making in pandemic situations. Experimental Results and Discussion In this section, the proposed model uses statistical prediction models, ARIMA and FBProphet, to forecast the upcoming cases. ARIMA can be used for prediction if the data is stationary. Therefore, various techniques have been applied to check the stationarity of data and transform the data if it is not stationary. In this experiment, data stationarity is checked using an augmented dicky-fuller and for deciding the optimal values of 'p' and 'q', PACF and ACF plots are used in the case of SARIMA model performance. The FBProphet model can be directly applied to the original dataset. The accuracy results are computed using both the prediction models for active COVID cases in India. The evaluation metrics and order used for the SARIMA model are mentioned in table 6. figure 18 show that the prediction errors using SARIMA are very small as compared to the FBProphet model, which has a high error. It is observed that the SARIMA model performs better for the correct prediction of infected, recovered, and death cases to provide proper movement of the services during the COVID pandemic situation. The estimated range of COVID-19 figures reported by the SARIMA model and FBProphet models for the upcoming days during the first wave of the pandemic can be visualized through figure 19 and 20. By observing the graph mentioned in figure 19 and 20, it can determine that hereafter the risk of contamination in India is likely to remain constant for a while and eventually decrease. The speculated overall recovery rate in India holds an optimistic trend for the upcoming days as the recovery rate is considerably higher than the number of new infections. Table 8 shows successful test cases of COVID patients in different complications like having COVID, no COVID etc. and computes the accuracy of every complication category and figure 21 analyses the same by doing comparative analysis of five defined classes as per complexity levels. Figure 1 . 1Architecture for fusion of Prediction Model to develop Ontology based decision support system Figure 2 : 2Workflow of the proposed model Figure 4 : 4The initial results of the ADF test From the results shown infigure 4, it can be observed that the data is non-stationary since the p-value crosses the 0.05 mark. Non-stationary data can be converted into stationary data through a sequence of transformations applied to the time series data. These transformations are as follows: 1) Taking Log 2) Subtracting simple Rolling Averages 3) Subtracting Exponential Rolling Averages 4) Subtracting Previous Values using shift() 5) Seasonal Decomposition Figure 8 . 8Core Classes, Data Property, Object Property and Individuals (Instances) of COVID-19 Ontology SWRL Rule of Body Mass Index: patient(?p)^has_weight(?p,?w)^has_height(?p,?h)swrlm:eval(?BMI, "703*w/(pow[h, 2])",? w,? h) -> hasBMI(?p,? BMI). Prediction Model 1 in [35] = -1.30 -(0.01 * age) + (0.44 * sex) + (1.75 * loss of smell and taste) + (0.31 * severe persistent cough) + (0.49 * severe fatigue) + (0.39 * skipped meals) ---(1) Figure 10 . 10SWRL rules for Model 1 and its working processInfigure 10, all six rules output is input for the second last rule and output of the second last rule is the input for the last rule which gives the output for equation 1.In Prediction model 2, 10 symptoms have been considered: chest pain, hoarse voice, abdominal pain, skipped meals, and delirium, and diarrhoea, shortness of breath, fatigue, persistent cough, and fever. Here, both models are based on Logistic regression.Prediction Model 2 in [36] = -2.30 + (0.01 * age) -(0.24 * sex) + (1.6 * loss of smell stage) + (0.76 * fever) + (0.33 * persistent cough) + (0.25 * fatigue) + (0.31 * diarrhoea) + (0.46 * skipped meals) -(0.48 + abdominal pain) -(2) As a result, precise SWRL rules are derived from equation 1 and equation 2 to estimate the probable chances of getting a COVID-19 score. It is further categorized in 3 different classes based on prediction score (1) "Higher Risk COVID-19 Case", if probability value is greater than 0.85. (2) "Medium-To-High Risk COVID-19 Case", if probability in between 0.5 and 0.85. (3) "Control-Needed Case", if probability less than 0.5. These all classes give risk factors for COVID-19 patients. Figure 12 . 12Moderate Case 03 Depends on Cough, fever and body pain 7. Finally, it informs you of all precautionary measures (https://www.who.int/emergencies/diseases/novelcoronavirus-2019/advice-for-public) as well as what to do in the current circumstances, based on recommendations for action. We can utilize SPARQL Query, which is covered in a subsequent part, to learn more about a patient. Figure 13 , 13tells about execution flow of patient 101 after running the reasoner using pellets and takes 196 ms time for the execution. Figure. 13 . 13Patient 101 execution flow console. Figure 15 . 15ProtégéVOWL visualization of ontologies[38] Figure 16 . 16SPARQL results have which patients have omicrons Also in Figure 17 .Figure 18 : 1718Comparative analysis of RMSE values Comparative analysis of MSE values Figure 17 and Figure 19 :Figure 20 : 1920Forecast Results of ARIMA Model Forecast Results of FB Prophet Model Later, Benevento et al. have used this method to generate the upcoming predictions of COVID-19 worldwide[17]. Table 1 : 1Fusion of predictive data analytics and Semantic (ontology) modelling in COVID-19 scenario Paper ID Problem Proposed solution Dataset/use case used Tool type Outcome [8] 2021 Early detection of COVID-19 patients having high level of complication Predict critically of any patients within 28 days of diagnosis including symptoms, isolation, demographics, treatment, comorbidities and hospitalization US electronic health records (IBM Explorys) Novel Web Platform Various performance metrics ROC AUC, PR AUC, Brier score, Log loss, Sensitivity, Specificity, and F1- score have been computed. [6] 2020 Prediction of COVID -19 pandemic cases A novel solution is proposed which combines AI, Big data and Semantic web services (SWS) Coronavirus prediction case study Protege (Standalon e Platform) Detection of suspected patients of COVID-19 to accurately identify the spreading rate. [7] 2021 Monitoring physiological parameters of students during COVID-19 An ontological framework CCOnto to describe situational behaviour in humans COVID- 19 use case COVID19 dashboard Analysing change in behaviour of university students due to COVID pandemic through ontological rules. [9] 2020 Critically analysing interconnection among hosts and different variants of coronaviruses Ontology based solution is proposed and classification of host- coronavirus interactions (HCI) and disease prediction 35 tested and originally collected HCI protein-protein interactions (PPIs) Python Proposed logical ontology with comput ational prediction model develops understanding of human patients of For making the ontology we took the data from the official website of the Ministry of Health and Family Welfare [REF website https://www.mohfw.gov.in/]. This dataset is collected by the researchers of Indian Statistical Institute //www.ncbi.nlm.nih.gov/sars-cov-2/]. The Cellfie Protégé plugin[ref: https://www.isibang.ac.in/~athreya/incovid19/data.html] also used SARS-CoV-2 data [REF: https: Table 2 : 2Transformation rule Table 3 . 3SWRL rules for COVID19 different case situation Table 4 : 4SPARQL query for patient diagnosis detected COVID positive who has age greater than 30 PREFIX rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#> PREFIX owl: <http://www.w3.org/2002/07/owl#> PREFIX rdfs: <http://www.w3.org/2000/01/rdf-schema#> PREFIX xsd: <http://www.w3.org/2001/XMLSchema#> SELECT DISTINCT ?patient ?gender ?age ?Diagnosis WHERE { ?patient rdf:type ?has_positive. ?patient owl:has_age ?age. ?patient owl:has_gender ?gender.?patients rdf: is_diagnose ?Diagnosis. FILTER(?age>30)} ORDER BY (?Patient) Table 5 : 5SPARQL query for each day's total active case PREFIXrdf:<http://www.w3.org/1999/02/22-rdf-syntax-ns#> PREFIX owl: <http://www.w3.org/2002/07/owl#> PREFIX rdfs: <http://www.w3.org/2000/01/rdf-schema#> PREFIX xsd: <http://www.w3.org/2001/XMLSchema#> SELECT ?datePosted (sum(?totalActiveCase) as ?nationalTotal) FROM <http://www.w3.org/1999/02/22-rdf-syntax-ns#> { ?prov rdf :Covid_Case_In_India_Karnatka ; rdf:totalActiveCase ?totalActiveCase ; xsd:datePosted ?datePosted. } GROUP BY ?datePosted ORDER BY desc(?datePosted) Table 6 : 6Evaluation Metrics for SARIMA Model The evaluation metrics computed using FBProphet model are given in table 7. Table 7 : 7Evaluation Metrics for FB Prophet ModelThe experimental results show the identified ARIMA order fits better and improves the performance due to the lower MAE, RMSE values. Table 8 : 8Test cases performed through COVID-19 Ontology S. No. AcknowledgementsThis research is supported by "Extra Mural Research (EMR) Government of India Fund by Council of Scientific & Industrial Research (CSIR)", Sanction letter no. -60(0120)/19/EMR-II. Authors are thankful to CSIR for providing the necessary equipment for the research. Authors are also thankful to the authorities of "Indian Institute of Information Technology, Allahabad at Prayagraj '', for providing us infrastructure and necessary support.Also check the ontology quality score through the metric of the ontology[40]based on the knowledge being represented infigure 22, which reflects the relationship and qualities of the domain knowledge according to equation 3.….Figure 22. Metrics of COVID-19 ontologyOn the basis of the efficiency with which base knowledge can be extracted[40], according to equation 4.Ontology score is based on the formula intable 9for score_rk and score_bk.Conclusion and Future ScopeThis paper explores the relevance and usefulness of various time series prediction models for analysing COVID-19 cases in India. A SARIMA-based prediction model, and FBProphet model are used for the prediction of the total number of daily new cases. The proposed model is fruitful in the current scenario for predicting future infected cases if the virus spread pattern does not change adversely. When the SARIMA model is used in the COVID-19 scenario, the predictions are generated on the basis of the series' prior values and error lags, which allows the model to alter its forecast values in the event of a sudden shift in trends. Since our interest is in creating short-term predictions using time series data and generating the outcome in the coming months, the SARIMA model appeared to be the best fit. A COVID -19 ontology is also designed to perform individual patient prediction into five designated classes such as Having COVID, No COVID, High Risk COVID case, Medium to High Risk case, and Control needed case. Apart from this, ontology based semantic modelling helps in predicting the chances of suffering from severe coronavirus infection depending on various individual factors like age, weight, gender, immune system, and having previously suffered from any other chronic disease. It helps in a better way to calculate the percentage score of getting affected by the virus and suggests what precautions and treatment should be taken while suffering from this infectious disease. 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M Saleem, G Szárnyas, F Conrads, S A C Bukhari, Q Mehmood, A C Ngonga Ngomo, The World Wide Web Conference. Saleem, M., Szárnyas, G., Conrads, F., Bukhari, S. A. C., Mehmood, Q., & Ngonga Ngomo, A. C. (2019, May). How representative is a sparql benchmark? an analysis of rdf triplestore benchmarks. In The World Wide Web Conference (pp. 1623-1633). Semantics and complexity of SPARQL. S Lohmann, S Negru, F Haag, T Ertl, J 39pérez, M Arenas, C Gutierrez, Semantic Web. 7Visualizing ontologies with VOWLLohmann, S., Negru, S., Haag, F., & Ertl, T. (2016). Visualizing ontologies with VOWL. Semantic Web, 7(4), 399-419. [39Pérez, J., Arenas, M., & Gutierrez, C. (2009). Semantics and complexity of SPARQL. ACM Transactions on Database Systems (TODS), 34(3), 1-45. Semantic rule Web-based Diagnosis and Treatment of Vector-Borne Diseases using SWRL rules. R Chandra, S Tiwari, S Agarwal, N Singh, arXiv:2301.03013arXiv preprintChandra, R., Tiwari, S., Agarwal, S., & Singh, N. (2023). Semantic rule Web-based Diagnosis and Treatment of Vector-Borne Diseases using SWRL rules. arXiv preprint arXiv:2301.03013. Author's Bibliography Sadhana Tiwari received the B. Tech. Degree in Computer with a focus in CSE and completed the. Author's Bibliography Sadhana Tiwari received the B. Tech. Degree in Computer with a focus in CSE and completed the Degree in Software Engineering from IIIT Allahabad, India. Currently I am working as a research scholar in IIIT Allahabad, India in the area of big data analysis and stream data mining in biomedical domain. M Tech, M. Tech. Degree in Software Engineering from IIIT Allahabad, India. Currently I am working as a research scholar in IIIT Allahabad, India in the area of big data analysis and stream data mining in biomedical domain. The main research interests are in the areas of Stream Analytics, Big Data, Stream Data Mining, Complex Event Processing System, Support Vector Machines and Software Engineering. Dr. Sonali Conducted many workshops and seminars related to the area of big data analytics and real time stream analytics. Csir Serb, Etc, Computer with a focus in CSE and completed the M. Tech. Degree in Information Technology from NIT. Patna, India; Allahabad, IndiaCurrently I am working as a research scholar in IIIT Allahabad, India in the area of Ontology Engineering and big data analysis in healthcare domain. Sonali Agarwal, working as an Associate Professor in the Information Technology Department of the Indian Institute of Information Technology (IIIT). She received many funded projects form DSTRitesh Chandra received the B. Tech. Degree in Computer with a focus in CSE and completed the M. Tech. Degree in Information Technology from NIT Patna, India. Currently I am working as a research scholar in IIIT Allahabad, India in the area of Ontology Engineering and big data analysis in healthcare domain. Sonali Agarwal, working as an Associate Professor in the Information Technology Department of the Indian Institute of Information Technology (IIIT), Allahabad, India. She received her Ph. D. Degree at IIIT Allahabad and joined as faculty at IIIT Allahabad since October 2009. The main research interests are in the areas of Stream Analytics, Big Data, Stream Data Mining, Complex Event Processing System, Support Vector Machines and Software Engineering. Dr. Sonali Conducted many workshops and seminars related to the area of big data analytics and real time stream analytics. She received many funded projects form DST, SERB and CSIR etc.
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[ "Entropy-and Distance-Based Predictors From GPT-2 Attention Patterns Predict Reading Times Over and Above GPT-2 Surprisal", "Entropy-and Distance-Based Predictors From GPT-2 Attention Patterns Predict Reading Times Over and Above GPT-2 Surprisal" ]
[ "Byung-Doh Oh \nDepartment of Linguistics\nDepartment of Linguistics\nThe Ohio State University\nThe Ohio State University\n\n", "William Schuler [email protected] \nDepartment of Linguistics\nDepartment of Linguistics\nThe Ohio State University\nThe Ohio State University\n\n" ]
[ "Department of Linguistics\nDepartment of Linguistics\nThe Ohio State University\nThe Ohio State University\n", "Department of Linguistics\nDepartment of Linguistics\nThe Ohio State University\nThe Ohio State University\n" ]
[ "Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing" ]
Transformer-based large language models are trained to make predictions about the next word by aggregating representations of previous tokens through their self-attention mechanism. In the field of cognitive modeling, such attention patterns have recently been interpreted as embodying the process of cue-based retrieval, in which attention over multiple targets is taken to generate interference and latency during retrieval. Under this framework, this work first defines an entropy-based predictor that quantifies the diffuseness of self-attention, as well as distance-based predictors that capture the incremental change in attention patterns across timesteps. Moreover, following recent studies that question the informativeness of attention weights, we also experiment with alternative methods for incorporating vector norms into attention weights. Regression experiments using predictors calculated from the GPT-2 language model show that these predictors deliver a substantially better fit to held-out self-paced reading and eye-tracking data over a rigorous baseline including GPT-2 surprisal. Additionally, the distance-based predictors generally demonstrated higher predictive power, with effect sizes of up to 6.59 ms per standard deviation on self-paced reading times (compared to 2.82 ms for surprisal) and 1.05 ms per standard deviation on eye-gaze durations (compared to 3.81 ms for surprisal).
10.48550/arxiv.2212.11185
[ "https://www.aclanthology.org/2022.emnlp-main.632.pdf" ]
254,926,669
2212.11185
cca8dd4332d13b3e8f3ece02ec0931449191aa4d
Entropy-and Distance-Based Predictors From GPT-2 Attention Patterns Predict Reading Times Over and Above GPT-2 Surprisal December 7-11, 2022 Byung-Doh Oh Department of Linguistics Department of Linguistics The Ohio State University The Ohio State University William Schuler [email protected] Department of Linguistics Department of Linguistics The Ohio State University The Ohio State University Entropy-and Distance-Based Predictors From GPT-2 Attention Patterns Predict Reading Times Over and Above GPT-2 Surprisal Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing the 2022 Conference on Empirical Methods in Natural Language ProcessingDecember 7-11, 2022 Transformer-based large language models are trained to make predictions about the next word by aggregating representations of previous tokens through their self-attention mechanism. In the field of cognitive modeling, such attention patterns have recently been interpreted as embodying the process of cue-based retrieval, in which attention over multiple targets is taken to generate interference and latency during retrieval. Under this framework, this work first defines an entropy-based predictor that quantifies the diffuseness of self-attention, as well as distance-based predictors that capture the incremental change in attention patterns across timesteps. Moreover, following recent studies that question the informativeness of attention weights, we also experiment with alternative methods for incorporating vector norms into attention weights. Regression experiments using predictors calculated from the GPT-2 language model show that these predictors deliver a substantially better fit to held-out self-paced reading and eye-tracking data over a rigorous baseline including GPT-2 surprisal. Additionally, the distance-based predictors generally demonstrated higher predictive power, with effect sizes of up to 6.59 ms per standard deviation on self-paced reading times (compared to 2.82 ms for surprisal) and 1.05 ms per standard deviation on eye-gaze durations (compared to 3.81 ms for surprisal). Introduction Much work in broad-coverage sentence processing has focused on studying the role of expectation operationalized in the form of surprisal (Hale, 2001;Levy, 2008) using language models (LMs) to define a conditional probability distribution of a word given its context (Smith and Levy, 2013;Goodkind and Bicknell, 2018). Recently, this has included Transformer-based language models (Wilcox et al., 2020;Merkx and Frank, 2021;Oh et al., 2022). However, expectation-based accounts have empirical shortcomings, such as being unable to fully account for garden-path effects (van Schijndel and Linzen, 2021) or predict the timing of delays in certain constructions . For this reason, some research has begun to focus on the effects of memory and attention using predictors calculated from language model representations. For example, Ryu and Lewis (2021) recently drew connections between the self-attention patterns of Transformers (Vaswani et al., 2017) and cue-based retrieval models of sentence comprehension (e.g. Lewis et al., 2006). Their attention entropy, which quantifies the diffuseness of the attention weights over previous tokens, showed patterns that are consistent with similarity-based interference observed during the processing of subject-verb agreement. However, these results relied on identifying one attention head specialized for the nsubj dependency, and an aggregated version of this predictor was not very strong in predicting naturalistic reading times in the presence of a surprisal predictor (Ryu and Lewis, 2022). This work therefore defines and evaluates several memory-and attention-based predictors derived from the self-attention patterns of the Transformerbased GPT-2 language model (Radford et al., 2019) on two naturalistic datasets, in the presence of a strong GPT-2 surprisal baseline. First, normalized attention entropy expands upon Ryu and Lewis's (2021) attention entropy by re-normalizing the attention weights and controlling for the number of tokens in the previous context. Additionally, three distance-based predictors that quantify the shift in attention patterns across consecutive timesteps are presented, based on the idea that the reallocation of attentional focus entails processing difficulty. Moreover, motivated by work on interpreting large language models that question the connection between attention weights and model predictions (e.g. Jain and Wallace, 2019), the norm-based analysis of the transformed vectors (Kobayashi et al., 2020(Kobayashi et al., , 2021 is newly applied to GPT-2 in this work to inform alternative formulations of attention weights. For example, it has been observed that while large language models tend to place high attention weights on special tokens (e.g. <|endoftext|> of GPT-2), these tokens contribute very little to final model predictions as their 'value' vectors have near-zero norms (Kobayashi et al., 2020). Attention weight formulations that incorporate the norms of the transformed vectors should therefore alleviate the over-representation of such special tokens and represent the contribution of each token more accurately. Results from regression analyses using these predictors show significant and substantial effects in predicting self-paced reading times and eye-gaze durations during naturalistic reading, even in the presence of a robust surprisal predictor. Additionally, alternative formulations of attention weights that incorporate the norms of the transformed vectors are shown to further improve the predictive power of these predictors. Background This section provides a mathematical definition of the self-attention mechanism underlying the GPT-2 language model and describes alternative normbased formulations of attention weights. Masked Self-Attention of GPT-2 Language Models GPT-2 language models (Radford et al., 2019) use a variant of a multi-layer Transformer decoder pro-posed in Vaswani et al. (2017). Each decoder layer consists of a masked self-attention block and a feedforward neural network: x l+1,i = FF(LN out (o l,i +x l,i ))+(o l,i +x l,i ),(1) where x l,i ∈ R d is the ith input representation at layer l, FF is a two-layer feedforward neural network, LN out is a vector-wise layer normalization operation, and o l,i is the output representation from the multi-head self-attention mechanism, in which H multiple heads simultaneously mix representations from the previous context. This output o l,i can be decomposed into the sum of representations resulting from each attention head h: o l,i = H h=1 V h X ′ l,i 1 ⊤ a l,h,i ,(2) where X ′ l,i def = [x ′ l,1 , ..., x ′ l,i ] ∈ R d×i is the sequence of layer-normalized input representations leading up to x ′ l,i from the previous context, and x ′ l,i def = LN in (x l,i ) is the layer-normalized version of x l,i . V h represents the head-specific valueoutput transformation, 1 and a l,h,i ∈ R i is the vector of attention weights: a l,h,i = SOFTMAX K h X ′ l,i 1 ⊤ ⊤ Q h x ′ l,i 1 √ d h ,(3) where Q h and K h represent the head-specific query and key transformations respectively, and d h = d/H is the dimension of each attention head. LN α is a vector-wise layer normalization operation (Ba et al., 2016) that first standardizes the vector and subsequently conducts elementwise transformations using learnable parameters c α , b α ∈ R d : LN α (y) = y − m(y) s(y) ⊙ c α + b α ,(4) where α ∈ {in, out}, m(y) and s(y) denote the elementwise mean and standard deviation respectively, and ⊙ denotes a Hadamard product. 1 As W b x 1 = Wx + b, the bias vectors are omitted from the equations. Additionally, for the simplicity of notation, the 'output' affine transform, which applies to the concatenated 'value-transformed' vectors from all attention heads in a typical implementation, is subsumed into V h . The bias vector of the 'output' transform is assumed to be distributed equally across heads. Weight-and Norm-Based Analysis of Previous Context Previous work that has studied the inner mechanism of Transformer models has focused on analyzing the relative contribution of each token to its final prediction. As a measure that quantifies the 'strength' of contribution, attention weights from the self-attention mechanism have been most commonly used. Similarly to recent work in cognitive modeling (e.g. Ryu and Lewis, 2021), this work also evaluates predictors calculated from attention weights (ATTN-W). a W,l,h,i = a l,h,i(5) While analysis using attention weights is common, the assumption that attention weights alone reflect the contribution of each token disregards the magnitudes of the transformed input vectors, as pointed out by Kobayashi et al. (2020). As an alternative, they proposed a norm-based analysis of the self-attention mechanism, which quantifies the contribution of each token as the norm of the transformed vector multiplied by the attention weight. In this work, in order to quantify the relative contribution of each token in the previous context, the norms of the transformed vectors are normalized across the sequence, resulting in 'norm-aware' weights that are comparable to attention weights (ATTN-N). a N,l,h,i = V h X ′ l,i 1 ⊤ ⊙ V h X ′ l,i 1 ⊤ ⊤ 1 1 2 ⊙ a l,h,i 1 ⊤ V h X ′ l,i 1 ⊤ ⊙ V h X ′ l,i 1 ⊤ ⊤ 1 1 2 ⊙ a l,h,i(6) More recently, Kobayashi et al. (2021) showed that the residual connection and the layer normalization operation (RESLN; RL) that follow the self-attention mechanism can also be conducted before aggregating representations over token positions. Motivated by this observation, the vector norms that take into consideration these subsequent operations are also examined in this work. Similarly to ATTN-N, the norms are normalized across the sequence to yield weights that are comparable (ATTNRL-N): a RL-N,l,h,i = ((g(V h X ′ l,i 1 ⊤ ) ⊙ g(V h X ′ l,i 1 ⊤ )) ⊤ 1) 1 2 ⊙ a l,h,i 1 ⊤ (((g(V h X ′ l,i 1 ⊤ ) ⊙ g(V h X ′ l,i 1 ⊤ )) ⊤ 1) 1 2 ⊙a l,h,i ) ,(7) where g(·) incorporates the residual connection (+x l,i ) and layer normalization (LN out ) of Eq. 1. Following the assumption that the residual connection serves to 'preserve' the representation at position i (Kobayashi et al., 2021) and that it is distributed equally across heads, x l,i is added to the representation at position i of each head after dividing it by the number of heads: g(Y) def = i j=1         Yδ j + x l,i Ha l,h,i[i] −m(Yδ j + x l,i Ha l,h,i[i] ) s(o l,i +x l,i ) ⊙cout + bout H if i = j Yδ j −m(Yδ j ) s(o l,i +x l,i ) ⊙cout + bout H if i ̸ = j   δ ⊤ j ,(8) where δ j is a Kronecker delta vector consisting of a one at element j and zeros elsewhere, and c out and b out refer to the learnable parameters of LN out . diffuseness of self-attention and distance-based predictors that capture the incremental change in attention patterns across timesteps can be defined. The first predictor defined in this work is normalized attention entropy (NAE): NAE π,l,h,i = a ⊤ π,l,h,i[1:i−1] log 2 (i − 1)1 ⊤ a π,l,h,i[1:i−1] (log 2 a π,l,h,i[1:i−1] 1 ⊤ a π,l,h,i[1:i−1] ), (9) where π ∈ {W, N, RL-N}. This is similar to the attention entropy proposed by Ryu and Lewis (2021) as a measure of interference in cue-based recall attributable to uncertainty about the target, with two notable differences. First, NAE controls for the number of tokens in the previous context by normalizing the entropy by the maximum entropy that can be achieved at timestep i. Furthermore, NAE also uses weights over x ′ l,1 , ..., x ′ l,i−1 that have been re-normalized to sum to 1, thereby adhering closer to the definition of entropy, in which the mass of interest sums to 1. 2 In addition to NAE, distance-based predictors are defined for capturing effortful change in attention patterns across timesteps. However, as it currently remains theoretically unclear how this distance should be defined, this exploratory work sought to provide empirical results for different distance functions. The first is ∆NAE, which quantifies the change in diffuseness across timesteps: ∆NAE π,l,h,i = |NAE π,l,h,i −NAE π,l,h,i−1 | (10) As with NAE, this predictor is insensitive to how the attention weights are reallocated between tokens in the previous context to the extent that the overall diffuseness remains unchanged. The second distance-based predictor is Manhat- tan distance (MD). 3 MD π,l,h,i = ||a π,l,h,i − a π,l,h,i−1 || 1(11) MD directly measures the magnitude of change in attention weights over all tokens at timestep i. MD is less sensitive to the linear distance between tokens and therefore makes it consistent with the predictions of McElree et al. (2003), who found that processing speed was not influenced by the 2 Preliminary analyses showed that regression models with attention entropy proper without these adjustments failed to converge coherently. 3 For the purpose of calculating this predictor, the ith element of a π,l,h,i−1 is assumed to be 0. amount of intervening linguistic material in the formation of a dependency. Finally, the Earth Mover's Distance (EMD; Rubner et al., 2000) is applied to quantify the shift in attention weights. EMD is derived from a solution to the Monge-Kantorovich problem (Rachev, 1985), which aims to minimize the amount of "work" necessary to transform one histogram into another. Formally, let P = {(p 1 , w p 1 ), ..., (p m , w pm )} be the first histogram with m bins, where p r represents the bin and w pr represents the weight of the bin; Q = {(q 1 , w q 1 ), ..., (q n , w qn )} the second histogram with n bins; and D = [d rs ] the distance matrix where d rs is the ground distance between bins p r and q s . The problem is to find an optimal flow F = [f rs ], where f rs represents the flow between p r and q s , that minimizes the overall work. WORK(P, Q, F) = m r=1 n s=1 d rs f rs(12) Once the optimal flow is found, the EMD is defined as the work normalized by the total flow. 4 EMD(P, Q) = m r=1 n s=1 d rs f rs m r=1 n s=1 f rs(13) To quantify the minimum amount of work necessary to 'transform' the attention weights, the EMD between attention weights at consecutive timesteps is calculated using EMD(a π,l,h,i−1 , a π,l,h,i ). The ground distance is defined as d rs = |r−s| i−1 in order to control for the number of tokens in the previous context. EMD can be interpreted as being consistent with Dependency Locality Theory (Gibson, 2000) in that reallocating attention weights to tokens further away in the context incurs more cost than reallocating weights to closer tokens. Code for calculating all of the predictors from GPT-2 under the different attention weight formulations is publicly available at https://github. com/byungdoh/attn_dist. Experiment 1: Evaluation of Predictors on Human Reading Times In order to evaluate the contribution of the entropyand distance-based predictors, regression models containing commonly used baseline predictors, surprisal predictors, and one predictor of interest were fitted to self-paced reading times and eye-gaze durations collected during naturalistic language processing. In this work, we adopt a statistical procedure that directly models temporal diffusion (i.e. a lingering reponse to stimuli) by estimating continuous impulse response functions and controls for overfitting by assessing the external validity of these predictors through a non-parametric test on held-out data. Response Data The first experiment described in this paper used the Natural Stories Corpus (Futrell et al., 2021), which contains self-paced reading times from 181 subjects that read 10 naturalistic stories consisting of 10,245 words. The data were filtered to exclude observations for sentence-initial and sentence-final words, observations from subjects who answered fewer than four comprehension questions correctly, and observations with durations shorter than 100 ms or longer than 3000 ms. This resulted in a total of 770,102 observations, which were subsequently partitioned into a fit partition of 384,905 observations, an exploratory partition of 192,772 observations, and a held-out partition of 192,425 observations. 5 The partitioning allows model selection (e.g. making decisions about baseline predictors and random effects structure) to be conducted on the exploratory partition and a single hypothesis test to be conducted on the held-out partition, thus obviating the need for multiple trials correction. All observations were log-transformed prior to regression modeling. Additionally, the set of go-past durations from the Dundee Corpus (Kennedy et al., 2003) also provided the response variable for regression modeling. The Dundee Corpus contains eye-gaze durations from 10 subjects that read 67 newspaper editorials consisting of 51,501 words. The data were filtered to remove unfixated words, words following saccades longer than four words, and words at sentence-, screen-, document-, and line-starts and ends. This resulted in a total of 195,507 observations, which were subsequently partitioned into a fit partition of 98,115 observations, an exploratory partition of 48,598 observations, and a held-out partition of 48,794 observations. All observations were log-transformed before model fitting. Predictors For each dataset, a set of baseline predictors that capture basic, low-level cognitive processing were included in all regression models. • Self-paced reading times (Futrell et al., 2021): word length measured in characters, index of word position within each sentence; • Eye-gaze durations (Kennedy et al., 2003): word length measured in characters, index of word position within each sentence, saccade length, whether or not the previous word was fixated. In addition to the baseline predictors, two surprisal predictors were also included in all regression models evaluated in this experiment. The first is unigram surprisal as a measure of word frequency, which was calculated using the KenLM toolkit (Heafield et al., 2013) with parameters estimated on the English Gigaword Corpus (Parker et al., 2009). The second is surprisal from GPT-2 Small (Radford et al., 2019), which is trained on ∼8B tokens of the WebText dataset. Surprisal from the smallest GPT-2 model was chosen because it has been shown to be the most predictive of self-paced reading times and eye-gaze durations among surprisal from all variants of GPT-2 (Oh et al., 2022). Finally, the entropy-and distance-based predictors defined in Section 3 were calculated from the attention patterns (i.e. a π,l,h,i where π ∈ {W, N, RL-N}) of heads on the topmost layer of GPT-2 Small. Contrary to previous studies that analyzed the attention patterns of all layers, this work focuses on analyzing those of the topmost layer out of the concern that the attention patterns of lower layers are less interpretable to the extent that they perform intermediate computations for the upper layers. Since the topmost layer generates the representation that is used for model prediction, the attention patterns from this layer are assumed to reflect the contribution of each previous token most directly. Subsequently, the by-head predictors were aggregated across heads to calculate by-word predictors. This assumes that each attention head contributes equally to model prediction, and is also consistent with the formulation of multi-head selfattention in Eq. 2. To calculate surprisal as well as the entropy-and distance-based predictors, each story of the Natural Stories Corpus and each article of the Dundee Corpus was tokenized according GPT-2's byte-pair encoding (BPE; Sennrich et al., 2016) tokenizer Figure 3: Improvements in CDR model log-likelihood from including each predictor on the exploratory partition of Natural Stories self-paced reading data (left) and Dundee eye-tracking data (right). and was provided as input to the GPT-2 model. In cases where each story/article did not fit into a single context window, the second half of the previous context window served as the first half of a new context window to calculate predictors for the remaining tokens. 6 Additionally, when a single word w i was tokenized into multiple subword tokens, negative log probabilities of subword tokens corresponding to w i were added together to calculate S(w i ) = − log P(w i | w 1..i−1 ). Similarly, the entropy-and distance-based predictors for such subword tokens were also added together. Procedures To evaluate the predictive power of each predictor of interest, a baseline regression model containing only the baseline predictors and full regression models containing one predictor of interest on top of the baseline regression model were first fitted to the fit partition of each dataset. In order to control for the confound of temporal diffusion, continuous impulse response functions (IRFs) were estimated using the statistical framework of continuous-time deconvolutional regression (CDR; Shain and Schuler, 2021). The predictors of interest were centered, and all regression models included a by-subject random intercept. 7 As a preliminary analysis, the predictive power of different predictors was compared on the exploratory partition by calculating the increase in log-likelihood (∆LL) to the baseline regression model as a result of including the predictor, following recent work (Goodkind and Bicknell, 2018;Wilcox et al., 2020;Oh et al., 2021). Subsequently, based on the preliminary exploratory results, the predictive power of one best entropy-based predictor and that of one best distance-based predictor were evaluated on the held-out partition of both datasets. More specifically, statistical significance testing was conducted using a paired permutation test (Demšar, 2006) of the difference in byitem squared error between the baseline regression model and the respective full regression models. Results The results in Figure 3 show that across both corpora, most of the entropy-and distance-based predictors make a notable contribution to regression model fit under all attention formulations. Given that the baseline model contains strong predictors such as unigram surprisal and GPT-2 surprisal, this may suggest their validity as predictors of comprehension difficulty. The exception to this is the EMD predictor, which did not show an increase in likelihood under ATTN-W on the Natural Stories Corpus and under ATTNRL-N on the Dundee Corpus. As EMD is sensitive to how the ground distance d rs between bins is defined (i.e. how costly it is to reallocate attention weights across input positions), a more principled definition of d rs may make EMD a more robust predictor. 8 Across the two corpora, ATTN-N+NAE and ATTNRL-N+MD appear to be the most predictive among the entropy-and distance-based predictors respectively. Additionally, the NAE and ∆NAE predictors showed different trends across the two corpora, where NAE contributed to stronger model fit than ∆NAE on the Natural Stories Corpus, while the opposite trend was observed on the Dundee Corpus. In contrast to various surprisal predictors that have shown a very similar trend in terms of predictive power across these two corpora (Oh et al., 2022), these two predictors may shed light on the subtle differences between self-paced reading times and eye-gaze durations. Finally, incorporating vector norms into attention weights (i.e. ATTN-N and ATTNRL-N) generally seems to improve the predictive power of these predictors, which provides support for the informativeness of input vectors in analyzing attention patterns (Kobayashi et al., 2020(Kobayashi et al., , 2021. Table 1 presents the effect sizes of ATTN-N+NAE and ATTNRL-N+MD on the held-out partition of the Natural Stories Corpus and the Dundee Corpus, which were derived by calculating how much increase in reading times the regression model would predict at average predictor value given an increase of one standard deviation. On both datasets, ATTNRL-N+MD appears to be a strong predictor of reading times, which contributed to significantly lower held-out errors. The entropy-based ATTN-N+NAE predictor showed contrasting results across the two corpora, showing a large effect size on the Natural Stories Corpus but not on the Dundee Corpus. This is consistent with the differential results between NAE and ∆NAE on the exploratory partition of the two corpora and may hint at differences between self-paced reading times and eye-gaze durations. In terms of magnitude, the two predictors showed large effect sizes on the Natural Stories Corpus, which were more than twice that of GPT-2 surprisal. On the Dundee Corpus, however, the effect size of ATTNRL-N+MD was much smaller compared to that of GPT-2 surprisal. Experiment 2: Do NAE and MD Independently Explain Reading Times? The previous experiment revealed that on the Natural Stories Corpus, the select entropy-and distancebased predictors from the attention patterns of GPT-2 contributed to significantly higher regression 9 The estimated IRF for ATTN-N+NAE showed a sign error, likely due to poor convergence. Therefore, we treated the effect of this predictor to be statistically non-significant. Corpus Predictor Effect Size (p-value) Natural Stories ATTN-N+NAE 6.87 ms (p < 0.001) GPT2SURP 2.56 ms ATTNRL-N+MD 6.59 ms (p < 0.001) GPT2SURP 2.82 ms Dundee ATTN-N+NAE N/A 9 (n.s.) GPT2SURP 4.22 ms ATTNRL-N+MD 1.05 ms (p < 0.001) GPT2SURP 3.81 ms model fit. Although they showed similarly large effect sizes on the held-out partition, the two predictors may independently explain reading times, as they are defined to quantify different aspects of attention patterns. The second experiment examines this possibility following similar procedures as the previous experiment. Procedures In order to determine whether the effect of one predictor subsumes that of the other, a CDR model including both ATTN-N+NAE and ATTNRL-N+MD was first fit to self-paced reading times of the fit partition of the Natural Stories Corpus. The CDR model followed the same specifications, random effects structure, and baseline predictors as described in Experiment 1. Subsequently, the fit of this regression model on the held-out partition was compared to those of the two regression models that contain only one of the two predictors from the previous experiment. More specifically, the ∆LL as a result of including the predictor(s) of interest were calculated for each regression model, and statistical significance testing was conducted using a paired permutation test of the difference in by-item squared error between the new 'ATTN-N+NAE & ATTNRL-N+MD' regression model and the respective 'ATTN-N+NAE' and 'ATTNRL-N+MD' regression models, which allowed the contribution of each predictor to be analyzed. Figure 4: Improvements in CDR model log-likelihood from including ATTN-N+NAE, ATTNRL-N+MD, and both on the held-out partition of Natural Stories selfpaced reading data. Results The results in Figure 4 show that the regression model including both ATTN-N+NAE and ATTNRL-N+MD achieves higher ∆LL on the held-out partition of the Natural Stories Corpus compared to regression models including only one of these predictors. Moreover, the difference in by-item squared error between the 'ATTN-N+NAE & ATTNRL-N+MD' regression model and the 'ATTNRL-N+MD' regression model was statistically significant (p < 0.001). In contrast, the significance between the 'ATTN-N+NAE & ATTNRL-N+MD' regression model and the 'ATTN-N+NAE' regression model was not statistically significant (p = 0.236). This indicates that the entropy-based ATTN-N+NAE contributes to model fit over and above the distance-based ATTNRL-N+MD and also subsumes its effect in predicting reading times. Correlation Analysis Between Predictors and Syntactic Categories Procedures In order to shed more light on the predictors newly proposed in this work, a series of correlation analyses were conducted. First, Pearson correlation coefficients were calculated between the entropyand distance-based predictors as well as the surprisal predictors to examine the similarity between predictors and the influence of different attention weight formulations. Subsequently, the most predictive ATTN-N+NAE and ATTNRL-N+MD predictors were analyzed with a focus on identifying potential linguistic correlates. This analysis used a version of the Natural Stories Corpus and the Dundee Corpus that had been manually annotated using a generalized categorial grammar annotation scheme (Shain et al., 2018). Results The correlation matrix in Figure 5 shows that within the same predictor, the different attention formulations did not make a very large difference, with 'intra-predictor' correlation coefficients at above 0.85 for most predictors. An exception to this trend was EMD, which showed a correlation coefficient of 0.74 on the Natural Stories Corpus and 0.72 on the Dundee Corpus between ATTN-W+EMD and ATTNRL-N+EMD. Such difference is to be expected, as EMD is the most sensitive to the change of location in attention weights among the distance-based predictors. This is also consistent with the exploratory regression results in Figure 3, where the ∆LL of EMD predictors varied the most as a function of different attention weight formulations. Additionally, the norm-based attention formulations seem to bring the entropy-based NAE closer to the distance-based predictors in terms of correlation coefficients. On both corpora, ATTN-N+NAE and ATTNRL-N+NAE show stronger correlations to distance-based predictors than ATTN-W+NAE. Interestingly, the highest correlation coefficient between NAE and a distance-based predictor is observed between ATTN-N+NAE and ATTNRL-N+MD at 0.90 on the Natural Stories Corpus and 0.91 on the Dundee Corpus, which were the two strongest predictors identified in Experiment 1. Such high correlation partially explains the results of Experiment 2, in which the influence of ATTN-N+NAE subsumed that of ATTNRL-N+MD. Finally, the entropy-and distance-based predictors showed moderate correlation to unigram surprisal at around 0.5 on both corpora. With regard to GPT-2 surprisal, these predictors showed weak correlation at around 0.3 on the Natural Stories Corpus, and around 0.4 on the Dundee Corpus. Together with the regression results, this further suggests that the proposed predictors capture a mechanistic process that is distinct from the frequency or predictability of the word. An analysis of the predictors according to syntactic categories showed that ATTNRL-N+MD may be sensitive to the transition between the subject constituent and the predicate constituent. Its average values for nouns in different contexts presented in Table 2 show that on both corpora, nouns occurring at the end of subject NPs were associated with a greater shift in attention patterns. This is consistent with the intuition that transitions between constituents entails cognitive effort during incremental processing. Discussion and Conclusion This work builds on recent efforts to derive memory-and attention-based predictors of processing difficulty to complement expectation-based accounts of sentence processing. Based on the observation that the self-attention patterns of Transformer-based language models can be interpreted as embodying the process of cue-based retrieval, an entropy-based predictor that quantifies the diffuseness of self-attention was first defined. Moreover, based on the idea that reallocation of attention may incur processing difficulty, distance-based predictors that capture the incremental change in attention patterns across timesteps were defined. Regression results using these predictors calculated from the GPT-2 language model showed that these entropy-and distance-based predictors deliver a substantially better fit to self-paced reading and eye-tracking data over a strong baseline including GPT-2 surprisal. To our knowledge, this work is the first to report robust effects of Transformer attention-based predictors in predicting reading times of broadcoverage naturalistic data. This provides support for Ryu and Lewis's (2021) observation that the self-attention mechanism of Transformers embodies the process of cue-based retrieval, and further suggests that representations that exhibit similaritybased interference can be learned from the selfsupervised next-word prediction task. Additionally, the strength of the distance-based predictors further demonstrates the potential to bring together expectation-and memory-based theories of sentence processing under a coherent framework. Limitations The connection between attention patterns of Transformer-based language models and human sentence processing drawn in this work is based on a model trained on English text and data from human subjects that are native speakers of English. Therefore, the connection made in this work may not generalize to other languages. Additionally, although the alternative formulations of self-attention weights resulted in stronger predictors of processing difficulty, they are more computationally expensive to calculate as they rely on an explicit decomposition of the matrix multiplication operation, which are highly optimized in most packages. Ethics Statement Experiments presented in this work used datasets from previously published research (Futrell et al., 2021;Kennedy et al., 2003), in which the procedures for data collection and validation are outlined. As this work focuses studying the possible connection between attention patterns of large language models and human sentence processing, its potential negative impacts on society seem to be minimal. Figure 1 : 1The predictors and attention weight formulations examined in this work. The entropy-based predictor (i.e. NAE) quantifies the diffuseness of attention over previous tokens at a given timestep, while the distance-based predictors (i.e. ∆NAE, MD, EMD) capture the change in attention patterns across consecutive timesteps (top row: weights at 'journey,' bottom row: weights at 'to'). These predictors can be calculated from attention weights formulated using different methods (i.e. ATTN-W, ATTN-N, ATTNRL-N). Figure 2 : 2Computations performed within the selfattention block of one head of the GPT-2 language model at a given timestep (i = 5). While the masked self-attention mechanism aggregates representations from the previous context in a typical implementation, the linear nature of the subsequent computations allows this aggregation to be deferred to after the residual connection and layer normalization, thereby allowing updated representations to inform alternative formulations of attention weights (i.e. ATTN-N, ATTNRL-N). Figure 5 : 5Pearson correlation coefficients between the predictors examined in this work on the Natural Stories Corpus (upper right triangle) and the Dundee Corpus (lower left triangle). Table 1 : 1The per standard deviation effect sizes of the predictors on the held-out partition of the Natural Sto- ries Corpus and the Dundee Corpus. Statistical signifi- cance was determined by a paired permutation test of the difference in by-item squared error between the base- line regression model and the respective full regression model containing the predictor of interest. The effect sizes of GPT-2 surprisal from the same regression mod- els are presented for comparison. Table 2 : 2The average ATTNRL-N+MD values for nouns at the end of subject NPs and nouns in other contexts. Entropy-and Distance-Based Predictors From Attention PatternsGiven the different formulations of self-attention weights, entropy-based predictors that quantify the The optimal flow can be found using the transportation simplex method. Additionally, due to the constraint that the total flow is equal to min( m r=1 wp r , n s=1 wq s ), the total flow is always equal to 1 in the context of attention weights. For both datasets, the fit partition, exploratory partition, and held-out partition contain data points whose summed subject and sentence number have modulo four equal to zero or one, two, and three respectively. In practice, most stories/articles fit within two windows. 7 Early analysis on the exploratory partition revealed that the training data does not support a richer random-effects structure, which led to severe overfitting. Please refer to Appendix A for details on IRF specifications, the optimization procedure of CDR models, as well as the transformations of baseline predictors. For example, a recency bias may be incorporated by defining the ground distance from tokens closer to the current timestep to be smaller than that from tokens that are farther back in the previous context. AcknowledgmentsWe thank the reviewers for their helpful comments. This work was supported by the National Science Foundation grant #1816891. All views expressed are those of the authors and do not necessarily reflect the views of the National Science Foundation.A CDR Implementation DetailsThe continuous-time deconvolutional regression (CDR) models used in this work were fitted using variational inference to estimate the means and variances of independent normal posterior distributions over all model parameters assuming an improper uniform prior. Convolved predictors used the threeparameter ShiftedGamma impuse response function (IRF) kernel:Posterior means for the IRF parameters were initialized at α = 0.2, β = 0.5, and δ = −0.2, which defines a decreasing IRF with a peak centered at t = 0 that decays to near-zero within about 1 s. Models were fitted using the Adam optimizer (Kingma andBa, 2015)with Nesterov momentum(Nesterov, 1983;Dozat, 2016), a constant learning rate of 0.001, and minibatches of size 1,024. For computational efficiency, histories were truncated at 128 timesteps. 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[ "THE EFFECT OF PROTON TEMPERATURE ANISOTROPY ON THE SOLAR MINIMUM CORONA AND WIND", "THE EFFECT OF PROTON TEMPERATURE ANISOTROPY ON THE SOLAR MINIMUM CORONA AND WIND" ]
[ "Alberto M Vásquez [email protected] \nHarvard-Smithsonian Center for Astrophysics\n60 Garden Street, MS 1502138CambridgeMA\n", "Adriaan A Van Ballegooijen \nHarvard-Smithsonian Center for Astrophysics\n60 Garden Street, MS 1502138CambridgeMA\n", "John C Raymond \nHarvard-Smithsonian Center for Astrophysics\n60 Garden Street, MS 1502138CambridgeMA\n" ]
[ "Harvard-Smithsonian Center for Astrophysics\n60 Garden Street, MS 1502138CambridgeMA", "Harvard-Smithsonian Center for Astrophysics\n60 Garden Street, MS 1502138CambridgeMA", "Harvard-Smithsonian Center for Astrophysics\n60 Garden Street, MS 1502138CambridgeMA" ]
[]
A semiempirical, axisymmetric model of the solar minimum corona is developed by solving the equations for conservation of mass and momentum with prescribed anisotropic temperature distributions. In the high-latitude regions, the proton temperature anisotropy is strong and the associated mirror force plays an important role in driving the fast solar wind; the critical point where the outflow velocity equals the parallel sound speed (v ¼ c k ) is reached already at 1.5 R from Sun center. The slow wind arises from a region with open-field lines and weak anisotropy surrounding the equatorial streamer belt. The model parameters were chosen to reproduce the observed latitudinal extent of the equatorial streamer in the corona and at large distance from the Sun. We find that the magnetic cusp of the closed-field streamer core lies at about 1.95 R . The transition from fast to slow wind is due to a decrease in temperature anisotropy combined with the nonmonotonic behavior of the nonradial expansion factor in flow tubes that pass near the streamer cusp. In the slow wind, the plasma is of order unity and the critical point lies at about 5 R , well beyond the magnetic cusp. The predicted outflow velocities are consistent with O 5þ Doppler dimming measurements from UVCS/ SOHO. We also find good agreement with polarized brightness ( pB) measurements from LASCO/SOHO and H i Ly images from UVCS/SOHO.
10.1086/379008
null
18,192,671
astro-ph/0310846
c30eef211ee5afa4765dceb58ec0f0e0cd00e763
THE EFFECT OF PROTON TEMPERATURE ANISOTROPY ON THE SOLAR MINIMUM CORONA AND WIND Alberto M Vásquez [email protected] Harvard-Smithsonian Center for Astrophysics 60 Garden Street, MS 1502138CambridgeMA Adriaan A Van Ballegooijen Harvard-Smithsonian Center for Astrophysics 60 Garden Street, MS 1502138CambridgeMA John C Raymond Harvard-Smithsonian Center for Astrophysics 60 Garden Street, MS 1502138CambridgeMA THE EFFECT OF PROTON TEMPERATURE ANISOTROPY ON THE SOLAR MINIMUM CORONA AND WIND Received 2003 April 11; accepted 2003 August 11Subject headings: MHD -solar wind -Sun: corona -Sun: magnetic fields -Sun: UV radiation A semiempirical, axisymmetric model of the solar minimum corona is developed by solving the equations for conservation of mass and momentum with prescribed anisotropic temperature distributions. In the high-latitude regions, the proton temperature anisotropy is strong and the associated mirror force plays an important role in driving the fast solar wind; the critical point where the outflow velocity equals the parallel sound speed (v ¼ c k ) is reached already at 1.5 R from Sun center. The slow wind arises from a region with open-field lines and weak anisotropy surrounding the equatorial streamer belt. The model parameters were chosen to reproduce the observed latitudinal extent of the equatorial streamer in the corona and at large distance from the Sun. We find that the magnetic cusp of the closed-field streamer core lies at about 1.95 R . The transition from fast to slow wind is due to a decrease in temperature anisotropy combined with the nonmonotonic behavior of the nonradial expansion factor in flow tubes that pass near the streamer cusp. In the slow wind, the plasma is of order unity and the critical point lies at about 5 R , well beyond the magnetic cusp. The predicted outflow velocities are consistent with O 5þ Doppler dimming measurements from UVCS/ SOHO. We also find good agreement with polarized brightness ( pB) measurements from LASCO/SOHO and H i Ly images from UVCS/SOHO. INTRODUCTION At the time of cycle minimum, the solar corona is more or less axisymmetric and stable for many months. The polar coronal holes are separated by an equatorial streamer belt that encircles the Sun. The fast solar wind originates from the coronal holes, while the slow wind originates from the vicinity of the streamer belt. The physical processes that drive the fast and slow winds are only partially understood. In particular, it is unclear why the Sun has such a bimodal outflow pattern with distinctly different physical conditions in the fast and slow winds. Does this distinction between fast and slow winds already exist at low heights in the corona, or does it arise only at larger distance from the Sun? What causes the transition from fast to slow wind as we approach the solar equator? What is the role of open and closed magnetic fields in this transition? To answer such questions, an empirical description of the corona is needed, i.e., a description of temperature, density, velocity, and magnetic field as functions of latitude, longitude, and radial distance from the Sun. Such modeling is a necessary step in the study of the physical processes by which the fast and slow winds are generated. Axisymmetric, semiempirical models of the solar corona have been developed by many authors (e.g., Pneuman & Kopp 1971;Yeh & Pneuman 1977;Steinolfson, Suess, & Wu 1982;Washimi, Yoshino, & Ogino 1987;Cuperman, Ofman, & Dryer 1990;Cuperman et al. 1993;Wang et al. 1993Wang et al. , 1998aLionello, Linker, & Mikic 2001). For example, Sittler & Guhathakurta (1999) used empirically derived electron density profiles to construct a model of the magnetic field, outflow velocity, effective temperature, and effective heat flux. These parameters are derived by solving the equations for conservation of mass, momentum, and energy and the magnetic induction equation. The model provides an estimate of the large-scale surface magnetic field at the Sun, which is estimated to be 12-15 G. The authors predict that the large-scale surface field is dominated by an octupole term. More recently, three-dimensional models of the corona have also been developed (e.g., Linker, van Hoven, & Schnack 1990;Wang et al. 1997a;Riley, Linker, & Mikic 2001). In this paper we develop a coronal model that synthesizes observational data obtained with instruments on the Solar and Heliospheric Observatory (SOHO) satellite, in particular, data from the Ultraviolet Coronagraph Spectrometer (UVCS; Kohl et al. 1995). UVCS observations have shown that the minor ions in coronal holes have kinetic temperatures that are much larger than those for protons and electrons Cranmer et al. 1999). Here '' kinetic temperature '' refers to the total velocity dispersion of the particles, including thermal and nonthermal components. Detailed analysis of kinetic temperatures for different ions has shown that some ions have higher temperature than others, indicating '' preferential heating '' of certain ions. Furthermore, Doppler dimming analysis of spectral lines such as O vi 1032 and 1037 has shown that some minor ions have '' anisotropic velocity distributions '': the velocity dispersion perpendicular to the (nearly radial) magnetic field in coronal holes is significantly larger than that parallel to the field. This high temperature and large temperature anisotropy of the ions is believed to be caused by dissipation of transverse waves (Dusenbery & Hollweg 1981;Isenberg & Hollweg 1983;Tu & Marsch 1997;Cranmer et al. 1999;Hu & Habbal 1999). A similar but smaller anisotropy may also exist for the protons ). In the presence of a diverging magnetic field, charged particles experience an outward force that causes the perpendicular motion of the particles to be converted into parallel motion. Therefore, if the perpendicular energy of the protons is continually replenished by wave dissipation, the protons can maintain an anisotropic velocity distribution (T p? > T pk ) and there is a net outward force on the protons. This socalled mirror force plays an important role in driving the fast solar wind (see review by Cranmer 2002). The purpose of the present paper is to include the effects of proton temperature anisotropy into a global model of the solar corona. The paper is organized as follows: In x 2 we discuss observations of streamers and the relationship between the streamer and the slow solar wind. In x 3 we present observational constraints on our global model, including temperature and density in the corona and magnetic flux at the coronal base. In x 4 we present a method for solving the coronal force balance equations, taking into account the mirror force. In x 5 we present our model results for coronal magnetic structure and outflow velocity. In x 6 we derive images of visible-light polarization brightness and H i Ly intensities from our model, and we compare our results with observations. The main results of this work are discussed in x 7. STREAMER STRUCTURE AND THE ORIGIN OF THE SLOW WIND The K corona is produced by Thomson scattering of photospheric white light by free electrons in the corona. In a study of the solar minimum corona using the Large Angle and Spectrometric Coronagraph (LASCO) on SOHO, Wang et al. (1997b) found that the large-scale structure of the coronal streamer belt at 3 R and beyond can be reproduced by a model in which the scattering electrons are concentrated around a single, warped current sheet that encircles the Sun. The angular width of this sheet is only a few degrees. The bright, narrow spikes seen in LASCO C2 and C3 images occur wherever the sheet is oriented edge-on in the plane of the sky (Wang et al. 1998b). In contrast, the latitudinal extent of the slow wind, as determined from in situ measurements, is about AE20 (Suess et al. 1999b), much larger than the thickness of the coronal plasma sheet observed with LASCO. This implies that the bulk of the slow wind originates from open-field lines outside the observed plasma sheet (Wang et al. 1998b). Sheeley et al. (1997) used time-lapse sequences of LASCO images to track the outward motion of small density enhancements ('' blobs '') in the plasma sheet. These blobs originate in the high corona above the top of the helmet streamer and slowly accelerate outward though the LASCO field of view (2.2-30 R ). Wang et al. (1998b) suggest that both the blobs and the plasma sheet represent closed-field material injected into the slow wind as a result of footpoint exchanges between the stretched helmet-streamer loops and neighboring open-field lines. According to this model, the ejection of the blobs does not cause any permanent disruption of the helmet streamer, which remains in a stretched, quasi-equilibrium state with its cusp at 3 R . Noci et al. (1997) and Raymond et al. (1997) observed the equatorial streamer with UVCS/SOHO at radii between 1.5 and 5 R . Images of the streamer in O vi 1032 show two bright legs separated by a dark lane. This lane extends radially along the streamer axis up to 3 R (also see Strachan et al. 2002;Frazin, Cranmer, & Kohl 2003). In contrast, H i Ly and white-light images do not show such a dark feature. This led Noci et al. (1997) to attribute the dark lane to a reduced oxygen abundance along the streamer axis. Raymond et al. (1997) further suggested that this abundance anomaly is due to gravitational settling of oxygen in the static proton/electron plasma of the closedfield streamer core. If this interpretation is correct, the closed-field lines must extend up to 3 R , consistent with the model of Wang et al. (1998b). The O vi images for 1996 July show that the streamer legs converge toward the equator in the range 2-2.5 R from Sun center. This suggests that the streamer cusp may be located at about 2 R , somewhat less than implied by the model of Wang et al. (1998b). We propose the following scenario: The blobs observed by Sheeley et al. (1997) originate at the cusp (2 R ) but become clearly visible only at a somewhat larger height (3-4 R ). The plasma within the blobs originates in the closed-field streamer core, which is affected by gravitational settling. Therefore, the blobs have low oxygen abundance compared with the surrounding slow solar wind. As the blobs move out into the region between 2 and 3 R , they contribute to reduced O vi emission along the axis of the streamer. In this way the effects of gravitational settling are transmitted to a larger height via the blobs. In contrast, the open-field lines farther away from the streamer axis always have a slow outflow and therefore are unaffected by settling. Strachan et al. (2002) measured O 5þ outflow velocities in streamers using the Doppler dimming effect (also see Habbal et al. 1997). In a latitudinal scan at 2.33 R , Strachan et al. found no measurable outflow velocity within the streamer (v < 20 km s À1 ) but a steep rise in outflow velocity occurs just beyond the bright streamer legs. The latitudinal width of the streamer, as defined by the point where the outflow velocity equals 100 km s À1 , is about AE20 (see Fig. 4d of Strachan et al. 2002), similar to the observed width of the slow wind at large distance from the Sun (Suess et al. 1999b). OBSERVATIONAL CONSTRAINTS ON GLOBAL MODEL This section describes the observational constraints on coronal density, temperature, and magnetic flux to be used in x 4. Electron Density The polarized brightness ( pB) of the K corona can be used to measure the coronal electron density (van de Hulst 1950). Here we use the results of Guhathakurta, Holzer, & MacQueen (1996) and Sittler & Guhathakurta (1999), who used Skylab data from 1973-1974 to derive electron density profiles for coronal holes and streamers at cycle minimum (see also Newkirk 1967;Allen 1973;Munro & Jackson 1977;Saito, Poland, & Munro 1977). Their results are expressed in terms of radial profiles, one for the polar coronal hole, another for the equatorial streamer. They used the following expression for the electron density within each component: N e ðrÞ ¼ a 1 e a 2 =r r À2 1 þ a 3 r þ a 4 r 2 þ a 5 r 3 ;ð1Þ where r is the heliocentric distance in units of solar radii and the parameters a 1 ; Á Á Á ; a 5 are given by Sittler & Guhathakurta (1999). The density profiles are shown in Figure 1. The Skylab data span the height range 1-5 R and were further extrapolated using data from the Ulysses mission (Phillips et al. 1995). These measurements correspond to selected days when the equatorial streamer belt was seen approximately end-on. This explains why their streamer densities are somewhat larger than those of Saito et al. (1977), who presented streamer densities averaged over many days. Electron Temperature One method for inferring the electron temperature T e is to use spectral line intensity ratios of two lines from the same ion. Wilhelm et al. (1998) used data from SUMER/ SOHO to estimate T e at radii 1.03-1.6 R in coronal holes at the last cycle minimum (1996)(1997). They use Mg ix 706 and 750, and they conclude that in both the plume and interplume regions the electrons barely reach the canonical temperature of 1 MK. Moreover, T e ðrÞ falls off rapidly with height. Although their temperature estimates depend on atomic data that, according to the authors, could be improved, their work suggests that in polar hole regions the electrons are significantly cooler than the ions (see Landi et al. 2001 for a quantitative analysis on uncertainties of T e estimates, derived from Be-like line ratios and using different theoretical methods). The rates of ionization and recombination of coronal ions decrease rapidly with distance from the Sun. Therefore, the charge state of the solar wind is determined in large part by the electron temperature T e in the inner corona, where the ionization and recombination times are still short compared to the solar wind expansion time. In situ measurements of the solar wind charge state can be used to estimate the coronal electron temperature. Ko et al. (1997) derived polar T e profiles from observations with SWICS/Ulysses. Their results can be approximated as a combination of two power laws (see Cranmer et al. 1999): T e ðrÞ ¼ 10 6 0:35r 1:1 þ 1:9r À6:6 À Á À1 K : Raymond et al. (1997) and Li et al. (1998) studied the ion-ization balance of various ions in a streamer observed with UVCS/SOHO in 1996 July. The results indicate that T e reaches a maximum value of about 1.6 MK in the streamer core. Because of the high density and low outflow velocity in streamers (e.g., Frazin 2002), we expect that protons and electrons are in thermal equilibrium with each other, so the electrons may be used as a proxy for the protons. Raymond et al. (1997) and Li et al. (1998) show that their observations are compatible with hydrostatic equilibrium in the streamer core (see also Gibson et al. 1999). In this paper we derive the electron temperature T e ðrÞ from the observed electron density N e ðrÞ, assuming hydrostatic equilibrium. In this paper we use a generalization of expression (2) to describe the radial variation of electron temperature at the pole and the equator: T e ðrÞ ¼ T 0 a þ 1 a þ br þ ð1 À bÞr À ;ð3Þ where T 0 is the temperature at the coronal base (r ¼ 1). The values of the parameters a, b, , and are given in Table 1, and the profiles are shown in Figure 2. The electron temperature model at the poles (dashed line) has been adjusted to approximately fit the observational data of Ko et al. (1997) and Cranmer et al. (1999) (triangles). The electrons are assumed to have a Maxwellian velocity distribution. The electron temperature at the equator (thick line) is assumed to be equal to the proton temperature, discussed in the next section. Proton Temperature Ion temperatures can be estimated by measuring the width of coronal emission line profiles. Because of rapid charge exchange between protons and neutral hydrogen, the latter can be used as a proxy for the protons (Withbroe et al. 1982). Allen, Habbal, & Hu (1998) studied the coupling between neutral hydrogen and protons by treating the hydrogen atoms as test particles in a proton-electron background. Their work indicates that the H i velocity distribution reflects the proton distribution at radii up to about 3 R in the polar regions and even higher in the equatorial regions. Above 3 R in the polar regions, the H i velocity distribution '' follows '' the proton distribution and reaches temperatures about 20% higher than the protons (see also Olsen, Leer, & Holzer 1994;Olsen & Leer 1996). These results strongly support the use of H i Ly profiles to measure the velocity spread of the protons along the line of sight (LOS). The observed velocity spread includes both thermal and nonthermal components (such as wave motions) and may also include a contribution from solar wind expansion in the direction along the LOS (most emission originates near the point of closest approach to the Sun, but there is some contribution from regions behind and in front of the plane of the sky where the solar wind velocity has a component along the LOS). Therefore, the velocity spread derived from the observed line width provides only an upper limit on the proton temperature: T p ðm H =k B ÞV 2 1=e =2, where V 1=e is the observed 1=e velocity spread, m H is the hydrogen mass, and k B is Boltzmann's constant. At the time of cycle minimum, the polar coronal holes are very large and polar observations at r > 1:5 R are unaffected by low-latitude streamers. UVCS/SOHO observations of heavier ions such as O 5þ show that the velocity distributions of these ions in coronal holes are highly anisotropic: the velocity spread in the radial direction (as derived from Doppler dimming analysis) is much smaller than that in the tangential direction (Kohl et al. 1998;Cranmer et al. 1999). It is unclear whether this anisotropy also exists for the protons; Doppler dimming analysis of H i Lyman lines does not provide strong constraints on the proton parallel velocity in coronal holes. The O 5þ anisotropy is believed to be due to the damping of transverse waves (e.g., due to ion-cyclotron resonance), and this perpendicular heating may also occur for the protons. In this paper we assume that the protons indeed have an anisotropic velocity distribution in the low-density polar regions (T p? > T pk ) but not in the equatorial region (T p? ¼ T pk ). Since the magnetic field over the pole is approximately radial and perpendicular to the LOS, the observed H i Ly line width provides an estimate for the proton perpendicular temperature. Furthermore, we assume that the proton parallel temperature equals the (isotropic) electron temperature, T pk ¼ T e , so that the radial variation of T pk at the pole and equator is given by equation (3) with parameter values given in Table 1. The anisotropy in proton temperature over the pole is consistent with in situ measurements that show that T p > T e in the solar wind (Marsh et al. 1982;Pilipp et al. 1987). Kohl et al. (1998) and Cranmer et al. (1999) analyzed H i Ly line profiles observed in polar coronal holes during the past cycle minimum (1996)(1997). Their results indicate proton perpendicular temperatures of up to 6 MK. The observed velocity width (V 1=e ) increases rapidly with height from about 190 km s À1 (T p? $ 2:2 MK) at 1.5 R to 240 km s À1 (T p? $ 3:5 MK) at 2.5 R and then slowly decreases to 250 km s À1 (T p? $ 3:8 MK) at 4 R . The results are consistent with earlier measurements from UVCS/SPARTAN by Kohl, Strachan, & Gardner (1996), who found peak temperatures of 5-6 MK at 2.25 R and 3.5 MK at 3.5 R in a polar coronal hole in 1993. From these several observations, measured temperatures at different heights are shown by asterisks in Figure 2. UVCS/SOHO observations in the equatorial regions indicate a roughly constant proton temperature within the core of the streamer belt. We reanalyzed UVCS data from a supersynoptic campaign during 1996 July (Raymond et al. 1997) and find Ly line widths of order 185 km s À1 (T p? $ 2:0 MK) at 1.5 R and 195 km s À1 (T p? $ 2:2 MK) at 2.6 R . At larger heights, the line widths decrease to 170 km s À1 (T p? $ 1:75 MK) at 3 R and 150 km s À1 (T p? $ 1:35 MK) at 4.5 R . The measurements are shown by the crosses in Figure 2. Streamer observations by Kohl et al. (1997), obtained about a month after the Raymond observations, exhibit very similar values (2.2 MK at 2 R , 1.5 MK at 4 R ). In this paper we use the following expression for the proton perpendicular temperature: T p? ðrÞ ¼ T 0 a þ 1 a þ br þ ð1 À bÞr À þ T 1 ðr À 1Þ 2 e ÀðrÀ1Þ=Dr ðr max À 1Þ 2 e Àðr max À1Þ=Dr :ð4Þ The first term is similar to that used for the electron temperature, but the second term allows us to impose a further increase in temperature over a limited range of heights (of order Dr), consistent with UVCS observations. The values of the parameters at the pole and at the equator are given in Table 1. In summary, our temperature models present the following main features: (1) at the pole, strong anisotropy for protons, and electron temperature much lower than T p? ; (2) at the equator, hydrostatic equilibrium isotropic velocity distributions and thermal equilibrium between species. Note that T 0 and are the same for all temperature models, so that T p? ¼ T pk ¼ T e at low heights in the corona. Figure 3 shows the longitude-averaged radial magnetic field B r as function of latitude at the solar surface (r ¼ 1). These data are derived from the Kitt Peak synoptic map for 1996 July, near the time of cycle minimum. There are no large active regions on the Sun at this time. The solid curve in Figure 3 is a fit of the form Magnetic Flux at Coronal Base B r ðÞ ¼ B 0 cos p ;ð5Þ where p ¼ 7 and B 0 ¼ þ10 G. Note that B r decreases monotonically from about +10 G in the north to À10 G in the south. At latitudes between AE50 , the average radial field is very small (less than 2 G). Therefore, the large-scale coronal field is dominated by the polar fields. CORONAL MODEL WITH ANISOTROPIC GAS PRESSURE At cycle minimum, the corona presents a relatively ordered structure with high-latitude coronal holes and an equatorial streamer belt. The corona is essentially axisymmetric and stable for many months (e.g., Gibson 2001). In the following we describe an axisymmetric model of the corona; an earlier version of this model was described by and . We use a spherical coordinate system ðr; ; Þ and assume that all scalar quantities are independent of azimuth . The magnetic field B is assumed to lie in the meridional plane: B ¼ D µ A ¼ 1 r sin 1 r @A @ê e r À @A @rê e ;ð6Þ where A ¼ A ê e is the vector potential and A ¼ A r sin is the field-line variable (note that A = constant along field lines). This variable increases from A ¼ 0 along the polar axis to A ¼ A eq at the equator (r ¼ R , ¼ =2). There is a critical field line A c that forms the boundary between open and closed magnetic regions; the field lines with 0 < A < A c are open, while those with A c < A < A eq are closed. Note that A c =A eq is the fraction of surface flux that is open. In this paper we assume A c =A eq ¼ 0:80. The magnetic configuration is illustrated in Figure 4. Rather than solving an energy equation for the coronal plasma, our semiempirical model is based on observed temperature distributions. The electron and proton temper-atures (T e , T pk , and T p? ) are interpolated between the pole and equator: Tðr; AÞ ¼ T pole ðrÞ þ T equa ðrÞ À T pole ðrÞ Â Ã ÈðAÞ ;ð7Þ where T pole ðrÞ and T equa ðrÞ are the observed profiles described in x 3 and ÈðAÞ is a function of the field-line variable that increases monotonically from Èð0Þ % 0 at the pole to ÈðA c Þ % 1 at the equator. We use the following expression for ÈðAÞ: ÈðAÞ ¼ 1 1 þ exp½ÀðA À A h Þ=A w ;ð8Þ where A h is the midpoint of the transition [ÈðA h Þ ¼ 1 2 ] and A w measures the width of the transition. In this paper we use A h =A c ¼ 0:70 and A w =ðA c À A h Þ ¼ 0:15. The function ÈðAÞ is shown in Figure 5. The coronal magnetic and velocity fields are computed by solving the momentum equation: D x " P P þ vv À Á ¼ ÀrÈ g þ 1 4 ð D µ BÞ µ B ;ð9Þ where " P P is the pressure tensor, is the mass density, v is the outflow velocity, and È g ðrÞ ¼ ÀGM =r is the gravitational potential. The steady-flow condition requires that v k B, and conservation of mass requires that v=B is constant along field lines. The pressure tensor is anisotropic: " P P ¼ p kŝ sŝ s þ p ? " I I Àŝ sŝ s À Á ;ð10Þ where " I I is the unit tensor,ŝ s is the unit vector along B, and p k and p ? are the parallel and perpendicular pressures: p k ¼ N e k B T e þ T pk À Á ;ð11Þp ? ¼ N e k B T e þ T p? À Á :ð12Þ Here we assume a pure hydrogen plasma. The left-hand side of equation (9) can be written as D x " P P þ vv À Á ¼ rp ? þ B d ds p B ŝ s þ p dŝ s ds ;ð13Þ where s measures distance along the field lines, B jBj, p p k À p ? þ v 2 , and we use (9) parallel toŝ s reads dp ? ds þ B d ds D x B ¼ 0. The componentp B ¼ À dÈ g ds ;ð14Þ and the solution of this equation will be further discussed in x 4.2. We now introduce the function P ? ðr; AÞ describing the dependence of p ? on the field-line variable: p ? ðr; Þ ¼ P ? ½r; Aðr; Þ. Then the gradient of p ? is rp ? ¼ @P ? @rê e r þ @P ? @A rA ;ð15Þ where the partial derivatives on the right-hand side are taken at constant A and constant r, respectively. It follows that dp ? =ds ¼ ð@P ? =@rÞ cos , where is the angle between s s andê e r , and similarly dÈ g =ds ¼ ðdÈ g =drÞ cos . Inserting these expressions into equation (14), we obtain @P ? @r þ dÈ g dr ¼ À B cos d ds p B ;ð16Þ and combining equations (9), (13), (15), and (16) yields @P ? @A rA þ B cos d ds p B ŝ s cos Àê e r ð Þþp dŝ s ds ¼ ð D µ BÞ 4 ðÀB ê e r þ B rê e Þ :ð17Þ In streamers the pressure anisotropy and outflow velocity are small, so the terms involving p can be neglected. The same is true in coronal holes because in these regions the magnetic field is nearly radial and the vectors ðŝ s cos Àê e r Þ and dŝ s=ds vanish for radial fields. Therefore, p plays only a minor role in the perpendicular force balance, and we neglect its effects in equation (17). Then the perpendicular force balance (17) reduces to ð D µ BÞ 4 ¼ r sin @P ? @A ;ð18Þ where we use equation (6). The solution of equation (18) is be discussed in x 4.1. We use an iterative method for solving equations (16) and (18). The method can be summarized as follows: We first construct a preliminary pressure model P ? ðr; AÞ by interpolating the observed perpendicular pressure between pole and equator, using an expression similar to equation (7). Then we compute the field-line variable Aðr; Þ by solving the perpendicular force balance as described below in x 4.1; this gives us a first guess for the shape of the magnetic field lines. Then we solve the parallel force balance separately along many field lines as described in x 4.2. This yields the mass density ðsÞ and velocity vðsÞ along each field line, which are then remapped to produce the density ðr; Þ and velocity vðr; Þ. We also update the perpendicular pressure P ? ðr; AÞ, which is then used in the next iteration to recompute the field-line variable Aðr; Þ. This process is repeated until convergence is achieved. Note that in the first iteration P ? ðr; AÞ is specified analytically, whereas in subsequent iterations P ? ðr; AÞ is obtained by numerical interpolation. Perpendicular Force Balance To solve equation (18) for the perpendicular force balance, we introduce the following integral over the coronal volume: L ¼ Z =2 0 Z R max R B 2 8 À P ? ðr; AÞ ! r 2 sin dr d ;ð19Þ where only one hemisphere of the Sun is considered and R max ¼ 10 R is the outer radius of the computational domain. The magnetic field B is related to A via equation (6), and A A r sin . At each step in our iterative procedure, P ? ðr; AÞ is a known function, so L depends only on the unknown function A ðr; Þ or, equivalently, Aðr; Þ. We assume the following boundary conditions for Aðr; Þ: AðR ; Þ ¼ B 0 R 2 p þ 1 1 À cos pþ1 À Á ; ð20Þ @A @r ðR max ; Þ ¼ 0 ;ð21Þ Aðr; 0Þ ¼ 0 ; ð22Þ A r; 2 ¼ A c for r > r cusp ;ð23Þ@A @ r; 2 ¼ 0 for r < r cusp :ð24Þ Equation (20) is derived from the surface flux distribution, equation (5), and the magnetic field is assumed to be radial at the outer boundary, r ¼ R max . Equations (23) and (24) describe the boundary conditions at the equator. For r > r cusp , there is a current sheet at the equator separating the open fields from the two hemispheres; hence, the magnetic field just above the equator is radial. For r < r cusp the magnetic field is closed over the equator, so the field is perpendicular to the equatorial plane. We now show that the function A ðr; Þ for which L reaches its minimum value is a solution of equation (18). Let A ðr; Þ be an arbitrary variation of A ðr; Þ; then the corresponding change in L is L ¼ Z =2 0 Z R max R ð D µ BÞ 4 À r sin @P ? @A !  A ðr; Þr 2 sin dr d ;ð25Þ where the boundary conditions (20)-(24) have been used. At the minimum, L vanishes for any function A ðr; Þ, so the quantity in square brackets in equation (25) must vanish. This is precisely the condition for perpendicular force balance, equation (18). The function A ðr; Þ is discretized on a nonuniform grid in r and h with 180 points in each direction. The minimization of L is performed using the Polak-Ribiere version of the conjugate gradient method (Press et al. 1992). This is an iterative method for adjusting the values of A at the 180 2 grid points until the minimum of L is reached. The cusp radius r cusp is also allowed to change in order to obtain the lowest possible L, but the amount of open magnetic flux (A c =A eq ) is held fixed. Parallel Force Balance The parallel component of the momentum equation (14) can be written in the following form: B d ds B ðc 2 k þ v 2 Þ ! þ c 2 ? 1 B dB ds ¼ À dÈ g ds ;ð26Þ where c k and c ? are the parallel and perpendicular sound speeds: c 2 k;? p k;? ¼ k B ðT p k;? þ T e Þ m p :ð27Þ Using v=B ¼ constant along field lines, we can eliminate the density from equation (26): 1 À c 2 k v 2 v dv ds ¼ df ds ; ð28Þ where f ðsÞ GM rðsÞ À c 2 k ðsÞ À Z s 0 c 2 ? ðs 0 Þ Bðs 0 Þ dB ds 0 ds 0 :ð29Þ At each step of our iterative procedure we have estimates for BðsÞ, rðsÞ, T pk ðsÞ, T p? ðsÞ, and T e ðsÞ on any open-field line, which allows us to compute the function f ðsÞ. According to equation (28), the sonic point (v ¼ c k ) is located at an extremum of f ðsÞ. The velocity vðsÞ is found by inward and outward integration of equation (28), starting at the sonic point. In practice we find that, in order to obtain a valid solution over the entire height range (R < r < R max ), the sonic point must be located at the global minimum of f ðsÞ. The density ðsÞ is computed from mass flux conservation and the boundary condition for the density at the coronal base (see x 3). This process is repeated for 180 different open-field lines, and the results are remapped to obtain the density ðr; Þ and velocity vðr; Þ on the ðr; Þ grid. RESULTS We iterated the parallel and perpendicular force balance equations as described in x 4. The process was repeated until the maximum change in P ? ðr; AÞ between iterations was less than 1%; this required 20 iterations. The results for the first and last iterations are shown in the left and right panels, respectively, of Figure 6. The thin curves represent magnetic field lines, and the thick curves are contours of ðr; Þ, the ratio of perpendicular gas pressure p ? and magnetic pressure B 2 =8. The cusp height in the final solution is r cusp % 1:95 R . Note that the magnetic structure changes little between the first and last iterations. Our model yields 5 1 in the polar regions but > 1 in the equatorial streamer, especially near the streamer cusp. This supports the idea that the streamer is magnetically contained by the strong polar fields that surround it on either side (Suess, Gary, & Nerney 1999a). We find that > 1 throughout the closed-field region of the streamer. However, the variation with height is not monotonic: has peaks at both the cusp and the streamer base, and lower values at Li et al. (1998), who estimated based on UVCS/SOHO and SXT/Yohkoh observations of streamers (1996 July), in combination with potential field extrapolation of the photospheric magnetic field. Their estimates indicate $ 5 at 1.15 R and $ 3 at 1.50 R , similar to the values found here. High values of were also found in MHD models that include heat and momentum deposition in the corona (Wang et al. 1998a;Suess, Wang, & Wu 1996). Figure 7 shows the outflow velocity for the final model. The bright region is the fast solar wind emanating from the coronal hole, and the dark region is the slow wind that flows along the open-field lines within the streamer. The maximum velocities are about 450 km s À1 for the fast wind and 190 km s À1 for the slow wind. This is somewhat smaller than the observed in situ values at 1 AU (800 and 400 km s À1 , respectively). We attribute this difference to the fact that our model does not include any momentum deposition effects other than the mirror force. However, the ratio of fast and slow wind speeds is about 2.5 in our model, consistent with in situ measurements. Figure 8 shows three quantities measured across magnetic field lines: the temperature anisotropy at 3 R , the radial position r s of the sonic point, and the asymptotic wind speed. Note that the variation of wind speed closely follows that of the temperature anisotropy (compare the top and bottom panels). This indicates that the fast wind is driven by the high perpendicular temperatures in the coronal hole, and the decrease in wind speed at the edge of the hole is mainly due to the decrease in temperature anisotropy. The middle panel shows that there is a sudden jump in sonicpoint radius once the transition from high to low temperature anisotropy is nearly complete. The jump occurs at A=A c ¼ 0:77 and is due to the appearance of a second (lower) minimum in the function f ðsÞ at a height well beyond the streamer cusp (see below). As a result, the sonicpoint radius changes discontinuously from about 1.5 R in the coronal hole to about 5.5 R in the streamer. Such large values for the sonic-point height in the streamer were found earlier by Wang (1994) and Chen & Hu (2001). Most of the decrease in wind speed occurs for A=A c between 0.6 and 0.77, well before the jump in sonic-point height, and the asymptotic wind speed does not change significantly at the jump. To show the transition between fast and slow wind more clearly, Figure 9 shows various quantities along field lines near the fast-slow boundary (the same line styles are used in all panels). Figure 9a shows the shapes of the field lines in the meridional plane. Figure 9b shows the function f ðsÞ defined in equation (29) and plotted as function of radial distance r for four field lines. Note that the triple-dotdashed curve has a minimum at about 1.5 R , whereas the solid curve has two minima, one at 1.7 R and another at 5.5 R . As shown in x 4.2, these minima indicate possible positions of the sonic point (v ¼ c k ). Figure 9d shows the radial positions r min of such minima as a function of fieldline variable A. In general, a global solution of the wind equation (28) can be found only when the integration is started at the global minimum, i.e., the minimum with the lowest value of f ðr min Þ. Therefore, as we move from the triple-dot-dashed field lines to the dot-dashed line, there is a discontinuous jump in the height of the sonic point. Figure 9c shows the outflow velocity vðrÞ along field lines near the fast-slow boundary. The dotted curve in this panel is the wind solution along the last open-field line, A ¼ A c . As we move from the coronal hole into the streamer, the wind speed at large height closely follows the decrease in perpendicular temperature, which mainly occurs between the two thick solid profiles. However, a further decrease occurs at lower heights (r $ 2 R ) between the triple-dot-dashed and dot-dashed field lines because of the jump in height of the sonic point. We conclude that, unlike the terminal speed, the low-height behavior of vðrÞ is strongly affected by the height of the sonic point. Such a region of stagnated flow was found earlier by many authors (e.g., Wang 1994;Suess & Nerney 1999, 2002Chen & Hu 2001). Figure 9e shows the nonradial expansion factor of the slow-wind flow tubes, f exp ðrÞ ¼ ðR =rÞ 2 BðR Þ=BðrÞ, where BðR Þ is the field strength at the coronal base for each field line. Note that f exp ðrÞ is a nonmonotonic function of radius for these field lines. The peak in f exp ðrÞ is because these field lines pass close to the streamer cusp where B % 0. In contrast, f exp ðrÞ increases monotonically for field lines in the coronal hole (not shown). It is remarkable to see that this '' cusp effect '' occurs at large distance from the cusp: the peak in f exp ðrÞ occurs for all field lines within about 1 R from the equatorial plane (see Fig. 9a). The triple-dotdashed field line at which the peak in f exp ðrÞ first develops is close to the field line where the jump in sonic-point height occurs. This is consistent with the suggestion by Suess & Nerney (1999, 2002, Chen & Hu (2002), and others that the existence of the slow wind is due to the nonmonotonic behavior of f exp ðrÞ. We suggest here that the decrease in perpendicular temperature as we approach the equatorial plane also plays an important role. To determine whether the '' cusp effect '' is due to the effects of gas pressure on the expansion of the flow tubes, we computed a partially open potential magnetic field with the same photospheric boundary conditions (see eq. [5]) and a thin current sheet in the equatorial plane for r > 1:95 R . The calculation is based on the work of Low (1986). First, the surface flux distribution is expressed in terms of four Legendre polynomials P n ðxÞ, with n ¼ 1, 3, 5, and 7. Then the field is extrapolated into the corona using equations (B3) and (B4) of Low (1986) for n ¼ 1 and 3 (for n ¼ 5 and 7 the effect of the current sheet can be neglected). We found that f exp ðrÞ has a peak for all open-field lines that pass below the point ðx; zÞ ¼ ð3:0; 1:4Þ R , where x is the distance from the rotation axis and z is the height above the equatorial plane. Therefore, the effect of the streamer cusp, located at the point ðx; zÞ ¼ ð1:95; 0Þ R , is present even in a potential field, and its spatial extent is similar to what was found in the numerical model (i.e., the effect is not due to the finite plasma ). For the value of A h chosen in this paper (A h =A c ¼ 0:7), the transition between fast and slow wind far from the Sun occurs at latitudes of AE20 . This is consistent with Ulysses observations taken at the time of cycle minimum (Suess et al. 1999b). At lower heights, the low speeds in the streamer legs found in our model are consistent with UVCS measurements of O 5þ outflow velocity using the Doppler dimming technique (Strachan et al. 2002). In a latitudinal scan at 2.33 R , Strachan et al. (2002) found no measurable outflow velocity in the streamer (v < 20 km s À1 ), but a steep rise in velocity occurs just beyond the bright streamer legs. The observed latitudinal half-width of the streamer at 2.33 R , as defined by the points where the outflow velocity equals 100 km s À1 , is about 20 . Our model predicts a halfwidth of 22 . Therefore, the predicted width of the streamer is roughly consistent with observations at both large and small heights. It should be noted that we use a realistic photospheric flux distribution that is peaked at the pole, B r ðR ; Þ ¼ 10 cos 7 G, whereas Suess et al. (1999a) and Chen & Hu (2001) assume a bipolar distribution, B r ðR ; Þ / cos . For a given fraction of open magnetic flux, the assumed flux distribution has a significant effect on the size of the polar coronal hole, the latitudinal width of the streamer core, and the magnitude of the nonradial expansion factor. However, potential field models indicate that the spatial extent of the cusp effect is similar in the two cases. COMPARISON WITH VISIBLE-LIGHT AND Ly OBSERVATIONS The coronal density model can be used to predict the polarization brightness ( pB) of the visible light that is scattered by free electrons in the corona (Thomson scattering). The pB is an integral along the LOS (van de Hulst 1950): pBðr 0 Þ ¼ 3 16 T " B B Z LOS N e ðxÞ r 0 r 2  ð1 À u 0 Þà AðrÞ þ u 0B BðrÞ 1 À u 0 =3 dx ;ð30Þ where x is the distance along the LOS, r 0 is the projected radial distance from Sun center, r ¼ ðr 2 0 þ x 2 Þ 1=2 is the true radial distance, T ¼ 6:65  10 À25 cm 2 is the Thomson scattering cross section, " B B ¼ 1:97  10 10 ergs cm À2 s À1 sr À1 is the mean disk intensity, and u 0 ¼ 0:63 is the limb-darkening coefficient. The functionsà AðrÞ andB BðrÞ are given by van de Hulst (1950). We used our axisymmetric model of the electron density distribution to compute synthetic pB images for various angles between the LOS and the solar equatorial plane. Figure 10 shows the results for ¼ 0 and ¼ 30 ; the latter represents a view of the Sun from above the ecliptic plane. We have overplotted selected field lines in the plane of the sky (white curves) and contours of the ratio of pB and its maximum value at the equatorial base (black curves). The equatorial streamer clearly stands out in the image with ¼ 0 but is less distinct for ¼ 30 . To compare these results with observations, the top panel of Figure 11 shows the radial pB profiles along the pole and equator and the bottom panel shows latitudinal profiles at three different heights (r 0 ¼ 1:15, 1.5, and 2.5 R ). The solid curves correspond to the edge-on view ( ¼ 0). The symbols represent pB measurements obtained with the Mauna Loa Mark III coronagraph and LASCO/SOHO Guhathakurta et al. 1999) and with the visible-light channel on UVCS/SOHO . The data were obtained in 1996 July and August when the equatorial streamer belt is seen approximately edge-on (Raymond et al. 1997). Our model fits the observations quite well, although the predicted latitudinal width of the streamer at 2.5 R is somewhat larger than observed. The dashed curve in the top panel of Figure 11 shows the radial profile along the pole for the out-of-ecliptic view ( ¼ 30 ). Note that there is a large increase in pB compared to the edge-on view. This is due to the fact that the LOS crosses the enhanced density equatorial streamer and is not representative of the available observations of the polar pB profile. We also computed the intensity of the strongest UV coronal emission line, H i 1216 (Ly), which is formed pB=B , as function of radial distance from sun center. The solid curves show the predicted equatorial (thick curve) and polar (thin curve) profiles for an edge-on view. The dashed curve is the predicted polar profile for a view from above the ecliptic plane ( ¼ 30 ). The symbols are measurements for the pole (triangles) and equator (diamonds) from Gibson et al. (1999), Guhathakurta et al. (1999), and Cranmer et al. (1999). Bottom: Predicted polarization brightness as function of colatitude at projected radii of 1.15, 1.5, and 2.5 R . The symbols are corresponding measurements from the above-cited references. almost entirely by resonant scattering of chromospheric Ly radiation by neutral hydrogen atoms in the corona (Noci, Kohl, & Withbroe 1987;Cranmer et al. 1999). The integrated intensity of the scattered Ly is I ¼ h 0 4 B 12 Z LOS N H ðxÞðrÞ Z I ð þ Þð À 0 Þ d dx ;ð31Þ where x measures distance along the LOS, N H ðxÞ is the neutral hydrogen density, r is the radial distance from Sun center at any point along the LOS, ðrÞ f1 À ½1 À ðR =rÞ 2 1=2 g=2 is a geometric dilution factor, I ðÞ is the chromospheric intensity (in ergs cm À2 s À1 Hz À1 sr À1 ) as function of frequency (in Hz), 0 ¼ 2:47  10 15 Hz is the line center frequency, 0 v r =c is the Doppler shift, v r is the radial component of the outflow velocity, ð À 0 Þ is the scattering profile of the coronal atoms (due to their thermal and nonthermal velocities), and B 12 is the Einstein coefficient. For a derivation of this expression and discussion of the relevant approximations, see Noci et al. (1987). The neutral-hydrogen density N H is computed using the collisional ionization rates of Scholz & Walters (1991) and the recombination rates of Hummer (1994). We computed Ly images for two different viewing angles, using densities, temperatures, and velocities from the coronal model. The results are shown in Figure 12. Although these images look similar to those for polarization brightness, the Ly intensity is somewhat sensitive to the Fig. 12.-Images of Ly intensity I as predicted from the coronal model for an edge-on view (left) and for a view 30 above the ecliptic plane (right). The black curves are contours of logðI=I max Þ, where I max is the maximum value of I at the coronal base. The white curves are selected field lines in the plane of the sky (labeled with A=A c ). The dark semicircle represents the solar disk. outflow velocity. This is shown more clearly in Figure 13, where we plot the radial profiles of Ly intensity with and without the Doppler dimming effect (solid and dashed curves, respectively). Note that there is a significant difference between predicted intensities with and without dimming for the polar region, but not for the equatorial region. Therefore, unlike pB measurements, Ly intensities provide constraints on the outflow velocity in the acceleration region of the solar wind. Figure 13 also compares our model predictions with measurements from UVCS/SOHO. The diamonds represent measurements taken along the streamer axis, based on our reanalysis of data obtained on 1996 July 26 (Raymond et al. 1997). The triangles are measurements along the polar axis based on observations from 1996 November to 1997 April Dobrzycka et al. 1999). Note that the predicted Ly intensities fit the observations quite well. This suggests that the model gives a reasonable representation of the velocity field in the inner part of the corona, where the main acceleration of the solar wind takes place. DISCUSSION We developed a stationary, axisymmetric MHD model for the global corona. The model includes a description of the temperature, density, velocity, and magnetic field as function of latitude and radius up to 10 R from Sun center. The velocity and magnetic fields are obtained by solving the parallel and perpendicular force balance equations, including the effects of inertia, anisotropic gas pressure, gravity, and Lorentz forces. The temperature models are based on observational data from UVCS/SOHO, and the magnetic flux distribution at the coronal base is taken from NSO/Kitt Peak synoptic maps. The model reproduces the main features of the global corona at the time of cycle minimum. We find that the high perpendicular temperature of the protons in the coronal hole plays a major role in driving the fast solar wind. In the streamer we find low outflow velocity and high plasma , consistent with earlier results (Suess et al. 1996;Wang et al. 1998a;Li et al. 1998). In our model the wind equation (28) is solved separately for many field lines. The sonic point along each field line occurs at a minimum of the function f ðsÞ that appears on the right-hand side of the wind equation (see eq. [29]). The transition from fast to slow wind occurs at an open-field line characterized by A h ¼ 0:7A c , where A c is the boundary between open and closed fields. The value of A h was adjusted to obtain the correct latitudinal width of the slowwind region both at large distance from the Sun (Suess et al. 1999b) and in the corona at 2.33 R (Strachan et al. 2002). In our model this transition is associated with two effects: (1) the decrease of proton perpendicular temperature T p? ðr; AÞ as we approach the equatorial plane, and (2) the appearance of a peak in the nonradial expansion factor f exp ðrÞ for field lines that pass close to the streamer cusp ('' cusp effect ''). The cusp effect is present at surprisingly large distances from the cusp ($1 R ) and is present even in potential-field models, so it is not a consequence of finite plasma . The combination of decreasing temperature anisotropy and cusp effect causes the global minimum of f ðsÞ (and therefore the sonic point) to occur well beyond the cusp and produces low outflow velocity near the cusp. These results are consistent with the suggestion by Noci et al. (1997) that the slow wind is due to special properties of the geometric spreading along the open-field lines that pass near the streamer core (also see Wang & Sheeley 1990;Wang 1994;Wang et al. 1997a;Chen & Hu 2001, 2002Suess & Nerney 1999, 2002. Our model does not provide a physical explanation for the temperature decrease at the fast-slow boundary and therefore cannot explain why the boundary occurs at A h ¼ 0:7A c . Understanding the physics of the fast-slow transition will require more detailed analysis of the energy balance of the coronal plasma, including the physical processes by which the temperature anisotropy of the protons is maintained. We speculate that proton perpendicular heating (by dissipation of transverse MHD waves) occurs in both the fast and slow winds, perhaps at roughly equal rates. However, the resulting temperature anisotropy T p? =T pk may be quite different in the two cases. In the fast wind, proton-proton collisions are less frequent because of the lower density, so the deviations from Maxwellian velocity distributions are larger than in the slow wind. Clearly, to understand why the fast-slow transition occurs at A h ¼ 0:7A c will require multidimensional models of wave heating and energy balance such a those developed by Chen & Hu (2001). We express our thanks to the anonymous referee for his/ her valuable comments on the paper, which helped to clarify the manuscript. This work was funded by the Smithsonian Astrophysical Observatory and by CONICET (the Argentinean National Council for Scientific and Technological Research). We also thank the Fundació n Antorchas (from Argentina) for partial support through grant 14056-20. Fig. 1 . 1-Electron density profiles derived from Skylab observations for the pole (thin curve) and the equator (thick curve). Fig. 3 . 3-Photospheric magnetic field as function of latitude at the time of the last cycle minimum (1996 July). The diamonds are measurements from Kitt Peak synoptic maps. The solid curve is a fit to the data. Fig. 4 . 4-Sketch of the dipolar configuration Fig. 5 . 5-Function ÈðAÞ used for interpolating temperatures between pole and equator. Fig. 6 . 6-Magnetic field and plasma pressure for the first iteration (left) and for the final model (right). The thin curves are magnetic field lines (contours of A), and the thick curves are contours of 8p ? =B 2 . Fig. 7 . 7-Outflow velocity in the meridional plane. The brightness level increases with magnitude of the outflow speed v, as shown by the scale on the left side of the figure (v in km s À1 ). The solar rotation axis is along the left axis of the plot, and the solar surface is indicated by the thick semicircle. The white curves are selected field lines labeled with their value of A=A c . intermediate heights. Our results are similar to those of Fig. 8 . 8-Various wind properties as functions of field-line variable A. Top: Temperature anisotropy ðTp? þ T e Þ=ðT pk þ T e Þ at r ¼ 3 R . Middle: Radius of sonic point where v ¼ c k (in R ). Bottom: Wind speed at R max ¼ 10 R (in km s À1 ).The thin solid vertical line in each panel indicates the field line A h where the latitudinal temperature gradient is largest, and the thick vertical lines indicate the width of the transition (A h AE A w ). The dot-dashed vertical line indicates the field line at which there is a sudden transition in the radius of the sonic point. Fig. 9 . 9-Transition between fast and slow wind. (a) Shape of various field lines in the meridional plane. (b) Functions f ðrÞ that appear on the right-hand side of the wind eq. (28) for different field lines. (c) Outflow velocity vðrÞ along field lines. The dotted curve corresponds to the last open-field line, A ¼ A c . (d ) Radial positions of the mimima of f ðrÞ as function of field-line variable A (diamonds). The vertical lines show the A-values of the field lines shown in other panels. (e) Nonradial expansion factors f exp ðrÞ for different field lines. The same line styles are used in all panels. Fig. 10 . 10-Images of visible-light polarization brightness ( pB) as predicted from the coronal model for an edge-on view (left) and for a view 30 above the ecliptic plane (right). The black curves are contours of logðpB=pB max Þ, where pB max is the value of pB at the coronal base. The white curves are selected field lines in the plane of the sky (labeled by A=A c ). The dark semicircle represents the solar disk. Fig. 11 . 11-Top: Visible-light polarization brightness of the corona, Fig. 13 . 13-Radial profiles of Ly intensity I. The solid curves are the predicted intensity along the pole (thin solid curve) and along the equator (thick solid curve) for the edge-on view ( ¼ 0). The dashed curves are the predicted profiles without Doppler dimming (i.e., with v r ¼ 0 in eq. [31]). The triangles and diamonds are intensity measurements from UVCS/ SOHO for the pole and equator, respectively (see text). TABLE 1 1Numerical Values of the Parameters for the Temperature Models T p? , T pk , T e at Equator T 0 (K) .................... T 1 (K) .................... a............................. b............................. ........................... ........................... Dr (R )...................Parameter T p? at Pole T pk , T e at Pole 8  10 5 8  10 5 8  10 5 3  10 6 0. 0. 0. 0. 0.1 0.23 0.47 0.33 .0.7 0.7 0.55 .6.6 6.6 6.6 1 . . . . . . 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[]
[ "A CHARACTERISATION OF C * -ALGEBRAS THROUGH POSITIVITY OF FUNCTIONALS", "A CHARACTERISATION OF C * -ALGEBRAS THROUGH POSITIVITY OF FUNCTIONALS" ]
[ "Marcel De ", "Jun Tomiyama " ]
[]
[]
We show that a unital involutive Banach algebra, with identity of norm one and continuous involution, is a C * -algebra, with the given involution and norm, if every continuous linear functional attaining its norm at the identity is positive.2010 Mathematics Subject Classification. Primary 46K05; Secondary 46H05.
10.15352/afa/1399899837
[ "https://export.arxiv.org/pdf/1205.6830v2.pdf" ]
2,531,246
1205.6830
70a676d8542a0e305fa5fc61e1bceb67033399b9
A CHARACTERISATION OF C * -ALGEBRAS THROUGH POSITIVITY OF FUNCTIONALS 22 Sep 2012 Marcel De Jun Tomiyama A CHARACTERISATION OF C * -ALGEBRAS THROUGH POSITIVITY OF FUNCTIONALS 22 Sep 2012and phrases Involutive Banach algebraC * -algebrapositive functional We show that a unital involutive Banach algebra, with identity of norm one and continuous involution, is a C * -algebra, with the given involution and norm, if every continuous linear functional attaining its norm at the identity is positive.2010 Mathematics Subject Classification. Primary 46K05; Secondary 46H05. If A is an involutive Banach algebra, then a linear map ω : A → C is called positive if ω(a * a) ≥ 0, for all a ∈ A. If the involution is isometric, and A has an identity 1 of norm one, then ω is automatically continuous, and ω = ω(1), see [4,Lemma I.9.9]. 1 For a unital C * -algebra A, there is a converse: if ω : A → C is continuous, and ω(1) = ω , then ω is positive (cf. [4,Lemma III.3.2]). Thus the positive continuous linear functionals on a unital C * -algebra are precisely the continuous linear functionals attaining their norm at the identity. Consequently, any Hahn-Banach extension of a positive linear functional, defined on a unital C * -subalgebra, is automatically positive again. As is well known, this is a basic characteristic of C * -algebras that makes the theory of states on such algebras a success. If A is a unital involutive Banach algebra with identity of norm one, but not a C * -algebra, then this converse, as valid for unital C * -algebras, need not hold: even when the involution is isometric, there can exist continuous linear functionals on A that attain their norm at the identity, but which fail to be positive. For example, for H ∞ (D), the algebra of bounded holomorphic functions on the open unit disk, supplied with the supremum norm and involution f * (z) = f (z) (z ∈ D, f ∈ H ∞ (D)), all point evaluations attain their norm at the identity, but only the evaluation in points in (−1, 1) are positive. As another example, consider ℓ 1 (Z), the group algebra of the integers. Then its dual can be identified with ℓ ∞ (Z), and the continuous linear functionals attaining their norm at the identity are then the bounded maps ω : Z → C, such that ω(0) = ω ∞ . Not all such continuous linear functionals are positive. 2 For example, if λ ∈ C, |λ| ≤ 1, then ω λ ∈ ℓ ∞ (Z), defined by ω(0) = 1, ω(1) = λ, and ω(n) = 0 if n = 0, 1, attains its norm at the identity of ℓ 1 (Z). However, if we define ℓ 0 : Z → C by ℓ 0 (0) = 1, ℓ 0 (1) = 1, and ℓ 0 (n) = 0 if n = 0, 1, then ℓ 0 ∈ ℓ 1 (Z), but ω 0 (ℓ * 0 ℓ 0 ) = 2 + λ need not even be real. It is the aim of this note to show that the existence of examples as above is no coincidence: there necessarily exist continuous linear functionals that attain their norm at the identity, yet are not positive, because the algebra in question has a continuous involution, but is not a C * -algebra. This is the main content of the result below which, with a rather elementary proof, follows from the far less elementary Vidav-Palmer theorem [1, Theorem 38.14]. We formulate the latter first for convenience. Theorem (Vidav-Palmer). Let A be a unital Banach algebra with identity of norm one. Let A S be the real linear subspace of all a ∈ A such that ω(a) is real, for every continuous linear functional ω on A such that ω = ω(1). If A = A S + i A S , then this is automatically a direct sum of real linear subspaces, and the well defined map (a 1 + ia 2 ) → (a 1 − ia 2 ) (a 1 , a 2 ∈ A S ) is an involution on A which, together with the given norm, makes A into a C * -algebra. As further preparation let us note that, if A is a unital involutive Banach algebra with identity of norm one and a continuous involution, and if a ∈ A is self-adjoint with spectral radius less than 1, then there exists a self-adjoint element b ∈ A such that 1 − a = b 2 . Indeed, using the continuity of the involution, the proof as usually given for a unital Banach algebra with isometric involution, cf. [4,Lemma I.9.8], which is based on the fact that the coefficients of the power series around 0 of the principal branch of √ 1 − z on D are all real, goes through unchanged. Theorem. Let A be a unital involutive Banach algebra with identity 1 of norm one. Then the following are equivalent: ( 1) The involution is continuous, and, if ω is a continuous linear functional on A such that ω = ω(1), then ω is positive; (2 ) The involution is continuous, and, if ω is a continuous linear functional on A such that ω = ω(1), and a ∈ A is self-adjoint, then ω(a 2 ) is real; (3) A is a C * -algebra with the given norm and involution. Proof. We need only prove that (2) implies (3). Suppose that a ∈ A is self-adjoint and that a < 1. Then, as remarked preceding the theorem, there exists a selfadjoint b ∈ A such that 1 − a = b 2 . If ω is a continuous linear functional on A such that ω = ω(1), then the assumption in (2) implies that 1−ω(a) = ω(1−a) = ω(b 2 ) is real. Hence ω(a) is real. This implies that ω(a) is real, for all self-adjoint a ∈ A, and for all continuous linear functionals ω on A such that ω = ω(1). Since certainly every element of A can be written as a 1 + ia 2 , for self-adjoint a 1 , a 2 ∈ A, this shows that A = A S + i A S . Then the Vidav-Palmer theorem yields that the involution in that theorem, which agrees with the given one, together with the given norm, makes A into a C * -algebra. In [2, Theorem 11.2.5], a number of equivalent criteria are given for a unital involutive Banach algebra-with a possibly discontinuous involution-to be a C *algebra, but positivity of certain continuous linear functionals is not among them. The proof above of such a criterion is made possible by the extra condition of the continuity of the involution. Although, given the Vidav-Palmer theorem, the proof is quite straightforward, we are not aware of a reference for this characterisation of C * -algebras through positivity of linear functionals. Since the result seems to have a certain appeal, we thought it worthwhile to make it explicit. Marcel de Jeu, Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands E-mail address: [email protected] Jun Tomiyama, Department of Mathematics, Tokyo Metropolitan University, Minami-Osawa, Hachioji City, Japan E-mail address: [email protected] Complete normed algebras. F F Bonsall, J Duncan, SpringerF.F. Bonsall and J. Duncan, Complete normed algebras, Springer, 1973. Banach algebras and the general theory of * -algebras. T W Palmer, Cambridge University PressII. * -algebrasCambridgeT.W. Palmer, Banach algebras and the general theory of * -algebras. Vol. II. * -algebras, Cam- bridge University Press, Cambridge, 2001. W R Rudin, Functional analysis. Second Edition. New YorkMcGraw-HillW.R. Rudin, Functional analysis. Second Edition, McGraw-Hill, New York, 1991. M Takesaki, Theory of operator algebras. I. New York-HeidelbergSpringerM. Takesaki, Theory of operator algebras. I, Springer, New York-Heidelberg, 1979.
[]
[ "Machine-learning approach for discovery of conventional superconductors", "Machine-learning approach for discovery of conventional superconductors" ]
[ "Huan Tran \nSchool of Materials Science & Engineering\nGeorgia Institute of Technology\n771 Ferst Dr. NW30332AtlantaGAUSA\n", "Tuoc N Vu \nInstitute of Engineering Physics\nHanoi University of Science & Technology\n1 Dai Co Viet Rd10000HanoiVietnam\n" ]
[ "School of Materials Science & Engineering\nGeorgia Institute of Technology\n771 Ferst Dr. NW30332AtlantaGAUSA", "Institute of Engineering Physics\nHanoi University of Science & Technology\n1 Dai Co Viet Rd10000HanoiVietnam" ]
[]
First-principles computations are the driving force behind numerous discoveries of hydride-based superconductors, mostly at high pressures, during the last decade. Machine-learning (ML) approaches can further accelerate the future discoveries if their reliability can be improved. The main challenge of current ML approaches, typically aiming at predicting the critical temperature Tc of a solid from its chemical composition and target pressure, is that the correlations to be learned are deeply hidden, indirect, and uncertain. In this work, we showed that predicting superconductivity at any pressure from the atomic structure is sustainable and reliable. For a demonstration, we curated a diverse dataset of 584 atomic structures for which λ and ω log , two parameters of the electronphonon interactions, were computed. We then trained some ML models to predict λ and ω log , from which Tc can be computed in a post-processing manner. The models were validated and used to identify two possible superconductors whose Tc 10 − 15K at zero pressure. Interestingly, these materials have been synthesized and studied in some other contexts. In summary, the proposed ML approach enables a pathway to directly transfer what can be learned from the high-pressure atomic-level details that correlate with high-Tc superconductivity to zero pressure. Going forward, this strategy will be improved to better contribute to the discoveries of new superconductors.
10.1103/physrevmaterials.7.054805
[ "https://export.arxiv.org/pdf/2211.03265v2.pdf" ]
253,383,991
2211.03265
34ab9012304ca49579d86e77240635dcc57dc33d
Machine-learning approach for discovery of conventional superconductors Huan Tran School of Materials Science & Engineering Georgia Institute of Technology 771 Ferst Dr. NW30332AtlantaGAUSA Tuoc N Vu Institute of Engineering Physics Hanoi University of Science & Technology 1 Dai Co Viet Rd10000HanoiVietnam Machine-learning approach for discovery of conventional superconductors First-principles computations are the driving force behind numerous discoveries of hydride-based superconductors, mostly at high pressures, during the last decade. Machine-learning (ML) approaches can further accelerate the future discoveries if their reliability can be improved. The main challenge of current ML approaches, typically aiming at predicting the critical temperature Tc of a solid from its chemical composition and target pressure, is that the correlations to be learned are deeply hidden, indirect, and uncertain. In this work, we showed that predicting superconductivity at any pressure from the atomic structure is sustainable and reliable. For a demonstration, we curated a diverse dataset of 584 atomic structures for which λ and ω log , two parameters of the electronphonon interactions, were computed. We then trained some ML models to predict λ and ω log , from which Tc can be computed in a post-processing manner. The models were validated and used to identify two possible superconductors whose Tc 10 − 15K at zero pressure. Interestingly, these materials have been synthesized and studied in some other contexts. In summary, the proposed ML approach enables a pathway to directly transfer what can be learned from the high-pressure atomic-level details that correlate with high-Tc superconductivity to zero pressure. Going forward, this strategy will be improved to better contribute to the discoveries of new superconductors. I. INTRODUCTION In the search for high critical temperature (T c ) superconductors, significant progress has been made during the last decade [1][2][3]. Among thousands of hydride-based superconducting materials computationally predicted [4][5][6][7][8][9][10][11][12], mostly at very high pressures, e.g., P 100 GPa, dozens of them, e.g., H 3 S [1], LaH 10 [2], and CSH [3], were synthesized and tested. This active research area is presumably motivated by Ashcroft, who, in 2004, predicted [13] that high-T c superconductivity may be found in hydrogen dominant metallic alloys, probably at high P . Another driving force is the development of firstprinciples computational methods to predict material structures at any P [14][15][16][17][18][19][20] and to calculate the electronphonon (EP) interactions [21,22], the atomic mechanism behind the conventional superconductivity, according to the Bardeen-Cooper-Schrieffer (BCS) theory [23]. While critical debates on some discoveries [24][25][26][27][28][29] are on-going, it seems that the one-day-realized dream of superconductors at ambient conditions may be possible. Readers are referred to some reviews [5,6,8,30] and a recent roadmap [9] for progresses, challenges, and future pathways of this research area. The central role of first-principles computations in the recent discoveries of conventional superconductors stems fromÉliashberg theory [31][32][33][34], of which the spectral function α 2 F (ω) characterizing the EP interactions could be evaluated numerically. The first inverse moment λ and logarithmic moment ω log of α 2 F (ω), together with an empirical Coulomb pseudopotential µ * , are the inputs to estimate T c by either solving theÉliashberg equations [31][32][33][34] or using the McMillan formula [35][36][37] (see Sec. * [email protected] II A for more details). In a typical workflow (Fig. 1), a search for stable atomic structures across multiple related chemical compositions is performed at a given pressure, usually with first-principles computations. Then, α 2 F (ω), λ, ω log , and finally T c are evaluated, identifying candidates with high estimated T c for possible new superconducting materials. Although structure prediction [14][15][16][17] and α 2 F (ω) computations [21,22] are extremely expensive and technically non-trivial, significant research efforts have been devoted to and shaped by this workflow. Machine-learning (ML) methods have recently emerged in the discoveries of superconductors [9,10]. As sketched in Fig. 1, existing ML efforts can be categorized into four lines, including (i) using some ML potentials to accelerate the structure prediction step [38], (ii) using some symbolic ML techniques to derive new empirical expressions for T c [39,40], (iii) developing some ML models to predict T c from a chemical composition at a given pressure P [41][42][43][44][45][46][47][48][49][50], and (iv) developing some ML models to predict λ, ω log , and α 2 F (ω) from the atomic structures [51]. While line (iii) is predominant, its role remains limited, presumably because the connections from the chemical composition and the target P to T c are deeply hidden. In fact, there are at least two "missing links" between the two ends of this approach. One of them is the atomic-level information while the other is the microscopic mechanism of the superconductivity, e.g., the EP interactions in conventional superconductors. The former is critical because for a given chemical composition, the properties of thermodynamically competing atomic structures can often be fundamentally different, e.g., one is insulating and another is conducting [18,52]. Therefore, ignoring the atomic structure is equivalent to adding an irreducible uncertainty into the ML predictions [53]. Likewise, the latter cannot be overemphasized. In fact, bypassing α 2 F (ω), λ, and ω log , and using an empirical value of µ * are intractable assumptions, and thus, uncontrollable approximations. In line (iv), initialized recently by Ref. 51 during the (independent) preparation of this work, these missing links are addressed in some ways. In this paper, we present an initial step to bring the atomic-level information into the ML-driven pathways toward new conventional (or BCS) superconductors, especially at ambient pressure. For this goal, we curated a dataset of 584 atomic structures for which more than 1,100 values of λ and ω log were computed at different values of P and reported, mostly in the last decade. The obtained dataset was visualized, validated, and standardized before being used to develop ML models for λ and ω log . Then, they were used to screen over 80, 000 entries of Materials Project database [54], identifying and confirming (by first-principles computations) two thermodynamically and dynamically stable materials whose superconductivity may exist at T c 10−15K and P = 0. We also proposed a procedure to compute λ and ω log , for which convergence are generally hard to attain [51]. This scheme relies on the direct connection between the atomic structures and λ and ω log , quantitatively described in Sec. II A. Pressure is an implicit input, i.e., P determines the atomic structures for which λ and ω log are computed/predicted. The design of this scheme has some implications. First, the ML models are trained on the atomic structures realized at high P and (computationally) proved to correlate with high-T c superconductivity. These structures can be considered "unusual" in the sense that their high-P atomic-level details, e.g., short bond lengths and distorted bond angles, are not usually realized at zero pressure. Therefore, we hope that the ML models can identify the atomic structures realized at P = 0 with relevant unusual atomic-level features, and thus, they may exhibit possible high-T c superconductivity. Second, massive material databases [55] like Materials Project [54], OQMD [56] and NOMAD with millions of atomic structures can now be screened directly with robust and reliable ML models. Given that only a small search space was explored in this demonstrative work, we expect more superconducting materials to be discovered in the next steps of our effort. II. METHODS A.Éliashberg theory and McMillan formula InÉliashberg theory [31][32][33][34], α 2 F (ω) is a spectral function characterizing the EP scattering, which is defined as α 2 F (ω) = 1 N 0 kk ijν |g ij,ν k,k | 2 δ(ε i k )δ(ε j k )δ(ω − ω ν k−k ). (1) Here, N 0 is the density of states at the Fermi level, g ij,ν k,k the electron-phonon matrix elements, ν the polarization index of the phonon with frequency ω, δ the delta-Dirac function, and (k and k )/(ε i k and ε j k ) the (electron wave vectors)/(band energies) corresponding to the band indices (i and j), respectively. The standard method to compute α 2 F (ω) is density functional perturbation theory (DFPT) [21,22], as implemented in major codes like Quantum ESPRESSO [57,58] and ABINIT [59][60][61]. Having α 2 F (ω), T c can be evaluated by numerically solving a set of (unfortunately, quite complicated)Éliashberg equations using, for example, Electron-Phonon Wannier (EPW) [62][63][64]. The much more frequent method to estimate T c is using some empirical formulas derived from theÉliashberg equations. Perhaps the most extensively used formula is T c = ω log 1.2 exp − 1.04(1 + λ) λ − µ * (1 + 0.62λ) ,(2) which was developed by McMillan [35] and latter improved by Allen and Dynes [36,37]. Here, λ = 2 ∞ 0 dω α 2 F (ω) ω(3) is the (averaged) isotropic EP coupling while ω log = exp 2 λ ∞ 0 dω ln(ω) α 2 F (ω) ω .(4) Following Ashcroft [13], the Coulomb pseudopotential µ * , which appears in Eq. 2 and connects with N 0 , was empirically chosen in the range between 0.10 and 0.15. Eq. (2) indicates that in general, high values of λ and/or ω log are needed for a high value of T c . Some new empirical formulas of T c were developed recently [39,40] using some symbolic ML techniques. Moving forward, developing a truly ab-initio framework for computing T c [65][66][67] is desirable and currently active. The McMillan formula (2) is believed to be good for λ ≤ 1.5 while additional empirical parameters are needed for larger λ [37]. Nevertheless, the exponential factor of Eq. 2 has a singular point at λ = µ * /(1 − 0.62µ * ), which could lead to unwanted/unphysical divergence. If we select µ * = 0.1 (or 0.15), T c → ∞ when λ approaches 0.1066 (or 0.1654) from below. Such values of λ have been realized in many computational works [68][69][70][71], although much larger values, e.g., λ ≥ 0.7, are generally needed for high-T c superconductors. Given these observations, we believe that a ML approach for discovering conventional superconductor should focus on λ, ω log , and perhaps α 2 F (ω), from which T c can easily be estimated using, for examples, Eq. (2). B. Basic idea and approach The ML approach used in this work focuses on predicting λ and ω log from the atomic structure of the considered materials. As visualized in Fig. 1, the role of P is embedded in the main input of this scheme, i.e., the atomic structure, which is determined from P . The rationale of this design is two fold. First, what this ML approach will learn is a direct and physics-inspired correlation from an atomic structure to λ and ω log through α 2 F (ω), as quantitatively described in Eqs. (1), (3), and (4). Second, the training data, which include the atomic environments/structures realized at multiple values of (sometimes very high) pressure P that could lead to very high values of λ and ω log , will be highly diverse and comprehensive. Consequently, the resulted ML models will thus be robust, reliable, and, more importantly, they can be used to recognize new high-T c superconductors that resemble unusual atomic-level details at any pressure, specifically P = 0 GPa. This approach involves some challenges, one of them is how to obtain good datasets for the learning scheme. Our solution is described below. C. Data curation This work requires a dataset of the atomic structures for which λ, ω log , and α 2 F (ω) were computed and reported. The curation of such a dataset is painstaking. Scientific articles published during the last 10 − 15 years, reporting computed superconducting properties of new or known materials, were collected. In majority of the articles, the atomic structures were reported in some Tables while electronic files of standard formats, e.g., crystallographic information file (CIF), were given in very few cases. In some cases, important information, e.g., angle β in a monoclinic structure, was missing from the Tables. When the provided information is sufficient, we used the obtained crystal symmetry/space group, lattice parameters, Wyckoff positions, and the coordinates of the inequivalent atoms, to manually reconstruct the reported structures. All the atomic structures obtained from electronic files and/or reconstructed from data Tables were inspected visually. During this step, a good number of them were found to be clearly incorrect, largely because of typos, number overrounding, and other possible unidentified reasons, when reporting the data. Incorrect structures were discarded. Superconducting-related properties, e.g., λ, ω log , and T c , which were computed and reported for the atomic structures at pressure P up to 800 GPa, were collected. These properties were mainly computed by some major workhorses like Quantum ESPRESSO [57,58], ABINIT [59][60][61], and EPW codes [62][63][64], employing different pseudopotentials, XC functionals, energy cutoffs, smearing width needed to compute the δ functions appearing in the expression (1) of α 2 F (ω), and more. We recognize that data of λ and ω log curated from scientific literature are not entirely uniform; they rather contain a certain level of uncertainty that will inevitably be translated into the (aleatoric) uncertainty of the predictions [53]. However, the demonstrated reproducibility of advanced firstprinciples computations [72] suggests that data carefully produced by major codes should still be consistent and reliable. To further improve the uniformity of the data, we used density functional theory (DFT) [73,74] calculations to optimize the obtained atomic structures at the pressures reported, employing the same technical details used for Materials Project database. The rationale behind this step is that the predictive ML models trained on the dataset will then be used to predict λ and ω log for the atomic structures obtained from Materials Project database. Therefore, the training data should be prepared at the same level of computations with the input data for predictions. In fact, a vast majority of our DFT optimizations were terminated after about a dozen steps or below, indicating that they were already optimized very well. Details of the optimizations are given in Sec. II E. Compared with the DFPT calculations for λ and ω log , the optimization step is computationally negligible. Our dataset includes 584 atomic structures for which at least λ was computed and reported. Among them, 567 atomic structures underwent ω log and thus, T c calculations (there is a trend in the community that computed λ is more likely to be reported than ω log when discussing the superconductivity). Our dataset, which is summarized in Figs. 2 (a), (b), (c), (d), and (e), contains 53 species and covers a substantial part of the Periodic Table. Five most frequently encountered species are H, B, Li, Mg, and Si, which were found in 505, 83, 57, 53, and 48 entries, respectively. The dominance of H in this dataset reflects the focus of the community on super hydrides when searching for high-T c superconductors. For λ, the smallest value is 0.089, reported in Ref. 68 for the P 4/mbm structure of LiH 2 at P = 150 GPa while the largest value is 5.81 reported in Ref. 44 for the Im3m structure of CaH 6 at P = 100 GPa. Likewise, the smallest value of ω log is 71 K reported in Ref. 75 for the I4/mmm structure of TiH at P = 50 GPa whereas the largest value is 2,234 K reported in Ref. 44 for the P 62m structure of CaH 15 at P = 500 GPa. Figs. 2 (d) and (e) provide two histograms summarizing the λ and ω log datasets. D. Data representation and machine-learning approaches Materials atomic structures are not naturally ready for ML algorithms. The main reason is that they are not invariant with respect to transformations that do not change the materials in any physical and/or chemical ways, e.g., translations, rotations, and permutations of alike atoms. Therefore, we used matminer [76], a package that offers a rich variety of material features, to convert (or featurize) the atomic structures into numerical vectors, which meet the requirements of invariance and can be used to train ML models. Starting from several hundreds components, optimal sets of features (the vector components) were determined using the recursive feature elimination algorithm as implemented in scikitlearn library [77]. The final version of the λ and ω log datasets have 40 and 38 features, respectively. In principle, two featurized datasets of λ and ω log can be learned simultaneously using a multi-task learning scheme so that the underlying correlations between λ and ω log may be exploited. However, the intrinsically deep correlations in materials properties require a sufficiently big volume of data to be revealed. We have tested some multi-task learning schemes and found that with a few hundreads data points, they are not significantly better than learning λ and ω log separately. In fact, similar behaviors are commonly observed in the literature [53]. Therefore, we examined six typical ML algorithms, including support vector regression, random forest regression, kernel ridge regression, Gaussian process regression, gradient boosting regression, and artificial neural networks two develop ML models for λ and ω log . For each algorithm, we created a pair of learning curves and used them to analyze the performance of the algorithm on the data we have. By carefully tuning the possible model parameters and examining the training and the validation curves, Gaussian process regression (GPR) [78,79] was selected. Details on the learning curves and the GPR models used for predicting λ and ω log are discussed in Sec. III A. E. First-principles calculations First-principles calculations are needed for two purposes, i.e., to uniformly optimize the curated atomic structures and to compute α 2 F (ω), λ, and ω log for those identified by the ML models we developed. For the first objective, we followed the technical details used for Materials Project database, employing vasp code [80,81], the standard PAW pseudopotentials, a basis set of plane waves with kinetic energy up to 520 eV, and the generalized gradient approximation Perdew-Burke-Ernzerhof (PBE) exchange-correlation (XC) functional. [82] Convergence in optimizing the structures was assumed when the atomic forces become < 10 −2 eV/Å after no more than 3 iterations. In the computations of α 2 F (ω), λ, and ω log , we used the version of DFPT implemented in ABINIT package [59][60][61], which also offers a rich variety of other DFT-based functionalities. Within this numerical scheme, we used the optimized norm-conserving Vanderbilt pseudopotentials (ONCVPSP-PBE-PDv0.4) [83] obtained from the PseudoDojo library [84] and the PBE XC functional [82]. The kinetic energy cutoff we used is 60 Hatree ( 1, 600eV), which is twice larger than the value suggested [83] for these norm-conserving pseudopotentials. The smearing width for computing α 2 F (ω) is 5 × 10 −6 Ha, i.e., 0.032 THz. This value was selected to be < 0.1% of the entire range of frequency while covering more than 4 (numerical) spacings of the frequency grid. Before entering the electron-phonon calculations with DFPT, the material structures under consideration were repeatedly optimized until the maximum atomic force is below 10 −5 Hatree/bohr, which is 5.1 × 10 −4 eV/Å, after no more than 3 iterations. Because the optimizations need the simulation box to change its shape, such a small number of iterations is required to minimize the cell volume change, thereby limiting the Pulay stress, and ultimately ensuring an absolute convergence of the force calculations. This level of accuracy is generally needed for phonon-related calculations. Eq. 1 indicates that α 2 F (ω) is evaluated on a q-point grid of q = k − k , which must be a sub-grid of the full k-point grid used to sample the Brillouin zone for regular DFT calculations. Therefore, calculations of α 2 F (ω) are extremely heavy while the convergence with respect to the q-point grid is critical and must be examined [44,51]. For this goal, we first computed α 2 F (ω), λ, and ω log using several q-point grids of q × q × q and k-point grids of k×k×k where q is as large as possible depending on the structure size and k ≥ 3 × q. Then, the computed values of λ and ω log are fitted to a linear function of 1/q. The values of the fitted functions at 1/q = 0, or, equivalently, at the limit of q → ∞, are the values assumed for λ and ω log . This procedure is visualized in Fig. 3 when λ and ω log of two atomic structures reported in this work were computed. Details on the q-point and k-point grids and the corresponding computed data used for the fitting procedure can be found in Supplemental Material [85]. A technique of similar philosophy has been demonstrated [86] in the computations of ring-opening enthalpy, the thermodynamic quantity that controls the ring-opening polymerizations. F. Candidates We obtained the Materials Project database [54] of 83,989 atomic structures and several properties uniformly computed at P = 0 using vasp [80,81]. Starting from this dataset, we selected a subset of 35 atomic structures that have energy above hull E hull < 0.03 eV/atom, zero band gap (E g = 0 eV), no more than 16 atoms in the primitive cell, and only the species included in the training data, specifically H (see Fig. 2). The first criterion "places" the selected atomic structures into the so-called "amorphous limit", a concept defined in an analysis of Materials Project database [87] and used to label the atomic structures that are (or nearly) thermodynamically stable and thus, they may be synthesized. In fact, some metastable ferroelectric phases of hafnia that are above the ground state of 0.03 eV/atom [18,88,89] have been stabilized and synthesized [90,91]. Next, E g = 0 eV was used to remove non-conducting materials while the third criterion aims at selecting small enough systems for which computations of λ and ω log are affordable. Finally, by considering only those having the species included in the training data, specifically H, we expect that the ML models will only be used in their domain of applicability. The procedure is summarized in Fig. 4. The set of 35 candidates has no overlap with the training data. This set is small because the requirement of having H is very strong. In fact, removing this requirement increases the candidate set size to 2,694. Given that the ML models are extremely rapid, there is in fact no time difference between predicting λ and ω log for 35 atomic structures and predicting these properties for 2,694 atomic structures. However, the dominance of H in the training dataset strongly suggests that the smaller set of 35 candidates is more suitable for the demonstration purpose of this work. In the next step, the training dataset will be augmented with λ and ω log computed FIG. 5. Learning curves obtained by learning two datasets of (a) λ and (b) ω log , a typical model trained on 90% of the data of (c) λ and (d) ω log and validated on the remaining unseen 10% data, and two ML models trained on 100% of the data of (e) λ and (f) ω log . In (a) and (b), each data point is associated with an errorbar obtained from 100 models that were independently trained. for materials having underrepresented species, and larger candidate sets will be examined. III. RESULTS A. Machine-learning models Given a learning algorithm and a dataset that has been represented appropriately, learning curves can be created using an established procedure. In this work, each dataset was randomly split into a training set and a (holdout) validation set. Next, a ML model was trained on the training set using standard 5-fold crossvalidation procedure [92] to regulate the potential overfitting. Then, the ML model was tested on the validation set, which is entirely unseen to the trained model. By repeating this procedure 100 times and varying the training set size, a training curve and a validation curve were produced from the mean and the standard devia- tion of the root-mean-square error (RMSE) of the predictions of the training sets and the validation sets. During the training/validating processes, randomness stems from the training/validation data splitting and the 5fold training data splitting for internal cross validation. As such random fluctuations are suppressed statistically by averaging over 100 independent models, the learning curves could provide some useful and unbiased insights into the performance of the data, the featurize procedure, the learning algorithm, and ultimately the ML models that are developed. Two learning curves obtained by using GPR to learn the (featurized) λ and ω log datasets are shown in Figs. 5 (a) and (b). In both cases, the training curves saturate at 0.15 (for λ) and 110 K (for ω log ). These values are small, i.e., they are 3 − 5% of the data range, implying that GPR can successfully capture the behaviors of the data. On the other hand, the validation curves of λ and ω log data do not saturate and keep decreasing. This behavior strongly suggests that if more data are available, the gap between the learning and the validation curves can further be reduced and the performance of the target ML models can readily be elevated. Figs. 5 (a) and (b) reveal that an error of 0.4 and 200 K can be expected for the predictions of λ and ω log , respectively. The expected errors are roughly 7 % of the whole range of λ and ω log data, which are significantly small compared to the results reported in Ref. 51. Figs. 5 (c) and (d) visualize two typical ML models trained on 90% of the λ and ω log datasets and validated on the remaining 10% of the datasets. Likewise, Figs. 5 (e) and (f) visualize two typical ML models, each of them was trained on the entire λ or ω log dataset using the exactly same procedure. In fact, each of them is one of 100 ML models that were trained independently and used to predict λ and ω log of the candidate set. B. Discovered superconductors and validations We used the developed ML models to predict λ and ω log of 35 atomic structures in the candidate set, and then to compute the critical temperature T c using the McMillan formula with µ * = 0.1. The predicted λ ranges from 0.31 to 0.88, and consequently, the predicted T c ranges from 0.16 K to 21.3 K. Six candidates with highest predicted T c are those with Materials Project ID of mp-24289, mp-1018133, mp-24081, mp-24287, mp-1008376, and mp-24208. Details of these candidates are summarized in Table I while comprehensive information of all 35 candidates can be found in Supplemental Material [85]. Examining the top six candidates, we found that mp-24081 is a trigonal structure of ScClH, whose primitive cell has 6 atoms and three very small lattice angles (α = β = γ = 21.38 • ). Computations of the EP interactions in such a structure are prohibitively expensive because the required k-point and q-point grids must be extremely large. In addition, Ce, the species showing up in mp-1008376, a cubic structure of CeH 3 , is not supported by ONCVPSP-PBE-PDv0.4 norm-conserving pseudopotential set [83]. Therefore, computations were performed for the remaining four candidates. Among them, mp-24289, a cubic structure of PdH and mp-1018133, a tetragonal structure of LiHPd, are dynamically unstable. In principles, each of them can be stabilized by following the imaginary phonon modes to end up at a dynamically stable structure with lower energy and symmetry [93]. Such heavy and cumbersome technical procedure was reserved for the next steps. The last two candidates are mp-24287, which is a cubic structure of CrH and mp-24208, which is also a cubic structure of CrH 2 . Both of them, visualized in Figs. 6 (a) and (b), are dynamically stable and thus, their λ and ω log were computed using the procedure described in Sec. II E. The phonon band structures, which prove the dynamical stability of mp-24289, mp-1018133, mp-24287, and mp-24208, can be found in the Supplemental Material [85]. Predicted and computed α 2 F (ω), λ, ω log , and T c (using the McMillan formula with µ * = 0.1) of mp-24287 and mp-24208 at P = 0 are given in Table I agree remarkably well with the ML predicted values. Given that magnesium diboride MgB 2 in its hexagonal P 6/mmm phase is the highest-T c conventional superconductor with T c 39 K [94], the examined materials have respectable (computed) critical temperature, i.e., T c = 15.7 K for mp-24287 and T c = 10.7 for mp-24208. By examining the electronic structures of mp-24287 and mp-24208 reported in Materials Project database, we confirmed that both of them are metallic in nature with a large density of states at the Fermi level. We extended our validation to the high-P domain by predicting and then computing λ and ω log of mp-24287 and mp-24208 after optimizing them at P = 50 GPa and P = 100 GPa. Both of them were found to be dynamically stable at these pressures while the computed superconducting properties are shown in Table I and Figs. 6 (e) and (f). We also found that the computed and the predicted values of λ and ω log at P = 50 GPa and P = 100 GPa are remarkably consistent. For both materials, computed λ and T c decrease while ω log increases from 0 to 100 GPa, and the ML models capture correctly these behaviors within the expected errors given from the analysis of the learning curves in Sec. III A. Specifically, predictions of λ at P = 50 GPa and P = 100 GPa are within 0.1 from the computed results, leading to a remarkably small error of 3 K in predicting T c . C. Further assessments on the predictions We attempted to verify our predictions in a few ways. First, additional calculations for λ and ω log of mp-24287 and mp-24208 using the local-density approximation (LDA) XC functional were performed at all the pressure values examined (see Sec. III B). The obtained results, as given in the Supplemental Material [85], are highly consistent with, i.e., within 2−3% of, the reported results using the PBE XC functional. Next, we used EPW code [62][63][64] to numerically solve theÉliashberg equations on the imaginary axis and then approximated the real-axis superconducting gap ∆ 0 of mp-24287 and mp-24208 using Páde continuation [95]. Within this scheme, the electron-phonon interactions were computed by Quantum ESPRESSO [57,58], using the ultra-soft pseudopotentials from PS Library [96], an energy cutoff of 120 Ry (which is 60 Ha, 1, 600 eV), a k-point grid of 24×24×24 and a q-point grid of 6×6×6. During the EPW calculations, we used a fine q-point grid of 12 × 12 × 12 and µ * = 0.1. The superconducting gap ∆ 0 (T ) computed for mp-24287 and mp-24208 and shown in Fig. 7 projects a T c 22 − 24 K for mp-24287 and a T c 7 − 8 K for mp-24208. These values are in good agreement with that reported in Fig. 6, providing a confirmation of the predicted superconductivity of mp-24287 and mp-24208 at P = 0 GPa. Finally, we turn our attention to the synthesizability of mp-24287 and mp-24208 by tracing their origin. Information from Materials Project database allows us to track them down to two entries numbered 191080 and 26630 of the Inorganic Crystalline Structure Database (ICSD), and finally to Refs. 97 and 98, respectively. In short, mp-24287 and mp-24208 were experimentally synthesized and resolved [97,98] sometimes in the past. Afterwards, some experimental [99,100] and computational [101,102] efforts followed, examining their magnetic, electronic, and mechanical properties. Perhaps because preparing them experimentally is challenging, little more is known about these materials. Given the documented evidence of the synthesizability of both mp-24287 and mp-24208 at 0 GPa, which is in contrast with the enormous challenges of performing experiments at hundreds of GPa, we hope that these materials will be resynthesized and tested for the predicted superconductivity in the near future. IV. REMARKS AND GOING FORWARD Predicting λ and ω log from the atomic structures has some advantages. First, the correlation between the atomic structures and λ and ω log , which will be learned, is direct, physics-inspired, and intuitive, while computing T c from λ and ω log is trivial. Second, the obtained ML models, which are accurate and robust, can be directly used not only for extant massive material databases with millions of atomic structures but also for any structure searches in an on-the-fly manner. Finally, by using pressure as an implicit input, the training data can be highly diverse and comprehensive, ultimately allowing the ML models to be able to handle unusual atomic envi-ronments, frequently encountered during unconstrained structure searches for new materials. The accuracy demonstrated in Sec. III B for the ML models of λ and ω log is rooted from a series of factors. The list includes at least a reliable training dataset, a featurizing procedure that can capture the essential information encoded in the atomic structures, a ML algorithm that can learn the featurized data efficiently, a careful justification of the domain of applicability of the ML models, and a good candidate set. On the other hand, these stringent factors limit the number of candidates used in this work, although the ML models are already very fast to make millions of predictions. In the next steps, we will improve the whole scheme in several ways. First, by enlarging and diversifying the dataset while maintaining its quality, the domain of applicability of the ML models will be systematically expanded. For examples, the candidate set obtained from the selection procedure described in Fig. 4 will jump to 2,694 atomic structures when we can remove the requirement of having H in the chemical composition. Coming that point, we believe that many more new superconductors can be identified and validated, at least by firstprinciples computations. Second, modern deep learning techniques will be used to improve and possibly to unify the featurizing and the learning steps. Third, the ML models will be integrated in an inverse design strategy to explore the practically infinite materials space in an efficient manner. Currently, (inverse) design of functional materials with targeted properties is a very active research area with many success stories [103][104][105][106][107][108][109][110]. We hope that superconducting materials discoveries can be added to this list in the near future. Finally, we will work with experimental experts to synthesize and test the superconducting materials discovered computationally, closing the loop of materials design. V. CONCLUSIONS We have demonstrated a ML approach for the discovery of conventional superconductors at any pressure. By exploring and learning the direct and physics-inspired correlation between the atomic structures and their possible superconducting properties, specifically λ and ω log , highly accurate and reliable ML models were developed. These models were validated against the standard firstprinciples calculations of λ and ω log , identifying two potential superconducting materials with respectable critical temperature T c at zero pressure. Interestingly, these materials have been synthesized and studied in some other contexts. The main implication of this approach is that by learning the high-P atomic-level details that are connected to high-T c superconductivity, the obtained ML models can be used to identify the atomic structures realized at zero pressure with possible high-T c superconductivity. Given that the models can be used directly for massive materials databases with millions of atomic configurations, more superconductors can be expected in near future. We plan to improve this strategy in multiple ways, hoping that it can better contribute to the search of high-T c superconductors that has been highly active during the last decade. FIG. 2 . 2A summary of computed λ, ω log , and Tc dataset of 584 superconducting materials reported and curated, including (a) the Periodic Table coverage, (b) 10 most-frequent species in the dataset, (c) 567 values of Tc computed and arranged at different pressures, and the distribution of (d) 584 computed values of λ, and (e) 567 values of ω log , are given. Among 53 species found in our dataset, 47 of them are shown in (a) and the other 6 species are Ac, Ce, La, Nd, Pm, and Pr. In (b), each solid circle represents a combination of a chemical composition and a pressure while errorbars are for cases Tc was computed for different atomic structures, using different methods, e.g., using McMillan formula and solvinǵ Eliashberg equations, and/or different values of µ * . FIG. 3 . 3Fitting procedure used to compute (a) λ and (b) ω log of mp-24287 and mp-24208, two atomic structures identified from Materials Project database. Solid symbols show λ and ω log computed with some finite q-point grids while stars represent the extrapolated values of λ and ω log at the limit of infinite q-point grid, i.e., 1/q = 0. FIG. 6 . 6Computed superconducting properties of mp-24287 and mp-24208, whose atomic structures are visualized in (a) and (b) and spectral function α 2 F (ω) and the accumulative λ(ω) are shown in (c) and (d). The k-point and q-point grids used for (c) are 24 × 24 × 24 and 8 × 8 × 8, respectively, while those used for (d) are 21 × 21 × 21 and 7 × 7 × 7, respectively. In (e) and (f), solid and dashed lines are used to show the computed (using the extrapolation procedure described in Sec. II E) and the predicted values λ, ω log , and Tc (computed from λ and ω log using McMillan formula with µ * = 0.1) at P = 0, 50, and 100 GPa. FIG. 7 . 7and Figs. 6 (c) and (d). Considering the expected errors of the ML models, it is obvious that the computed λ and ω log Distribution of the zero-pressure superconducting gap function ∆0(T ) computed by numerically solving theÉliashberg equations for mp-24287 and mp-24208. The dashed curves, joining the middle point of the distributions, serve as the guide to the eyes. The critical temperature Tc is estimated to be at the middle point of the downward-sloping segment of the ∆0(T ) curves. TABLE I . ISix hydrogen-containing materials that have highest predicted Tc among 35 materials in the candidate set, given in the top part. For each of them, the ID and the energy above hull E hull obtained from Materials Project are given (the pressure P and computed band gap are all zero). Predicted λ, ω log , and Tc were obtained from the ML models and computed using McMillan formula with µ * = 0.1. Among the 6 materials, computations were performed for 4 materials, two of them (mp-24287 and mp-24208) are dynamically stable, thus computed λ, ω log , and Tc are available. In the bottom part of the Table, predicted and computed values of λ, ω log , and Tc are reported for two dynamically stable materials, i.e., mp-24287 and mp-24208, at 50 GPa and 100 GPa. MP ID Chemical Space P E hull Predicted Computation Computed formula group (GPa) (eV/atom) λ ω log (K) Tc (K) performed Dyn. stable λ ω log (K) Tc (K) mp-24289 PdH F m3m 0 0.02 0.88 377.2 21.3 Yes No − − − mp-1018133 LiHPd P 4/mmm 0 0 0.79 321.0 14.5 Yes No − − − mp-24081 ScClH R3m 0 0 0.65 445.9 13.0 No − − − − mp-24287 CrH F m3m 0 0 0.63 446.6 11.9 Yes Yes 0.89 276.2 15.7 mp-1008376 CeH3 F m3m 0 0 0.60 418.5 9.5 No − − − − mp-24208 CrH2 F m3m 0 0 0.60 352.5 8.0 Yes Yes 0.75 264.4 10.7 mp-24287 CrH F m3m 50 − 0.57 540.3 10.6 Yes Yes 0.67 362.8 11.3 CrH F m3m 100 − 0.54 601.4 9.6 Yes Yes 0.61 413.7 10.1 mp-24208 CrH2 F m3m 50 − 0.52 477.6 6.9 Yes Yes 0.65 323.2 9.4 CrH2 F m3m 100 − 0.53 561.6 8.4 Yes Yes 0.64 348.0 9.7 . A Drozdov, M Eremets, I Troyan, V Ksenofontov, S Shylin, Nature. 52573A. Drozdov, M. Eremets, I. Troyan, V. Ksenofontov, and S. Shylin, Nature 525, 73 (2015). . 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[ "APPLICATION OF DE-SHAPE SYNCHROSQUEEZING TO ESTIMATE GAIT CADENCE FROM A SINGLE-SENSOR ACCELEROMETER PLACED IN DIFFERENT BODY LOCATIONS", "APPLICATION OF DE-SHAPE SYNCHROSQUEEZING TO ESTIMATE GAIT CADENCE FROM A SINGLE-SENSOR ACCELEROMETER PLACED IN DIFFERENT BODY LOCATIONS" ]
[ "Hau-Tieng Wu ", "Jaroslaw Harezlak " ]
[]
[]
Objective: Commercial and research-grade wearable devices have become increasingly popular over the past decade. Information extracted from devices using accelerometers is frequently summarized as "number of steps" (commercial devices) or "activity counts" (research-grade devices). Raw accelerometry data that can be easily extracted from accelerometers used in research, for instance ActiGraph GT3X+, are frequently discarded. Approach: Our primary goal is proposing an innovative use of the de-shape synchrosqueezing transform to analyze the raw accelerometry data recorded from a single sensor installed in different body locations, particularly the wrist, to extract gait cadence when a subject is walking. The proposed methodology is tested on data collected in a semi-controlled experiment with 32 participants walking on a one-kilometer predefined course. Walking was executed on a flat surface as well as on the stairs (up and down). Main Results: The cadences of walking on a flat surface, ascending stairs, and descending stairs, determined from the wrist sensor, are 1.98±0.15 Hz, 1.99±0.26 Hz, and 2.03±0.26 Hz respectively. The cadences are 1.98±0.14 Hz, 1.97±0.25 Hz, and 2.02±0.23 Hz, respectively if determined from the hip sensor, 1.98±0.14 Hz, 1.93±0.22 Hz and 2.06±0.24 Hz, respectively if determined from the left ankle sensor, and 1.98±0.14 Hz, 1.97±0.22 Hz, and 2.04±0.24 Hz, respectively if determined from the right ankle sensor. The difference is statistically significant indicating that the cadence is fastest while descending stairs and slowest when ascending stairs. Also, the standard deviation when the sensor is on the wrist is larger. These findings are in line with our expectations. Conclusion: We show that our proposed algorithm can extract the cadence with high accuracy, even when the sensor is placed on the wrist.
10.1088/1361-6579/accefe
[ "https://export.arxiv.org/pdf/2203.10563v2.pdf" ]
247,594,765
2203.10563
7ca593a895d88bea41709ea0f3678e6d08d24c95
APPLICATION OF DE-SHAPE SYNCHROSQUEEZING TO ESTIMATE GAIT CADENCE FROM A SINGLE-SENSOR ACCELEROMETER PLACED IN DIFFERENT BODY LOCATIONS Hau-Tieng Wu Jaroslaw Harezlak APPLICATION OF DE-SHAPE SYNCHROSQUEEZING TO ESTIMATE GAIT CADENCE FROM A SINGLE-SENSOR ACCELEROMETER PLACED IN DIFFERENT BODY LOCATIONS actinogramcadencede-shapesynchrosqueezing Objective: Commercial and research-grade wearable devices have become increasingly popular over the past decade. Information extracted from devices using accelerometers is frequently summarized as "number of steps" (commercial devices) or "activity counts" (research-grade devices). Raw accelerometry data that can be easily extracted from accelerometers used in research, for instance ActiGraph GT3X+, are frequently discarded. Approach: Our primary goal is proposing an innovative use of the de-shape synchrosqueezing transform to analyze the raw accelerometry data recorded from a single sensor installed in different body locations, particularly the wrist, to extract gait cadence when a subject is walking. The proposed methodology is tested on data collected in a semi-controlled experiment with 32 participants walking on a one-kilometer predefined course. Walking was executed on a flat surface as well as on the stairs (up and down). Main Results: The cadences of walking on a flat surface, ascending stairs, and descending stairs, determined from the wrist sensor, are 1.98±0.15 Hz, 1.99±0.26 Hz, and 2.03±0.26 Hz respectively. The cadences are 1.98±0.14 Hz, 1.97±0.25 Hz, and 2.02±0.23 Hz, respectively if determined from the hip sensor, 1.98±0.14 Hz, 1.93±0.22 Hz and 2.06±0.24 Hz, respectively if determined from the left ankle sensor, and 1.98±0.14 Hz, 1.97±0.22 Hz, and 2.04±0.24 Hz, respectively if determined from the right ankle sensor. The difference is statistically significant indicating that the cadence is fastest while descending stairs and slowest when ascending stairs. Also, the standard deviation when the sensor is on the wrist is larger. These findings are in line with our expectations. Conclusion: We show that our proposed algorithm can extract the cadence with high accuracy, even when the sensor is placed on the wrist. INTRODUCTION Wearable physical activity (PA) monitors based on accelerometers have been widely used to characterize the free-living movement of individuals in large scale observational studies, including The National Health and Nutrition Examination Survey (NHANES) [42], UK Biobank [17], Women's Health Initiative [19], Study of Latinos (SoL) [2], among many others. Accelerometers are small, non-invasive and body-worn devices that can collect data continuously in three orthogonal axes for several days at a time and provide high-density measurements of movement ranging from 1 to 100 Hz [22]. Such ecological data collection is potential for providing PA characteristics that can be considered objective and free of a recall bias compared to the traditional questionnaire-based assessment. The most commonly considered PA characteristic is unarguably the PA volume (or PA level). It has been linked to improved cardiorespiratory fitness [18] and reduced risk of chronic conditions [8], including cardiovascular disease [41], stroke [28], diabetes [26], breast [34] and colon cancers [46], osteoporosis [6], depression [38], and declining physical [16] and cognitive functioning [27]. While daily PA volumes are important themselves and have been extensively studied, high-fidelity data collected with wearable accelerometers create new and exciting opportunities for healthcarers to obtain more detailed information about PA. It has been shown in several controlled experiments that accelerometry data can be used to estimate kinematic movement characteristics with particular emphasis on ambulation, as its characteristics, traditionally measured in the controlled clinical environment, are related to many health conditions including Parkinson's disease [5] and obesity [24], and survival in older adults [39]. Among many such characteristics, what we are concerned with in this work is the gait cadence. The gait cadence is defined as the number of steps taken during a fixed time unit while walking, usually expressed in steps-per-minute or steps-persecond and directly related to gait speed, which is a temporal metric that is well-reflected in high-density accelerometry data [14]. Accelerometry-based monitoring of gait cadence has successfully migrated from in-thelab to free-living settings, with the use of devices worn on the lower-back [12], hip [44], thigh [13], and ankle [31]. While lab-based assessment of gait cadence has been widely used and well-studied, it is still challenging to implement in real-life settings given the time and space limitations of health facilities. Ideally, we would like to measure mobility with easyto-access, inexpensive, and multipurpose equipment, like the wrist-based accelerometry measurements. Clearly, using a watch-like monitor that does not need to be removed for sleep and does not require burdensome body adhesives [12] or bandages [25] can increase the participant comfort levels resulting in improved compliance and diminished recruitment hardship [43]. Methods dedicated for accelerometry data collected from wrist-located devices are still in active development. For example, relatively simple methods are based on zerocrossings [50] and windowed zero-crossings [21] and can be used to estimate cadence based on data collected by trunk-and hip-worn devices. Fasel et al. [20] proposed the comb filtering approach to estimate the fundamental frequency of the accelerometry signal observed during walking and related it to the gait cadence, while Karas et al. [23] used dictionary-based pattern recognition to segment walking strides in the accelerometry data to assess the cadence based on the estimated duration of strides. These works, among others, suggest that walking-originated accelerometry data can be modeled within a wide frame of non-stationary time series analysis. In [10], the angular data recorded from one inertial motion unit (IMU) attached to the thigh was used to evaluate various activities in the controlled and uncontrolled environment and transitions between activities. Other works that utilize angular motion data recorded by the single IMU worn on the thigh in the controlled environment are [3,33]. In [33], the gait speed is also estimated via the polar radius of the phase portrait. Analogously, Shin and colleagues [35], proposed to estimate the averaged walking distance from data collected by sensors located on the waist using a linear combination of "walking frequency" and "acceleration variance" detected by zero crossing. While there have been many works, such accelerometry data collected in a free-living environment still impose major methodological and computational challenges [22]. We recognize the growing need for robust analytical methods for wrist-worn devices, which motivates the focus of this manuscript on wearable accelerometry data, particularly those collected in a semi-controlled and free-living environment. To illustrate the complexity and non-stationarity of the accelerameter data, we consider an example collected from a wrist-worn accelerometer with a sampling rate of 80 Hz in one adult during short bouts of walking followed by a continuous 400-meter walking session. The top panel of Figure 1 shows an example of the vector magnitude of accelerometry data collected during walking and standing still. The bottom panel represents two instances of 0.8 sec 0.9 sec 0.7 sec 0.7 sec Signal (g) FIGURE 1. A typical accelerometer signal recorded from one subject that performs different physical activities lasting for one minute. The trend has been removed. The subject is walking during the first part and third parts of the signal and is stationary in the middle. The 3-second-long segments were zoomed in and plotted in the bottom panels. zoomed-in strides. We can observe that portions of the signal, corresponding to walking bouts, manifest a quasi-periodic character. Additionally, three rest breaks, between 135 and 153 seconds, can be seen in the form of almost flat lines in the signal, followed by an arm position change between 156 and 158 seconds. Furthermore, both the magnitude and frequency of accelerometry data in walking bouts changes throughout the walk, likely due to the variation in gait cadence. Here we list the most evident non-stationary characteristics of the data that we will further address in our modeling framework: (F1) The frequency and magnitude of walking accelerometry data undergo smooth variations. (F2) Walking occurs in short-time bouts between resting and non-walking activities. (F3) The walking patterns in accelerometry data are non-sinusoidal and consist of a fundamental frequency with multiple harmonic components. (F4) Accelerometry data contain a random noise component. The goal of this work is to harness the above-mentioned non-stationary characteristics and faithfully estimate the time trajectory of gait cadence in accelerometry data using time-frequency representation (TFR). To that end, we propose a novel use of the deshaped synchrosqueezing transform (dsSST), previously introduced by Wu et al. [29] and develop an algorithm to track the walking cadence (twice of the time-varying fundamental frequency of the accelerometer data). DsSST has been applied to various challenging biomedical signals, like extracting fetal electrocardiogram from the transabdominal maternal electrocardiogram [40] and recycling cardiogenic artifacts in impedance pneumography [30]. We further demonstrate how the proposed algorithm addresses challenges F1 to F4 and validate our approach for data collected independently with four accelerometers in 32 healthy adults. The rest of this manuscript is organized as follows. In Section 2, we describe the mathematical model of the accelerometer signals. In Section 3, we detail the dsSST and the proposed cadence estimation algorithm. In Section 4, we describe the considered database and report the results of the performed analysis. Finally, in Section 5, we provide the conclusions and the discussion. MATHEMATICAL MODEL Motivated by characteristics (F1)-(F4), we propose the following phenomenological walking model to model the walking activity in the accelerometer signal. We denote the data collected from the triaxial accelerometer signals as f [loc] x (t), f [loc] y (t), and f [loc] z (t), where the placement (location) of the sensor is indicated by loc, and x, y and z indicates the three axes. In this work, we consider the PA signal determined by the signal magnitude, which is defined as Y [loc] (t) = [ f [loc] x (t)] 2 + [ f [loc] y (t)] 2 + [ f [loc] z (t)] 2 . Below, we ignore the superscript [loc] when we model the signal. First, we set up notation to describe the "walking bout". Assume there are L ∈ N nonoverlapping nontrivial intervals I 1 , . . . I L ⊂ R that the subject is walking. The PA signal determined from the accelerometer is modeled by (1) Y (t) = g(t) + Φ(t), where g(t) is the deterministic signal describing the walking activity and Φ is the inevitable random noise. We make the following assumptions about g and Φ. Fix ε > 0 a small constant. (2) g(t) = L ∑ l=1 a l (t)s l (φ l (t))χ I l , where χ I l is the indicator function (that is, χ I l (t) = 0 when t / ∈ I l and χ I l = 1 when t ∈ I l ) and for each l = 1, . . . , L, (C1) φ l (t) is a C 2 function that is strictly monotonically increasing denoting the phase function of the l-th walking interval; (C2) φ l (t) > 0 is the instantaneous frequency (IF), or momentary stride speed, of the l-th walking interval so that and |φ l (t)| ≤ εφ l (t) for all t ∈ I l ; (C3) a l (t) > 0 is a C 1 function denoting the amplitude modulation (AM) of the l-th walking interval so that |a l (t)| ≤ εφ l (t) for all t ∈ I l ; (C4) s l is a continuous 1-period function so that |ŝ l (1)| > 0 and s l 2 = 1, which may not be sinusoidal; and (C5) Φ(·) is a random noise that is stationary in the wide sense with finite variance and short range dependence. The cadence is defined as the double of the momentary speed of stride; that is, 2φ l (t) over I l . Note that compared with the traditional consideration of cadence that is measured steps-per-minute or steps-per-second from the high-density accelerometry data [14], in this work we consider the cadence in the instantaneous sense. Plainly, the cadence is defined on the sub-second level, which captures the fact that the walking speed changes over time, even within one stride. Physically, a l means the "strength" of the walking activity and s l is understood as the wave-shape function describing the walking pattern of the l-th walking interval. Both strength and pattern might change from one step to another. Obviously, since the walking activity is a synergy of different activities in different body locations, a l and s l depend on the sensor location. The phase φ l deserves some more discussion. While physically φ l (t) should be the same across different sensor locations, the phase φ l might be different due to the global phase shift [1]; that is, the phase of the l-th walking bout recorded from two different locations might be different by a global constant, while their derivatives are the same. On the other hand, even over the same sensor location, s l , φ l (t), φ l (t) and a l (t) might be different for different walking bout. For example, the walking pattern on the road during I i might be different from the stair climbing pattern during I j , even if the signal is recorded from the hip sensor. We shall mention that the assumptions |a l (t)| ≤ εφ l (t) and |φ l (t)| ≤ εφ l (t) for all t ∈ I l for ε > 0 a small constant in C2 and C3 are for the sake of identifiability of the model and carrying out a theoretical analysis. Since theoretical development is not the focus of this paper, we refer readers with interest to [9,11] for technical details. Finally, we put an assumption on the lengths of walking intervals. To our knowledge, there does not exist a general consensus about the definition of "walking" [45]. Precisely, should we consider a subject is "walking" if he/she only moves one or two steps? In other words, how many continuous steps should exist before we consider the subject is walking over that interval? We do not plan to provide a solution in this work. Instead, we provide a definition based on the property of time-frequency (TF) analysis, like short-time Fourier transform (STFT). Empirically, to stably estimate the instantaneous frequency from an oscillatory signal by STFT, the window function should span long enough to cover 8-10 oscillatory cycles. Thus, only if a subject moves continuously for more than 10 steps shall we call such process a walking process. Mathematically, this definition is imposed by the following assumption: (C6) For each i = 1, . . . , L, we have |I i | > δ i , where δ i > 0 is assumed to be long enough to cover at least 10 walking cycles. Note that by definition of φ i (t), it means that δ i > φ −1 i (10 × 2π) − φ −1 i (0) , since when a subject finishes one walking cycle the phase grows by 2π. Depending on the cadence, δ i clearly vary; the faster the cadence over I i is, the shorter the δ i will be. Clearly, Y (·) is a random process. We call the random process satisfying (C1)-(C6) the phenomenological walking model. In practice, we only obtain one realization of Y (or recorded accelerometer signal) with the main goal of estimating the cadence over each walking interval from the recorded signal. It is worth noting that the phenomenological walking model is a generalization of the adaptive non-harmonic model [29,47] that captures the walking bout effect. The main signal processing mission in this paper is estimating the cadence 2φ l (t) from one noisy realization of Y (t) over I l . Moreover, if I l is unknown, we need to estimate it. Last but not the least, we shall mention that this definition is consistent with the definition of sustained harmonic walking considered in [45], where the sustained harmonic walking is defined as walking for at least 10s (aligned with C6) with low variability of step frequency (aligned with C2). Let us take a deeper look at the wave-shape function (the walking pattern). Assume L = 1 and I 1 = R to simplify the discussion. Under this assumption, we have (3) f (t) = a 1 (t)s 1 (φ 1 (t)) = a 1 (t)α 0 + ∞ ∑ j=1 (α j a 1 (t)) cos(2π jφ 1 (t) + β j ) , where α 0 = 1 0 s 1 (t)dt, and {α j } ⊂ R ≥0 and {β j } ⊂ [0, 2π) are from Fourier coefficients of s 1 . Usually, we call (α 1 a 1 (t)) cos(2πφ 1 (t)+β 1 ) the fundamental component of f (t), and for j ≥ 1, we call (α j a 1 (t)) cos(2π jφ 1 (t) + β j ) the j-th harmonic of f (t). In the spectrogram (and the associated TFRs determined by the synchrosqueezing transform (SST)) shown in Figure 3, we see that the fundamental component and the harmonics are represented as horizontal curves (indicated by arrows). Visually we can see that both signals oscillate about once per second, while their oscillatory patterns are different. The difference of oscillatory pattern is reflected in the intensities of the harmonics. For the signal in Figure 3(a), we see that α 1 is smaller than α j for any j > 1 in Figure 3(b). However, for the signal in Figure 3(d), α 1 is stronger than other α j for all j > 1, which we can see via the intensities of the horizontal curves in Figure 3(e). It is clear that different walking patterns could lead to vary different spectrograms. PROPOSED ALGORITHM In this section, we describe our proposed cadence estimation algorithm. The overall flowchart of the proposed algorithm is shown in Figure 2. We shall briefly describe the motivation behind the dsSST development before introducing the algorithm. When the signal oscillates non-sinusoidally, as discussed above with Figure 3, there is no guarantee that the fundamental component is stronger than its multiples, even if the rectification technique [37] is applied. As a result, it is often difficult to directly apply the ridge detection method [7,15] to the TFRs. We thus want a reliable method to filter out its harmonics. The dsSST contains as key ingredients two nonlinear operators for this purpose -short-time cepstrum and SST [29]. In short, cepstrum can be viewed as the Fourier transform of the "manipulated Fourier transform", which reflects the period of the oscillation. Given the reciprocal relationship between the frequency and period, the fundamental frequency could be preserved by the masking technique. SST is a nonlinear-type TF analysis tool. Its main purpose is sharpening the TFR determined by STFT or CWT for the sake of improving the readability of oscillatory signals by nonlinearly twisting the phase information [9,11]. 3.1. Cadence estimation algorithm. 3.1.1. Step 0: Input signal and preprocessing. Take the PA signal that is uniformly sampled over a discrete set of time points with the sampling interval ∆ t > 0 and the sampling rate f s = ∆ −1 t . Suppose the recording starts at time t = 0. Write the uniformly sampled signal as a column vector f 0 ∈ R N , where N is the number of samples and the -th entry of f 0 is the signal sampled at time ∆ t , where = 1, 2, . . . , N. Then, detrend the signal by the standard median filter with the order 10 f s , rectify the detrended signal (that is, take the magnitude of each sample), and denote the resulting signal as f. In the real life scenario, it is usually unknown when a subject is walking. To apply the proposed cadence estimation algorithm, we need to estimate I l in the phenomenological walking model described in Section 2. We suggest to apply the comb filter based algorithm [45] to achieve this goal, and continue with the following algorithm. 3.1.2. Step 1: Short time Fourier transform (STFT). Choose a discrete window function h ∈ R 2K+1 , a discretization of a chosen window h, which satisfies h(K + 1) = 1. Define 2K + 1 to be the window length. Write h ∈ R 2K+1 for the discretization of the derivative of the window function. For example, a discrete Gaussian window (and its derivative) with standard deviation σ > 0 sampled over the interval [−0.5, 0.5] at a sampling interval of 1 2K is defined as, h(k) = e −( k−1 2K −0.5) 2 2σ 2 , h (k) = − k − 1 2K − 0.5 h(k) σ 2 ,(4) where k = 1, . . . , 2K + 1. Introduce the parameter M so that 2M is the chosen number of points on the frequency axis of our time-frequency representation. Evaluate the STFT of f, a matrix V f ∈ C N×(M+1) with entries (d) (c) (f) (e) (g) (h) (i) (j) (k) (l) Signal (g) Signal (g)(5) V f (n, m) = 2K+1 ∑ k=1 f(n + k − K − 1)h(k)e −i2π(k−1)(m−1) The entries of the inverted STCT are calculated by interpolation: U f (n, m) = g n m − 1 2M .(8) See Figure 2 for an example of STCT and iSTCT. 3.1.4. Step 3: De-shape STFT. The dsSTFT of f, a matrix W f ∈ C N×(M+1) , is given by the pointwise product (9) W f (n, m) = V f (n, m)U f (n, m) , where n = 1, . . . , N and m = 1, . . . , M + 1. See Figure 2 for an example of dsSTFT. 3.1.5. Step 4: Sharpen STFT and dsSTFT.. To sharpen the TFR determined by STFT or dsSTFT, we calculate the frequency reassignment rule. First, calculate (10) V f (n, m) := 2K+1 ∑ k=1 f(n + k − K − 1)h (k)e −i2π(k−1)(m−1) 2M . Choose a threshold υ > 0 and calculate the reassignment operator Ω Ω Ω υ f (n, m) = −ℑ V f (n,m) V f (n,m) N 2π(2K+1) when |V f (n, m)| > υ −∞ when |V f (n, m)| ≤ υ.(11) The traditional SST of f, denoted as SV υ f ∈ C N×(M+1) , is implemented by SV υ f (n, m) = ∑ l; m=l−Ω Ω Ω υ f (n,m) V f (n, l).(12) The dsSST of f, a matrix SW υ f ∈ C N×(M+1) , is given by the formula SW υ f (n, m) = ∑ l; m=l−Ω Ω Ω υ f (n,m) W f (n, l). See Figure 3 for an illustration of SST. The weak line in the spectrogram associated with the fundamental component in Figure 3(b) is clearly enhanced in Figure 3(c). In the illustration of dsSST in Figure 4, we can easily see that the harmonics are gone. 3.1.6. Step 5: Cadence extraction. In the last step, we estimate the cadence over an interval I i that the subject is walking. Assume the subject is walking over the interval spanning from 1 ≤ N 1 to N 1 < N 2 ≤ N. We first fit a curve to the dsSST via c * = max c∈Z N 2 −N 1 +1 M+1 N 2 −N 1 +1 ∑ m=1 log |SW υ f (c(m), m)| ∑ n i=1 ∑ N 2 −N 1 +1 j=1 |SW υ f ( j, i)| − λ N 2 −N 1 +1 ∑ m=2 |c(m) − c(m − 1)| 2 ,(14) where Z M+1 = {1, 2, . . . , M + 1} and λ > 0 is the penalty term controlling the regularity of the curve c. Here, c ∈ Z N 2 −N 1 +1 M+1 indicates a curve in the TFR SW υ f restricted on the interval spanning from N 1 to N 2 and ∑ N 2 −N 1 +1 m=2 |c(m) − c(m − 1)| 2 quantifies the regularity of c. Based on the robustness property of SST [9], the extracted curve c * is a robust estimator of the IF of the fundamental component φ 1 (t). As a result, φ 1 at time (N 1 + − 1)∆ t is estimated as φ 1 ((N 1 + − 1)∆ t ) := f s 2M c * ( ) , where = 1, . . . , N 2 − N 1 + 1 and f s 2M is the discretization interval in the frequency axis. The cadence at time (N 1 + − 1)∆ t is then estimated as cadence((N 1 + − 1)∆ t ) := 2 φ 1 ((N 1 + − 1)∆ t ) . A similar procedure could be applied to estimate the cadence over other walking intervals. RESULTS 4.1. Database. We consider the Indiana University Walking and Driving Study (IUWDS). Data were collected in 2015 on n = 32 (19 females) physically healthy individuals in a wide age range between 21 and 51 years of age in a partially controlled environment. The individuals were given instructions for a walking route during the experiment that included walking on a flat surface and climbing stairs up and down, but otherwise the behavior was unrestricted. Data were collected using four tri-axial ActiGraph GT3X+ accelerometers at a frequency of 100 Hz. An example of the collected data is presented in Figures 1 and 3. The study was approved by the Institutional Review Board (IRB) of the Indiana University, and all participants provided written informed consent to participate. In this database, there are four locations that the tri-axial accelerometer signals are simultaneously recorded from, including wrist (wr), hip (hi), left ankle (la), and right ankle (ra); that is, we have Y [wr] (t), Y [hi] (t) Y [la] (t) and Y [ra] (t) recorded simultaneously from each subject. We only focus on the data with experts' annotation. Parameter choices. We choose the Gaussian window when we run STFT and dsSST with the window span of 12 seconds. Canonical frequency bins are chosen in the discretization of STFT; that is, M = N/2 is chosen in (5) and γ = 0.3 in (6) and υ in (11) is set to 10 −9 . The penalty term in (14) is chosen to be λ = 1. These parameters are chosen based on physiological needs and not optimized by any grid search, and the results are not sensitive to a small perturbation of these parameters (results not shown). Figure 4, we illustrate a typical example of Y [ra] (t) from one of the study participants, where we show its spectrogram, SST and dsSST. The experts' labels are superimposed as the red curve on the presented signal, with "1" denoting walking, "2" denoting descending stairs, "3" denoting ascending stairs, and "0" denoting clapping, separating different walking activities. It can be clearly seen that there is a dominant curve around 1Hz in the spectrogram, SST and dsSST, when the subject is walking. As described in the methodology section, physically, the dominant curve indicates the IF of the fundamental component, which quantifies the "momentary speed of strides", and the doubled value of this IF is the cadence. The multiples are inevitable in the spectrogram and SST, while the multiples disappear in the dsSST. When the subject ascends or descends stairs, the IF fluctuates around 1 Hz and the TFR is blurred; when the subjects clap their hands, the IF is totally blurred. This suggests that the accelerometry signal and its spectral content during walking are different from those during other activities. The preceding observation agrees with the intuition that when a subject is walking on a flat surface, the cadence is more stable than when ascending or descending stairs. To appreciate the challenge caused by the feature (F4), we show another typical example of f [wr] (t) in Figure 6. In this case, we see that the strength of fundamental frequency multiples is greater than the fundamental component on the spectrogram. This challenge is alleviated after applying the de-shape algorithm. To analyze data from all study participants, we quantitatively evaluate the cadence over different walking intervals, where the walking intervals are provided by the experts' labels. We set p = 0.005 [4] as the statistical significance level in the following statistical inference and apply the Bonferroni correction if multiple testing is carried out. The results are shown in Table 1. Visualization. In (a) (b) (c) (d) (e) (f) (g) (g) First, we compare the cadence of different walking activities. For each sensor location, the estimated cadence of ascending stairs is lower than that of walking on the flat surface, and further lower than descending stairs, where the both differences are with statistical significance under the one-sided Wilcox rank sum test, except comparing the cadence during descending stairs and walking on the flat surface from the hip sensor (p = 0.027). Also, when a subject is walking on the flat surface, the standard deviation of the cadence is the smallest compared with ascending and descending stairs with statistical significance under the F test. Second, we compare different sensor locations. The one-way analysis of variance (ANOVA) shows that during walking on the flat surface, there is no difference (p = 0.377) when the cadence is estimated from different sensor locations. However, during ascending stairs and descending stairs, there is at least one sensor location that leads to significantly different cadence estimation. This suggests that when a subject is climbing stairs, the cadence estimation agreement among different locations might be lower. The variability of cadence estimation from the wrist is larger with statistical significance under the F test. To have a deeper look, we show the mean and standard deviation of the estimated cadence for all subjects in Figure 7, where we sort the subjects according to the mean cadence estimated from the hip sensor. As reported above, there is little cadence estimation discrepancy among different sensor locations when a subject is walking on the flat surface. The figure suggests that it is not guaranteed that a faster cadence of walking on the flat surface implies a faster cadence of ascending or descending stairs. However, the trend exists with statistical significance by fitting a linear regression model (that is, cadence(i) = β 0 +β 1 i, where cadence(i) is the mean cadence over the walking interval and i is the sorted subject index). However, the trend is less obvious when descending stairs and the trend of the estimated cadence recorded from wrist and hip is not statistically significant. The above findings are in line with our expectations. To further study if cadence can be reliably estimated from a single sensor placed in different locations, we apply the Bland-Altman plot analysis. See Figures 8 and 9 for a comparison. We see that no matter where we put the sensor, overall the agreement rate is accurate up to two digits in the mean, and the limits of agreement (LoA) are between −0.6 and 0.6. The comparison between the wrist and other locations shows a larger mean and wider LoA for non-wrist locations. When comparing the results from the two ankles, the mean and LoA are both smaller. We also see that when a subject is ascending or descending stairs, the LoA is larger compared with when the subject is walking. We suggest that we could accurately and reliably estimate the cadence up to the sub-second level, while the agreement is worse when a subject is climbing stairs. DISCUSSION In this work, we propose to apply dsSST to accelerometer data collected during walking to reduce non-sinusoidal oscillation in the TFR and estimate the fundamental oscillatory The unit is steps-per-second. All values are shown as mean ± standard deviation. * : the p-value of running the one-way ANOVA is less than 10 −7 ; * * : the p-value of running the one-way ANOVA is less than 10 −10 . component. We then linked this component to a stride-to-stride frequency and obtained estimates of instantaneous gait cadence. Utilized technique is a de-shape extension of STFT aiming to handle noisy and non-stationary characters of bio-signals, which is specifically designed to alleviate the impact of harmonics and enhance the fundamental component of the biological signal that does not oscillate sinusoidally [30,40]. Application of the proposed cadence estimation method to the IUWDS database shows that the estimated cadence during the predefined walking intervals agrees with the intuition It is clear that the multiples are stronger than the fundamental components, and they are suppressed in the dsSST. To enhance the visualization, the dynamical range of each plot is set to be between the 0 and the 99% percentiles of all entries. that descending stairs is faster than walking on the flat surface, which in turn is faster than ascending stairs. The proposed technique handles the inevitable time-varying magnitude and cadence and performs well even when walking is interrupted by rest breaks and when the accelerometer signal is recorded from the wrist. In this work, we suggest to apply the comb filter based approach proposed by one of the authors [45] to detect walking bout. From the statistical perspective, walking detection is equivalent to detecting I i . This could be viewed as a special case of the change point detection problem. In the statistical literature, usually it is the change point of the trend that is detected. However, what we encounter in the walking detection problem is detecting the change point point of an oscillatory component while facing the time-varying nature of cadence, as well as the variation of wave-shape functions associated with the walking patterns. To our knowledge, the only work in detecting the change point of an oscillatory component is [49], and it is the sinusoidal oscillatory component with fixed amplitude and frequency that is considered. How to generalize such change point detection algorithm to the walking detection problem is under study and will reported in our future work. In short, we believe that the greatest strength of the presented work is its focus on a single sensor measurements and good performance even for wrist-worn devices. It is potential to be implemented in the real-life setting to enhance subjects' compliance. When combined with the easy-to-access, inexpensive, and multipurpose properties of the accelerometer, like 5 10 15 20 25 30 Sorted subject index Besides the above encouraging results, we also recognize several limitations of our approach. First, data were collected on a relatively small sample of healthy adults, which warrants further validation efforts. Also, while the experiment was designed to best mimic outdoor and free-living walking environment, using real-world and multi-day data is necessary to fully understand the utility of our algorithm. From the statistical perspective, the noise is in general not possible to be stationary. A theoretical justification of the algorithm under the non-stationary noise warrants a further exploration. Also, while we propose to choose K in (??) by a grid search approach, how to develop a more principled and faster approach would be an interesting topic that we will explore and report in our future work. ACKNOWLEDGMENTS The authors thank Dr. Jacek Urbanek for fruitful discussion and the initial participation of this manuscript preparation. The research of Jaroslaw Harezlak was partially supported by the NIMH grant R01MH108467. FIGURE 2 . 2The overall flowchart of the de-shape algorithm. (a) is the input signal. (b) is the rectified signal. (c) is the slice of the spectrogram of (b) at the 10th sec, and (d) is the spectrogram. (e) is the slice of the short-time cepstral transform (STCT) of (b) at the 10th sec, and (f) is the STCT. (g) is the inverse STCT (iSTCT) at the 10th sec, which when combined with the spectrogram leads to the de-shape STFT (dsSTFT) at the 10th sec. The associated iSTCT and dsSTFT are shown in (h) and (j) respectively. If the synchrosqueezing transform (SST) is applied, the result at the 10-th sec is shown in (k), and the de-shape SST (dsSST) is shown in (l). 2M ,, 2Mwhere f(l) := 0 when l < 1 or l > N, n = 1, . . . , N is the time index and m = 1, . . . , M + 1 is the frequency index. The spectrogram is thus an N × (M + 1) matrix where the (n, m)-th entry is |V f (n, m)| 2 . See Figure 3 for two illustrative examples of a spectrogram. 3.1.3. Step 2: Short-time cepstral transform and inverse short-time cepstral transform. The short-time cepstral transform (STCT) of f is represented by a matrix C f ∈ C where γ > 0 is the chosen power parameter and m = 1, . . . , 2M is the quefrency index. We crop C f and consider only the first M + 1 columns associated with the positive quefrency axis. The inverse STCT (iSTCT) of f is represented by a matrix U f ∈ R N×(M+1) . For each time index n, consider the function g n : [0, ∞] → R whose known values are g n 1 m − 1 = C f (n, m) m = 1, ..., M + 1. FIGURE 3 . 3The accelerometer signals in panels (a) and (d) were recorded from two subjects. The spectrograms of these signals are shown in panels (b) and (e), respectively and the synchrosqueezed spectrograms of these signals are shown in (c) and (f) respectively. In panels (b), (c), (e) and (f), the fundamental component is indicated by the blue arrow, and its multiples are indicated by the red arrows.4.4. Statistical analysis of the obtained results. FIGURE 4 . 4From top to bottom: f [ra] (t) superimposed the labels, the spectrogram, the SST, and the dsSST. The red arrow indicates the instantaneous frequency (IF) of the fundamental component (the cadence is the double of the IF), and the blue arrow indicates the IF of the multiple. It is clear that the multiples are suppressed in the dsSST. To enhance the visualization, the dynamical range of each plot is set to be between 0 and the 99% quantile of all entries. FIGURE 5 . 5A zoom-in ofFigure 4. From top to bottom: f [ra] (t) superimposed on the labels, the spectrogram, the SST, and the dsSST. The red arrow indicates the instantaneous frequency (IF) of the fundamental component (the cadence is the double of the IF), and the blue arrow indicates the IF of the multiple. It is clear that the multiples are suppressed in the dsSST. To enhance the visualization, the dynamical range of each plot is set to be between 0 and the 99% quantile of all entries. FIGURE 6 . 6From top to bottom: f [wr] (t) superimposed on the labels, the spectrogram, the SST, and the dsSST. The red arrow indicates the instantaneous frequency (IF) of the fundamental component (the cadence is the double of the IF), and the blue arrow indicates the IF of the multiple. JH: idea, literature review, data analysis and write-up.8. COMPETING INTERESTS STATEMENTNo competing interests. FIGURE 8 .FIGURE 9 . 89The Bland-Altman plots when comparing different sensor locations. For left to right columns: the comparison of wrist and hip, left ankle and right ankle, respectively. From top to bottom rows: when a subject is walking, ascending stairs and descending stairs, respectively. The mean and limits of agreement (LoA) and their 95% confidence intervals are shown. The Bland-Altman plots when comparing different sensor locations. The caption is the same as that for Figure 8, except that for left to right columns: the comparisons of hip and left ankle, hip and right ankle, and left ankle and right ankle respectively. where ξ ≥ 0 and δ denotes the Dirac measure, and Ω TIENG WU, DEPARTMENTS OF MATHEMATICS AND DEPARTMENT OF STATISTICAL SCIENCE, DUKE UNIVERSITY, DURHAM, NC, UNITED STATES OF AMERICA Email address: [email protected] JAROSLAW HAREZLAK, DEPARTMENT OF EPIDEMIOLOGY AND BIOSTATISTICS, INDIANA UNIVERSITY SCHOOL OF PUBLIC HEALTH, BLOOMINGTON, IN, UNITED STATES OF AMERICA Email address: [email protected] FIGURE 7. A summary plot of means and standard deviations of 32 subjects. The subjects are sorted by the cadence measured from the hip sensor. From left to right: when a subject is walking, ascending stairs and descending stairs. From top to bottom: when the sensor is put in wrist, hip left ankle and right ankle. the wrist-based accelerometry measurements, the proposed algorithm might open a new path toward future digital health environment.1.5 2 2.5 WRIST Cadence (bout) Walking 5 10 15 20 25 30 Sorted subject index 1.5 2 2.5 Cadence (bout) Ascending stairs 5 10 15 20 25 30 Sorted subject index 1.5 2 2.5 Cadence (bout) Descending stairs 5 10 15 20 25 30 Sorted subject index 1.5 2 2.5 HIP Cadence (bout) 5 10 15 20 25 30 Sorted subject index 1.5 2 2.5 Cadence (bout) 5 10 15 20 25 30 Sorted subject index 1.5 2 2.5 Cadence (bout) 5 10 15 20 25 30 Sorted subject index 1.5 2 2.5 Left Ankle Cadence (bout) 5 10 15 20 25 30 Sorted subject index 1.5 2 2.5 Cadence (bout) 5 10 15 20 25 30 Sorted subject index 1.5 2 2.5 Cadence (bout) 5 10 15 20 25 30 Sorted subject index 1.5 2 2.5 Right Ankle Cadence (bout) 5 10 15 20 25 30 Sorted subject index 1.5 2 2.5 Cadence (bout) 5 10 15 20 25 30 Sorted subject index 1.5 2 2.5 Cadence (bout) 10. APPENDIX 10.1. Technical details. In this subsection, we provide some technical reasons for the proposed algorithm. To avoid distracting from the main idea, we only describe the basic ideas of the de-shape methodology and refer the interested reader to[29]for rigorous mathematical details of dsSTFT, and[9,11]for the squeezing part of dsSST. STFT is a widely applied tool aiming to analyze non-stationary signals. Basically, STFT divides the observed signal into pieces, analyzes the spectrum of each piece separately, and patches those spectral results together. Specifically, for a given signal f , with a chosen window function h, such as a Gaussian function centered at the origin, the STFT is defined aswhere t ∈ R indicates time and ξ ∈ R indicates frequency. We call |V f (t, ·)| 2 the spectrogram of the signal f at time t, since it represents the power spectrum of the truncated signal f (·)h(· − t) around t. Note that when the wave-shape function is non-sinusoidal, as was explained in Section 2 and shown inFigures 3 and 4, there is an oscillatory pattern in the power spectrum |V f (t, ·)| 2 caused by the harmonics. A naive idea leading to the cepstrum idea[32]is that the frequency of this oscillation provides information about the period of the signal f . To handle the time-varying frequency and amplitude nature of the biomedical signals, the short-time cepstral transform (STCT) was proposed in[29]and is defined bywhere γ > 0 is sufficiently small and q ∈ R is called the quefrency (its unit is seconds or any feasible unit in the time domain). The reason for taking the γ th power of |Vf (t, ξ )| 2 does oscillate, the amplitude of this oscillation changes from one cycle to another. To remove the influence of this amplitude modulation, we can take the natural logarithm of |V (h) f (t, ξ )| so that the amplitude modulation is decoupled as a "low-frequency component." However, taking the natural logarithm might be numerically unstable , so we use the approximation |V f (t, ξ )| γ , called the "soft logarithm." Systematic exploration of this idea can be found in[29]. Ultimately, we obtain the fundamental period and its multiples in C (h,γ) f (t, ·). The main step in the de-shape algorithm is the inverse STCT (iSTCT), which takes into account the inverse relationship between two main quantities describing oscillation, the period and frequency:where ξ > 0 is given in Hz. Since C (t, ·) as a nonlinear "mask" and remove multiples from the STFT bywhere ξ > 0 is interpreted as frequency. The final TF representation is |Wwhich is a nonlinearly filtered spectrogram. This step can be viewed as applying a nonlinear filter to the signal to remove the influence of the wave-shape function. SeeFigure 3for the squared magnitude of W (h,γ) f . It is clear that the multiples associated with the non-sinusoidal oscillation disappear, and the TFR suggests the cadence.If sharpening the time-frequency representation (TFR) could help, we can further consider the optional step by taking the synchrosqueezing transform (SST) idea[11]into account. This nonlinear operator is produced by applying the reassignment rule[9,11]. Consider(19)SVwhere ξ ≥ 0 and δ is the Dirac measure, and the reassignment rule ΩHere, Dh(t) is the derivative of the chosen window function h, ℑ denotes the imaginary part, and υ > 0 gives a threshold so as to avoid instability in the computation when |V (t, ·) provides a sharpened spectrogram of the oscillatory signal at time t. We refer interested readers to a non-mathematical tutorial[48], and several recent developments[9,36]. The deSTFT can be sharpened by Reconsider phase reconstruction in signals with dynamic periodicity from the modern signal processing perspective. 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[ "On the Mean Square Error Optimal Estimator in One-Bit Quantized Systems", "On the Mean Square Error Optimal Estimator in One-Bit Quantized Systems", "On the Mean Square Error Optimal Estimator in One-Bit Quantized Systems", "On the Mean Square Error Optimal Estimator in One-Bit Quantized Systems" ]
[ "Benedikt Fesl ", "Michael Koller ", "Fellow, IEEEWolfgang Utschick ", "Benedikt Fesl ", "Michael Koller ", "Fellow, IEEEWolfgang Utschick " ]
[]
[]
This paper investigates the mean square error (MSE)-optimal conditional mean estimator (CME) in one-bit quantized systems in the context of channel estimation with jointly Gaussian inputs. We analyze the relationship of the generally nonlinear CME to the linear Bussgang estimator, a well-known method based on Bussgang's theorem. We highlight a novel observation that the Bussgang estimator is equal to the CME for different special cases, including the case of univariate Gaussian inputs and the case of multiple pilot signals in the absence of additive noise prior to the quantization. For the general cases we conduct numerical simulations to quantify the gap between the Bussgang estimator and the CME. This gap increases for higher dimensions and longer pilot sequences. We propose an optimal pilot sequence, motivated by insights from the CME, and derive a novel closed-form expression of the MSE for that case. Afterwards, we find a closed-form limit of the MSE in the asymptotically large number of pilots regime that also holds for the Bussgang estimator. Lastly, we present numerical experiments for various system parameters and for different performance metrics which illuminate the behavior of the optimal channel estimator in the quantized regime. In this context, the well-known stochastic resonance effect that appears in quantized systems can be quantified.
10.1109/tsp.2023.3282063
[ "https://export.arxiv.org/pdf/2212.04470v2.pdf" ]
254,408,559
2212.04470
4c244ad5fea815df3125d58a8a93f2c980f1e58f
On the Mean Square Error Optimal Estimator in One-Bit Quantized Systems 27 Apr 2023 Benedikt Fesl Michael Koller Fellow, IEEEWolfgang Utschick On the Mean Square Error Optimal Estimator in One-Bit Quantized Systems 27 Apr 2023arXiv:2212.04470v2 [cs.IT] 1Index Terms-Bussgang theoremchannel estimationcondi- tional mean estimationone-bit quantizationmean square error This paper investigates the mean square error (MSE)-optimal conditional mean estimator (CME) in one-bit quantized systems in the context of channel estimation with jointly Gaussian inputs. We analyze the relationship of the generally nonlinear CME to the linear Bussgang estimator, a well-known method based on Bussgang's theorem. We highlight a novel observation that the Bussgang estimator is equal to the CME for different special cases, including the case of univariate Gaussian inputs and the case of multiple pilot signals in the absence of additive noise prior to the quantization. For the general cases we conduct numerical simulations to quantify the gap between the Bussgang estimator and the CME. This gap increases for higher dimensions and longer pilot sequences. We propose an optimal pilot sequence, motivated by insights from the CME, and derive a novel closed-form expression of the MSE for that case. Afterwards, we find a closed-form limit of the MSE in the asymptotically large number of pilots regime that also holds for the Bussgang estimator. Lastly, we present numerical experiments for various system parameters and for different performance metrics which illuminate the behavior of the optimal channel estimator in the quantized regime. In this context, the well-known stochastic resonance effect that appears in quantized systems can be quantified. I. INTRODUCTION D IGITAL signal processing has great impact on modern communication systems, in which the analog signals undergo quantization through analog-to-digital converters (ADCs) . The resolution of the ADCs determines how much information is preserved after the quantization of the analog signal. In many applications, the extreme case of one-bit quantization is of particular interest, e.g., in the context of lossy compression and rate distortion theory [1], [2], wireless sensor networks [3], or audio coding [4]. In recent years, one-bit quantization gained a lot of interest in multiple-input multiple-output (MIMO) communication systems, where transmitter and receiver are possibly equipped with a large number of antennas. Since the power consumption of the ADCs, which are needed for every antenna, is growing exponentially with their resolution, one-bit ADCs are considered as a power-efficient solution [5]. A major drawback of a system with one-bit ADCs is the severe information loss which diminishes the capacity [6]- [11] and achievable rate [12], [13], and causes signal processing tasks like channel estimation to Benedikt Fesl and Michael Koller are with Professur für Methoden der Signalverarbeitung, Technische Universität München, 80333 München, Germany (e-mail: [email protected]; [email protected]). Wolfgang Utschick is with Professur für Methoden der Signalverarbeitung, Technische Universität München, 80333 München, Germany (e-mail: [email protected]) and with the Munich Data Science Institute (MDSI). ©This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. become challenging [14]- [16]. However, in contrast to multibit considerations, the study of one-bit quantization allows for closed-form solutions in many applications which we discuss in the following. Thereafter, the found exact solutions can be leveraged to approximate solutions for higher resolutions. Remarkably, it is shown in [6], [7] that the capacity is not severely reduced by the coarse quantization at low signal-to-noise ratios (SNRs). The capacity was further analyzed for high SNRs in [8] and for a large number of antennas in [9]. In [12], [13], the achievable uplink throughput and rate is discussed. In order to achieve high data rates in quantized systems, accurate estimation of the channel state information (CSI) is particularly important. In consideration of the highly nonlinear quantization, channel estimation is considerably more demanding, leading to an extensive collection of approaches in the literature, e.g., [11], [17]- [32]. The work in [11] studies least squares (LS) estimation. In [17]- [20], iterative maximum likelihood (ML) methods are proposed, whereas in [21]- [23], the maximum a posteriori (MAP) estimator is analyzed. In [21], [24] the Cramér-Rao bound (CRB) is investigated. In [23], [25]- [27], compressive sensing (CS) is studied for channel estimation, and deep learning approaches were investigated in [28]- [32]. Apart from that, the Bussgang theorem turned out to be helpful for channel estimation [33]. The theorem, a special case of the more general Price theorem [34], states that the auto-correlation and cross-correlation of a zero-mean Gaussian signal before and after it has passed through a nonlinear operation is equal up to a constant. A direct consequence of the theorem is the fact that the signal after the quantization can be decomposed in a linear fashion into a desired signal part and an uncorrelated distortion. This property was investigated for various nonlinearities [35] and further extended to non-zero quantization thresholds [36]. In the context of signal processing, the theorem was successfully used to design channel estimators. To this end, the linearized model is combined with the linear minimum mean square error (MMSE) estimator [37]- [40]. Despite this extensive variety of channel estimation algorithms, there is so far no detailed discussion about the CME for the one-bit quantization case. The CME is well-known to be the minimizer for the MSE loss function, which is the most frequently considered criterion for the channel estimation performance. More generally, the CME is the optimal predictor for all Bregman loss functions, of which the MSE is a special case [41]. On the one hand, there are approximations of the CME, e.g., in wideband systems with many channel taps and antennas where the quantization noise becomes approximately Gaussian [42], or with neural networks that are trained on unquantized data [29]. On the other hand, the CME is known for the simple one-dimensional case for a Gaussian input with a single pilot observation [1,Sec. 13.1], [43]. However, to our knowledge, there exists no general discussion about the multivariate case with correlated channel entries, or with longer pilot sequences that can not be optimally decorrelated. In [40], a two-stage (concatenated) MMSE estimator is proposed, where in the first stage the product of the pilot matrix and the channel is estimated with the CME. Afterwards, it is assumed that the remaining error is Gaussian, which is unlikely to hold in general, such that the channel can be recovered by decorrelating with the (pseudo-)inverse pilot matrix. This corresponds to a second stage consisting of a linear MMSE solution. Interestingly, this approach is shown to be equal to the Bussgang estimator in the one-bit quantization case. Contributions: We discuss the CME in systems with one-bit quantization for the general case of multivariate Gaussian channels in the presence of additive white Gaussian noise (AWGN) and multiple pilot observations. We present closed-form solutions for special cases and numerical computations for the general case. In that respect, we discuss the optimal pilot sequence from the perspective of the CME that was independently suggested in [29] for learning a neural network based channel estimator. We identify cases in which the Bussgang estimator is equivalent to the CME by proving that their closed-form MSE expressions are equal and we find asymptotic MSE limits for the large number of pilots regime. We then quantify the gap between the linear Bussgang estimator and the nonlinear CME in the general case by numerical experiments and show that the gap increases for higher dimensions and longer pilot sequences in the presence of correlations between the channel entries. In this context, the well-known stochastic resonance effect [44], which allows for outperforming the asymptotic limits in the presence of noise, is numerically quantified. We finally discuss further performance metrics such as the cosine similarity and a lower bound on the achievable rate for a matched filter. The remainder of this work is structured as follows. In Section II, the problem formulation is described, in Section III, the Bussgang estimator is summarized, and in Section IV, the findings on the CME are presented. Section V then discusses further performance metrics, Section VI presents the numerical results, and Section VII concludes the work. Notation: The (element-wise) signum function is denoted by sign(·). The Kronecker product is denoted by ⊗ and the trace of matrix is written as tr(·). The real and imaginary part of a complex scalar/vector/matrix is stated as ℜ(·) and ℑ(·), respectively. The (column-wise) vectorization of a matrix which stacks the columns is denoted by vec(·). We denote the probability density function (PDF) of a continuous random variable (RV) x as f x and the probability mass function (PMF) of a discrete RV y as p y . We denote by N C (x; µ, C) the complex Gaussian distribution with mean µ and covariance C, evaluated at x, which yields a circularly symmetric RV after subtracting its mean. Similarly, the real-valued Gaussian density is denoted by N (x; µ, C). We denote the Gaussian error function as erf(x) = 2 √ π x 0 exp(−t 2 )dt. We denote the uniform distribution between a and b as U(a, b). The function ∠(·) gives the angle in [0, 2π) of a complex number. The k, ℓth entry of a matrix X is denoted by [X] k,ℓ . II. PROBLEM FORMULATION We consider the following generic system equation R = Q(Y ) = Q(ha T + N ) ∈ C N ×M(1) where R = [r 1 , r 2 , . . . , r M ] contains M quantized observations of the vector of interest h ∈ C N with the known pilot vector a ∈ C M that fulfills the power constraint a 2 2 = M . We denote by Y the unquantized received signal. Let the vector h be a zero-mean Gaussian RV, i.e., h ∼ N C (0, C h ). The desired signal is distorted with additive zero-mean Gaussian noise N = [n 1 , n 2 , . . . , n M ] where n i ∼ N C (0, C n ) and is afterwards quantized with the complex-valued one-bit quantization function Q(·) = 1 √ 2 (sign(ℜ(·)) + j sign(ℑ(·)))(2) which is applied element-wise to the input vector/matrix. The system model in (1) can be equivalently described in its (column-wise) vectorized form as r = Q(y) = Q(Ah + n) ∈ C N M(3) with A = a ⊗ I, r = vec(R), y = vec(Y ), and n = vec(N ). The system model describes, e.g., a single-input multiple-output (SIMO) wireless communication scenario where a base station equipped with N antennas receives M pilot signals from a single-antenna user terminal. The ADCs at the receiver have one-bit resolution, which is modeled by the quantization function in (2). Although we especially focus on the application of channel estimation in this work, the analysis is not limited to this instance and might yield insights to various applications such as lossy compression [2], wireless sensor networks [3], audio coding [4], or control theory [45], to name a few. The overall goal is to estimate the signal h in an optimal sense where the vector a is known and the channel and noise are both zero-mean Gaussian with known covariances. The optimality depends on the metric of choice, however, the most frequently used criterion in the context of channel estimation is the MSE between the true channel h and its estimateĥ, i.e., MSE = E[ h −ĥ 2 2 ](4) for which the CME, defined aŝ h = E[h|r] = hf h|r (h|r)dh,(5) is known to yield optimal estimates. The CME is generally not available since it requires full knowledge of the conditional PDF f h|r . Furthermore, even if the PDF is known, the CME does generally neither have a closed-form solution nor is it feasible to compute numerically in real-time systems. For this reason, the CME is usually approximated by sub-optimal approaches with a practically reasonable trade-off between performance and complexity, e.g., using the linear MMSE estimator. In a jointly Gaussian setting, the linear MMSE estimator is well-known to be the CME. However, in the quantized case, this statement is no longer true in general. Nonetheless, for a zero-mean jointly Gaussian input, the Bussgang decomposition, based on Bussgang's theorem [33], can be leveraged to design a linear MMSE estimator in the quantized case. The Bussgang estimator is sub-optimal in general due to its linearity. However, up to now there has been no analysis of the connection between the Bussgang estimator and the CME in the case of one-bit quantization. In this work, we investigate the MSE optimality of the Bussgang estimator and identify cases in which the Bussgang estimator is equal to the CME and we show when the Bussgang estimator differs from the CME. To this end, after revising the Bussgang estimator, we derive the CME for onebit quantization in the multivariate case. Simulation results based on the derived quantities verify the discussion. III. BUSSGANG ESTIMATOR In this section, we briefly revise the linear MMSE estimator based on the Bussgang decomposition which is a direct consequence of Bussgang's theorem [33]. In particular, the Bussgang decomposition implies that the system in (3) can be written as a linear combination of the desired signal part and an uncorrelated distortion q as r = Q(y) = By + η = BAh + q,(6) where B is the Bussgang gain that can be obtained from the linear MMSE estimation of r from y as B = C ry C −1 y = 2 π diag(C y ) − 1 2 ,(7) cf. [46,Sec. 9.2], and where the distortion term q = Bn + η contains both the AWGN n and the quantization noise η. As the statistically equivalent model (6) is linear, one can formulate the linear MMSE estimator h buss = C hr C −1 r r.(8) The cross-correlation matrix between the channel and the received signal is calculated as C hr = E[h(BAh + q) H ] = C h A H B H(9) which follows from the fact that the noise term q is uncorrelated with the channel h, see [12, Appendix A]. The autocorrelation matrix C r is calculated via the so-called arcsine law [47] as C r = 2 π (arcsin (Ψ ℜ(C y )Ψ ) + j arcsin (Ψ ℑ(C y )Ψ )) (10) where Ψ = diag(C y ) − 1 2 . Further reading on the Bussgang estimator can be found in [12], [37]. In [40], it is discussed that the Bussgang estimator is equal to a two-stage estimator that first computes the CME of the product Ah of the pilot matrix and the channel, and afterwards recovers the channel h. Since the resulting error after the first step is non-Gaussian, the second step is non-optimal from the CME perspective. In the following, we discuss cases in which the Bussgang estimator is indeed equal to the CME, and we show the differences between the CME and the Bussgang estimator in the general case. IV. CONDITIONAL MEAN ESTIMATOR In this section, we derive the CME for one-bit quantization for the general system in (3). Afterwards, we discuss the CME for different simplifications, i.e., with and without AWGN or multiple pilots, and for the uni-as well as multivariate case, where these cases are differently combined to provide a general overview. For each instance, we discuss possible closed-form solutions of the CME and its relationship to the Bussgang estimator, where we identify cases in which equality between these estimators holds. We start by rewriting the CME from (5) as E[h|r] = h p r|h (r|h)f h (h) p r (r) dh (11) = 1 p r (r) hf h (h)p r|h (r|h) dh(12) by making use of Bayes' theorem. It is known that f h = N C (0, C h ). In the following, we also find expressions for the prior and conditional probabilities. We first define the set of all vectors y which are mapped onto a given observation r as Q(r) = y ∈ C MN : Q(y) = r .(13) Integrating the PDF f y over this set yields the prior probability of observing the respective quantized observation, i.e., p r (r) = Q(r) f y (y) dy,(14) where due to the joint Gaussianity we have f y = N C (0, C y ) with C y = AC h A H +C n , cf. (3) . The conditional probability p r|h (r|h) is found similarly, however, now the integrand changes to N C (Ah, C n ), where h is no longer a RV: p r|h (r|h) = Q(r) N C (x; Ah, C n ) dx.(15) Plugging (14) and (15) into (12) yields a generally computable expression of the CME for the model in (3). A. Numerical Integration For the most general case, the integrals in (12), (14), and (15) do neither have a closed-form solution, nor can they be simplified. In this case, the CME has to be found by numerical integration. However, the number of solutions for the discrete number of possible observations r ∈ C MN grows exponentially with 4 MN , not to mention the increasing complexity for each individual solution. Since in (14) and (15) we have integrals over Gaussian PDFs, there exist computationally efficient algorithms. In this paper, the numerical integration of the Gaussian density is based on an algorithm from [48] which first transforms the integral into an integration over the unit hypercube by combining a Cholesky decomposition of the covariance matrix with a priority ordering of the integration variables. The transformed integral is then approximated based on a Monte Carlo method which is implemented in [49]. In the following, we discuss special instances of the general model (3) that allow for a simplified computation of the CME (12). B. Univariate Case with a Single Observation In the following, the case of a scalar N = 1 system with a single pilot observation, i.e., M = 1 and a = 1, is considered. Since the quantization function in (2) is applied independently to the real and imaginary part of its input, deriving only the real part formula of the CME is sufficient here since the imaginary part is computed analogously. To this end we consider h ℜ ∼ N (0, σ 2 /2), n ℜ ∼ N (0, η 2 /2), and for the ease of notation we define r ℜ = sign(h ℜ + n ℜ ). Because of the symmetry of the zero-mean Gaussian density, we get p r ℜ (r ℜ ) = 1/2. Furthermore, the integral in (15) can be computed in closed-form to, cf. [50], p r ℜ |h ℜ (r ℜ = 1|h ℜ ) = ∞ 0 N (x; h ℜ , η 2 /2) dx (16) = 1 2 1 + erf h ℜ η ,(17)p r ℜ |h ℜ (r ℜ = −1|h ℜ ) = 0 −∞ N (x; h ℜ , η 2 /2) dx (18) = 1 2 1 − erf h ℜ η .(19) We can summarize (17) and (19) as p r ℜ |h ℜ (r ℜ |h ℜ ) = 1 2 1 + r ℜ erf h ℜ η .(20) Plugging this result into (12) E[h ℜ |r ℜ ] = ∞ −∞ hf h ℜ (h) 1 + r ℜ erf h η dh (21) = 1 √ π σ 2 σ 2 + η 2 r ℜ .(22) This can now easily be extended to the complex case with h ∼ N C (0, σ 2 ), n ∼ N C (0, η 2 ), and r = Q(h + n) which reads as E[h|r] = 2 π σ 2 σ 2 + η 2 r.(23) The closed-form MSE from (4) is then computed to MSE = σ 2 1 − 2 π σ 2 σ 2 + η 2 .(24) Interestingly, if we evaluate the Bussgang estimator in (8), we get the exact same CME and MSE expressions. To conclude, for this case the Bussgang estimator is indeed the CME. Note that in the multivariate case with an identity pilot matrix A = I and uncorrelated channels C h = diag(c h ) as well as uncorrelated noise C n = diag(c n ), the PDFs in (14) and (15) can be decomposed into the product of their marginals. Thus, the CME from (23) can be applied elementwise and is therefore also equal to the Bussgang estimator. The same expression as in (23) was found in [43], more generally for non zero-mean channels. However, the Bussgang theorem from [33] can not be applied for non zero-mean RVs and thus, the equality does not hold in general. Additionally, we can evaluate the well-known linear MMSE estimator for unquantized observations where it is the CME. There, the closed-form MSE is simply MSE = σ 2 1 − σ 2 σ 2 + η 2 .(25) We can now compute the necessary SNR for the unquantized system to achieve the same MSE as the one-bit quantized system without AWGN which we will analyze later in the numerical experiments section. C. Multivariate Noiseless Case with a Single Observation In this case, we consider M = 1 with n = 0. Similarly as in Section IV-B, it is sufficient to derive only the real part of the CME. To this end, we consider r ℜ = sign(h ℜ ) with h ℜ ∼ N (0, C h ) where C h ∈ R N ×N is a full matrix. We abuse the notation for Q(r ℜ ) = {h ℜ ∈ R N : sign(h ℜ ) = r ℜ }. In the following Theorem 1, a simplified expression of the general CME (12) is derived whose advantage is discussed afterwards. Theorem 1. The CME for the above system can be computed element-wise as [E[h ℜ |r ℜ ]] i = 1 p r ℜ (r ℜ ) N n=1 [r ℜ ] n [C h ] i,n N (0; 0, [C h ] n,n ) Q(r n ℜ ) N n (x; 0, C h ) dx,(26) where N n (0, C h ) is the (N − 1)-dimensional Gaussian PDF where the nth row and column of C −1 h is deleted. The superscript n of a vector truncates its nth element. Proof: See Appendix A. The advantage of the integral in (26) is that, in contrast to the double integral that appears in (12), cf. (15), it can be efficiently computed by means of [48] as described in Section IV-A because there appear only integrals over Gaussian PDFs. Note that the expression (26) simplifies to the elementwise CME as computed in (23) for diagonal covariances since then [C h ] i =n i,n = 0 and the (N −1)-dimensional integral equals 2 −(N −1) due to symmetry. This analysis can be interpreted as an asymptotic SNR analysis in which the AWGN goes to zero. The extension to the complex-valued case is straightforward by stacking the real and imaginary parts of each complex vector and using the real-valued representationČ h ∈ R 2N ×2N of the covariance matrix C h ∈ C N ×N , cf. [52,Sec. 15.4], given asČ h = ℜ(C h ) −ℑ(C h ) ℑ(C h ) ℜ(C h ) .(27) In contrast to the univariate case in Section IV-B, the resulting CME expression (26) is nonlinear and no longer in closed-form, which is clearly a difference to the linear Bussgang estimator. We will present numerical evaluations for this case that reveal a performance gap between the estimators. Fig. 1: Illustration of the iterative restriction of the angular range of the channel's phase and determination of the subsector by means of φ low (r) and φ high (r) for the case of M = 3 pilots. D. Univariate Noiseless Case with Multiple Observations In this case, we consider N = 1 with n = 0, i.e., the system from (3) simplifies to r = Q(ah) with h ∼ N C (0, σ 2 ). Note that in this case it is not sufficient to compute the real-valued CME because of the product of channel and pilot vector. This was the reason for the two-stage approach in [40]. Opposite to the unquantized case, having multiple pilot observations of the same realization is helpful in the context of quantization as more information can be recovered. Because the amplitude of the received signal is lost after the one-bit quantizer, having varying amplitude in the pilot sequence a has no influence on the observed signal. However, one can introduce phase shifts in order to increase the obtained information. In the following, we discuss the optimal pilot sequence and the resulting CME based on it. We start by rewriting the CME in terms of the magnitude and phase of the channel h = αe jθ where, due to the circular symmetry, the phase is uniformly distributed as θ ∼ U(0, 2π) and is independent of the Rayleigh distributed magnitude, cf. [ since the magnitude information is lost due to the one-bit quantization and its estimate is thus independent of the received signal. Therefore, we can set the magnitude of each pilot to one without loss of generality in order to fulfill the power constraint a 2 2 = M . It is shown in the following that a recovery of the phase information of the channel depending on the number of pilots M up to a certain resolution can be achieved. The key consideration thereby is to consecutively send pilots with an increasing phase shift and detect whether or not the quantization label is changing, which allows to narrow down the angular range of the channel's phase. We first note that there exist four quantization points in the complex plane, each representing a quadrant with an angle of π/2. We further assume without loss of generality that the first pilot is sent without a phase shift, which determines in which of the four quadrants with an angular range of π/2 the channel lies. In turn, if the subsequent pilots have a phase shift up to π/2 the quantization label will ultimately change once the rotated channel ends up in the neighboring main quadrant. Building on this, an optimal pilot sequence is of the form [a] m = exp(jψ m ), ψ m ∈ [0, π 2 ), m = 1, . . . , M,(29) Algorithm 1 Finding the boundary angles φ low (r) and φ high (r). which we assume to be ordered as ψ 1 < · · · < ψ M . Put differently, the pilots m = 2, ..., M divide each of the four main sectors into M circular sub-sectors if ψ m = 0 ∀m = 2, ..., M . After observing a receive signal r, we would like to determine in which of the sub-sectors the channel lies. More precisely, we would like to associate a sub-sector, described by the angles φ low (r) and φ high (r), to every possible receive signal r such that if we receive r, we know that the channel lies in the sub-sector described by φ low (r) and φ high (r). In Fig. 1, we exemplarily show how to determine the subsector of interest for M = 3 pilots. We therefore assume that the pilots have equidistant phase shifts in [0, π/2) which we prove to be MSE-optimal later in Theorem 2. The leftmost picture shows that the first unquantized receive signal [y] 1 = [a] 1 h lies in the positive quadrant and thus the corresponding quantized receive signal is [r] 1 = Q([y] 1 ) = q 1 which determines the main sector by means of φ low and φ high . The middle picture shows the situation after the second pilot is sent. Since the rotated channel [y] 2 = [a] 2 h yields the same quantization label [r] 2 = Q([y] 2 ) = q 1 , one can restrict the sector by reducing the angle φ high by the angle of ψ 2 , due to fact that if the channel would lie in the cropped area the rotated channel would yield the neighboring quantization label. The rightmost picture shows that the last receive signal is again [r] 3 = q 1 , which means that the sector can be further restricted by the same argumentation. This determines the final boundary angles φ low and φ high . In the same way as in the example of Fig. 1 for a different sequence where a change of the quantization label occurs for a certain pilot, the lower boundary angle φ low is increased, ultimately yielding a sub-sector of the same size. This procedure is summarized formally in Algorithm 1. It remains to show that the equidistant pilot sequence minimizes the MSE in combination with the CME. Therefore, the CME of the normalized channel can be obtained as Due to the absence of AWGN, similarly as in (47), the conditional probability p r|θ is an indicator function p r|θ (r|θ) = 1 φ low (r) ≤ θ < φ high (r) 0 else(32) where the boundary angles φ low (r) and φ high (r) are determined via the above discussed procedure. Given these preparations, we can now find the MSE-optimal pilot sequence. Theorem 2. The MSE-optimal pilot sequence for the above system contains equidistant phase shifts ψ m = π(m−1) 2M ∀m = 1, ..., M, such that [a] m = exp j π 2M (m − 1) , m = 1, ..., M.(33) Proof: See Appendix B. For the optimal pilot sequence (33), the boundary angles from (32) can now be compactly written as φ low (r) = 1 M M m=1 ∠([r] m ) − π 4(34) and φ high (r) = φ low (r) + π 2M . Interestingly, the same pilot sequence was proposed in [29] with the purpose to achieve a bijective mapping between a fixed dataset of channels and corresponding observations with long enough pilot sequences. We can now find a closed-form solution for the CME based on the optimal pilot sequence from Theorem 2. Theorem 3. The CME for the above system is computed to E[h|r] = 2M σ √ π sin π 4M exp j π 4M + φ low (r) . (35) Proof: See Appendix C. The closed-form MSE for the CME from (35) is then MSE = σ 2 1 − 4M 2 π sin 2 π 4M .(36) In contrast, the MSE for the Bussgang estimator from (8) for the present system is computed to MSE = σ 2 1 − 2 π a H C −1 r a .(37) Although the MSE terms in (36) and (37) seem to be different, we show their equality in the following. (37) is equal to the MSE of the CME from (36) for an arbitrary number of pilot observations M . Theorem 4. The MSE of the Bussgang estimator from Proof: See Appendix D. Theorem 4 surprisingly shows that the linear Bussgang estimator achieves the same MSE as the CME and is thus an MSE-optimal estimator for that case. Interestingly, as a byproduct of the proof in Appendix D we have found a closedform expression for the inverse covariance matrix C −1 r in (37). Yet again, this case can be interpreted as an asymptotically high SNR analysis. It should be noted here that a similar proof is not possible in the presence of AWGN as shown through numerical experiments later. With the closed-form MSE expression in (36), we can now study the asymptotically large number of pilots regime where we find the following result which, due to Theorem 4, also holds for the Bussgang estimator. Proof. We compute Theorem 5. For M → ∞ the MSE from (36) is lim M→∞ MSE = σ 2 1 − π 4 .(38)lim M→∞ M sin π 4M = lim M→∞ π 4 cos π 4M = π 4(39) where the rule of L'Hôpital was used. Since this limit exists, we can conclude that lim M→∞ M 2 sin 2 π 4M = π 2 16 .(40) Plugging (40) into (36) verifies (38). E. Summary So far, we have discussed the CME and its relationship to the Bussgang estimator for different system instances. Remarkably, in Section IV-B and Section IV-D the Bussgang estimator is shown to be equal to the CME which has a closed-form. This is a novel result that has not been stated in the literature so far. Because of the closed-form MSE expression of the CME, an analysis of the large number of pilots regime was possible which resulted in a closed-form limit. What is more, we have discussed in Section IV-C that the equality might not hold for correlated channel entries, even in the absence of AWGN. However, we have derived a computationally efficient expression for the CME in that case. An overview of whether the Bussgang estimator is equal to the CME or not for the different cases can be found in Table I. Therein, we show the cases where the Bussgang estimator is MSE-optimal with respect to the dimension (Dim.) N , the presence/absence of AWGN, the correlation (Corr.) between channel entries and the number of pilot observations M . For convenience, we referenced the subsections with the corresponding derivations. The first row refers to the general case where none of the discussed special cases applies. In the following, after introducing different performance metrics besides the MSE, we discuss the above results in numerical experiments which verify the theoretical findings. V. DIFFERENT PERFORMANCE METRICS A. Cosine Similarity The amplitude of the received signal is lost due to the onebit quantization. For this reason, one might solely be interested in how well an estimator can recover the angle of the signal of interest. A measure for the difference of the angle ψ between the true and estimated channel is the cosine similarity which we define as cos ψ = ℜ(h Hĥ ) h ĥ .(41) Note that the cosine similarity lies between −1 (opposite direction) and 1 (same direction). B. Achievable Rate Lower Bound The achievable rate is of great interest in quantized systems [12], [13]. For our considerations, we are aiming at a comparison between the CME and the Bussgang estimator via a lower bound on the corresponding achievable rate of a respective data transmission system that is taking the CSI mismatch into account. To this end, after estimating the channel with the pilot transmission in (3), the data symbol s is transmitted over the same channel, i.e., r = Q(hs + n) = Bhs + q (42) where in the second equation the linearized model with Bussgang's decomposition is used where q = Bn + η. We make the worst-case assumption that the aggregated noise is Gaussian, i.e., q ∼ N C (0, C q = C r − BC h B H ), cf. [54], and use a matched filter g H MF =ĥ H B H (C r − BC h B H ) −1(43) at the receiver. Note that the variance of the data symbol s is assumed to be one without loss of generality. We further assume that the SNR is the same during pilot and data transmission. As in [55], we split up the received data symbol asŝ (45) with Monte Carlo simulations. VI. NUMERICAL RESULTS In this section, we perform numerical experiments that verify our theoretical derivations and allow for quantifying the performance gap between the Bussgang estimator and the CME in the general case. We consider a diagonal noise covariance C n = η 2 I. For the case of N = 1, we choose without loss of generality the variance of the channel to σ 2 = 1. In the multivariate case, we randomly sample channel covariance matrices with the following procedure, cf. [56]. We first generate a matrix S where both the real and imaginary parts of all entries are i.i.d. as U(0, 1). Then, we compute the eigenvalue decomposition (EVD) of S H S = V ΣV H . Finally, U(0, 1). Afterwards, we normalize each covariance matrix by first dividing by its trace and afterwards multiplying by its dimension, such that tr(C h ) = N . For each covariance matrix we draw a single channel sample. This procedure averages out all possible structural features of the covariance matrix. For all simulations we draw a total of 10,000 samples. With the property a 2 2 = M and E[ h 2 ] = N we define the SNR as we construct C h = V diag(1 + ξ)V H , where [ξ] m ∼SNR = a 2 2 E[ h 2 ] E[ n 2 ] = 1 η 2 .(46) The MSE from (4) Normalized MSE quantized noiseless quantized noisy unquantized noisy Fig. 2: MSE performance of the CME from Section IV-B for N =M =1 for unquantized and one-bit quantized observations. A. Univariate Case with a Single Observation We first verify the findings from Section IV-B with N = M = 1 in Fig. 2 where the normalized MSE is plotted against the SNR. It can be observed that the CME converges to the noiseless MSE from (24) for η = 0, which equals to 1 − 2 π , for high SNR values. Additionally, we have plotted the wellknown linear MMSE estimator for unquantized observation, where it is the CME, from (25). Comparing a one-bit quantized system without AWGN with an unquantized system, we can compute their MSE intersection point by means of (24) and (25) with the SNR definition (46) to SNR = 2 π−2 ≈ 2.435dB which is verified by the simulation. B. Multivariate Noiseless Case with a Single Observation Next, we discuss the system from Section IV-C with M = 1 and n = 0. As discussed above, for this case there exists no closed-form solution, but we can use a computationally efficient algorithm from [48] to evaluate the integrals from (14) and (26). In Fig. 3 it can be seen that the normalized MSE decreases for larger dimensions N which is a consequence of the existing cross-correlations that are exploited for the estimation. Furthermore, the gap between the Bussgang estimator and the CME increases for higher dimensions. That is, the sub-optimality of the Bussgang estimator seems to have greater impact especially in higher dimensions. Note that in the case of N = 1 there exists the closed-form solution from before where the Bussgang estimator is exactly the CME. In order to further investigate the performance gap between the CME and the Bussgang estimator in this case, in Fig. 4, we evaluate the cumulative distribution function (CDF) curves of the MSE of both estimators for N = 24 dimensions. The vertical axis shows the probability that the MSE of the respective estimators is smaller than or equal to the value on the horizontal axis for each channel sample. It can be seen that the CME estimator is consistently better than the Bussgang estimator, especially in the region around the mean value of the MSE which we plotted for the ease of reference. In contrast, both estimators perform almost equally well for outliers, i.e., for channel samples which yield a high estimation error. decreases. However, the CME achieves a higher cosine similarity than the Bussgang estimator with an increasing gap for higher dimensions where a saturation is observed. This is in accordance with the results from the MSE investigations. In Fig. 5 (bottom), we analyze the achievable rate lower bound from (45) in bits/s/Hz. The lower bound is very similar for both estimators which increases from fewer than 1 bit/s/Hz for N = 1 up to over 3 bits/s/Hz for N > 14. A possible explanation for this small gap is, together with the results from the analysis of the cosine similarity, that the Bussgang estimator is able to provide a good estimate of the channel directions which seems to be especially important for the achievable rate. Only for higher dimensions a small gap between the lower bounds can be observed which agrees with the results from the MSE analysis. C. Univariate Noiseless Case with Multiple Observations In this subsection, we evaluate the results from Section IV-D with N = 1 and n = 0. Fig. 6 shows the normalized MSE over the number of pilots M . First, it can be seen that the MSE of the estimators in (8) and (35) are indeed equal for the conducted simulation, verifying Theorem 4. Second, the MSE converges from 1 − 2 π for M = 1 to 1 − π 4 , which is the limit for M → ∞ as shown in Theorem 5, already at a moderate number of M = 10 pilots. D. General Case In the following, we investigate more general cases which are evaluated with numerical integration as described in Section IV-A. First, we consider the univariate case of N = 1 for different numbers of pilot observations (M ∈ {1, 2, 5, 10}) and with AWGN for varying SNRs in Fig. 7 where we compare the derived optimal pilot sequence from (33), labeled "opt", with an all-ones pilot sequence a = 1, labeled "ones" which was used occasionally in recent works [18], [24]. For convenience, we have also plotted the closed-form solutions for the noiseless case with M = 1 and M → ∞ as dashed lines. As expected, for M = 1, the CME equals the Bussgang estimator and converges to the closed-form MSE 1 − 2 π for high SNR values. Interestingly, for an increasing number of pilot observations 1 − π 4 1 − 2 π -M=1 -M=2 -M=5 | M=10 SNR [dB] Normalized MSE CME opt Bussgang opt CME ones Bussgang ones Fig. 7: MSE performance comparison between the Bussgang estimator and the CME from Section IV-A for the univariate case with multiple pilot observations. there is an increasing gap between the Bussgang estimator and the CME, especially in the mid to high SNR regime. What is more, the optimal pilot sequence achieves a lower MSE than the all-ones pilot sequence with a large gap especially for high SNR values, validating the analysis in Section IV-D and indicating that the optimal pilot sequence performs well even in the case of finite SNR values and for the sub-optimal Bussgang estimator. In contrast, at low SNR values, both estimators perform almost equally well which can be explained by the dominating effect of the AWGN in this regime. It can also be seen that for higher numbers of pilots the MSE has a minimum at a certain SNR after which it increases again. What is more, the asymptotic limit of M → ∞ for infinite SNR (n → 0) can be outperformed with a finite number of pilots and finite SNR. These observations, seeming counterintuitive at first, are a consequence of the stochastic resonance effect [44]. This is a well-known effect related to quantization where noise can improve the performance of a system with quantization. Hence, this analysis allows for a numerical quantification of the stochastic resonance effect, which is the reason why the asymptotic limit can be outperformed and thus also describes the effect of the increasing MSE after a certain SNR value. Interestingly, the influence of the stochastic resonance increases for higher numbers of pilot observations and the optimal pilot sequence from (33) is less prone to the effect of an increasing MSE in the high SNR regime than the all-ones sequence. Next, in Fig. 8, we analyze the behavior of the Bussgang estimator and the CME for N = 1 and an increasing number of pilots M at a fixed SNR of 10dB. Similar as described before, the performance gap increases for higher numbers of pilots. Whereas the MSE of the Bussgang estimator saturates with a high error floor, the MSE of the CME decreases further for higher numbers of pilots. Finally, in Fig. 9, we depict the MSE for the case of multiple dimensions with AWGN over the SNR for correlated channel entries. For convenience, we have also plotted the closedform solutions for N = 1 with and without AWGN. Due to the exponentially increasing complexity of the numerical integration, we were only able to plot until N = 3 to show SNR [dB] N = 1 noiseless N = 1 noisy CME Bussgang Fig. 9: MSE performance comparison between the Bussgang estimator and the CME from Section IV-A for the multivariate case with a single pilot observation. the tendency of the results. First, an offset of the normalized MSE for higher dimensions can be observed which seems to decrease with the number of dimensions. Moreover, the Bussgang estimator is very close to the CME which is in accordance with the prior results for low dimensions. The curve for N = 2 (N = 3) intersects with the noiseless case of N = 1 at about 15dB (12dB) SNR. VII. CONCLUSION AND FUTURE WORK In this work, we have investigated the CME for one-bit quantized systems in a jointly Gaussian setting. We derived a novel closed-form solution for multiple pilot observations and a transformation into a computationally efficient expression where only Gaussian integrals have to be evaluated for the case without AWGN. Additionally, we have proposed an MSEoptimal pilot sequence which we motivated by the structure of the CME. It turned out that a successive shift of the transmit signals' angle is optimal which was already proposed in prior work with a different motivation in the context of deep learning. We analyzed the well-known Bussgang estimator and showed the equivalence of the Bussgang estimator and the CME in the univariate case with a single pilot observation, which extends to the multivariate case when there is no correlation between the channel or noise entries. In addition, we showed that the closed-form MSE terms of the Bussgang estimator and the CME are equal in the case of multiple pilots without AWGN. To this end, we found a closed-form expression for the inverse of the covariance matrix of the quantized receive signal which is derived with the arcsine law. For the general case where numerical integration is necessary, we found that the performance gap between the Bussgang estimator and the CME increases for higher dimensions, longer pilot sequences, or higher SNR values. The findings of this work may facilitate extensions to more general input distributions, e.g., Gaussian mixture models (GMMs), different nonlinearities such as the clipper function instead of the onebit quantization, or to quantizers with higher resolutions. APPENDIX A. Proof of Theorem 1 Proof. First, it can be observed that due to the absence of noise the expression in (15) simplifies to an indicator function p r ℜ |h ℜ (r ℜ |h ℜ ) = 1, h ℜ ∈ Q(r ℜ ) 0, h ℜ ∈ Q(r ℜ )(47) which means that the probability of receiving the quantized observation r ℜ is one if the underlying channel lies in Q(r ℜ ) since the quantization is deterministic. With this, we get E[h ℜ |r ℜ ] = 1 p r ℜ (r ℜ ) Q(r ℜ ) h ℜ f h ℜ (h ℜ ) dh ℜ .(48) Second, we can interpret the integral in (48) as the (scaled) mean of a truncated multivariate Gaussian, i.e., Q(r ℜ ) h ℜ f h ℜ (h ℜ ) dh ℜ = p r ℜ (r ℜ )E[h ℜ ](49) whereh ℜ ∼ fh ℜ is a truncated Gaussian RV with PDF fh ℜ (h ℜ ) = 1 pr ℜ (r ℜ ) N (h ℜ ; 0, C h )h ℜ ∈ Q(r ℜ ) 0h ℜ ∈ Q(r ℜ )(50) and p r ℜ (r ℜ ) is the normalization such that fh ℜ is a valid PDF that integrates to one. In [57], [58], the mean of a onesided truncated Gaussian RV is derived, which in our case is computed to p r ℜ (r ℜ )E[[h ℜ ] i ] = (51) N n=1 [r ℜ ] n [C h ] i,n N (0; 0, [C h ] n,n ) Q(r n ℜ ) N n (x; 0, C h ) dx. Plugging (51) into (49) together with (48) yields the expression in (26) and thus finishes the proof. B. Proof of Theorem 2 Proof. We first observe that with the procedure in Algorithm 1 the angle domain can be split into at most 4M regions by choosing ψ m = 0 ∀m = 2, ..., M because any of the four quadrants can be divided into M distinct regions which are identified via (32). Now, we write the CME asĥ =αe jθ such that we can reformulate the MSE as E[|αe jθ −αe jθ | 2 ] = E[α 2 +α 2 − 2αα cos(θ −θ)](52) by using the cosine law. Since we are solely interested in the phase estimate for the design of the optimal pilot sequence and the cosine is monotonically decreasing between [0, π/2), the minimization of the MSE between the phases E[(θ −θ) 2 ] is equivalent to the minimization of the MSE from (52) for fixed magnitudes. Note that because of the uniform distribution f θ the CMEθ is exactly the midpoint between the boundary angles which can also be seen later in Theorem 3. The optimization of the pilot sequence is thus given as where in (55) the integral over the whole angle domain is split up into the 4M subsets that are defined via the boundary angles from (32) which fulfill φ 1 < · · · < φ 4M where φ 1 = 0 and φ 4M = 2π. The optimization problem in (55) is identical to the well known problem of finding the quantization regions of a non-uniform quantizer which minimize the distortion [59]. Because the angles are uniformly distributed, the optimal solution is found by equidistant integration regions, cf. [59,Sec. 5.6], which is achieved by equidistant phase shifts in [0, π 2 ), i.e., ψ m = π(m−1) 2M ∀m = 1, ..., M . Plugging this solution into (29) results in (33) and finishes the proof. C. Proof of Theorem 3 Proof. We use the simplified representation of (28) where the mean of the Rayleigh distributed magnitudes is known to be, cf. [53, Appendix A.1.3], E[α] = √ π 2 σ.(56) Furthermore, due to the equidistant phase shifts which divide the input space into 4M distinct regions, cf. Appendix B, and the symmetry of the Gaussian PDF, we have the prior probability p r (r) = 1 4M in this case. Plugging this result and the definition of the uniform distribution f θ into (31), the CME of the normalized channel is computed to E[e jθ |r] = 4M 2π φlow(r)+ π 2M φlow(r) e jθ dθ. It is sufficient to solve the integral for the first angle sector [0, π 2M ) and afterwards shift the solution by the angle of φ low (r). We thus compute π 2M 0 e jθ dθ = sin π 2M + j 1 − cos π 2M (58) = 2 sin π 4M exp j π 4M .(59) After plugging (56) together with the generalized solution from (57) and (59) into (28) yields (35) which finishes the proof. e jθ f θ (θ)p r|θ (r|θ) dθ. is normalized by E[ h 2 ] = N . Fig. 5 ( 5top) evaluates the cosine similarity from (41) over different dimensions N . For an increasing number of dimensions it can be seen that the cosine similarity generally Fig. 3 : 3MSE performance comparison between the Bussgang estimator and the CME from Section IV-C for the multivariate noiseless case with a single pilot observation. Fig. 4 : 4CDF curves of the MSE performance comparison between the Bussgang estimator and the CME from Section IV-C for the multivariate noiseless case with a single pilot observation for N = 24 dimensions. Fig. 5 : 5Cosine similarity (top) and achievable rate lower bound (bottom) comparison between the Bussgang estimator and the CME from Section IV-C for the multivariate noiseless case with a single pilot observation. Fig. 6 : 6MSE performance comparison between the Bussgang estimator and the CME from Section IV-D for the univariate noiseless case with multiple pilot observations. Fig. 8 : 8MSE performance comparison between the Bussgang estimator and the CME from Section IV-A for the univariate case with SNR = 10dB. yields, cf. [51, eq. (2.7.2.4)], 53, Appendix A.1.3]. Consequently, we obtain E[αe jθ |r] = E[α|r]E[e jθ |r] = E[α]E[e jθ |r], TABLE I : IOverview of different system parameters and the MSE-optimality of the Bussgang estimator. [T ] n,n =Theorem 4Proof. With the vector in(33), its outer product is computed asand thusFor the considered case, the unquantized observation is given as y = ah, and thus its covariance matrix can be written as C y = σ 2 aa H . Furthermore, from the property that |[a] m | 2 = 1, it follows that diag(C y ) = σ 2 I. Consequently, the expression for the covariance matrix from (10) simplifies to C r = 2 π arcsin(ℜ(aa H )) + j arcsin(ℑ(aa H )) .After plugging in(61)and(62)we getwhere the equation in (65) holds since the arguments in (64) satisfy − π 2 < π 2M (m − n) < π 2 . The following Lemma computes C −1 r in closed form. Proof. To prove that (66) is the inverse of (63) we define the matrix T = C −1 r C r . If we show that T is the identity matrix, i.e., T = I, then Lemma 1 holds. The m, nth element of T is given asDue to the sparsity of C −1 r , only three summands are non-zero. For example, we can compute [T ] m,n for m = n = 1:We show that the diagonal elements of T , i.e., m = n, are one in (67) and that all off-diagonal elements of T , i.e., m = n, are zero in (68).With Lemma 1, we can now find the expression ofPlugging this into (37) results in the same MSE term as in(36)and thus finishes the proof. 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[]
[ "On Quantum Information Before the Page Time", "On Quantum Information Before the Page Time" ]
[ "Jonah Kudler-Flam [email protected] \nSchool of Natural Sciences\nInstitute for Advanced Study\n08540PrincetonNJUSA\n\nPrinceton Center for Theoretical Science\nPrinceton University\n08540PrincetonNJUSA\n", "Yuya Kusuki [email protected] \nWalter Burke Institute for Theoretical Physics\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n\nRIKEN Interdisciplinary Theoretical and Mathematical Sciences\n351-0198WakoSaitamaJapan\n" ]
[ "School of Natural Sciences\nInstitute for Advanced Study\n08540PrincetonNJUSA", "Princeton Center for Theoretical Science\nPrinceton University\n08540PrincetonNJUSA", "Walter Burke Institute for Theoretical Physics\nCalifornia Institute of Technology\n91125PasadenaCAUSA", "RIKEN Interdisciplinary Theoretical and Mathematical Sciences\n351-0198WakoSaitamaJapan" ]
[]
While recent progress in the black hole information problem has shown that the entropy of Hawking radiation follows a unitary Page curve, the quantum state of Hawking radiation prior the Page time is still treated as purely thermal, containing no information about the microstructure of the black hole. We demonstrate that there is significant quantum information regarding the quantum state of the black hole in the Hawking radiation prior to the Page time. By computing of the quantum fidelity in a 2D boundary conformal field theory (BCFT) model of black hole evaporation, we demonstrate that an observer outside of an evaporating black hole may distinguish different black holes via measurements of the Hawking radiation at any time during the evaporation process, albeit with an exponentially large number of measurements. Furthermore, our results are universal, applicable to general BCFTs including those with large central charge and rational BCFTs. The techniques we develop for computing the fidelity are more generally applicable to excited states in CFT. As such, we are able to characterize more general aspects of thermalization in 2D conformal field theory.
10.1007/jhep05(2023)078
[ "https://export.arxiv.org/pdf/2212.06839v2.pdf" ]
254,636,293
2212.06839
f560dc7cb2f87d26ce710eb7e584d8793ced6912
On Quantum Information Before the Page Time 23 Apr 2023 Jonah Kudler-Flam [email protected] School of Natural Sciences Institute for Advanced Study 08540PrincetonNJUSA Princeton Center for Theoretical Science Princeton University 08540PrincetonNJUSA Yuya Kusuki [email protected] Walter Burke Institute for Theoretical Physics California Institute of Technology 91125PasadenaCAUSA RIKEN Interdisciplinary Theoretical and Mathematical Sciences 351-0198WakoSaitamaJapan On Quantum Information Before the Page Time 23 Apr 2023 While recent progress in the black hole information problem has shown that the entropy of Hawking radiation follows a unitary Page curve, the quantum state of Hawking radiation prior the Page time is still treated as purely thermal, containing no information about the microstructure of the black hole. We demonstrate that there is significant quantum information regarding the quantum state of the black hole in the Hawking radiation prior to the Page time. By computing of the quantum fidelity in a 2D boundary conformal field theory (BCFT) model of black hole evaporation, we demonstrate that an observer outside of an evaporating black hole may distinguish different black holes via measurements of the Hawking radiation at any time during the evaporation process, albeit with an exponentially large number of measurements. Furthermore, our results are universal, applicable to general BCFTs including those with large central charge and rational BCFTs. The techniques we develop for computing the fidelity are more generally applicable to excited states in CFT. As such, we are able to characterize more general aspects of thermalization in 2D conformal field theory. Introduction The black hole information problem was first characterized by Stephen Hawking as a "breakdown of predictability in gravitational collapse" [1,2]. The essence of the problem is that the information describing the matter that travels beyond the black hole horizon is lost forever; the black hole evaporates, radiating a spectrum of particles that only depends on the macroscopic thermodynamic parameters, such as mass, charge, and angular momentum, with no dependence on the microscopic details of the quantum state of the collapsing matter. Strictly speaking, this is not a breakdown in predictability in the forward time direction because knowing the state of the universe prior to gravitational collapse deterministically fixes, in semi-classical gravity, the quantum state of the universe after evaporation. However, in the backwards time direction one is unable to predict the early-time quantum state of the collapsing matter even with complete knowledge of the final state due to the entropy-generating, many-to-one map of the forward time evolution. The quantum states of the radiation emitted from distinct black holes are entirely indistinguishable. The original computations of the quantum state of the radiation were done on fixed spacetime geometries with only minor adjustments included to account for the gravitational backreaction of the radiation [1,3]. Remarkably, recent developments [4][5][6][7] have shown that the inclusion of certain wormhole configurations in the gravitational path integral, macroscopically different than the original background geometry, lead to the conclusion that the Hawking radiation "purifies itself" after the so-called "Page time" defined as the time at which the coarse-grained entropy of the radiation equals the coarse-grained entropy of the black hole [8] which is given by the area of the horizon in Planck units [1,9] S BH = A 4 . (1. 1) This demonstrates that the late-time state is not thermal, restoring hope that predictability may survive in the backwards time direction. It is of course crucial to understand not only if predictability is restored but when and how the information of the early-time quantum state escapes out of the black hole and into the radiation. A precise and operationally meaningful way to quantify this escape of information is to consider two microscopically distinct, but macroscopically indistinct evaporating black holes e.g. two black holes formed from collapse of quantum matter in orthogonal quantum states with identical total conserved quantities. 1 If an observer has access to the radiation of one of the black holes, they may make a quantum measurement. If information has escaped from the black hole, then a judiciously chosen measurement will allow the observer to determine which of the two quantum states formed the black hole. The larger the amount of information that has escaped, the easier it is for the observer to distinguish the black holes using measurements on the radiation. To quantify this, we will study the quantum fidelity, a useful measure of distinguishability. For two distinct evaporating black holes, Hawking's prediction corresponds to the fidelity equaling one at all times, meaning perfect indistinguishability. Our goal is to determine if, when, and how the fidelity can be found to be less than one, re-establishing predictability in quantum gravity (in simple toy models). The aforementioned progress on the entropy of Hawking radiation suggests that quantum information escapes the black hole only after the Page time. This is the time at which a region behind the horizon called the "island" forms, denoting the region whose quantum state is "reconstructible" on the radiation. This is consistent with the standard belief that there is no information in early Hawking radiation. The goal of this paper is to show that this belief receives important corrections and there is indeed genuine information regarding the black hole microstructure emanating within the radiation even before the Page time when no island is present. This work builds upon previous hints regarding nontrivial information in early Hawking radiation. 2 The computation of fidelity that we seek was first computed in [12] for the PSSY model [7] involving two-dimensional Jackiw-Teitelboim gravity decorated with end-of-world branes that are entangled with an auxiliary "radiation" system. By summing over particular replica wormhole configurations in the Euclidean path integral, it was found that before the Page time, the fidelity was F = 1 − 1 2 e −(S BH −S rad ) + . . . , which is very close to, though less than one, proving that in principle different black hole microstates may be distinguished prior 1 These two states may be thought of as living in the same microcanonical energy band as we generally do not expect exact degeneracies in the spectrum. 2 A complementary analysis of information in the Hawking radiation prior to the Page time was analyzed in [10,11] where it was argued that there is significant quantum entanglement in the radiation starting at a much earlier time scale referred to as t b . to the Page time. Had replica wormholes not been included in the calculation, the fidelity would have been exactly one at all times, consistent with Hawking's analysis. The deviations from F = 1 in this model were more systematically studied in the context in the JLMS formula [13] in [14]. While this model is certainly illuminating, it has certain drawbacks such as not describing genuine time evolution of the evaporation process as well as lacking matter fields. We are motivated to investigate more complicated evaporation models like those in [6,15,16]. These are built from a two dimensional conformal field theory whose boundary is coupled to a "quantum dot" or conformal boundary that is holographically dual to a two-dimensional gravitational system. Such a system can support non-trivial time evolution and a spectrum of matter fields. When prepared in the thermofield double state, the time evolution of this system may be interpreted as a two-dimensional black hole radiating into a bath 3 that is "bulk" conformal field theory. This conformal field theory thus comprises the "radiation" (see Figure 1). The time evolution of the entanglement entropy (Page curve) for this model was considered from both the bulk and boundary perspectives in [15][16][17] where it was found that at the Page time, an island forms that encompasses the black hole interior. We analyze the quantum fidelity between two distinct black holes in this model, modeled by placing particles of different flavors behind the black hole horizon, and find that there is significant information in the Hawking radiation prior to the formation of the island, with the fidelity between two evaporating black holes being F = 1 − (C ppr − C qqr ) 2 Γ 1 2 + ∆ r 4 √ πΓ (1 + ∆ r ) e −2∆r(S BH −S rad ) , (1.2) where ∆ r is the conformal dimension of the lightest primary field in the CFT spectrum and C ppr /C qqr are the OPE coefficients between the operator placed behind the horizon and the lightest primary. As expected, this is not an identical answer to the PSSY model due to the increased complexity in the theory, though it shares the key qualitative feature that it is a finite amount less than one, albeit non-perturbatively suppressed in the black hole entropy, implying the escape of information prior to the Page time. We believe that this is a general feature of evaporating black holes. After the Page time, we find the fidelity to be nearly zero, implying that the black holes are easily distinguishable, a notion we make precise in the following section. We discuss the similarities and differences in the mechanisms leading to information leakage in the BCFT model and the PSSY model. The Model We consider a two-dimensional conformal field theory with a conformal boundary. If this CFT is "holographic," the three-dimensional dual has the conformal boundary extend into the bulk geometry as a "Cardy brane." We treat the two-dimensional theory of this brane as the black hole system while the remaining geometry may be considered the radiation. To provide nontrivial evaporation dynamics, we prepare the boundary conformal field theory in the thermofield double state |T F D = 1 Z(β) i e −βE i /2 |E i 1 ⊗ |E i 2 (1.3) and evolve with H = H 1 ⊗ 1 2 + 1 1 ⊗ H 2 |T F D(t) = 1 Z(β) i e −(β+4it)E i /2 |E i 1 ⊗ |E i 2 . (1.4) Under this time evolution, there will be a nontrivial Page curve. In order to add flavor to the black hole, we include a boundary excitation. More specifically, we act with an operator on the boundary that has been evolved by β/4 in imaginary time T F D(t) (1) ∝ i e −(β+4it)E i /2 O 1 (0, −β/4) |E i 1 ⊗ |E i 2 , (1.5) T F D(t) (2) ∝ i e −(β+4it)E i /2 O 2 (0, −β/4) |E i 1 ⊗ |E i 2 . (1.6) With this choice of imaginary time evolution, the gravitational picture includes a particle behind the horizon at all times. Overview In Section 2, we provide background on the distinguishability measures that we compute, particularly the quantum fidelity. This makes precise how one should interpret our results in terms of an observers ability to know the details of the black hole formation from measurements only on the radiation. In Section 3, we compute the entropy in the island model using conformal field theory techniques. In Section 4, we compute the fidelity in the island model, presenting the mechanisms leading to our main result and comparing these to the PSSY model. While of a different general motivation, in Section 5, we discuss how similar computations can be done to characterize eigenstate thermalization for extended subsystems in two-dimensional conformal field theory. Certain technical details are left to the appendices. 2 What do distinguishability measures measure? In the midst of technical calculations, it is easy to lose sight of the concrete meanings of the quantities we are computing. We emphasize in this section the precise meaning of the quantum fidelity, which fortunately quantifies a very natural and concrete task that an experimentalist may perform. The Uhlmann fidelity (which we will simply call the fidelity) measures the distinguishability between two quantum states, ρ and σ and is defined as F (ρ, σ) = Tr √ ρσ √ ρ. (2.1) This quantity obeys several nice properties such as Jozsa's axioms [18] and is bounded as 0 ≤ F (ρ, σ) ≤ 1, (2.2) where the upper (lower) bound is saturated if and only if ρ = σ (ρσ = 0). A fundamental task in quantum information theory is that of quantum hypothesis testing. Quantum hypothesis testing is the scenario where Alice is given a quantum state with the promise that it is either ρ or σ. It is her task to make a quantum measurement to determine which state she was given. There is an error probability of her determining she has state ρ when she really has state σ. Similarly, there is an error probability of Alice determining she has state σ when she really has state ρ. The sum of these two error probability, P err , for the optimized measurement 4 is bounded from both above and below by the fidelity as 1 − 1 − F (ρ, σ) 2 ≤ P err ≤ F (ρ, σ). (2.3) Moreover, if Alice is given n copies of the state, she can make a more complicated measurement acting on all copies such that P (n) err ≤ F (ρ, σ) n . (2.4) Clearly, if the fidelity is less than one, if given a sufficient number of copies of the state, Alice may identify the state with high probability. This will be the case in the black hole setting where the sufficient number of copies will be exponentially large in the entropy of the black hole prior to the Page time. After the Page time, we find that only a single copy of the state is needed. The square roots present in the fidelity make it challenging (though still possible) to compute. Simpler quantities to analyze only involve the second moments of the density matrices. For illustration, we will consider the super-fidelity [20] F S (ρ, σ) := Tr ρσ + (1 − Tr ρ 2 )(1 − Tr σ 2 ) (2.5) and the geometric mean (GM) fidelity [21] F GM (ρ, σ) := Tr ρσ Tr ρ 2 Tr σ 2 . (2.6) The super-fidelity is useful in that it upper bounds the fidelity. Therefore before the Page time, when we see that the super-fidelity is less than one, the fidelity must also be less than one. The GM fidelity is a useful proxy for the fidelity as it generally follows the same qualitative behavior even though it is neither an upper nor lower bound. This will be useful near the Page time where we do not have analytic control over the fidelity. It is also useful after the Page time because we find it to be non-zero which necessarily means that ρ = σ so the fidelity itself must also be non-zero. Von Neumann Entropy We warm up by considering the entropy of the radiation, deriving a Page curve. This calculation will also set our definition of the coarse-grained black hole and radiation entropies. The radiation system consists of the bulk conformal field theory from position d to ∞ on both sides of the thermofield double (the red region in Figure 1), where d is playing the role of a cutoff between the black hole and radiation subsystems. The coordinates, z, on the half cylinder have Re[z] ∈ [0, ∞) and Im[z] ∈ [0, 2π). We first conformally map the half cylinder to the disk using w(z) = e 2π β z . The endpoints of the radiation subsystem are mapped to w 1 =w 1 = −e − 2π β d and w 2 =w 2 = e − 2π β d on the disk. The moments of the density matrix may be computed using a two-point correlation function of twist operators Tr (ρ n A ) = 2π β 4hn σ n (w 1 ,w 1 )σ n (w 2 ,w 2 ) disk . (3.1) We may evolve in imaginary time and subsequently analytically continue the correlation function to real time, such that w 1 = −e − 2π β (d+t) , w 2 = e − 2π β (d−t) ,w 1 = −e − 2π β (−d+t) ,w 2 = e − 2π β (−d−t) . (3.2) For simplicity, we take the high-temperature limit (β → 0). In this limit, the answer is independent of the specific details of the CFT. Later on, this limit will allow us to compute subleading corrections to the fidelity reliably. The correlation function on the disk can be naively "unfolded" to a correlation function without a boundary, an approximation that has been leveraged in the past to simplify replica calculations in boundary conformal field theory [22,23]. Due to the unfolding, the correlation function is defined on a chiral CFT. The unfolding is not precise because we neglect a potential interface operator corresponding to the boundary. This simplification replaces bulk-bulk-boundary OPE coefficients with the corresponding bulk-bulk-bulk OPE coefficients, though importantly does not change the qualitative behavior that we intend to isolate. One may prefer to call the brane in the bulk a Z 2 brane or Hartman-Maldacena brane [24] instead of a Cardy brane. The high-temperature limit then corresponds to an OPE limit of the chiral twist fields such that at early times (t < d), when w 1 ∼ w 2 andw 1 ∼w 2 Tr (ρ n A ) = 2π β 4hn (w 1 − w 2 ) −2hn (w 1 −w 2 ) −2hn ,(3.3) and at late times (t > d), when w 1 ∼w 1 and w 2 ∼w 2 Tr (ρ n A ) = 2π β 4hn (w 1 −w 1 ) −2hn (w 2 −w 2 ) −2hn . (3.4) Both expressions may be analytically continued in n to find the von Neumann entropy S vN (A) = lim n→1 1 1 − n log Tr (ρ n A ) = 2πc 3β t + c 3 log β 2π , t < d 2πc 3β d + c 3 log β 2π , d < t , (3.5) where is a UV cutoff. There are subleading (in β) terms that also contribute including the boundary entropy, though we neglect these due to β being our expansion parameter. In comparing to the Page curve, we can now identify 2πc 3β t as corresponding to the coarse-grained entropy of the radiation, S rad , and 2πc 3β d as the coarse-grained entropy of the black hole, S BH . A large boundary entropy may be added to the boundary but this will not significantly change the following formulas. Fidelity in the island model Super and GM Fidelities PSSY Model To gain intuition, we first evaluate the super-fidelity and GM fidelity between two radiation states in the PSSY model. The analog of (1.6) is |Ψ 1 := 1 √ k k i |ψ i,1 B |i R , ρ(1) R := Tr B |Ψ 1 Ψ 1 | , (4.1) |Ψ 2 := 1 √ k k i |ψ i,2 B |i R , ρ(2)R := Tr B |Ψ 2 Ψ 2 | ,(4.2) where, as described in [7], |ψ i,a may be interpreted as a black hole state with the EOW brane in state i and a small excitation with flavor a propagating behind the horizon. The purity of the radiation is the same for both states Tr ρ (1) R 2 = Tr ρ (2) R 2 = 1 k 2 i,j | ψ i,a | ψ j,a B | 2 , a = 1, 2. (4.3) Crucially, the inner product appearing in the sum is not proportional to a Kronecker delta. Instead, the inner product defines the following boundary conditions in the gravitational path integral (4.4) where the black lines represent the asymptotic boundaries and the dotted blue lines enforce that an EOW brane with a particular flavor intersects the boundary. These boundary conditions can be filled in in two topologically distinct ways, 5 corresponding to a factorized bulk path integral and a wormhole solution (4.5) Completing the sums over i and j and including the normalization of the state, Z −2 1 , we find Tr ρ (1) R 2 = Tr ρ (1) R 2 = 1 k + Z 2 Z 2 1 . (4.6) We interpret the first term as e −S rad and the second term as e −S (2) B where S (n) B is the coarsegrained Rényi entropy of the black hole. The overlap is given by Tr ρ (1) R ρ (2) R = 1 k 2 i,j ψ i,1 | ψ j,1 B ψ i,2 | ψ j,2 B ,(4.7) which sets the boundary conditions as (4.8) Now, only the disconnected solution contributes to the path integral because the brane indices are incompatible, not allowing a wormhole. Summing over i and j, we find Tr ρ (1) R ρ (2) R = e −S rad . (4.9) The super-fidelity and GM fidelity are subsequently F S (ρ (1) R ρ (2) R ) = 1 − e −S (2) B , F GM (ρ (1) R ρ (2) R ) = 1 1 + e −(S (2) B −S rad ) . (4.10) The super-fidelity implies that the fidelity is less than one before the Page time by an amount at least exponentially small in the black hole entropy. This implies that an observer may distinguish the two black holes given an exponentially large number copies of the state of the radiation. The GM fidelity displays a Page curve that transitions between close to one and close to zero at the Page time (see Figure 2). Using heavier machinery involving higher moments, one finds that the GM fidelity faithfully mimics the behavior seen in the fidelity [12]. The BCFT Model We now demonstrate the analogs of these calculations in the BCFT model by considering the operators in (1.6) to have conformal dimensions 1 ∆ c and the CFT to be "holographic." Later, we will provide more general calculations. In this regime of dimensions, the operators behave as generalized free fields and their correlations functions may be computed using Wick contractions. Moreover, their bulk duals are quantum fields that are massive enough to follow classical trajectories but light enough as to not backreact on the geometry. The Wick contractions between different replicas are somewhat analogous to the replica wormhole contributions from (4.5). Because φ p and φ q are orthogonal fields (their two-point function is trivial), their contractions do not contribute to the path integral. This mimics the phenomenon where replica wormholes in (4.8) connecting different flavored EOW branes do not contribute to the gravitational path integral. Using the replica path integral approach, the purities and overlaps of the two states can be seen to be given by the following correlation functions where Σ 2,2 is the manifold with two replica sheets glued along two intervals with endpoints (after analytic continuation to Lorentzian time) at Tr ρ (a) R ρ (b) R = X † ab (i)X ab (−i) Σ 2,2 , X ab = O a ⊗ O b ,(4.z 1 = −e − 2π β (d+t) , z 2 = e − 2π β (d−t) , z 3 = e − 2π β (−d−t) , z 4 = −e − 2π β (−d+t) . (4.12) The operator X ab is bi-local, being a tensor product of primary operators located at the same point but on separate sheets. This should not be confused with an operator in the orbifolded theory. This manifold has the topology of a torus. For the purity (a = b), there are two Wick contractions, one contracting operators on the same sheet and the other contracting operators on opposite sheets. While the difference in magnitude of these terms is large, it is important to include both in order to find the leading corrections to the fidelity. For the overlap (a = b), there is only a single Wick contraction, just as in the PSSY model. Famously, the torus partition function undergoes a first-order phase transition when the two cycles have equal length, corresponding in gravity to the Hawking-Page transition [25]. In our model this signals the Page transition and there is a change in the functional form of the Wick contractions. The precise form of the GM and super fidelities are somewhat complicated and can be found in Appendix A. Here, we emphasize the fact that they mimic the results from the PSSY model in that they imply that the fidelity is less than one by an amount exponentially suppressed by the black hole entropy before the Page time and are non-zero but very small after the Page time. We plot the GM fidelity in Figure 2, comparing the curve to the simple answer from the PSSY model. The fidelities are seen to have very similar behavior. Uhlmann fidelity in the BCFT Model To compute the full Uhlmann fidelity, we need to compute higher moments of the density matrices by using a replica trick [26][27][28], F (ρ, σ) = lim m,n→ 1 2 Tr (ρ m σρ m ) n . (4.13) To leading order, the fidelity was computed between two states excited above the vacuum with heavy operators [28]. Using the same techniques, we may compute the leading order Page curve for the fidelity. This will simply lead to a step function with the fidelity equalling one before the Page time and zero after the Page time. The subleading corrections to this calculation are difficult to control in generality, so we instead use a separate method in the following section. The moments may be evaluated using a 2k-point function on, Σ k,2 , the k-sheeted replica manifold glued along two intervals Tr ρ (1) R m ρ (2) R ρ (1) R m n = X † k (i)X k (−i) Σ k,2 φ † p (i)φ p (−i) 2mn C φ † q (i)φ q (−i) n C , X k = φ ⊗m p ⊗ φ q ⊗ φ ⊗m p ⊗n . (4.14) where k := (2m + 1)n. Using the Riemann-Hurwitz formula, one can see that the replica manifold is a (k − 1)-genus surface and thus extremely difficult to evaluate in generality. We only consider the leading Wick contraction. Prior to the Page time, this contraction connects operators on the same replica sheet which is identical to the normalization factor, so the fidelity is trivially one. After the Page time, the Wick contraction of the closest pairs of operators connects operators on cyclically permuted sheets. Because these involve contracting φ p with φ q , we find that the fidelity is zero. To find a non-zero contribution, one needs sum contractions involving distant operators. The denominator will then dominate significantly over this contribution giving a very small fidelity. We already understood from the previous subsection that the fidelity is not exactly zero because otherwise the GM fidelity would be exactly zero. We now characterize this leading contribution. The high-temperature limit We proceed to a more general calculation that may be done in the high-temperature (β → 0) limit. The following calculations are more general in that they are do not place requirements on the dimensions of the boundary primary operators and the CFT is unconstrained. For example, the CFT can be a free boson. It is convenient to describe the replica partition function using twist operators on the complex plane rather than a correlation function of primary fields on the (k −1)-genus surface Z n,m = X k (i)σ k (z 1 )σ k (z 2 )X k (−i) disk,C ⊗k /Z k , X k := φ ⊗m p ⊗ φ q ⊗ φ ⊗m p ⊗n , (4.15) This expression is purely formal because the operator X k is not included in the spectrum of the orbifold theory C ⊗k /Z k because X k is not symmetric under cyclic permutations. Nevertheless, this notation is useful as explained in [29]. This is because we can take the OPEs of operators on each sheet in (4.14). Grouping like-terms among the sheets, we find operators that are indeed symmetric under cyclic permutation, and under orbifolding reside in the untwisted sector. The effective OPE coefficients with an operator T in the untwisted sector are dependent on how we take the OPE. We will write these explicitly shortly. Again, we avoid the technical complications of the conformal boundary without discarding any of the essential physics by using the doubling trick and neglecting the contribution from the potential interface operator Z n,m = X k (i)σ k (z 1 )σ k (z 2 )σ k (z 3 )σ k (z 4 )X k (−i) C ⊗k /Z k , (4.16) where the insertion points are z 1 = −e − 2π β (d+t) , z 2 = e − 2π β (d−t) , z 3 = e − 2π β (−d−t) , z 4 = −e − 2π β (−d+t) . (4.17) Before the Page time Let us evaluate the correlation function before Page time. For convenience, we apply the following conformal map, w = z + i z − i . (4.18) The correlation function is transformed to Z n,m = (conformal factor) X k (0)σ k (w 1 )σ k (w 2 )σ k (w 3 )σ k (w 4 )X k (∞) C ⊗k /Z k . (4.19) We leave the conformal factor implicit because the factors corresponding to the local operators cancel via the denominator of (4.14) and the factors corresponding to the twist fields are trivial in k → 1 limit because the dimensions of the twist fields go to zero. In the high-temperature limit, the insertion points can be approximated at early times (t < d) as w 1 = −1 − 2ie − 2π β (d+t) , w 2 = −1 + 2ie − 2π β (d−t) , w 3 = 1 + 2ie − 2π β (d+t) , w 4 = 1 − 2ie − 2π β (d−t) . (4.20) We expand the correlation function in this limit, taking the OPEs of pairs of operators on each sheet Z n,m = 2e − 2π β (d−t) −4h k 1 + 2 T ∈SĈ X k X k TĈσ kσk T (−1) l T 2 2e − 2π β (d−t) ∆ T + · · · , (4.21) where S is the set of the lightest non-vacuum fields. We denote the scaling dimension by ∆ and the spin by l. Any field in the untwisted sector of the orbifold CFT can be described as T := 1 k (T 0 ⊗ · · · T k−1 + (cyclic permutations)) . (4.22) For these untwisted states, we define the OPE coefficients bŷ C σ kσk T := σ k (0)T (1)σ k (∞) C ⊗k /Z k ,(4.23) and C X k X k T := n−1 k=0 m−1 i=0 C ppT i+(2m+1)k C ppT i+m+(2m+1)k C qqT m+(2m+1)k + (cyclic permutations). (4.24) The lightest non-vacuum contributions have the following form, Ψ i = 1 k ψ r ⊗ I ⊗i ⊗ ψ r ⊗ I ⊗k−i−1 + (cyclic permutations) , i ≤ k 2 . (4.25) For these states, the OPE coefficients may be evaluated as T ∈SĈ X k X k TĈσ kσk T (−1) l T 2 = k/2 i=1 (2m+1) i n(2m − 1)C 2 ppr + 2nC ppr C qqr C σ kσk Ψ i r (−1) lr + k/2 i=1 (2m+1)|i 2nmC 2 ppr + nC 2 qqr C σ kσk Ψ i r (−1) lr = k/2 i=1 n(2m − 1)C 2 ppr + 2nC ppr C qqr C σ kσk Ψ i r (−1) lr + k/2 i=1 (2m+1)|i n (C ppr − C qqr ) 2 C σ kσk Ψ i r (−1) lr (4.26) We first take the m → 1/2 limit 2nC ppr C qqr 2n−1 i=1 1 4n sin πi 2n 2∆r + n 2 (C ppr − C qqr ) 2 n−1 i=1 1 4n sin πi n 2∆r (4.27) where we use the relation [30], 6 (−1) lr k/2 i=1 C σ kσk Ψ i r = 1 2 k−1 i=1 1 2k sin πi k 2∆r := 1 2 f (k, ∆ r ). (4.28) We then take the n → 1/2 limit 1 2 2∆r+2 (C ppr − C qqr ) 2 f 1 2 , ∆ r . (4.29) We conjecture using numerical calculations 7 that the analytic continuation n → 1 2 is given by 8 f 1 2 , h = − Γ 1 2 + h 2 √ πΓ (1 + h) < 0. (4.30) 6 In [30], there is an additional factor n, which comes from the cyclic permutation. We do not need this factor here because we already included all terms of the cyclic permutation in the summand of (4.26). 7 In practice, this means evaluating f (n, h) for fixed integer h and general n, identifying (using Mathematica) an analytic function of n that reproduces the sequence at fixed h, analytically continuing to n = 1/2, then identifying an analytic function that reproduces the sequence in h. 8 We would like to thank Nathan Benjamin for informing us of this conjectured expression. This is consistent with all explicitly summable examples, such as h = 0, 1. Crucially, this function is everywhere negative. For consistency, this had to be the case because the fidelity is bounded above by one. Finally, we obtain F (ρ p A , ρ q A ) = 1 − (C ppr − C qqr ) 2 Γ 1 2 + ∆ r 4 √ πΓ (1 + ∆ r ) e −2∆r(S BH −S rad ) . (4.31) For descendants of the vacuum, the OPE coefficients for the two primary fields are identical C ppr = C qqr ,(4.32) where we use the assumption h p = h q . Therefore, the first nontrivial contribution is given by the first non-vacuum primary state. After the Page time After Page time (t > d), the insertion points can be approximated as w 1 = −1 − 2ie − 2π β (d+t) , w 2 = 1 + 2ie − 2π β (t−d) , w 3 = 1 + 2ie − 2π β (d+t) , w 4 = −1 − 2ie − 2π β (t−d) . (4.33) In this case, the high-temperature limit corresponds to operators on neighboring replica sheets to become close, so we wind up with a different OPE limit. The correlation function is correspondingly modified to X k (i)σ k (z 1 )σ k (z 2 )σ k (z 3 )σ k (z 4 )X k (−i) C ⊗k /Z k , X k := φ ⊗(m+1) p ⊗ φ q ⊗ φ ⊗(m−1) p ⊗n . (4.34) We can expand the correlation function in the high-temperature limit as 2e − 2π β (t−d) −4h k 1 + 2 T ∈SĈ X k X k TĈσ kσk T (−1) l T 2 2e − 2π β (t−d) ∆ T + · · · ,(4.35) where S is a set of the first non-vacuum states. The OPE coefficient is given bŷ C X k X k T := n−1 k=0 m−1 i=0 C ppT i+(2m+1)k C ppT i+m+(2m+1)k C pqT m+(2m+1)k C pqT m+1+(2m+1)k + (cyclic pernutations). (4.36) The first non-vacuum contribution is again given by (4.25). Consequently, we obtain T ∈SĈ X k X k TĈσkσkT (−1) l T 2 = k/2 i=1 (2m+1) (i+l) l=−1,0,1 n(2m − 3)C 2 ppr + 4nC ppr C pqr C σ kσk Ψ i r (−1) lr + k/2 i=1 (2m+1)|i n(2m − 1)C 2 ppr + 2nC 2 pqr C σ kσk Ψ i r (−1) lr + 2 k/2 i=1 (2m+1)|(i+1) n(2m − 2)C 2 ppr + 2nC ppr C pqr + nC 2 pqr C σ kσk Ψ i r (−1) lr = k/2 i=1 n(2m − 3)C 2 ppr + 4nC ppr C pqr C σ kσk Ψ i r (−1) lr + k/2 i=1 (2m+1)|i 2n (C ppr − C pqr ) 2 C σ kσk Ψ i r (−1) lr + k/2 i=1 (2m+1)|(i+1) 2n (C ppr − C pqr ) 2 C σ kσk Ψ i r (−1) lr (4.37) In the m → 1/2 limit, this equals −2nC 2 ppr + 4nC ppr C pqr which disappears when taking n → 1/2, so the fidelity vanishes. We find this to be a somewhat miraculous cancellation, given the complexity of the replica calculation. Of course, we know that the fidelity cannot be exactly zero due to our calculation of the GM fidelity. However, this calculation shows that the non-zero contributions must be more subleading than O e −2∆r(S rad −S BH ) . We have found that before the Page time, the fidelity is close to one (4.31). Using (2.4), we see that ones needs an O(e #(S BH −S rad )) number of copies of the state of the radiation prior to the Page time in order to distinguish microstates. This is qualitatively the same conclusion as in [12] in the simpler PSSY model. Moreover, the lighter the operators in the spectrum of the given CFT, the more easily the states can be distinguished. The light bulk fields appear to mediate the transfer of information from inside the black hole into the radiation system. This is a new feature which has no analog in [12]. After the Page time, the fidelity is close to zero F (ρ p A , ρ q A ) = o e −2∆r(S rad −S BH ) . (4.39) Therefore, a single judiciously chosen measurement of the radiation distinguishes the two different states of the black hole. While we have demonstrated that these measurements are effective, we have not shown that they are feasible in practice. Indeed one may expect that they are exponentially complex. While a disjoint concept from the unitarity of black hole evaporation, it would be interesting to understand if and when simpler measurements of the radiation may distinguish black holes. Fidelity between primary states We conclude with a further application of the techniques we developed for computing the fidelity in general CFTs by evaluating the fidelity between distinct primary states in CFT. In the case where the two primary operators, φ p and φ q , have similar and large dimensions, this probes eigenstate thermalization. If a given system exhibits eigenstate thermalization, the matrix elements of (few-body) observables, O, in the energy eigenstate basis behave as [31] p| and the functions f O (E) and g O (E p , E q ) are smooth and O(1). The matrix R pq is a pseudo-random variable with zero mean and unit variance. We recall that if the fidelity between two state in a subregion is close to one, the trace distance must be small [32] and then so are properly normalized matrix elements [33,34]. This "subsystem eigenstate thermalization" thus implies (5.1). O |q = f O (E)δ pq + e − S(E) 2 g O (E p , E q )R pq ,(5. We consider a primary state on a circle of circumference 2π reduced to a subsystem A of size 2πx (see Figure 3) ρ (p) A := trĀ |p p| , ρ (q) A = trĀ |q q| . (5.2) In the short interval (x 1) limit, the calculation is essentially the same as that in Section 4.2 such that F (ρ (p) A , ρ (q) A ) = 1 − (C ppr − C qqr ) 2 Γ 1 2 + ∆ r 8 √ πΓ (1 + ∆ r ) (sin πx) 2∆r ,(5.3) where, as before, the subscript r represents the lightest primary field in the OPE of φ p × φ p and φ q × φ q . This answer holds for any CFT. We compare now with the one example in the literature that has been explicitly computed, the c = 1 free boson [26]. The fidelity between the vacuum state and the vertex state corresponding to the operator V = e iαφ , where φ is the boson field, is given by [26] F (ρ In the short interval limit, this is approximated by F (ρ (0) A , ρ (V ) A ) 1 − παx 4 2 . (5.5) The first non-vacuum state is given by the U (1) current i∂φ(z) of conformal weight (1, 0), so using the fact that C V V i∂φ(z) = −α, we find complete consistency with our general answer (5.3). Of course, there is no exponential suppression in this fidelity, which may be understood both because the free boson theory is integrable and that here we have only considered lowlying states. In contrast, for high-energy states and irrational (such as holographic) CFTs, the OPE coefficients may be expected to lead to exponential suppression [35][36][37]. where ω 1 + ω 2 + ω 3 = 0. The lattice roots can be expressed in terms of the Jacobi theta function θ i or the elliptic integral of the first kind K as e 1 = π 2 12ω 2 1 θ 2 (τ ) 4 + 2θ 4 (τ ) 4 = K(x) 2 3ω 2 1 (2 − x), e 2 = π 2 12ω 2 1 θ 2 (τ ) 4 − θ 4 (τ ) 4 = K(x) 2 3ω 2 1 (2x − 1), e 3 = − π 2 12ω 2 1 2θ 2 (τ ) 4 + θ 4 (τ ) 4 = − K(x) 2 3ω 2 1 (1 + x), (A.3) where the moduli parameter is defined by τ ≡ ω 3 ω 1 . Here we take the following convention, K(x) = π 2 0 dθ 1 − x sin 2 θ = 2 F 1 1 2 , 1 2 , 1; x . (A.4) For this convention, the relation between the cross ratio and the moduli parameter is x = e 2 − e 3 e 1 − e 3 = θ 2 (τ ) θ 3 (τ ) 4 , τ = i K(1 − x) K(x) . (A.5) One can see using (A.2) that the edges of the intervals in Σ 2,2 come from the generators, (ω 1 , ω 2 , ω 3 , 0) T 2 → (1, x, 0, ∞) Σ 2,2 . (A.6) The Weierstrass elliptic function is related to the Jacobi elliptic function as ℘(t + ω 3 ) − e 3 = K(x) ω 1 sn K(x)t/ω 1 , √ x 2 x. (A.7) Using this relation, we can give the conformal map g(z, x) from Σ 2,2 to T 2 as t = ω 3 + ω 1 K(x) sn −1 z x , √ x = ω 3 + ω 1 K(x) √ z x 0 dη (1 − η 2 )(1 − xη 2 ) (≡ ω 3 + g(z, x)). (A.8) Now we can evaluate the geometric-mean fidelity as a correlation function on a torus. We assume that the CFT is holographic and the operators φ p , φ q are light enough to not backreact on the gravity i.e. 1 h p , h q c. More precisely, we assume that the conformal dimension of φ p , φ q behaves like c with 1 in the large c limit. Then, we can evaluate the correlation function by using Wick's theorem. As a result, we obtain that before Page time (corresponding the Hawking-Page transition of the torus partition function) I(ρ p A , ρ q A ) |sin (π (g(y, x) − g(y * , x)))| 2hp+2hq × |sin (π (g(y, x) − g(y * , x)))| 2hp+2hq + |sin (π (g(y, x) + g(y * , x)))| 2hp+2hq −1 , (A.9) and after Page time I(ρ p A , ρ q A ) sin π 2τ (g(y, x) − g(y * , x)) 2hp+2hq × sin π 2τ (g(y, x) − g(y * , x)) 2hp+2hq + sin π 2τ (g(y, x) + g(y * , x)) 2hp+2hq −1 , (A.10) where we have set 2ω 1 = 1. The super-fidelity is related to the GM fidelity as Figure Figure 1. Left: the Euclidean path integral that prepares our state of interest. There are two copies of a boundary conformal field theory. The red, semi-infinite subregions account for the radiation while the black boundary together with cutoff region accounts for the black hole. An operator shown in green is inserted on the boundary. Right: the Lorentzian continuation of this state with the boundary degrees of freedom holographically represented as a 2D eternal black hole. The excitation intersects the bifurcation point and remains behind the horizon. After the Page time, an island (blue) forms that includes the bulk excitation. Figure 2 . 2The GM fidelity in the PSSY (left) and BCFT (right) models. n (C ppr − C pqr ) Figure 3 . 3The configuration considered for fidelity between primary states. The circumference of the system is 2π and we consider the reduced state on A = [0, 2πx]. F S 1. Left: the Euclidean path integral that prepares our state of interest. There are two copies of a boundary conformal field theory. The red, semi-infinite subregions account for the radiation while the black boundary together with cutoff region accounts for the black hole. An operator shown in green is inserted on the boundary. Right: the Lorentzian continuation of this state with the boundary degrees of freedom holographically represented as a 2D eternal black hole. The excitation intersects the bifurcation point and remains behind the horizon. After the Page time, an island (blue) forms that includes the bulk excitation. 1 ) 1where S(E) is the microcanonical entropy at E =Ep+Eq 2 The term "bath" is a bit of a misnomer as it suggests Markovian dynamics which are not unitary by construction. We stick with tradition in calling this non-Markovian system a bath. Such an optimized measurement can be explicitly constructed[19]. We ignore higher genus solutions because these are exponentially suppressed in the large ground state entropy S0. A Torus four-point functionTo evaluate a correlation function (4.11) on the replica manifold Σ 2,2 , it is useful to find a conformal transformation from a torus T 2 to Σ 2,2 . We focus on a particular map from T 2 with generators of the lattice, (2ω 1 , 2ω 3 ) to Σ 2,2 with two branch cuts A = [0, x] ∪ [1, ∞]. This conformal map can be expressed in terms of the Weierstrass elliptic function ℘ as (see e.g.[38])where z, t describe the coordinate of Σ 2,2 and T 2 . Here the constants e i (called the lattice roots) are given by Particle creation by black holes. S W Hawking, World Scientificin Euclidean quantum gravityS.W. Hawking, Particle creation by black holes, in Euclidean quantum gravity, pp. 167-188, World Scientific (1975). Breakdown of predictability in gravitational collapse. S W Hawking, Physical Review D. 142460S.W. Hawking, Breakdown of predictability in gravitational collapse, Physical Review D 14 (1976) 2460. On particle creation by black holes. R M Wald, Communications in Mathematical Physics. 459R.M. Wald, On particle creation by black holes, Communications in Mathematical Physics 45 (1975) 9. 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[ "Optimal dynamic insurance contracts * †", "Optimal dynamic insurance contracts * †" ]
[ "Vitor Farinha Luz [email protected] \nUniversity of British Columbia\n\n", "Brian Baisa \nUniversity of British Columbia\n\n", "Eduardo Souza Rodrigues \nUniversity of British Columbia\n\n", "Jaqueline Oliveira \nUniversity of British Columbia\n\n", "Bruno Badia \nUniversity of British Columbia\n\n", "Sofia Moroni \nUniversity of British Columbia\n\n", "Marina Carvalho \nUniversity of British Columbia\n\n", "Marcelo Sant&apos;anna \nUniversity of British Columbia\n\n", "Sabyasachi Das \nUniversity of British Columbia\n\n", "Eduardo Faingold \nUniversity of British Columbia\n\n", "Li Hao \nUniversity of British Columbia\n\n", "Mike Peters \nUniversity of British Columbia\n\n", "Alex \nUniversity of British Columbia\n\n" ]
[ "University of British Columbia\n", "University of British Columbia\n", "University of British Columbia\n", "University of British Columbia\n", "University of British Columbia\n", "University of British Columbia\n", "University of British Columbia\n", "University of British Columbia\n", "University of British Columbia\n", "University of British Columbia\n", "University of British Columbia\n", "University of British Columbia\n", "University of British Columbia\n" ]
[]
I analyze long-term contracting in insurance markets with asymmetric information.The buyer privately observes her risk type, which evolves stochastically over time. A long-term contract specifies a menu of insurance policies, contingent on the history of type reports and contractable accident information. The optimal contract offers the consumer in each period a choice between a perpetual complete coverage policy with fixed premium and a risky continuation contract in which current period's accidents may affect not only within-period consumption (partial coverage) but also future policies.The model allows for arbitrary restrictions to the extent to which firms can use accident information in pricing. In the absence of pricing restrictions, accidents as well as choices of partial coverage are used in the efficient provision of incentives. If firms are unable to use accident history, longer periods of partial coverage choices are rewarded, leading to menus with cheaper full-coverage options and more attractive partial-coverage options; and allocative inefficiency decreases along all histories.These results are used to study a model of perfect competition, where the equilibrium is unique whenever it exists, as well as the monopoly problem, where necessary and sufficient conditions for the presence of information rents are given. * This paper supersedes an earlier paper entitled "Dynamic Competitive Insurance". † I would like to thank Dirk Bergemann, Johannes Hörner and Larry Samuelson for encouragement and feedback on this project. I would also like to thank
null
[ "https://export.arxiv.org/pdf/2208.14560v1.pdf" ]
251,953,214
2208.14560
37e6fe293c0aba051cd20962b1e0b6430dc11d73
Optimal dynamic insurance contracts * † September 1, 2022 Vitor Farinha Luz [email protected] University of British Columbia Brian Baisa University of British Columbia Eduardo Souza Rodrigues University of British Columbia Jaqueline Oliveira University of British Columbia Bruno Badia University of British Columbia Sofia Moroni University of British Columbia Marina Carvalho University of British Columbia Marcelo Sant&apos;anna University of British Columbia Sabyasachi Das University of British Columbia Eduardo Faingold University of British Columbia Li Hao University of British Columbia Mike Peters University of British Columbia Alex University of British Columbia Optimal dynamic insurance contracts * † September 1, 2022Frankel and the seminar participants at the Yale Micro lunch and Summer workshop, 2022 EUI Alumni Conference, NYU, Olin Business School, University of British Columbia, Columbia Business School, Notre Dame, Boston University, Warwick, EPGE/FGV, EESP/FGV and the Catholic University of Rio de Janeiro for helpful comments. I analyze long-term contracting in insurance markets with asymmetric information.The buyer privately observes her risk type, which evolves stochastically over time. A long-term contract specifies a menu of insurance policies, contingent on the history of type reports and contractable accident information. The optimal contract offers the consumer in each period a choice between a perpetual complete coverage policy with fixed premium and a risky continuation contract in which current period's accidents may affect not only within-period consumption (partial coverage) but also future policies.The model allows for arbitrary restrictions to the extent to which firms can use accident information in pricing. In the absence of pricing restrictions, accidents as well as choices of partial coverage are used in the efficient provision of incentives. If firms are unable to use accident history, longer periods of partial coverage choices are rewarded, leading to menus with cheaper full-coverage options and more attractive partial-coverage options; and allocative inefficiency decreases along all histories.These results are used to study a model of perfect competition, where the equilibrium is unique whenever it exists, as well as the monopoly problem, where necessary and sufficient conditions for the presence of information rents are given. * This paper supersedes an earlier paper entitled "Dynamic Competitive Insurance". † I would like to thank Dirk Bergemann, Johannes Hörner and Larry Samuelson for encouragement and feedback on this project. I would also like to thank Introduction A vast majority of insurance contracts cover risks that are present over many periods, such as auto, auto or home insurance. This allows insurance firms to benefit from the use of dynamic pricing schemes, where consumers premium and coverage evolve over time and incorporate any information observed over the course of their interaction. The dynamics of coverage and premium are a central issue in several insurance markets including auto insurance (Dionne and Doherty (1994), Cohen (2005)), health insurance (Handel et al. (2015), Atal (2019), Atal et al. (2020)) and life insurance (Hendel and Lizzeri (2003), Daily et al. (2008)). This observation raises the fundamental theoretical question of what dynamic pricing schemes arise as a result of profit maximization and what are their consequences for coverage and premium dynamics. The theoretical insurance literature has focused on scenarios with either symmetric risk information 1 or permanent risk types. 2 In this paper, I characterize profit maximizing long-term contracts in repeated interactions with evolving, persistent and private risk information, and apply this characterization to study equilibrium outcomes under perfect competition and monopoly environments. In my model, a risk-averse consumer (she) may incur incidental losses in each period, such as a car accident (auto insurance) or expenditures from medical procedures (health insurance); and the probability distribution over losses is determined by her risk-type, which is privately known by the consumer and follows a persistent Markov process. A high (low) type has a higher (lower) expected expected income, net of losses, in each period. I assume that an insurance firm offers a long-term contract to the consumer after she has observed her initial risk-type. A long-term contract represents a commitment to a schedule, in which a menu of insurance policies (referred to as flow contracts) with different premiums and coverage levels is offered to the consumer in each period. Crucially, the menu offered on a given period may depend on previous choices made by the consumer as well as any available information regarding accident history. Motivated by common regulatory policies that limit the use of accident history in pricing (Handel et al. (2015), Farinha Luz et al. (2021)), I allow for restrictions on the amount of information about past accidents that can be explicitly used by firms in setting insurance policy offers to be made to the consumer. These restrictions include as special cases both fully contingent contracts, where the whole history of accidents can be used, and realization-independent contracts, where the firm cannot use any past accident information in pricing. The first part of the paper (Sections 3-7) considers fixed discounted utility levels for the consumer, dependent on her initial risk-type, and characterizes the profit maximizing longterm contract that delivers these utility levels. This allows for the derivation of qualitative properties of optimal contracts that hold in both the competitive (Section 8) and monopolistic (Section 9) settings studied in the second part of the paper, where each market setting corresponds to different equilibrium utility levels for the consumer. I show that the optimal contract features a simple pricing scheme: in every period the consumer chooses between a complete coverage insurance policy with a perpetual fixed premium (efficient in this environment), and a partial coverage policy, in which case future offers will depend on additional accident realizations. The consumer is induced to choose the full coverage option when having a low-type realization, and to choose partial coverage when having successive high-type realizations. In each period, an insurance firm has access to two sources of information: accident information -which is directly observed by the firm -and the consumer's choice (or announcement in a direct mechanism) which provides endogenous information about their type. In the optimal mechanism, policy offers use both pieces of information to efficiently screen different risk types. Accident information that is indicative of high-types is rewarded with higher continuation utility, in the form of more attractive future contracts. In a similar fashion, the choice of partial coverage in a given period is indicative of a high-type and is rewarded with higher continuation utility. The characterization of distortion and coverage dynamics is challenging due to two technical challenges. First, flow contracts are multi-dimensional objects, describing a premium and coverage for each possible loss level. Second, the presence of risk aversion leads to a non-separability issue absent in quasi-linear environments. The separation of consumers with different types requires the introduction of distortions, in the form of partial coverage. The marginal efficiency loss from the introduction of distortions potentially depends on the underlying utility level obtained by the consumers in a given period. As a consequence, the optimal contract requires a joint consideration of the issues of (i) intertemporal allocation of utility to be provided ot the consumer as well as (ii) the efficient spreading of distortions over time. I tackle both of these issues by introducing an auxiliary static cost minimization problem in Section 6, which finds the optimal flow contract that (i) delivers a certain utility level if consumed by a high-type, and (ii) provides a fixed punishment if chosen by a low-type pretending to be high-type. In Section 7 two optimality conditions are derived which charac-terize both the efficient spreading of utility and distortions over time, in terms of the solution to this auxiliary cost minimization problem. Under a mild condition on the consumer's Bernoulli utility function, 3 the auxiliary cost function is supermodular, i.e., the marginal cost of distortions in a flow contract increases in the utility level generated by it. Under cost-supermodularity, the intertemporal optimality conditions can be used to obtain sharp characterization of the dynamics of utility and distortions. For the case of realization-independent contracts, we show that distortions -measured in terms of the consumer's exposure to risk -are strictly decreasing over time. As a consequence, the optimal mechanism has decreasing distortions along all paths. When it comes to the flow utility dynamics, we show that consecutive high-type announcementswhich are revealed by the choice of partial coverage -are rewarded by leading to the offer of partial contract offers that generate higher flow utility, as well as lower distortion. For the case of fully contingent contracts, distortions and utility depend on the history of income realizations and hence are stochastic, even conditioning on types. I present results for two special cases. First, I focus on the case of two periods and show that the firm has an incentive to reward high-type announcements and reduce distortions. I recast the firm's contract design problem as a cost minimization one and show that, when averaging over possible first-period income realizations, the expected marginal cost of distortions in flow contracts decrease over time, while the expected marginal cost of flow utility increases over time following the choice of partial coverage. Second, we focus on the illuminating knife-edge case of constant relative risk aversion utility u(c) = √ c, which corresponds to a separable auxiliary cost function, and an arbitrary number of periods. Consecutive choices of partial coverage -which corresponds to high-type announcements -lead to a path of insurance policies that depend on the history of income realizations. I show that these policies have lower distortions and higher flow utility levels over time, when averaging over possible income realizations. In Section 8, we consider a competitive model in which multiple firms make long-term contract offers to a consumer who is able to commit to a long-term contract. I extend the characterization of Rothschild and Stiglitz (1976) to our environment and show that a unique outcome, featuring the flow utility and distortion dynamics aforementioned, can be sustained by a pure strategy equilibrium, and obtain necessary and sufficient conditions for such an equilibrium to exist. The assumption of consumer commitment is reasonable in markets with high search or switching costs which preclude or inhibit the consumer's consideration of or transition to new firms. In the absence of such frictions, the commitment assumption is potentially with loss. If the low-type is an absorbing state of the types' Markov process (such as a chronic condition in health insurance), I propose a simple competitive model with consumer reentry in the market and show that firms' endogenous beliefs about the type of consumer searching for a new contract discourages consumer firm switching and, as a consequence, the commitment outcome can be sustained in a Perfect Bayesian Equilibrium of the nocommitment model. This follows from the fact that, the optimal contract with commitment is back-loaded: both the partial-and the full-coverage policy options within the optimal menu become more attractive over time. Section 9 studies the case of monopoly, where the consumer's (type-dependent) outside option is given by zero coverage. The optimal contract is always separating and hence features the flow utility and distortion dynamics aforementioned. The consumer with initial high-type has no information rent, having total utility equal to that in the absence of insurance. I present a condition that is necessary and sufficient for the optimal contract to leave information rents to the consumer with initial low-type, meaning that her utility is strictly higher than the no-insurance option. Related literature This paper contributes to the literature on competitive screening, initiated by the seminal contributions of Rothschild and Stiglitz (1976) and Wilson (1977), which studies situations in which private information leads to inefficiencies in competitive markets. Rothschild and Stiglitz (1976) considers insurance markets in which customers have private information regarding their risk characteristics. They show that competition leads to a unique equilibrium in which high-type consumers, who have lower accident probabilities, are screened by the choice of partial insurance at better premium rates (per unit of coverage). Cooper and Hayes (1987) extends the analysis of Rothschild and Stiglitz (1976) to a multi-period setting in which consumers have fixed risk types and full commitment. They find that optimal long-term contracts use experience rating as an efficient sorting device in addition to partial coverage. The optimal contract features a single type announcement in the first period and customers make no further choices. Dionne and Doherty (1994) allows for renegotiation and competition in the same fixed-types model and find equilibria with semi-pooling. My analysis is also related to the dynamic mechanism design literature (see Courty and Li (2000), Battaglini (2005), Eső and Szentes (2007), Pavan et al. (2014)), which differs from the insurance model considered here in two relevant ways. First, two sources of information exist in each period: the consumer's coverage choice (or type announcement) as well as the accident information that can be directly used in future offers. This is in contrast with standard screening models in which the only source of information to the mechanism designer over the course of an interaction is the series of choices or announcements made by the consumer. Second, the presence of curvature in preferences implies that the premium dynamics are completely pinned down in the optimal contract. This is in contrast with quasi-linear environments considered in the literature, where transfers are only pinned down up to their total ex-post present value. 4 A notable exception on both fronts is Garrett and Pavan (2015), which studies the optimal compensation scheme for a manager who is privately informed about his persistent productivity type in two periods. Production in each period is observable and corresponds to the sum of productivity and unobservable effort. Instead of studying a relaxed problem, their approach is to construct two sets of perturbations which retain implementability in the original mechanism design problem and exploit these perturbations to characterize the dynamics of distortions, which correspond to under-provision of effort, in the optimal mechanism. In their model, even the implementation of constant efforts (which only depend on first period productivity) requires the introduction of additional consumption variation in the second period, which is costly for a risk averse manager. This is a crucial difference to my framework, where distortions correspond to consumption variability within a period. As a consequence, they show that, if the manager is risk neutral and types are sufficiently persistent, distortions in the optimal mechanism increase over time on average. The reverse is true if the manager is sufficiently close to risk neutrality. Another closely related paper is Battaglini (2005), which considers the design of dynamic selling mechanisms in a monopoly setting where customer's valuation follows a two-state Markov chain. In the optimal mechanism, production becomes efficient when the customer obtains his first high valuation and converges to the efficient level along the history path of consecutive low valuations, which is analogous to our characterization of distortions for realization-independent contracts (Proposition 3). Even though the productive allocation is fully characterized, the pricing scheme is not unique. Hendel and Lizzeri (2003) study long-term life insurance contracts with symmetric information and one-side commitment. The optimal contract features front-loaded payments, which remedies the consumer's inability to commit by locking in consumers who expect to pay lower premiums later in the relationship. Using data from the U.S. life insurance market, they show that front-loading is a common feature of contracts, with more front-loading being negatively correlated with net present value of premiums. Ghili et al. (2021) study the same contract design problem in health insurance markets. They characterize the optimal dynamic contract assuming symmetric risk information and one-sided commitment, which features full insurance in each period. The presence of onesided commitment precludes complete consumption smoothing over time, hence the consumer is offered a consumption floor, which is adjusted upwards whenever the consumer's outside option is attractive enough so that the participation constraint becomes binding. Finally, they estimate a stochastic Markovian model of health status and medical expenses using data from the state of Utah and numerically find the optimal long-term contract for the estimated model parameters. The analysis of commitment in Subsection 8.1 shows that the presence of adverse selection consumers may serve as a lock-in device, allowing for implementation of the commitment solution. The argument relies on firms making a negative inference about consumer's risk level when observing a switching consumer. The presence of such pessimistic beliefs is in line with the findings in Cohen (2005) who shows, in the context of Israeli auto insurance, that consumers switching insurance companies have disproportionally bad accident histories and are high risk. In a related paper, de Garidel-Thoron (2005) studies a two-period model with ex-ante symmetric information and learning, and shows that keeping firms from observing the accident history of switching consumers serves as a commitment device and leads to welfare improvements. Model Types A consumer (she/her) lives for T ≤ ∞ periods. At the beginning of each period, she privately observes her type θ t ∈ Θ ≡ {l, h}, which determines a probability distribution over realized income y t ∈ Y , with Y ⊂ R + finite. The occurrence of higher losses or damages is represented by a lower level of final income. 5 I refer to type h as the high-type. In each period t = 1, . . . , T , the probability distribution of income level y t depends on type θ t and is represented by p i ∈ ∆Y , for i = l, h, satisfying y∈Y p l (y) y < y∈Y p h (y) y. I assume types {θ t } T t=1 follow Markov process. The time-invariant transition probabilities are denoted as π ij ≡ P (θ t+1 = j | θ t = i) , while the distribution of initial type θ 1 is denoted by π i ≡ P(θ 1 = i). Types are persistent, i.e., having a given type in period t < T leads to a higher probability of having the same type in period t + 1. In short, we assume that π ii > π ji , for i, j ∈ {l, h}. Note that π ll = π hh = 1 is included as a special case. 6 Preferences Consumer preferences over final consumption flows are determined by Bernoulli utility function u : R + → R,, assumed to be twice continuously differentiable, strictly concave and strictly increasing. 7 The inverse of utility function u (·) is denoted as ψ (·). The consumer discounts the future according to factor δ ∈ (0, 1). The utility obtained from deterministic consumption stream {c t } T t=1 is given by T t=1 δ t−1 u (c t ) . I study the profit maximization of a risk-neutral firm with the same discount factor. The payoff (or profit) obtained by a firm is determined by its net payments made to the consumer, if its contract is accepted, and zero otherwise. Hence the payoff obtained by a firm, for a given realization of the income and consumption paths {y t , c t } T t=1 is given by T t=1 δ t−1 (y t − c t ) . 5 If the customer has fixed per period flow income y 0 > 0 and losses l ∈ L, her realized income is y = y 0 − l. 6 The time-invariance assumptions made here can be significantly relaxed, as discussed in Section 10. 7 This formulation rules out the relevant case of logarithmic utility, as its domain excludes zero. For finite T , the results extend to preferences satisfying lim c↓0 u (c) = −∞. Both the firm's and consumer's preferences are extended to random outcomes by using expected payoffs. Flow contracts A flow contract is a standard single period insurance policy: it specifies a premium and coverage for all possible income realizations. I assume that the income realization is observable and contractable and hence the fulfillment of a flow contract does not involve incentive considerations. For tractability, a flow contract is described here through the induced final consumption of the consumer, i.e., the set of flow contracts is given by Z ≡ R Y + . Each contract z ∈ Z specifies that the final consumption of the consumer if income y ∈ Y is realized, which is equal to the income y plus any policy coverage minus the premium paid, is equal to z (y). If a flow contract z is provided to a consumer with type θ, the consumer's utility from flow contracts can be described by function v (z, θ) ≡ y∈Y p θ (y) u [z (y)] , while the profits obtained by a firm can be described by ξ (z, θ) ≡ y∈Y p θ (y) [y − z (y)] . Long-term contracts A long-term contract is a mechanism which specifies at the initial period t = 1 the flow contracts to be offered to the consumer at each period, which may depend on messages sent by the consumer and, potentially, information about the history of past income realizations. In several insurance markets, regulatory restrictions limit the extent to which firms can explicitly use the history of accidents or losses in pricing contracts (see Handel et al. (2015), for example). I allow for such restrictions by assuming that firms are only able to use a coarsening of the history of past income realizations. These restrictions are represented by a signal structure, which is composed of a finite set of signals Φ and a surjective signal function φ : Y → Φ. The signal structure is exogenously fixed. It imposes restrictions on the firm's contract design problem, as the income realization y t can only impact future offers via φ t ≡ φ(y t ). One special case is that of fully-contingent mechanisms, with Φ = Y and φ (y) = y for all y ∈ Y . In this case, prices depend explicitly on both the history of reports by the consumer as well as the observed history of accidents, both of which are informative to firms. Another extreme case of interest is that of realization independent mechanisms, which corresponds to Φ being a singleton. In this case, firms are completely unable to use the history of previous income realizations in pricing. While the results in Subsections 7.1 and 8.1 focus on the case of realization-independent mechanisms, all other results apply for an arbitrary signal structure. From the revelation principle, we can restrict attention to direct mechanisms, where the set of possible messages in each period coincides with the set of types Θ, and truthful equilibria, in which the consumer finds it optimal to truthfully report her type. I denote the history of announcements and signals up to period t as the observable history η t ∈ H t ≡ Φ t × Θ t . 8 I denote a history of types (θ 1 , . . . , θ t ) up to period t as θ t , the expanded history (θ 1 , . . . , θ t , θ t+1 ) as (θ t , θ t+1 ), the sub-history (θ τ , . . . , θ τ ), for τ ≤ τ ≤ t, as [θ t ] τ τ , and, for a history θ t and τ ≤ t, refer to θ τ as [θ t ] τ . Also define h t as the t-period history (h, . . . , h). The same notation is used for income realizations y t , signal realizations φ t and histories η t . A direct mechanism is defined as M = {z t } T t=1 , with z t : H t−1 × Θ → Z, and the set of direct mechanisms is M. A direct mechanism M = {z t } T t=1 specifies the flow contract z t η t ,θ t to be provided to the consumer at each period t, which depends on the history of signals and announcements up to period t − 1, as well as the new reportθ t made at period t. The flow contract at period t determines the level of coverage obtained by the consumer within period t, and hence her realized consumption in period t, which is given by z t y t | η t ,θ t and also depends on the realized income y t . Figure 1 illustrates the sequence of events within a direct mechanism for a particular period. I assume that flow contracts with arbitrary income-contingent transfers can be executed without frictions. In the example of health insurance, this means that can specify and enforce transfers/coverage that depends directly on health shocks within a given period, but may -depending on φ -not be able to use this information explicitly when determining future offers to be made to the consumer. A private history in period t includes all information available to the consumer, and is 8 For completeness, we define H 0 ≡ {∅}. Figure 1: Timing of events of a particular period t within a direct mechanism described as η t p = y t ,θ t , θ t ∈ H t p , which includes the history of income realizations y t , type realizations θ t and reported typesθ t . The set of private histories in period t is denoted as H t p . A reporting strategy is denoted by r = {r t } T t=1 with r t : H t−1 p × Θ → Θ. The truth-telling strategy is denoted by r * and satisfies r * t η t−1 p , θ = θ, for all η t−1 p ∈ H t−1 p and θ ∈ Θ. The history of reports up to period t generated on-path by strategy r -which depends solely on realized history types θ t and income levels y t−1 -is denoted as r t (y t−1 , θ t ). The set of reporting strategies is denoted as R. The consumer's payoff from a reporting strategy r ∈ R is denoted as V r 0 (M ) ≡ E δ t−1 v z t φ t−1 , r t y t−1 , θ t , θ t . Definition 1. A direct mechanism M = {z t } T t=1 is incentive compatible if, for all r ∈ R, V r * 0 (M ) ≥ V r 0 (M ) . The set of incentive compatible mechanisms is denoted as M IC . Finally, for an observable history η t−1 = (θ t−1 , y t−1 ) ∈ H t−1 and period t type θ t ∈ Θ, we denote the continuation utility of a consumer following reporting strategy r as V r t (M | η t−1 , θ t ), or simply as V t (M | η t−1 , θ t ) for reporting strategy r * . When the mechanism M is clear -such as the profit maximizing one -we may simply use V t (η t−1 , θ t ) for brevity. Profit maximization I assume throughout that the consumer privately learns θ 1 prior to contracting with the firm. An insurance firm's mechanism design problem can be separated into two parts. First, a firm must decide on how attractive its offer is for different types of potential customers, which is described by the total expected utility from participation (utility choice). Second, firms must choose the mechanism details in order to maximize profits within all mechanisms that deliver the same utility to the consumer (feature design). I start by focusing on the feature design problem and provide a characterization of profit maximizing mechanisms, for fixed discounted expected utility to be provided to the consumer with each initial type. The study of the feature design problem allows for the characterization of qualitative properties of optimal mechanisms that hold under multiple market structures which lead to different equilibrium utility levels for the consumer. Sections 8 and 9 analyze the cases of perfect competition and monopoly, respectively. The total discounted expected profits from a direct mechanism M ∈ M is given by Π (M ) ≡ E T t=1 δ t−1 ξ z t η t−1 , θ t , θ t .(1) I am interested in studying market settings where consumers receive offers by firms after learning their initial state θ 1 ∈ {l, h}. The set of feasible utility pairs for both initial types consistent with finite profits is given by 9 V ≡ {(V 1 (M | l) , V 1 (M | h)) | M ∈ M IC , Π(M ) > −∞} . I refer to the profit maximization problem of the firm, for any V = (V l , V h ) ∈ V as Π * (V ) ≡ sup M ∈M IC Π (M ) ,(2) subject to, for θ ∈ {l, h}, V 1 (M | θ) = V θ .(3) If this problem has a unique solution, we denote it as M V . Complete information benchmark In the absence of private information, efficient long-term contracts provide complete coverage to the consumer, with final consumption that does not depend on their realized history of incomes and types, except for initial type θ 1 . I denote a full coverage flow contract with consumption level c ∈ R + as z c , i.e., z c (y) = c, for all y ∈ Y . I define the complete information problem as that of maximizing profits, given by (1), subject to payoff constraint (3), with choice set M. The solution to the complete information problem is denoted by z CI t T t=1 . The following lemma, whose proof is standard and hence omitted, formally states that the complete information solution is perfect consumption smoothing. Lemma 1. (Complete information benchmark) The solution to the full information problem satisfies z CI t η t−1 p , θ t = z c CI (θ 1 ) , where c CI (·) is defined by T t=1 δ t−1 u c CI (θ 1 ) = V θ 1 . If V l = V h , the complete information solution is incentive compatible and hence solves Π * (V ). Our analysis of distortions will focus on utility pairs V ∈ V satisfying V h > V l , i.e., where the utility delivered to an initially high-type consumer is higher than that of a low type. The focus on this case is justified in Sections 8 and 9, where this inequality is shown to hold in equilibrium for both the competitive and monopoly settings. In the reverse case where V l > V h , all the results presented hold when interchanging the role of types h and l. In particular, the relevant relaxed problem would consider "upward" incentive constraints. Incentives and distortions I now characterize the solution to the profit maximization problem (2). The main result in this section, Lemma 3, provides a characterization of the solution of the relaxed problem, which is used to show that the solution to the relaxed problem also solves the firm's original profit maximization problem. In an optimal mechanism, the impact of any information revealed over time -be it signal realizations or type announcements -is determined by its likelihood ratio: the ratio of its probability when coming from either a low-or high-type. For example, income realizations that are relatively more likely for high-types are rewarded with higher flow and higher continuation utilities. Additionally, the assumption of type persistence implies that future high-type announcements, under truth-telling, are relatively more likely to come from a consumer with initially high-type. As a consequence, high-type announcements are rewarded with more attractive continuation contracts. More precisely, a solution to the relaxed problem is shown to satisfy three monotonicity properties that provide insights into the the efficient provision of dynamic incentives. These monotonicity properties are also used, in Subsection 5.3, to guarantee that the solution to the relaxed problem solves the original profit maximization problem. Relaxed problem Define a reporting strategy r to be a one-shot critical deviation (OSCD) from truth-telling at period t with signal history φ t−1 if the consumer reports a high-type when having a low-type for the first time at period t, but otherwise reports her type truthfully. In other words, it satisfies r t (φ t−1 , h t−1 , h t−1 , l) = h, for some period t ∈ {1, . . . , T } and signal history φ t−1 ∈ Φ t−1 , but is otherwise identical to the truth-telling strategy. I say that a mechanism is one-shot incentive compatible (OSIC) if the consumer has no profitable OSCD, and the set of OSIC direct mechanisms is defined as M OSIC . One-shot deviations are restrictive in three ways. It involves a misreport in a single contingency, and truth-telling otherwise. This single deviation occurs along a truth-telling path, i.e., the consumer considers a single misreport at a given period t after t − 1 periods of truth-telling. Finally, this deviation involves the consumer announcing to have a low-risk type within period t, when in fact having a high-risk type. Figure 2 represents graphically the potential deviations considered in the relaxed problem if the set of possible signals is Φ = {φ a , φ b , φ c }. The blue arrows representing the direction of misreports, where the lowtype pretends to have high-type in a given period. The relaxed problem is the following: Figure 2: This figure represents the set of binding incentive constraints in the relaxed problem. The black branches represent possible type realizations, red branches correspond to possible signal realizations, and the blue curved arrows represent the upward incentive constraints considered in the relaxed problem. Π R (V ) ≡ sup M ∈M OSIC Π (M ) ,(4)subject to, for θ ∈ {l, h}, V 1 (M | θ) = V θ . Assumption 1. The relaxed problem has a solution, for all V ∈ V. As shown below, existence of a solution is guaranteed in the presence of a finite horizon (T < ∞), or in the infinite horizon case as long as utility function u(·) is bounded. I do not provide an exhaustive analysis of the issue of existence of a solution for the infinite horizon case, but the characterization results in this paper apply more generally whenever an optimal mechanism exists. Lemma 2. If T < ∞ or u(·) is bounded, the relaxed problem has a solution for all V ∈ V. Proof. Consider any V ∈ V. The definition of V implies that Π R (V ) ≥ Π * (V ) > −∞. Consider a sequence of mechanisms M n = {z n t } t≤T n∈N feasible in problem Π R (V ) such that Π(M n ) > −∞ and Π(M n ) → n→∞ Π R (V ). The fact that Π(M n ) > −∞ implies that, for each period t and history (φ t−1 , θ t ) ∈ Φ t−1 × θ t , sequence {z n t (φ t−1 , θ t )} n∈N is uniformly bounded by a constantz(φ t−1 , θ t ) < sup c≥0 u(c) and, as a consequence, we can assume without loss (potentially using a subsequence) that {z n t (φ t−1 , θ t )} n∈N converges for any t, φ t−1 , θ t . Now let M * ≡ {lim n z n t } t≤T . If u(·) is bounded or T < ∞, the dominated convergence theorem implies that V r (M * ) = lim n→∞ V r (M n ) for any reporting strategy r, which implies that M * is feasible in problem Π R (V ). Fatou's lemma implies that (the profits of any two mechanisms only differ in the consumption paid out to the agent): − Π(M * ) ≤ lim inf n [−Π(M n )] = −Π R (V ),(5) which means that M * solves Π R (V ). Define τ (η t−1 , θ t ) as the first period at which the consumer has a low type (if any), i.e., τ (η t−1 , θ t ) = min{t ∈ {0, . . . , t} | [θ t−1 , θ t ] t = l}, with inf ∅ = ∅. The example τ (φ t , (l, h, . . . , h)) = 1 represents a first period low-type, and τ = ∅ represents the absence of a low-type realization, i.e., τ (φ t , (h, . . . , h)) = ∅. In the rest of this section, we omit the dependence of τ on the history of signals and types for brevity. For any income level y ∈ Y and signal φ 0 ∈ Φ, define likelihood ratios (y) ≡ p l (y) p h (y) , and (φ 0 ) ≡ y∈φ −1 (φ 0 ) p l (y) y∈φ −1 (φ 0 ) p h (y) . As is standard in contract theory, these ratios are useful because they represent how useful each income/signal realization is in screening different types. A realization with high likelihood ratio is more "indicative" of a low-type. The following Lemma states that the relaxed problem has a unique solution and outlines key properties of its solution. This result is proved in the Appendix and its proof exploits the recursive structure of problem Π R . Lemma 3. For any V ∈ V, the relaxed problem has a unique solution. Moreover, if V h > V l , its solution satisfies: (i) All OSIC constraints hold as equalities. (ii) No distortions following low-type: there exists {c t } T 1 , with c t : Φ t−1 → R + such that z t (φ t−1 , θ t−1 , θ t ) = z cτ (φ τ −1 ) , whenever τ = ∅. Additionally, there exist {µ t , λ t } T t=1 with (µ t , λ t ) : Φ t → R×R + such that, for any (η t−1 , θ t ) such that τ = ∅, (iii) Flow contracts following high-types have partial coverage: µ t−1 (φ t−1 ) − λ t−1 (φ t−1 ) (y t ) ≤ 1 u (z t (y t | η t−1 , h)) (6) with (6) holding as a equality if z t (y t | η t−1 , θ t ) > 0. (iv) Future type reward: V t (η t−1 , h) > V t (η t−1 , l) (v) Future signal effect: for any φ, φ such that (φ ) ≤ (φ): π lh V t ((φ t−2 , φ , h t−1 ), h) + π ll V t ((φ t−2 , φ , h t−1 ), l) ≥ π lh V t ((φ t−2 , φ, h t−1 ), h) + π ll V t ((φ t−2 , φ, h t−1 ), l) Property (i) state that all upward incentive constraints considered in the relaxed problem hold as equalities. The optimal contract is supposed to provide higher utility to a consumer with high initial type, while deterring deviations from low-types. Given the presence of type persistence, one way to achieve this goal is to use a future high-type realization as a signal that the consumer's initial type is θ 1 = h. In other words, the utility gap between consumers with initially high and low types is propagated to future periods, with consecutive hightype announcements being "rewarded" with higher utility. For this reason, upward incentive constraints bind not only at t = 1, but in all periods. Property (ii) corresponds to a "no distortion at the bottom" result. This follows directly from the fact that the relaxed profit maximization problem ignores "downward" incentive constraints. In Subsection 5.3 we show that those ignored constraints are guaranteed to hold in the solution to the relaxed problem. Properties (iii)-(v) of Lemma 3 illustrate how exogenous signal information (φ t ) and endogenous reports (θ t ) are used in an optimal mechanism to efficiently screen consumers with different types using both within-period coverage as well as continuation utility. Given the crucial role of this properties, we now discuss in detail the interpretation of each such property and show, in Lemmas 4 and 5, how these properties imply that incentives within the mechanism are properly aligned. These results are akin to the standard role played by the monotonicity property in mechanism design, which are used to show that mechanisms solving an appropriately defined relaxed problem ignoring certain incentive constraints satisfies the ignored incentive constraints. Monotonicity I now introduce a new set of monotonicity conditions on mechanisms which, together with binding "upward" incentive compatibility constraints, imply incentive compatibility. Lemma 3 implies that the solution to the relaxed problem satisfies all three monotonicity conditions and all OSIC constraints as equalities, and hence solves the firm's original problem. As discussed in the Appendix, these monotonicity properties allows us to exploit the recursive structure of the firm's relaxed problem, which can be broken into a series of oneperiod problems in which the designer chooses in each period a pair of flow contracts as well as promised continuation utilities to the agent. In this subsection, I discuss how the monotonicity conditions provide insights into the the efficient provision of dynamic incentives and their role in connecting the relaxed and original profit maximization problems. Flow monotonicity Flow monotonicity is akin to standard static monotonicity notions, in that it guarantees that within-period incentives are aligned by making sure that the partial coverage contract tailored to the (current) high-type consumer rewards accident realizations that are indicative of a high type, according to a likelihood ratio condition. More precisely, flow monotonicity corresponds to property (iii) in Lemma 3. For any income level y ∈ Y , the likelihood (y) provides a measure of how much income y is indicative of low-types. As a consequence, it is expected that profit maximizing contracts seeking to attract high-type consumers will use flow contracts that punish income realizations that are indicative of low-types (high (y)) and reward realizations that are indicative of high-types (low (y)). Definition 2. A flow contract z ∈ Z satisfies flow monotonicity if it satisfies (y ) > (y) =⇒ z(y ) ≤ z(y). The use of flow contracts satisfying flow monotonicity implies that consumers with hightypes have a higher benefit from choosing them, relative to a full-coverage contract. Lemma 4. For any pair of flow contracts z, z c ∈ Z, with z satisfying flow monotonicity and c ≥ 0, v(z, h) − v(z c , h) ≥ v(z, l) − v(z c , l) Proof. First notice that v(z c , h) = v(z c , l). Consider an ordering y [k] #Y k=1 of income realizations such that (y [k] ) is weakly decreasing in k. Flow monotonicity implies that both z(y k ) and 1 − (y [k] ) are increasing in k. Hence we have that v(z, h) − v(z, l) = K k=1 p h (y k )u(z(y k )) [1 − (y k )] ≥ K k=1 p h (y k )u(z(y k )) K k=1 p h (y k ) [1 − (y k )] , which is equal to zero since the last term in this expression is zero. Continuation monotonicity The notion of flow monotonicity is inherently static. However, in a dynamic environment firms have extra instruments to screen consumers, namely they can use future contracts as additional screening instruments. From the consumer's incentive perspective, the use of such dynamic incentive schemes are represented by changes to their expected continuation utility within the mechanism as a response to either their choices or realized income within a period. The term continuation monotonicity relates to how, in any given period, the consumer's continuation utility depends on her previous announcements and signal realizations, which correspond to respectively to properties (iv) and (v) in Lemma 3. The first such notion is continuation-signal-monotonicity (CSM), which states that signal realizations that are indicative of low-types are punished in future periods. From the point of view of period t, the consumer's continuation payoff from period t + 1 onwards depends no only on period t's realized signal but also on her realized type in period t + 1. CSM compares different signal realizations, while taking averages over possible future types using the probability distribution of a low-type in period t, who should be discouraged from misreporting their type in period t. Definition 3. For any t < T , a mechanism M satisfies CSM at period t and history η t−1 if, for any y, y ∈ Y (y ) > (y) implies π lh V t (M | (φ t−2 , φ , h t−1 ), h) + π ll V t (M | (φ t−2 , φ , h t−1 ), l) ≥ π lh V t (M | (φ t−2 , φ, h t−1 ), h) + π ll V t (M | (φ t−2 , φ, h t−1 ), l) In a similar vein to the previous notions of monotonicity, we now introduce the notion of continuation-type-monotonicity (CTM), which considers the impact of the consumer's type on their continuation utility. As discussed in Section 3, we restrict attention to mechanisms that deliver a higher discounted utility to a consumer with initial type θ 1 = h. CTM requires that this ordering holds for a specific period and history. Definition 4. For any t < T , a mechanism M satisfies continuation type monotonicity in period t and history η t−1 if, for any η t = (η t−1 , φ, h), V t+1 (M | η t , h) > V t+1 (M | η t , l). These monotonicity properties allows us to show that incentives are aligned since the continuation contract following an announcementθ = h is more attractive to a consumer whose true type is indeed θ t = h. For a fixed mechanism M , period t < T and history η t−1 = (φ t−1 , θ t−1 ), define the continuation utility obtained by a consumer with type i ∈ {l, h} announcing to have a high type asV t+1 (i | η t−1 ) ≡ φ∈Φ p i (φ) j=l,h π ij V t+1 (φ t−1 , φ, θ t−1 , h), j(7) Lemma 5. If mechanism M satisfies CSM and CTM in period t, given history η t−1 , then V t+1 (h | η t−1 ) ≥V t+1 (l | η t−1 ). Proof. Rearranging summation terms gives us: V t+1 (h | η t−1 ) −V t+1 (l | η t−1 ) = φ∈Φ p h (φ)(π hh − π lh ) V t+1 (η t−1 , φ, h, h) − V t+1 (η t−1 , φ, h, l) + φ∈Φ p h (φ) j=l,h π lj V t+1 (η t−1 , φ, h, j) − φ∈Φ p l (φ) j=l,h π lj V t+1 (η t−1 , φ, h, j), Persistence of types and CTM implies that the first term is non-negative. The second and term terms can be rewritten as φ∈Φ p h (φ)(1 − (φ)) j=l,h π lj V t+1 (η t−1 , φ, h, j) , which is positive since it is an average of the product of two terms that are positively correlated (from CSM), with the first term having zero expectation: φ∈Φ p h (φ)(1 − (φ)) = 0. Sufficiency of OSIC Besides providing insights into the efficient design of incentives, the three monotonicity notions introduced also allows us to guarantee that the solution of the relaxed problem is feasible in the original profit maximization problem, and hence solves it. Lemma 6. If M solves the relaxed problem, the consumer has no profitable one-shot deviation from truth-telling in M . Proof. Consider a period t with private history η t−1 p = (y t−1 ,θ t−1 , θ t−1 ) and current type θ t . Since types follow a Markov process, the consumer's preferences over continuation reporting strategies are identical for (i) private history η t−1 p = (y t−1 ,θ t−1 , θ t−1 ) with period t type θ t , and (ii) private historyη p t−1 = (y t−1 , θ t−1 , θ t−1 ) with period t type θ t . In other words, we only need to look for deviations for private histories without past misreports. If θ t−1 includes any low-type realization, the result holds trivially since the optimal mechanism allocates constant consumption from period t onwards. I focus now on the case θ t−1 = h t−1 . If θ t = l, a one-shot deviation is a special case of the constraints considered in the OSCD concept and hence is satisfied in any mechanism that is feasible in the relaxed problem. If θ t = h, using Lemma 3-(ii) we can represent the net gain from a one-shot deviation as 1 − δ T −t+1 1 − δ u(c t (φ t−1 )) − v(z t (η t−1 , h)) + δV t+1 (h | η t−1 ) ,(8) where η t−1 = (φ t−1 , θ t−1 ) is the public history connected with the private history in focus andV t+1 is defined as in (7). From Lemma 3, items (iii)-(v), we know that the optimal mechanism satisfies flow and continuation monotonicity and hence, using Lemmas 4 and 5, the net gain in (8) is weakly lower than 1 − δ T −t+1 1 − δ u(c t ) − v(z t (η t−1 , l)) + δ φ∈Φ p l (φ) j=l,h π ljVt+1 (l | η t−1 ) ,(9) which is the net gain from truth-telling for a consumer with type θ t = l relative to a misreport in period t. From Lemma 3-(i), we know this is zero. The absence of profitable one-shot deviations is enough to guarantee that a mechanism M = {z t } T t=1 is incentive compatible as long as the consumer's reporting problem satisfies continuity at infinity, which is guaranteed as long as flow utilities are bounded: sup |v(z t (η t−1 ), θ)| | t = 1, . . . , T, η t−1 ∈ H t−1 , θ ∈ Θ < ∞, in which case we say that the mechanism M is bounded. This condition is automatically guaranteed if time is finite or the Bernoulli utility function u(·) is bounded. It holds more generally if consumption in the solution to the relaxed problem is uniformly bounded. Corollary 1. If the solution to the relaxed problem is bounded, it is optimal. Proof. If a profitable deviation r from truth-telling r * exists in mechanism M , boundedness of M implies -through standard arguments -that a finite profitable deviation r only involving misreports up to period t < ∞ also exists. Since one-shot deviations from truthtelling are not profitable, then modifying r by dictating truth-telling in period t, regardless of the history, is a weak improvement from r . Repeated application of this argument implies that r * weakly dominates r , a contradiction. Hence M solves the relaxed problem and is feasible in the firm's original problem, so it is optimal. Auxiliary problem The characterization of coverage and price dynamics in an insurance setting poses two technical challenges. First, the flow contract space, Z = R Y + , is multi-dimensional and does not have a natural notion of distortions, such as as under-provision in standard screening models (Mussa and Rosen (1978), Myerson (1981)). The underlying inefficiency in this model is the exposure of the consumer to risk, or partial coverage, which is introduced as a way of screening hightypes. I introduce a notion of distortion that is directly tied to the source of distortions, which is the need to preclude low-type consumers from misreporting their types. Second, the introduction of risk aversion in a dynamic screening environment leads to a non-separability issue absent from linear environments. The marginal efficiency loss of from the introduction of distortions potentially depends on the underlying utility level obtained by the consumers. As a consequence, the optimal contract jointly chooses both the intertemporal allocation of utility to be provided ot the consumer as well as the spreading of distortions over time. I tackle both of these issues by introducing a tractable auxiliary static cost minimization problem. This auxiliary cost function is then used to study the original dynamic profit maximization problem. Definition From the point of view of incentives, flow contracts fulfill two roles: to provide a certain utility level to a consumer with currently high-type, while also discouraging misreports by a low-type consumer. Hence we consider, for (ν, ∆) ∈ R 2 , the problem of finding an optimal flow contract z ∈ Z satisfying a promise keeping constraint: v(z, h) = ν,(10) as well as a threat-keeping constraint introducing a penalty for misreporting l-type consumers (we will later focus on the case ∆ < 0) v(z, l) = ν − ∆. I denote the set of pairs (ν, ∆) ∈ R 2 such that a flow contract satisfying both (10) and (11) exists as A ⊂ R 2 . The utility wedge ∆ between consumers with different types is the source of distortions in the contract since exposing the consumer to risk is the only way to discourage low-type consumers from misreporting. I refer to it directly as the level of distortion. Define the following auxiliary static cost minimization problem P A , for any (ν, ∆) ∈ A: χ (ν, ∆) ≡ inf z∈Z y∈Y p h (y) z (y) , subject to constraints (10) and (11). Lemma 7. For any (ν, ∆) ∈ A, problem P A has a unique solution and χ is strictly increasing in both coordinates if ∆ ≥ 0. Moreover, if the solution at (ν, ∆) is interior then χ is twice differentiable in an open neighborhood of (ν, ∆). Proof. Follows directly from Lemma 15 in the Appendix. For any (ν, ∆) ∈ A, we denote the solution of P A as ζ(ν, ∆). In the remainder of the analysis, I assume that this cost function is supermodular, i.e., that the marginal cost of distortions is increasing in the utility level delivered to the agent. Assumption 2. (Cost supermodularity) Function χ(·) satisfies ∂ 2 χ(ν, ∆) ∂ν∂∆ ≥ 0, whenever P A has an interior solution. The supermodularity of function χ(·) can be written in terms of utility function u(·), and is equivalent to the requirement that the coefficient of absolute risk aversion r u does not decrease with consumption "too quickly". This is guaranteed if u (·) has non-decreasing absolute risk aversion (IARA), which includes constant absolute risk aversion (CARA) utility as a special case. It also holds for the case of constant relative risk aversion (CRRA) utility with coefficient above 1/2. More formally, defining the coefficient of absolute risk aversion as r u (c) ≡ − u (c) u (c) , we can state the following result, which is proved in the Appendix. Lemma 8. Assume r u : R + → R + is differentiable, then the following are equivalent: (i) Assumption 2 holds, (ii) r u (x) u (x) + 2 [r u (x)] 2 ≥ 0, (iii) ψ (x) > 0. Proof. In the Appendix. Interpretation The auxiliary problem provides a way to study feature of the flow contracts offered along the optimal mechanism. In this subsection, we discuss how properties of the cost function χ(·) translate into risk exposure and consumption within the cost-minimizing flow contract. The optimal intertemporal allocation of consumption and distortions, to be studied in Section 7, is characterized in terms of the marginal costs of flow utility and distortions. For any (ν, ∆) ∈ R × R + for which problem P A has an interior solution ζ(·), we show in the Appendix (Lemma 16) that marginal costs relate to consumption in cost-minimizing contract according to: 1 u (ζ(y)) = χ ν (ν, ∆) + χ ∆ (ν, ∆) [1 − (y)] ,(12) where we use notation χ j (·) ≡ ∂χ(·) ∂j . The inverse marginal utility, which is strictly increasing in consumption, represents the marginal cost for the firm to deliver an additional infinitesimal amount of utility to the consumers for each specific income realization. Given that χ(·) is a convex function, an increase in flow utility ν (or in distortion ∆) leads to an increase in χ ν (or in χ ∆ ). From (12), we can separately relate both marginal costs with the consumer's consumption and risk exposure. First, the marginal cost of flow utility is related to the consumer's marginal utility through the following equation: y∈Y p h (y) 1 u (ζ(y)) = χ ν (ν, ∆), This is a standard inverse marginal utility optimality condition in contract theory (see Rogerson (1985), for example). As I illustrate in the examples below, since the function [u (c)] −1 is strictly increasing in consumption, an increase in χ ν can be seen as an overall increase in consumption, with the exact connection being dependent on the shape of utility u(·). Second, the following condition relates the distortion level ∆ to the responsiveness of consumption to income realizations within a period: for any y, y ∈ Y 1 u (ζ(y )) − 1 u (ζ(y)) = −χ ∆ (ν, ∆) [ (y ) − (y)] . Intuitively, cost minimizing flow contracts reward income realizations with low likelihood ratio (·). These two marginal cost equations imply that χ 2 ∆ (ν, ∆) φ∈Φ p h (y) [1 − (y)] 2 = φ∈Φ p h (y) [u (ζ(y))] −1 − y ∈Y p h (y )[u (ζ(y ))] −1 2 . The right-hand side corresponds to the variance of the inverse marginal utility of the consumer. Hence an increase in distortion ∆, and hence in χ ∆ , generates a larger dispersion in the inverse marginal utility of consumer across income realizations. A larger distortion requires that the flow contract expose the consumer to more risk, so that low-risk consumer finds this contract less attractive. This condition shows that this additional dispersion is monotonically related to χ ∆ (·). A clearer connection can be made for parametric families of utility functions. Example: CRRA preferences Consider the case of CARA preferences, with u(c) = c 1−ρ − 1 1 − ρ , with coefficient of relative risk aversion ρ > 0. In this case, an increases in the flow utility ν is directly related to an increase in the ρ-th moment of the consumption distribution: χ ν (ν, ∆) = y∈Y p h (y)ζ(y) ρ , while an increase in distortion ∆ leads to a larger variance of the ρ-th power of final consumption: χ 2 ∆ (ν, ∆) = y∈Y [ζ(y) ρ − E h (ζ ρ )] 2 y∈Y p h (y) [1 − (y)] 2 , where E h (ζ ρ ) ≡ y∈Y p h (y)ζ(y). Two particular cases are of special interest. In the case ρ = 1, where the utility becomes u(c) = log(c), the marginal costs χ ν and χ ∆ are multiples of the mean and variance, respectively, of the within-period consumption of the consumer. Alternatively, if ρ = 1/2, the marginal costs χ ν and χ ∆ are multiples of the mean and variance, respectively, of the within-period ex-post utility of the consumer -u(ζ(y)) -in the cost minimizing contract ζ(·). Hence distortion parameter ∆ represents the consumer's exposure to risk, but evaluated from the point of view of her utility level. Example: binary outcomes In Section 7, I provide results on the dynamics of utility flow and distortions. In the special case of binary outcomes, with Y = y,ȳ satisfying (y) > (ȳ), conditions (10) and (11) imply that both are connected to consumer's income-contingent utility in a simple way: u(ζ(ȳ)) = ν + p h (y) p h (ȳ) − p l (ȳ) ∆, u(ζ(y)) = ν − p h (ȳ) p h (ȳ) − p l (ȳ) ∆. In which case ν and ∆ correspond directly to the utility level and dispersion of the consumer within a period. Distortion and consumption dynamics In this section, we use the auxiliary cost problem introduced in Section 6 to shed light on the dynamics of distortions and consumption in the optimal mechanism. The firm's profit maximization is directly linked with problem P A since each within-period flow contract provided by the firm must be optimal within the set of flow contracts that deliver the same utility level for truth-telling consumers (which corresponds to ν) and discourages misreporting consumers by the same amount (which corresponds to ∆). In other words, each contract solves the auxiliary problem P A for a particular pair (ν, ∆). We can then separate the problem of the firm into two "sub-problems": one in which flow utility and distortions are allocated across periods, which we study in this section, and another one in which the chosen flow utility and distortion levels must be delivered in a cost-minimizing fashion, as discussed in Section 6. Our analysis now restricts attention to the case of finitely many periods. The following proposition formalizes this separation, by showing that all flow contracts delivered in the optimal mechanism are indeed solutions to the auxiliary cost minimization problem, with suitably chosen pairs of utility flow and distortion. Given optimal mechanism M = {z t } T t=1 , define the following: ν t (φ t−1 ) ≡ v z t (h t−1 , φ t−1 , h), h , ∆ t (φ t−1 ) ≡ ν t (φ t−1 ) − v z t (h t−1 , φ t−1 , h), l , which correspond respectively to the flow utility and distortion in the partial coverage contract offered in period t following type announcements h t . Proposition 1. The optimal mechanism M = {z t } T t=1 satisfies, for any t = 1, . . . , T , z t (h t−1 , φ t−1 , h) = ζ(ν t (φ t−1 ), ∆ t (φ t−1 )). Proof. Fix any period t < T and signal history φ t . For any mechanismM = {z t } T t=1 , the flow contractz t (h t−1 , φ t−1 , h) only affects the OSIC constraints in the relaxed problem via v z t (h t−1 , φ t−1 , h), h and v z t (h t−1 , φ t−1 , h), l , while it only affects the firm's profits via y∈Y p h (y)z t (h t−1 , φ t−1 , h) (y) . Hence modifying mechanism M by substituting flow contract z t (h t−1 , φ t−1 , h) by ζ(ν t (φ t ), ∆ t (φ t )) still satisfies OSIC and (given uniqueness of solution in P A ) strictly increases profits in the relaxed problem if z t (h t−1 , φ t−1 , h) = ζ(ν t (φ t ), ∆ t (φ t )). From now on, we refer to {ν t , ∆ t } T t=1 simply as the flow utility and distortion in the optimal mechanism following a sequence of high-type announcements. I use a similar notation to refer to the flow per-period utility derived by the consumer following a first low-type announcement in period t: ν l t (φ t−1 ) ≡ u(c t (φ t−1 )). We can now use Proposition 1 and focus solely on the problem of intertemporal allocation of flow utility ν and distortion ∆ within the optimal contract. The following proposition displays the optimality conditions connected with intertemporal allocation of utility and distortions. Since all terms in Proposition 2 depend on φ t−1 , we omit this dependence for brevity. Proposition 2. For any t = 1, . . . , T and φ t−1 ∈ Φ t−1 , if the solution to the cost minimization problem P A has an interior solution, i.e., (ν t , ∆ t )(φ t−1 ) ∈ int(A), (ν t , ∆ t )(φ t−1 , φ ) ∈ int(A), for all φ ∈ Φ, then the following optimality conditions hold χ ν (ν t , ∆ t ) = φ∈Φ p h (φ) π hh χ ν [(ν t+1 , ∆ t+1 )(φ t )] + π hl χ ν [(ν l t+1 (φ t ), 0)] ,(13)χ ∆ (ν t , ∆ t ) = φ∈Φ p h (φ) π hh π hh − π lh χ ∆ [(ν t+1 , ∆ t+1 )(φ t )] +π hl χ ν [(ν t+1 , ∆ t+1 )(φ t )] − χ ν [(ν l t+1 (φ t ), 0)] .(14) The Proposition above relies on the use of local optimality conditions of the firm's profit maximization problem and hence assumes its solution is interior. The interiority condition in Proposition 2 holds if the optimal mechanism has strictly positive consumption flows. Conditions (13) and (14) characterize the dynamics of flow utility and distortions along the non-trivial histories that involve distortions. Equation (13) represents the efficient intertemporal allocation of flow utilities, or consumption. It states that the marginal cost of providing a high-type with higher flow utility in period t must be equalized to the expected marginal cost of higher flow utility in period t + 1. The marginal cost of flow utility coincides with the expectation of the inverse of the consumer's marginal utility (see Subsection 6.2), which shows that this condition is a special case of the inverse Euler equation studied in dynamic allocation problems (see Rogerson (1985), Farhi and Werning (2012)). Equation (14), on the other hand, represents the efficient intertemporal allocation of distortions. Let's consider the problem, in period t < T , of discouraging a consumer with low period t type from pretending to have a high type. Due to the presence of type persistence, this can be done in two ways: by exposing the consumer to risk in period t, which is represented by a larger ∆ t ; or by exposing the consumer to more risk in period t + 1 as long as the consumer still claims to be of high type (θ t+1 = h). Of course, given the convexity of the cost function χ(·), the optimal mechanism uses both screening methods in a balanced way. However, (14) illustrates the differences in using current versus future period distortions. First, notice that future distortions are only useful due to the persistence of types. As type persistence is reduced, or (π hh − π lh ) is smaller, the optimal mechanism relies mostly on current period distortions for screening purposes. In the limit where π hh = π lh , the use of future distortions in screening is useless and, as a consequence, the optimal contract only features distortions in the first period. Second, reducing the distortion in period t while increasing it in period t + 1 not only has a direct cost impact as it requires exposing the consumer to risk (this is captured by the χ ∆ term), but it also implies that next period's type realization now has a larger impact on the consumer's final utility. This additional utility dispersion has a marginal cost which depends on the gap of the marginal cost of flow utility in period t + 1 for both possible types (represented by the second and third terms on the right-hand side of (14)). Realization-independent contracts I start by focusing on the case of realization-independent contracts, i.e., long-term contracts that do not use the history of past income realizations when determining the menu to be offered to the consumer within a period. As discussed in the introduction, this represents an extreme case of price restrictions that limit how much information can be used explicitly by firms when pricing consumption. This subsection also restricts attention to the finite-horizon case, i.e., T < ∞. Since flow contracts now only depend on the history of announcements, we drop the dependence on history φ t . In this case, equations (13) and (14) become χ ν (ν t , ∆ t ) = π hh χ ν (ν t+1 , ∆ t+1 ) + π hl χ ν (ν l t+1 , 0),(15) and χ ∆ (ν t , ∆ t ) = π hh π hh − π lh χ ∆ (ν t+1 , ∆ t+1 ) + π hl χ ν (ν t+1 , ∆ t+1 ) − χ ν (ν l t+1 , 0) .(16) The supermodularity guaranteed by Assumption 2 allows us to characterize the dynamic behavior of utility and distortions. Let's focus on the intertemporal allocation between a given periods t and t + 1, following a sequence of high-type realizations h t . Given that the optimal contract rewards high-type announcements, the flow utility following an additional high-type realization in period t + 1, ν t+1 , is higher relative to that of a consumer with a low-type realization, ν l t+1 . Together with the presence of distortions following a high-type announcement (∆ t+1 > 0) and supermodularity of χ(·) (Assumption 2), we can then conclude that χ ν (ν t+1 , ∆ t+1 ) > χ ν (ν l t+1 , 0). From equation (15), we can already conclude that consecutive high-type announcements lead to an increase in the marginal cost of flow utility: χ ν (ν t+1 , ∆ t+1 ) > χ ν (ν t , ∆ t ) > χ ν (ν l t+1 , 0).(17) Now, using intertemporal distortion allocation condition (16), we have that χ ∆ (ν t , ∆ t ) > π hh π hh − π lh χ ∆ (ν t+1 , ∆ t+1 ) > χ ∆ (ν t+1 , ∆ t+1 ) In other words, profit maximization mandates that, along high-type path h T , the marginal cost of flow utility is increasing while the marginal cost of distortions is decreasing. Using convexity of χ and, once again, Assumption 2 (see Lemma 17) these properties can be translated into statements about the dynamic behavior of flow utility and distortions, which are summarized in the following Proposition. Proposition 3. If V h > V l , the optimal mechanism satisfies: (i) High-type utility flows increase, i.e., {ν t } T t=1 is strictly increasing, (ii) Distortions decrease, i.e., {∆ t } T t=1 is strictly decreasing. Proof. In the Appendix. Realization-dependent contracts Considering a general signal structure, we are able to use Proposition 2 to extend the monotonicity result of Proposition 3 in two cases. First, we use the case of constant absolute risk aversion with coefficient 1/2 as an illustrative example. This knife-edge example simplifies the analysis since it is the only one in which the marginal costs of flow utility and distortions are separable. We also look at the case of general preferences with two periods. Quadratic cost I now assume that the consumer's utility function takes the following form: u(c) = 2 √ c. This example is particularly tractable since the auxiliary cost function χ (·) is (i) separable in ν and ∆, i.e., ∂ 2 χ(ν, ∆) ∂ν∂∆ = 0, whenever twice differentiable, and (ii) quadratic, which implies that χ ν (·) and χ ∆ (·) are linear in ν and ∆, respectively. In this case, intertemporal optimality conditions (13) and (14) in Proposition (2) can be rewritten as statements in terms of utility flow and distortions: ν t (φ t−1 ) = φ∈Φ p h (φ t ) π hh ν t+1 (φ t ) + π hl ν l t+1 (φ t ) ,(18)∆ t (φ t−1 ) = φ∈Φ p h (φ t ) π hh π hh − π lh ∆ t+1 (φ t ) + N ν t+1 (φ t ) − ν l t+1 (φ t ) ,(19) for constant N ≡ ∂ 2 χ ∂ν 2 > 0. The first equation implies that flow utilities form a martingale in the optimal mechanism. In the optimal mechanism, the continuation utility following a high-type realization is always larger than the one following a low-type announcement, and the martingale property of flow utilities implies that flow utilities in any given period t are also larger following a hightype realization θ t = h, everything else equal. Together with (18) and (19), this implies that distortions follow a supermartingale, conditional on the type path θ T . Additionally, the change in flow utility across periods is similar to the realization-independent case: a high-type realization leads to higher utility flow while a low-type realization leads to lower utility flows -when integrating over the possible signal realizations. These statements are formalized below. Proposition 4. If V h > V l and the optimal mechanism is interior, then: (i) distortions follow a supermartingale, conditional on θ T , i.e., ∆ t (φ t−1 ) > φ∈Φ p h (φ t )∆ t+1 (φ t ) and (ii) within period flow utilities are increasing in past periods' type announcements, i.e., φ∈Φ p h (φ t )ν t+1 (φ t ) > ν t (φ t−1 ) > φ∈Φ p h (φ t )ν l t+1 (φ t ) Proof. First notice that the flow utility in period t if the consumer's first low-type announcement is in period t is strictly lower than that if the consumer has one more high-type realization in period t since: V t (φ t−1 , h t−1 , l) =ν l t (φ t−1 ) T τ =t δ τ −t =ν t (φ t−1 ) + δ φ∈Φ p l (φ t ) π lh V t+1 (φ t , h t , h) + π ll V t+1 (φ t , h t , l) − ∆ t (φ t−1 ) <ν t (φ t−1 ) + δ φ∈Φ p h (φ t ) π hh V t+1 (φ t , h t , h) + π hl V t+1 (φ t , h t , l) − ∆ t (φ t−1 ) =ν t (φ t−1 ) T τ =t δ τ −t − ∆ t (φ t−1 ). The first equality follows from ν l t being defined as the constant flow utility obtained by the consumer following a first low-type realization in period t. The second equality corresponds to the binding incentive constraint in period t. The inequality follows from Lemma 3-(iv). Finally, the last equality follows from the martingale property (18). Hence we conclude that ν t (φ t−1 ) > ν l t (φ t−1 ), for all t and φ t−1 ∈ Φ t−1 . The result then follows directly from equations (18) and (19). General preferences Now consider an arbitrary signal structure and focus on the case of two periods. As mentioned before, in maximizing profits the firm has an incentive to reward subsequent high-type announcements (as long as V h > V l ), and to rely less on later periods' distortions in order to provide incentives. As a consequence, the marginal costs of flow utility must be increasing, while the marginal cost of distortions must be decreasing following consecutive high-type announcements (which correspond to partial coverage), as shown below. As discussed in Subsection 6.2, the marginal cost terms χ ν and χ ∆ are related to the consumption level and risk exposure in a given flow contract. Hence the interpretation of Proposition 5 is that the consecutive choices of partial coverage are rewarded, when averaging over signal realizations, with more consumption and less distortions. Proposition 5. Assume T = 2, the optimal mechanism satisfies χ ∆ (ν 1 , ∆ 1 ) > φ∈Φ p h (φ)χ ∆ [(ν 2 , ∆ 2 )(φ)] and φ∈Φ p h (φ)χ ν [(ν 2 , ∆ 2 )(φ)] > χ ν (ν 1 , ∆ 1 ) > φ∈Φ p h (φ)χ ν [(ν l 2 (φ), 0)], Proof. Lemma 3 implies that V 2 (φ, h) = ν 2 (φ, h) > V 2 (φ, l) = ν l 2 (φ), which implies that χ ν [(ν 2 , ∆ 2 )(φ)] > χ ν [ν l 2 (φ)] , for all signals φ ∈ Φ. The result then follows directly from (13) and (14). In the case of realization independent contracts, monotonicity properties of the marginal cost functions (which are guaranteed by supermodularity) allow us to translate statements about marginal costs directly into statements about the levels of flow utility and distortion, as in Proposition 3. Once we allow for richer signal structures, statements about the expectations of marginal costs cannot be translated into statements about the expectations of flow utility and distortion, except for the special case in which the marginal cost functions are linear in the flow utility and distortion pair (ν, ∆) covered in Subsection 7.2.1. Competitive analysis I now consider a competitive model that extends the analysis of Rothschild and Stiglitz (1976) and Cooper and Hayes (1987) in allowing for both persistent non-constant risk types and offers containing dynamic mechanisms. For now, I assume a single contracting stage between consumers and firms in the first period, which means that both firms and consumers can commit. The role of the commitment assumption and possible ways of relaxing it are discussed in Subsection 8.1. Consider the following extensive form. A finite number of firms simultaneously offer a mechanism to a consumer. The consumer observes her initial type θ 1 and decides which firm's mechanism to accept, if any. I assume exclusivity, i.e., the consumer can choose at most one mechanism. If the buyer does not accept any offer, she obtains no insurance coverage and gets discounted utility V i ≡ E T t=1 δ t−1 u (y t ) | θ 1 = i .(20) If the consumer accepts a contract, in each period she observes type θ t ∈ Θ, then announces a message to the chosen firm. At the end of the period the income realization y t is observed and the customer receives (or pays) transfers from the firm as described in the chosen mechanism. I study (weak) Perfect Bayesian Bayesian (PBE) of this extensive form. If the consumer's types were observed by firms, the consumer would obtain actuarily fair full insurance in equilibrium, smoothing consumption both across income realizations within a period as well as across periods. In other words, a consumer with initial type θ ∈ {l, h} would receive a time-and income-independent flow consumption with total discounted utility V F I i ≡ u(c F I i ) T t=1 δ t−1 ,(21) where c F I i represents the discounted average expected lifetime income of a consumer with initial type θ 1 = i: c F I i ≡ E T t=1 δ t−1 y t T t=1 δ t−1 | θ 1 = i . Equilibrium outcomes are reminiscent of RS, with (initially) low-type consumers receiving their full-information utility level efficiently, while (initially) high-type consumers receive an inefficient partial coverage contract which delivers utility in the interval V F I l , V F I h . I assume that the full information continuation utility vector is feasible in the presence of private information, i.e., that (V F I l , V F I h ) ∈ V. This assumption insures that the following critical payoff vector, which will be shown to correspond to the equilibrium utility level of the consumer, is well-defined. Define as Π * i (V ), for i = l, h, the expected discounted profit obtained by the firm conditional on the consumer's initial type being θ 1 = i. Lemma 9. There exists a unique pair V * ≡ (V * l , V * h ) ∈ V satisfying V * l = V F I l , Π h (V * l , V * h ) = 0. Proof. Utility level V * l can be defined as V F I l . The existence and uniqueness of V * h follows from the fact that Π h (V * l , ·) is strictly decreasing and continuous (since it is convex) and satisfies Π h (V * l , V * l ) > 0, Π h V * l , V F I l < 0. For simplicity, we also focus on equilibria with two properties. 10 First, the consumer's strategy is symmetric, i.e., the probability she accepts the offer of firm j, when firms' offered mechanisms are (M, M ), is equal to the probability of accepting the offer of firm j = j when firms' offered mechanisms are exchanged. Second, we assume that the consumer follows the truth-telling reporting strategy whenever it is optimal. I say that a direct mechanism M = {z t } T t=1 constitutes an equilibrium outcome if a PBE exists in which the on-path net consumption of the consumer corresponds exactly to their consumption in mechanism M when following a truth-telling reporting strategy. The following result -proven in the Appendix -shows that the unique equilibrium outcome is characterized by the solution of problem Π (·) studied in Sections 5-7. 11 Proposition 6. Any pure strategy PBE has outcome M V * . Moreover, a pure strategy equilibrium exists if, and only if, u u −1 c F I l ∂ + Π h (V * ) ∂V l < µ l µ h .(22) The left-hand side of condition (22) does not depend on initial type distribution (µ l , µ h ), and hence a pure strategy equilibrium exists if, and only if, the share of (initially) low-types in the population is sufficiently large. This is in line with the classical analysis in RS, which requires that the share of high-risk consumers be sufficiently high. Since V * l < V * h , the equilibrium outcome in the competitive model -described in Proposition 6 -is the solution of a particular instance of the problem studied in Sections 5-7. 12 10 The last two restrictions do not affect equilibrium outcomes but substantially simplify the analysis. 11 We use notation ∂+Π ∂Vi (V ) to denote the right-derivative of Π with respect to V i at V . 12 The interiority of the equilibrium outcome, which is assumed in multiple characterization results, can be guaranteed as long as the income distribution induced by both types is not "too" distinct. To see this notice that, if p h = p l , we have that V * l = V * l and as a consequence the equilibrium outcome is the full information one, which is interior. The role of commitment I have assumed so far that consumers are able to commit to a long-term mechanism. The goal of this section is to discuss the validity of this assumption, its role in the formal analysis, and the impact of relaxing it. If consumers have the option to renege on a mechanism at any given period and an available offer dominates their anticipated continuation contract, they would choose to do so and start a relationship with a new firm. The assumption of consumer commitment is reasonable in the presence of institutional features that prevent or hinder consumers' contract switching. For example, the presence of switching costs leads to lock-in effects allowing firms, at the outset, to credibly offer contracts that, at some future point in the interaction, may lead to lower continuation utility to the consumer relative to the offers available in the market (see Honka (2014) and Handel and Schwartzstein (2018) for discussions of switching costs in auto and health insurance, respectively) An alternative justification for commitment is informational: since consumers have private information about their types, their decision to search for a new contract may be interpreted as a negative signal by potential new firms and, as a consequence, lead to less attractive offers for switching consumers. I now show that, if consumer's low-type is an absorbing state, firm's negative inference from consumer's switching decision may serve as a commitment device and allow for the equilibrium outcome in the model with commitment to be robust to consumer reentry. Absorbing state This section assumes that the low-type is an absorbing state of the consumer type's Markov process, i.e., π ll = 1. I also restrict attention to the case of realization-independent contracts. Consider the following extension of the extensive form considered in Section 8. For simplicity, assume that a different pair of firms can potentially make offers to the consumer in each period, i.e., the set of firms is F ≡ F j t | j ∈ {1, 2} , t = 1, . . . , T . In the first period, the consumer and firms {F 1 1 , F 2 1 } interact as described in the baseline competitive model. 13 However, in each period t = 2, . . . , T the consumer decides whether to stay with her current mechanism or reenter the market searching for a new contract. If the consumer decides to reenter the market, firms F 1 t and F 2 t observe the consumer reentry decision and then simultaneously offer a mechanism to this consumer. The consumer then decides whether to accept any of the new offers made or to remain uninsured. Regardless of period t's choices, the same set of moves occurs in period t + 1, if t < T . The outcome described in Proposition 6 can arise in a PBE of this extended model, as long as firms' off-path beliefs are pessimistic. Firms' offers upon reentry in periods t = 2, . . . , T depends on their beliefs about the type of a consumer that decides to reenter the market. The most pessimistic belief firms may hold is to assume that this consumer has a low type in the current period for sure. In this case, it is optimal for firms F 1 t and F 2 t to behave as if they were in a market without information asymmetry and offer an efficient contract which provides perfect consumption smoothing for a consumer with low-type in period t, i.e., with consumption flow c o ≡ E ỹ t |θ t = l .(23) Define V o t ≡ u(c o ) T τ =t δ τ −t to be the continuation utility from taking such an offer. Notice that, while extreme, these strategies are sequentially optimal for firms given their beliefs. Consider the strategy profile for firms such that, for j = 1, 2, firm F j 1 use the strategy described in the commitment model, while firm F j t for t ≥ 2 makes the constant consumption offer described in (21). The following lemma shows that, given firms' strategy profile, consumers never want to reenter the market. Lemma 10. If the equilibrium contract M V * is interior, then the consumer's continuation utility satisfies V t (θ t ) ≥ V o t for all periods t = 1, . . . , T , and almost all θ t ∈ Θ t . Proof. First notice that, from Proposition 6 and π ll = 1 we have that V 1 (l) = T τ =1 δ τ −1 u(c o ). Now consider period t = 2, . . . , T and history (h t−1 , l). The consumer's continuation utility in this case is given by V t (h t−1 , l) = T τ =t δ τ −t (ν τ − ∆ τ ) . Since, from Proposition 3, {ν t } T t=1 is increasing and {ν t } T t=1 is decreasing, it follows that V t (h t−1 , l) > V 1 (l) T τ =t+1 δ τ −t−1 T τ =1 δ τ −1 , and hence the left-hand side is larger than V 1 (l) = V o t . The only possible histories remaining are h t , for t = 1, . . . , T and the proof is concluded since V t (h t ) > V t (h t−1 , l). Monopoly Now consider an interaction between a Monopolist and a consumer. While the consumer's utility vector (V * l , V * h ) in the competitive model is an equilibrium object determined by firms' zero profit and no-deviation conditions, in the Monopolist problem, it is the result of an additional layer of optimization by the seller, taking into account the consumer's typedependant participation constraint. To be more precise, we consider the Monopolist's problem of designing a mechanism to maximize revenue, with the assumption that the consumer is privately informed about their initial type at the contracting stage. All future type realizations are also privately observed by the consumer. The consumer's outside option are given by V θ 1 defined in (20). Intuitively, the Monopolist's problem can be divided in two parts. First, for any utility level is delivered to the consumer, conditional on her initial type, the offered contract must provide these utility levels in a cost minimizing way. In other words, the optimal mechanism is the solution to Π(V ), for some V ∈ V. Second, the utility vector to be offered to the consumer, conditional on her initial type, must be chosen optimally. This gives rise to the following problem, which we denote as P M : max V ∈V Π(V ), subject to V i ≥ V i , for i = l, h. The next proposition shows that the Monopolist's optimal contract solves Π * (V l , V h ), with V l < V h . The participation constraint for the initially high-type buyer necessarily binds. However, the low-type buyer's participation constraint might be slack, i.e., the consumer may have information rents. This can be optimal because increasing the utility offered to the low-type buyer relaxes the binding incentive constraint in the profit maximization problem, leading to higher profits from the high-type buyers (Π h (·) increases with V l ). Proposition 7. The Monopolist's optimal offer is M V M , where V M is the solution to P M . Moreover, V M satisfies: V M h = V h , V M l ∈ [V l , V h ). Proof. The first part of the proposition is trivial. We now prove that V M h = V h . By way of contradiction, suppose this is not the case, i.e., (0) T t=1 δ t−1 for γ > 0 sufficiently small is feasible and a strict improvement given concavity of V and Π(·). V M h > V h . If, additionally, V M l ≥ V M h , then mechanism M V , with V = V M (1−γ)+γu On the other hand , if V M l < V M h , then mechanism M V , with V = (V M l , V M h − ε), for ε > 0 sufficiently small is feasible, since it is in the convex hull of V M , (V M l , V M l ) ⊂ V. It also strictly increases profits, which contradicts optimality of V M . We now show that V M l < V h . If, by way of contradiction, V M = (V h , V h ) then a reduction in the utility offered to the low-type is feasible and profitable since ∂Π ∂V l (V h , V h ) = −µ l ψ (c h ), where c h is the constant consumption flow that generates discounted utility V h (see Lemma 13 in the Appendix). Proposition 7 implies that the Monopolist's optimal mechanism is also a special instance of the problem proposed in Section 3 and characterized in Sections 5-7. Additionally, Proposition 7 allows us to obtain a simple characterization of information rents in the optimal contract. Given the concavity of Π(·), the utility vector V ≡ (V l , V h ) solves problem P M if, and only if, ∂Π ∂V l (V ) = − µ l u [u −1 (c l )] + µ h ∂ + Π h ∂V l (V ) ≤ 0,(24) where c l is the constant consumption flow that generates discounted utility V l . The first term in (24) corresponds to ∂Π l ∂V l (V ), which follows from the fact that the lowtype receives an efficient continuation contract in mechanism M V , for any utility vector V close to V . This analysis can be summarized in the following result. Proposition 8. The Monopolist's optimal mechanism leaves no information rents (i.e., V M = V ) if, and only if, u u −1 (c l ) ∂ + Π h ∂V l (V ) ≤ µ l µ h . In summary, the high-type receives no information rent and a distorted allocation as the least willing to pay for coverage. On the other hand, the low-type, who is most willing to pay for coverage, receives an efficient continuation contract. Additionally, if an initial lowtype is sufficiently likely, this type receives no information rent in the optimal mechanism. It is a key observation that the optimal mechanism always separates different initial types into different continuation contracts, even if both participation constraints bind. This is a consequence of the type-dependent outside options in this model. Conclusion This paper studies a natural dynamic extension of the workhorse insurance theoretical models proposed in the literature following Rothschild and Stiglitz (1976), Wilson (1977) and Stiglitz (1977). The analysis introduces new tools to the dynamic mechanism design literature to deal with the presence of curvature and exploit the recursive structure of the profit maximization problem, namely the use of an auxiliary cost minimization problem and the characterization of continuation-monotonicity. I show that the optimal contract uses both signals about consumer accidents as well as partial coverage as informative signals, affecting future offers made to consumers. In the case of realization-independent contracts, a strong efficiency result hods: distortions decrease along all histories. It is also shown that partial coverage contracts become more attractive over time, which implies -if T = ∞ or types are fully persistent -that the partial insurance contract offer becomes cheaper following a longer spell of partial coverage. The assumption of time-invariance on the types and income processes can be significantly relaxed. The results in Section 5 can be extended to time-dependent transitions π t ij i,j,t≤T and income distributions {p t i (·)} i,t≤T , as long as types are persistent, i.e., π t ii > π t ji for all periods. The time-invariance of income distributions p i (·) is necessary for the results in Section 7, as the relevant cost-minimization problem P A must be time-invariant. The analysis of competition focuses on pure strategies. If condition (22) does not hold, an equilibrium will necessarily involve randomization over mechanisms, following Farinha Luz (2017). In that paper, firms randomize over the utility level offered to both types, however all offers on the equilibrium path have the property that the utility delivered to the high-type is higher that the one delivered to the low-type. 14 It is a natural conjecture that a similar mixed strategy equilibrium may exist in the setting considered here, which would imply that on-path outcomes still correspond to special instances of the problem studied here. Multiple related directions for research remain open. The extension to multiple types leads to technical difficulties present in other dynamic mechanism design problems related to the relevance of non-local incentive constraints (see Battaglini and Lamba (2019)). A natural conjecture that follows from my characterization is that the optimal contract features a finite menu of flow contracts with different levels of coverage, with reductions in coverage at a given period being rewarded, leading to subsequent menus that are more attractive. The recursive characterization of monotonicity, in terms of flow allocation and continuation utilities, may prove useful in the study of other dynamic mechanism design problems with curvature and persistence, such as screening risk averse buyers or Mirrleesian taxation problems. A crucial question in dynamic contracting is the issue of commitment, especially on the side of consumers. The discussion in Subsection 8.1 illustrates the role that information asymmetry across firms may play as a commitment device. However, it is important to understand how the optimal contract structure changes as we consider more complex models where consumer reentry may be driven by preference or exogenous separation shocks leading to presence of switching consumers with high-types, improving firms' beliefs about switching consumers and hence leading to more attractive outside offers. firm, conditional on continuation utility levels V , as Π t (V ) ≡ sup M u t ∈M u t,OSIC E T τ =t δ τ −t [y t − ψ [ϑ τ ([θ τ ] τ t , [y τ ] τ t )]] | θ t−1 = h t−1 ,(25) subject to, for i ∈ {l, h}, V i = E T τ =t δ τ −t ϑ τ ([θ τ ] τ t , [y τ ] τ t ) | θ t = h t−1 , i ,(26)with Π T +1 (0, 0) = 0 if T < ∞. Denote as M * ,V t ∈ M u t, OSIC the solution to this problem, whenever is exists and is unique. Finally, for v ∈ 1−δ T −t+1 1−δ u (R + ), define the full information continuation profit function as Π F I t,i (v) ≡ T τ =t δ τ −t E [y t | θ t = i] − ψ 1 − δ 1 − δ T −t+1 v . It is easy to show that Π t (V ) ≤ π hl Π F I t,l (V l ) + π hh Π F I t,h (V h ). If V l ≥ V h , both functions coincide since offering constant utility 1−δ 1−δ T −t+1 V i , following type θ t = i , is feasible (incentive constraints of relaxed problem do not bind). Recursive representation The following preliminary result is useful in studying the recursive structure of the relaxed problem. Lemma 11. If problem Π t (·) has a solution then: (i) It is strictly concave, (ii) Its solution is unique. Proof. Proof of (i). For any t, consider utility pairs V 1 , V 2 ∈ V t , with V 1 = V 2 , and α ∈ (0, 1). For k = 1, 2, take optimal period t continuation UDMs M u,k t ≡ ϑ k τ T τ =t in problem Π t (V i ). The mechanism M u,α t = {αϑ 1 τ + (1 − α) ϑ 2 τ } T τ =t is feasible in Π t (V α ) and, since the objective function in (25) is strictly concave, generates profits strictly above αΠ t (V 1 ) + (1 − α) Π t (V 2 ). Since all incentive constraints as well as (26) are linear in utility flows, we have that M u,α t ≡ ϑ k τ T τ =t is feasible in Π t (V α ). Hence Π t (V α ) > αΠ t (V 1 )+(1 − α) Π t (V 2 ). Proof of (ii). Follows similarly from strict concavity of objective function and linearity of constraints in problem Π t (·). We extend the definition of ξ by defining, for any function ϑ : Y → u (R + ), ξ (ϑ, θ) ≡ y∈Y p θ (y) [y − ψ [ϑ (y)]] . Define a period t policy as a any pair (ϑ, N ) such that ϑ : Y → u (R + ) and N : Φ → V t+1 , and the set of period t policies as N t . Finally, define Γ t ≡            (V l , V h ) | ∃ (ϑ, N ) ∈ N t such that V h = y∈Y p h (y) [ϑ (y) + π hh N h (φ (y)) + π hl N l (φ (y))] V l ≥ y∈Y p l (y) [ϑ (y) + π hh N h (φ (y)) + π hl N l (φ (y))] V l ∈ 1−δ T −t+1 1−δ u (R + )            , and the following optimization problem, choosing the optimal period t policy: P t (V ) ≡ sup (ϑ,N )∈Nt π hh ξ (ϑ, h) + δ y∈Y p h (y) Π t+1 (N (φ (y))) + π hl Π F I t,l (V l ) s.t. V l ≥ y∈Y p l (y) {ϑ (y) + δ [π lh N h (φ (y)) + π ll N l (φ (y))]} V h = y∈Y p h (y) {ϑ (y) + δ [π hh N h (φ (y)) + π hl N l (φ (y))]} ,(27) with solution, whenever it exists, denoted as ϑ V t , N V t . Lemma 12. For any t = 1, ..., T and V ∈ V t , let {ϑ * τ } T τ =t be the solution to problem (25). The following hold, for any t ≤ T : (i) (Full insurance following low type) if UDM {ϑ * t } T τ =t solves Π t (V ), then, for any 1 ≤ τ ≤ T − t + 1 and y τ −1 ∈ Y τ −1 , ϑ t+τ (θ τ , y τ ) = ϑ t+τ (θ τ , y τ ) if θ τ , θ τ (h τ −1 , l) and y τ , y τ y τ −1 . 15 (ii) Recursive utility set representation: V t = Γ t , (iii) Π t (·) satisfies the following recursive equation, i.e., Π t (·) = P t (·), (iv) Flow contract and utilities following high type: for any y ∈ Y , V ∈ V t and i ∈ {l, h}, ϑ * t (h, y) = ϑ V t (y) , T τ =t+1 δ τ −t−1 ϑ ([θ] τ t , [y] τ t ) | θ t = h t , i , y t = y = N V i (φ (y)) . Proof. We start by proving (i). Consider problem Π t (V ), 1 ≤ τ ≤ T − t + 1, and y τ −1 ∈ Y τ −1 . Utility flows for histories (θ τ , y τ ) satisfying θ τ (h τ −1 , l) and y τ y τ −1 only affect the objective function through E T τ =t+τ −1 δ τ −t y τ − ψ ϑ τ θ τ τ t , y τ τ t | [θ T ] t+τ −1 t = (h τ −1 , l), [y T ] t+τ −2 t = y τ −1 ,(28) and they only affect incentive constraints through E T τ =t+τ −1 δ τ −t ϑ τ θ τ τ t , y τ τ t | [θ T ] t+τ −1 t = (h τ −1 , l), [y T ] t+τ −2 t = y τ −1 ,(29) Convexity of ψ implies that a constant utility flow, following type realizations (h τ −1 , l) and income realizations y τ −1 uniquely maximizes (28), subject to (29). Hence any solution to the profit maximization problem Π t must satisfy (i), in which case (28) is equal to Π F I t,l (V l ). Hence, we can restrict the choice set in the profit minimization problem to the set M u t,OSIC , which is the subset of M u t,OSIC satisfying (i). As a consequence, the continuation mechanism following a low-type announcement is pinned down by the continuation utility of the consumer: it corresponds to the constant consumption amount that delivers such continuation utility. We now show (ii)-(iv). Fix a period any t. Any OSIC mechanism inM u t,OSIC can be described by the continuation utility obtained conditional on θ t = l, the within-period t utility flows generated to a consumer with θ t = h and the continuation utility flows provided from period t + 1 onwards for a consumer with θ t = h (which constitutes a period t + 1 OSIC mechanism). This statement is formalized below. DefineĀ t ≡ R × [u (R)] Y × M u t+1,OSIC Φ as the set of triples (V l , ϑ(·),N (·)) where, in period t with type announcement h and income realization y ∈ Y , ϑ(y) represents flow utility in period t andN (φ(y)) ∈ M u t+1,OSIC represents the continuation mechanism offered from period t + 1 onwards. The flow utility in a continuation mechanismN (φ(y)), if history (θ τ , y τ ) are observed starting in period t + 1 is denoted asN (θ τ , y τ ; φ (y)). So a one-to-one mapping exists between the setM u t,OSIC and the set A t defined as                  V l , ϑ (·) ,N (·) ∈Ā t | ∃ (V l , V h ) ∈ R 2 s.t. V i (φ) = E T τ =t+1 δ τ −t−1N [θ τ , y τ ] τ t+1 ; φ | θ t+1 = i , V l ≥ y∈Y p h (y) [ϑ (y) + δ(π lh V h (φ (y)) + π ll V l (φ (y)))] , V l ∈ 1−δ T −t+1 1−δ u (R + ) .                  . (30) The one-to-one mapping a : A t →M u t,OSIC assigns, for each V l , ϑ (·) ,N (·) ∈ A t , the mechanism a V l , ϑ (·) ,N (·) = {ϑ τ } T τ =t satisfying: ϑ τ (θ τ −t , y τ −t ) = 1−δ 1−δ T −t+1 V l , for any (θ τ −t , y τ −t ) (l, y) and any y ∈ Y ; ϑ t (h, y) = ϑ (y) for any y ∈ Y ; and for any τ ≥ t + 1 and (θ τ −t−1 , y τ −t−1 ), ϑ τ ((h, θ τ −1 ) , (y, y τ −1 )) =N (θ τ −1 , y τ −1 | φ (y)). The mechanism a V l , ϑ (·) ,N (·) satisfies OSIC since: (i) the inequality in expression (30) implies that the period t one-shot incentive constraint is satisfied, and (ii) the one-shot incentive constraints for periods τ ≥ t + 1 are guaranteed since the continuation mechanismN (φ (y t )), for any y t ∈ Y , is contained in M u t+1,OSIC . It is easy to show that, for any M u t ∈M u t,OSIC , the vector a −1 (M u t ) is an element of A t . Moreover, the set of utilities generated by mechanisms a V l , ϑ (·) ,N (·) , for all V l , ϑ (·) ,N (·) ∈ A t coincides with Γ t , which implies (ii). Now consider any V ∈ V t and any mechanism a V l , ϑ (·) ,N (·) ∈M u t,OSIC feasible in the problem defining Π t (V ). Define N = (N l , N h ) : Φ → R 2 , for each i ∈ {l, h}, by N i (φ) ≡ E T τ =t+1 δ τ −t−1N [θ τ , y τ ] τ t+1 ; φ | θ t+1 = i . The new mechanism a V l , ϑ (·) , M * ,N (·) t+1 , where M * ,N (φ) t+1 ∈ M u t+1,OSIC is the optimal mechanism in the problem P t+1 (N l (φ) , N h (φ)), for all φ ∈ Φ, is also in A t , generates the same continuation utility in period t, for any θ t ∈ {l, h}, and generates strictly higher profits if N * ,N (·) t+1 =N . Hence, without loss of optimality in problem P t (V ), we can focus on mechanisms indexed by utility level V l and mappings (ϑ, N ) : Y → u (R + ) × V t+1 , which are given by a V l , ϑ (·) , M * ,N (·) t+1 . From now on, we refer to such mechanisms via the triple (V l , ϑ, N ) ∈ u (R + ) × [u (R + )] Y × [V t+1 ] Φ . The mechanism connected with (V l , ϑ, N ) satisfies OSIC and constraint (26) if, and only if, it satisfies V l ≥ y∈Y p l (y) {ϑ (y) + δ [π hh N h (φ (y)) + π hl N l (φ (y))]} ,(31) holding as an equality for ε = 0. Since the right-hand side (RHS) is differentiable and the lefthand side is strictly concave, their derivative coincides. The derivative of the RHS at ε = 0 is π hh ψ (u 0 ) = π hh d dv Π F I t,h (V 0 ), which pins down ∂ ∂V h Π t (V 0 , V 0 ). A similar argument can be used to find ∂ ∂V l Π t (V 0 , V 0 ), with an ε-perturbation that is feasible in problem P t (V 0 , V 0 + ε), for ε small. The following result shows what type of distortions arise in the profit maximizing mechanism and the dynamic behavior of continuation utilities. Lemma 14. For any V ∈ int (V t ), the solution ϑ V , N V of (27) satisfies: there exists multipliers λ ≥ 0 and µ > 0 such that: (i) Multiplier signal: λ > 0 and both constraints in (27) hold as equalities if, and only if, V h > V l , (ii) Within-period distortions: ϑ V t satisfies −ψ ϑ V (y) + µ − λ (y)    ≤ 0 ϑ V (y) = u (0) , = 0 , if ϑ V (y) > u (0) . (iii) Continuation utility rewards high types: if V h > V l , t < T , then N V h (φ) > N V l (φ) . for any φ ∈ Φ such that N V l (φ) , N V h (φ) > 1−δ T −t+1 1−δ u 0 , (iv) Continuation utility signal rewards: for any φ, φ ∈ Φ (φ ) ≤ (φ) =⇒ π ll N V l (φ ) + π lh N V h (φ ) ≥ π ll N V l (φ) + π lh N V h (φ) , with this inequality holding strictly if V h > V l and N V (φ) ∈ int(V t+1 ). Proof. Consider arbitrary period t and V ∈ int (V t ). Step 1. There exists period t policy (ϑ, N ) ∈ N t feasible in problem P t (V ) such that inequality constraint in (27) holds strictly: just consider a feasible policy in problem P t (V l − ε, V h ), for ε > 0 small. Step 2. The optimization problem P t (V ) has a concave objective, convex choice set N t and linear constraints. Step 1 implies that a feasible policy ϑ V , N V ∈ N t for problem P t (V ) is optimal if, and only if, there exists multipliers λ ≥ 0 and µ ∈ R such that ϑ V , N V , (λ) ∈ N t × R + form a saddle point of the Lagrangian ξ (ϑ, h) + y∈Y p h (y) ϑ (y) [µ − λ (y)] + λV l − µV h + φ∈Φ p h (φ) Π t+1 (N (φ (y))) + [µπ hh − λ (φ) π lh ] N h (φ (y)) + [µπ hl − λ (φ) π ll ] N l (φ (y)) .(32) Moreover, the necessary condition for optimality of N (·) is 0 ∈ ∂ i Π t+1 (N (φ(y))) + µπ hi − λ (φ)π li ,(33) while (ii) is a necessary condition for optimality of ϑ(·). Since ∂ − h Π t+1 (N (φ(y))) < 0, (33) implies µ > 0. Step 3 (item i). If λ = 0, (32) becomes the Lagrangian of the problemP t (V ), where the period t incentive constraint is ignored, which has as unique solution constant flow utility equal to 1−δ 1−δ T −t+1 V i , following announcement θ = i, which only satisfies the period t one-shot incentive constraint if V h ≤ V l . Alternatively, if V l ≥ V h , the optimal mechanism in P t (V ) involves constant utilities that do not depend on income y, which is only optimal in (32) if λ = 0. Step 4 (item ii). Property (ii) is equivalent to local optimality of ϑ V in (32). Step 6 (item iii). If V l < V h and N V h (φ 0 ) ≤ N V l (φ 0 ) for some φ 0 ∈ Φ, a contradiction follows. Necessary condition (33) implies ∂ + h Π t+1 N V (φ 0 ) + [µπ hh − λ (φ 0 ) π lh ] ≤ 0 ≤ ∂ − l Π t+1 N V (φ 0 ) + [µπ hl − λ (φ 0 ) π ll ] which, using λ > 0, gives us ∂ + h Π t+1 N V (φ 0 ) π hh ≤ λ (φ 0 ) π lh π hh − µ < λ (φ 0 ) π ll π hl − µ ≤ ∂ − l Π t+1 N V (φ 0 ) π hl .(34)Finally, N V h (φ 0 ) ≤ N V l (φ 0 ) and Lemma 13 imply that Π t+1 is differentiable at V and ∂ i Π t+1 N V (φ 0 ) π hi = d dV i Π F I t+1,i N V i (φ 0 ) = d dV l Π F I t+1,l N V i (φ 0 ) but, given convexity of Π F I t+1,l (·), we have ∂ h Π t+1 N V (φ 0 ) π hh > ∂ l Π t+1 N V (φ 0 ) π hl , which contradicts (34). Step 7 (item iv). For any φ ∈ Φ, equation (33) implies λ (φ)π ll − µπ hl λ (φ)π lh − µπ hh ∈ ∂Π t+1 (N V (φ(y))) Considering any two φ, φ ∈ Φ, convexity of Π t+1 (·) implies: Π t+1 (N V (φ )) − Π t+1 (N V (φ)) ≤ λ (φ)π ll − µπ hl λ (φ)π lh − µπ hh T [N V (φ ) − N V (φ)] Π t+1 (N V (φ)) − Π t+1 (N V (φ )) ≤ λ (φ )π ll − µπ hl λ (φ )π lh − µπ hh T [N V (φ) − N V (φ )] Summing up both equations gives us: 0 ≤ λ[ (φ) − (φ )] π ll π lh T [N V (φ ) − N V (φ)], which implies (iv). Proof of Lemma 3 Proof. Uniqueness follows from Lemma 11. The remaining statements are proved in the order: ii → iv → i → iii → v. Statement (ii) follows directly from Lemma 12-(i), which has Π 1 = Π * as a special case. All other properties are derived from Lemma 14. For any t = 1, . . . , T and η t−1 = (h t−1 , φ t−1 ), we have that: z t (y | η t−1 , h) = ϑ(y), V t+1 (η t−1 , φ t , h, i) = N i (φ t ), for i = l, h, where (ϑ, N ) is the solution to problem P t V t (η t−1 , l), V t (η t−1 , h) .(35) Proof of (iv). Follows from V l < V h and Lemma 14-(iii). Proof of (i). For any history η t−1 = (h t−1 , φ t−1 ), the result follows from (iv), which states that the continuation utility following θ t = h is strictly higher than that following θ t = l, and Lemma 14-(i), which implies that the within-period t upward incentive constraint binds as a result. Let µ t−1 (φ t−1 ) and λ t−1 (φ t−1 ) be the Lagrange multipliers of the problem (35), with η t−1 = (h t−1 , φ t−1 ). Proof of (iii). Follows from Lemma 14-(ii), using the following relationship to the solution ϑ(·) of problem (35): ψ (ϑ(y t )) = 1 u (z t (y t | η t−1 , h)) . Proof of (v). Follows directly from Lemma 14-(iv). Results on the auxiliary problem For simplicity, I write the auxiliary problem in terms of utility levels: χ (ν, ∆) = inf x:Y →u(R + ) y p h (y) ψ [x (y)] , subject to y p h (y) x (y) = ν, and y p l (y) x (y) = ν − ∆. The following statement provides important properties of cost function χ that are used in the analysis. Lemma 15. The problem P A satisfies the following: (i) It has a unique solution, (ii) χ(·) is strictly convex, moreover, if x (·) ∈ int [u (R + )] Y solves P A , then: (iii) χ (ν, ∆) is twice continuously differentiable in an open neighborhood of (ν, ∆), (iv) sign ∂χ(ν,∆) ∂∆ = sign (∆), (v) cross derivative sign: sign ∂ 2 χ ∂ν∂∆ =          sign (∆) , if ψ > 0 −sign (∆) , if ψ < 0 = 0, if ψ = 0, (vi) ψ (x(y)) = χ ν (ν, ∆) + χ ∆ (ν, ∆) [1 − (y)]. Proof. Existence of solution follows from the fact that x ∈ [u (R + )] Y | y p h (y) ψ [x (y)] ≤ K is compact, for any K ∈ R + . Uniqueness and convexity (items i-ii) follows from the strict convexity of the objective function and linearity of the constraints in (x, ν, ∆). The following are necessary and sufficient conditions for x (·) ∈ int [u (R + )] Y to be interior are: ∃λ, µ ∈ R such that ψ (x (y)) − λ + µ (y) = 0, y∈Y p h (y) x (y) = ν, y∈Y p l (y) x (y) = ν − ∆. for all y ∈ Y . Consider {y i } i∈I an ordering of Y such that { (y i )} i∈I is increasing. Then distributions {p h (y i )} i∈I and {p l (y i )} i∈I are ordered in terms of the monotone likelihood ratio property (MLRP). It then follows that {x (y i )} i∈I is decreasing (increasing) if µ > 0 (µ < 0), which implies that ∆ must be strictly positive (negative). As a consequence, sign (µ) = sign (∆) (item iv). If (λ, µ, x (·)) solve (36) − (38), then by the implicit function theorem the system has a unique continuously differentiable solution λ ν ,∆ , µ ν ,∆ , x (· | ν , ∆ ) for (ν , ∆ ) in an open neighborhood of (ν, ∆). Therefore χ is continuously differentiable at (ν, ∆), and its derivative is given by ∂ ∂ν χ (ν, ∆) ∂ ∂∆ χ (ν, ∆) = λ ν,∆ − µ ν,∆ µ ν,∆ . Continuous differentiability of λ ν,∆ , µ ν,∆ implies that χ (·) is twice continuously differentiable at (ν, ∆) (item iii). Finally, simple differentiation implies ∂ 2 χ (ν, ∆) ∂ν∂∆ = y p l (y) ψ (x(y)) − y p h (y) ψ (x(y)) y p h (y) ψ (x(y)) y [p l (y)] 2 p h (y)ψ (x(y)) + y p l (y) ψ (x(y)) 2 . Now assume ψ > 0, 1 ψ (x(y i )) i∈I is increasing (decreasing) in i if, only if, ∆ > 0 (∆ < 0). Since {p h (y i )} i∈I and {p l (y i )} i∈I are MLRP ordered, we have that sign ∂ 2 χ (ν, ∆) ∂ν∂∆ = sign (∆) . An analogous argument proves the cases ψ = 0 and ψ < 0 (item v). Item (vi) follows from (36) and (39). Lemma 16. The solution to P A for the pair (ν, ∆) satisfies 1 u (ζ(y)) = χ ν (ν, ∆) + χ ∆ (ν, ∆) [1 − (y)] . Proof. Follows from Lemma 15, with the solution in terms of utility flows and consumption being connected via 1 u (ζ(y)) = ψ (x(y)). The following statement connects variation in marginal cost of utility and distortions to the levels of utility and distortions and will be instrumental in the analysis of Subsection 7.1. Lemma 17. Suppose χ (·) is strictly convex and ∂ 2 χ ∂ν∂∆ > 0, then χ ν (ν, ∆) ≥ χ ν (ν , ∆ ) , χ ∆ (ν, ∆) ≤ χ ∆ (ν , ∆ ) ⇒ ν ≥ ν , ∆ ≤ ∆ . Additionally, if the left conditions hold with strict inequalities, then the implications also hold with strict inequalities. Proof. Since χ is twice continuously differentiable, the following holds: χ ν (ν, ∆) − χ ν (ν , ∆ ) = (ν − ν ) 1 0 χ vv (ι (α)) dα + (∆ − ∆ ) 1 0 χ ν∆ (ι (α)) dα ≥ 0, χ ∆ (ν, ∆) − χ ∆ (ν , ∆ ) = (ν − ν ) 1 0 χ ν∆ (ι (α)) dα + (∆ − ∆ ) 1 0 χ ∆∆ (ι (α)) dα ≤ 0, where ι (α) = α (ν, ∆) + (1 − α) (ν , ∆ ), for α ∈ [0, 1]. Using ∂ 2 χ ∂ν∂∆ > 0, these imply (∆ − ∆ ) 1 0 χ ν∆ (ι (α)) dα 1 0 χ vv (ι (α)) dα − 1 0 χ ∆∆ (ι (α)) dα 1 0 χ ν∆ (ι (α)) dα ≥ 0. However, convexity of χ (·) second order continuous differentiability, implies that the function Γ defined over a neighborhood of (0, 0), given by (v 0 , ∆ 0 ) → 1 0 χ (f (α) + (v 0 , ∆ 0 )) dα is also convex and twice continuously differentiable. Convexity implies that |Γ (0, 0)| = 1 0 χ vv (ι (α)) dα 1 0 χ ∆∆ (ι (α)) dα − 1 0 χ ν∆ (ι (α)) dα 2 > 0. This implies that ∆ ≤ ∆ , which together with χ ν (ν, ∆) ≥ χ ν (ν , ∆ ) implies ν ≥ ν . Moreover, if the left inequalities in the Lemma hold strictly, we have similarly that ∆ < ∆ and ν > ν . Proof of Lemma 8 Proof. The equivalence of (i) and (iii) follows from . The second order derivative of ψ = u −1 is given by ψ (x) = − u (ψ (x)) u (ψ (x)) 1 u (ψ (x)) 2 . The absolute risk aversion of utility u (·) at c ∈ R + is given by r u (c) ≡ − u (c) u (c) . Therefore direct derivation implies that ψ (x) = r u (x) u (x) + 2 [r u (x)] 2 [u (x)] 5 , where x = u (x). This implies the equivalence between (ii) and (iii). Utility and distortion dynamics In this section, we focus throughout on a particular t = 1, . . . , T − 1 and φ t−1 ∈ Φ t−1 and study the problem of reallocating both flow utilities and distortions across periods t and t + 1, following series of announcementsθ t = h t and signals φ t−1 . For notational brevity, we will omit the dependence of distortions and flow utilities on φ t−1 , denoting ν t (φ t−1 ) and ν t+1 (φ t−1 , φ) simply as ν t and ν t+1 (φ), for example. Define problem P I min (ν,∆,(ν (φ),∆ (φ),ν l (φ)) φ∈Φ )∈Ā χ(ν, ∆) + δ φ∈Φ p h (φ) π hh χ (ν (φ), ∆ (φ)) + π hl χ(ν l (φ), 0) subject to: ν + δ φ∈Φ p h (φ) π hh ν (φ) + π hl ν l (φ) = V t (h t−1 , h),(40)ν − ∆ + δ φ∈Φ p l (φ) π lh ν (φ) + π ll ν l (φ) = V t (h t−1 , l),(41) and ν (φ) − ∆ (φ) + δ φ ∈Φ p l (φ ) π lh V t+2 (h t+1 , (φ, φ ), h) + π ll V t+2 (h t+1 , (φ, φ ), l) = ν l (φ) + T t =t+2 δ t −t−1 u(c t+1 (φ)). (42) With the setĀ defined as A ≡ (ν, ∆, (ν (φ), ∆ (φ), ν l (φ)) φ∈Φ ) | (ν, ∆), (ν (φ), ∆ (φ)), (ν l (φ), 0) ∈ A, ∀φ ∈ Φ . Lemma 18. The vector ν t , ∆ t , ν t+1 (φ), ∆ t+1 (φ), ν l t+1 (φ) φ∈Φ , solves problem P I . Proof. First notice that, from Proposition 1, vector ν t , ∆ t , ν t+1 (φ), ∆ t+1 (φ), ν l t+1 (φ) φ∈Φ is feasible in P I . For any (ν, ∆, (ν (φ), ∆ (φ), ν l ) φ∈Φ ) ∈Ā, consider a new mechanism that is identical to optimal mechanism M except for changing: (i) z t (h t−1 , h) to ζ(ν, ∆), (ii) z t+1 (h t , φ, h) to ζ(ν (φ), ∆ (φ)), (iii) consumption in t + 1 following signals (φ t−1 , φ) and announcements (h t , l) to ψ(ν (φ)). All incentive constraints in relaxed problem for announcements in t ≥ t+2 are unaffected. All incentive constraints in relaxed problem for announcements in t ≤ t − 1 are unaffected since (40) guarantees that the continuation payoff of the consumer at the start of period t is unchanged. The period t incentive constraint, following φ t−1 , is guaranteed by (41), while all period t + 1 incentive constraints are guaranteed by (42). Finally, the mechanism modification only affects the firm's expected discounted profits via the expression in the objective function of problem P I . Optimality of M implies the result. Proof of Proposition 2. Using equations (40)-(42), we can simplify problem P I to one where only flow utilities in period t+1 are chosen by finding expressions for distortions (∆, (∆ (φ)) φ∈Φ ) and period t flow utility ν, in terms of (ν (φ)) φ∈Φ . By assumption, the solution of this problem is interior and hence the following local optimality conditions must be satisfied: χ 1 v + χ 1 ∆ 1 − π lh π hh (φ) = χ 2 v (φ) + χ 2 ∆ (φ),(43) and χ 1 v + χ 1 ∆ 1 − π ll π hl (φ) = χ 2,l v (φ) − π hh π hl χ 2 ∆ (φ),(44) where χ 1 k ≡ χ k (ν t , ∆ t ), χ 2 k (φ) ≡ χ k (ν t+1 (φ), ∆ t+1 (φ)), for k ∈ {ν, ∆}, and χ 2,l v ≡ χ v (ν l t+1 (φ)). Multiplying (43) by π hh p h (φ) and (44) by π hl p h (φ), adding both equations, and finally summing across signals φ ∈ Φ gives us (13). Subtracting (43) from 44, multiplying the result by p h (φ) and summing over φ ∈ Φ gives us (14). (15) implies that χ t,l ν > χ t ν = π hh χ t+1 ν + π hl χ t+1,l ν > χ t+1,l ν =⇒ ν l t > ν l t+1 . Combining (48), (49) and (50) we have ∆ t < δπ lh V d t+1 , and, using (45) once again, we have that V d t ≤ δπ hh V d t+1 , which contradicts property (ii), which holds at period t + 1 by our inductive assumption. We now prove property (ii). Since property (i) holds for any t ≥ t, Lemma 19 implies that ∆ t −1 > ∆ t for all t ≤ t ≤ T . Now notice that V d t = T τ =t [δ(π hh − π lh )] τ −t ∆ τ , and hence we have V d t−1 − V d t = T −t s=0 [δ(π hh − π lh )] s (∆ t+s−1 − ∆ t+s ) + [δ(π hh − π lh )] T −t ∆ T , which is strictly positive since the series {∆ τ } T τ =t−1 is strictly increasing. Proof of Proposition 3. Follows directly from Lemmas 19 and 20 Competitive analysis The two firms are labeled A and B. Let V E = (V E l , V E h ) denote the equilibrium utility level of the consumer conditional on her initial type, and V j = (V j l , V j h ) denote the vector describing the utility level obtained the consumer with both possible initial types when accepting the equilibrium offer of firm j = A, B. Finally, we denote the equilibrium profit of firm j = A, B as Π * j . Lemma 21. Any pure strategy PBE has outcome M V * . Finally, if V E See Hendel and Lizzeri (2003) andGhili et al. (2021). 2 SeeCooper and Hayes (1987) andDionne and Doherty (1994). This condition requires that the consumer's absolute risk aversion does not decrease "too quickly" with consumption, which is guaranteed if the consumer has non-decreasing absolute risk aversion. This statement assumes equal discount rates for both involved parties.Krasikov et al. (2021) show that, if the mechanism designer is more patient than the consumer, the efficiency gain from front-loading completely pins down transfers in the optimal mechanism. Restricting attention to mechanisms generating finite profits is a technical condition that avoids the analysis of transversality conditions when studying the firm's problem and is without loss for the applications considered in Sections 8 and 9. It is innocuous in the case T < ∞. The assumption of two firms is purely for notational convenience. A high-type corresponds to a low-risk in the notation used in Farinha Luz (2017). For completeness, we define Y 0 ≡ {∅} and impose that all income histories succeed ∅. Appendix 1 Recursive formulationIn this section, I provide a recursive representation of the firm's relaxed problem described in (4) This representation is then used to prove Lemma 3. To explore the recursive structure of the problem at hand, we define, for any period, the set of feasible continuation utilities that can be delivered to a consumer for any given current type as well as the maximal continuation profit that can be obtained by the firm.DefinitionsUtility mechanismsFor tractability, I will represent direct mechanisms through the utility flow generated for each period and history. Consider any direct mechanism M = {z t } T t=1 , period t and sequence of types and incomes (θ t , y t ) ∈ Θ t × Y t . The utility flow generated in period t isIn other words, we impose that the utility flow in period t only depend on the history of incomes y t−1 through the observable signal history φ t−1 . Similarly, a period t continuation UDM assigns utility flows beyond period t which depend on the history of types and income levels starting at period t, i.e., it is equal to, θ τ , y τmeasurable. I define the set of period t continuation UDM such that no one-shot OSCD with a misreport in periods τ ≥ t is profitable as M u t,OSIC .ProfitsDefine, for any t = 1, ..., T , the set of type-contingent continuation utilities generated by incentive compatible continuation mechanisms:It is easy to show that these sets are convex. For any V ∈ V t , define the problem P t (V ) of finding the maximal continuation profit obtained by aand it generates profitsHence, the maximal profit Π t (·) must satisfy (27) and the profit maximizing continuation UDM must be generated by the solution to(27). This implies (iii) and (iv).PropertiesWe use notation ∂ + i Π t (·) and ∂ − i Π t (·) to represent the right-and left-derivatives of Π t (·) with respect to V i , for i = l, h. For any time t, continuation utility vector V ∈ int(V t ) and constant c, we denote the expression1−δ u 0 and u 0 > u (0). The optimal period t policy ϑ V , N V induces constant utility flow u 0 , i.e., ϑ V (y) = u 0 , for all y ∈ Y . Now for ε sufficiently small, define an alternative policy θ , N V with utility flow given bỹThis alternative policy is feasible in problem P t (V 0 + ε, V 0 ), which impliesProof. We prove this statement by induction. First, consider t = T . In this case we have thatand, since Proposition 1 guarantees that ∆ T > 0, supermodularity implies (i) as:Lemma 19 then implies that ∆ T −1 > ∆ T . Property (ii) follows since V d T = ∆ t and V d T −1 = ∆ T −1 + δ(π hh − π lh )∆ T . Now suppose that properties (i)-(ii) hold for all t = t + 1, . . . , T . We start by showing that property (i) holds. The continuation utility of a high-type consumer satisfieswhile the continuation utility of a lot-type consumer satisfiesSubstituting equations(46)and(47)into(45)gives us the following, after some manipulation:Suppose, by way of contradiction, thatBut, given supermodularity of χ, this inequality requiresAlso, from our inductive hypothesis, we know that χ t+1 ν > χ t+1,l ν , which together withProof. The proof is divided into four parts. I) Π(V E ) = 0 and both firms make zero profits in equilibrium. Optimality of the consumer's acceptance strategy implies that, for j = A, B and i = l, h, the following hold:s offer is accepted with positive probability by type i. Hence, if firm j's offer is accepted by consumer with type i with positive probability, its continuation profits are at mostas increasing the utility of type i = i increases the profit opportunities from the consumer with type i. This means that the total profits obtained by firms is at mostHowever, by offering mechanism M V with V = V E + ( , ), for > 0 sufficiently small, each firm can guarantee profits Π(V ). In equilibrium, the offer made by each firm must dominate M V . Since profit function Π is continuous, we have lim →0 Π(V E +( , )) = Π(V E ).Combining the two implications, we haveThis implies that Π(V E ) = 0 and that both firms make zero profits.II) V E i ≥ V F I l , for i = l, h. First suppose that, by way of contradiction, V E l < V F I l . In this case firm A could guarantee positive profits by offering a mechanism with non-contingent constant consumption satisfyingfor > 0 sufficiently small, since it makes positive profits from the l-type consumer and, if the h-type consumer were to choose this contract, it would also generate positive profits since the discounted average income of the h-type consumer is strictly higher than that of the l-type consumer. h < V F I l , a similar profitable deviation exists.Suppose, by way of contradiction, that Π h (V E ) > 0. This implies that V E h < V F I h and hence concavity of the feasible utility set V implies that, for > 0 sufficiently small,and hence incentive compatibility implies that both types' equilibrium utility are strictly above T t=1 δ t−1 u(0). As a consequence, an incentive compatible mechanism in which the utility of both consumers is reduced exists, i.e., there exists V ∈ V such that V i < V E i , for i = l, h. The existence of feasible utility vectors (V E l , V E h + ), for > 0 and V , together with convexity of V, implies that: for ε > 0 sufficiently small, there exists utility vectorṼ ∈ V such thatConsidering ε sufficiently small, a deviation to offer MṼ leads to profits approximately equal towhich contradicts point I.IV) Points II and III imply that total profits are non-positive, conditional on any realization of the consumer's initial type. Combined with I, it implies thatLemma 22. A pure strategy equilibrium exists if, and only if,Proof. (If) Consider the following strategy profile. Both firms offer mechanism M V * and consumers follow an optimal equilibrium that follows truth-telling whenever optimal and treats firms symmetrically. Given the definition of V * , no firm has a profitable deviation in which a single type is attracted to its offer. This is trivially the case for type l. For type h, such a deviation would require that it offer utility pairHence, this strategy profile constitutes a pure strategy equilibrium if, and only if, no alternative mechanism can attract both types and generate strictly positive expected profits. But given concavity of Π(·), we can find V ≥ V * such thatif, and only if, we can find a pair (d h , d l ) ≥ 0 such thatwhere we have used the fact that Π(·) is differentiable in V l , whenever V l < V h . Since ∂ + Π ∂V h (V * ) < 0, such a pair exists if, and only if, ∂Π ∂V l (V * ) > 0. Condition ∂Π ∂V l (V * ) > 0 coincides with the expression in the lemma since, for V with V l < V h Π(V ) = µ l Π F I (V l ) + µ h Π h (V l , V h ).(Only if) If the condition in the Lemma fails, we can find V ≥ V * such that Π(V ) > Π(V * ). Now consider any pure strategy equilibrium with outcome M V * . Offer M V is a profitable deviation by any firm. Lock-in in dynamic health insurance contracts: Evidence from Chile. 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Optimal Auction Design. R B Myerson, Mathematics of Operations Research. 6Myerson, R. B. (1981): "Optimal Auction Design," Mathematics of Operations Research, 6, 58-73. Dynamic mechanism design: A myersonian approach. A Pavan, I Segal, J Toikka, Econometrica. 82Pavan, A., I. Segal, and J. Toikka (2014): "Dynamic mechanism design: A myersonian approach," Econometrica, 82, 601-653. Repeated moral hazard. W P Rogerson, Econometrica: Journal of the Econometric Society. Rogerson, W. P. (1985): "Repeated moral hazard," Econometrica: Journal of the Econo- metric Society, 69-76. Equilibrium in competitive insurance markets: An essay on the economics of imperfect information. M Rothschild, J Stiglitz, The Quarterly Journal of Economics. Rothschild, M. and J. Stiglitz (1976): "Equilibrium in competitive insurance markets: An essay on the economics of imperfect information," The Quarterly Journal of Economics, 629-649. Monopoly, non-linear pricing and imperfect information: The insurance market. J E Stiglitz, The Review of Economic Studies. Stiglitz, J. E. (1977): "Monopoly, non-linear pricing and imperfect information: The insurance market," The Review of Economic Studies, 407-430. A model of insurance markets with incomplete information. C Wilson, Journal of Economic Theory. 16Wilson, C. (1977): "A model of insurance markets with incomplete information," Journal of Economic Theory, 16, 167-207. Realization-independent mechanisms For brevity, we now define χ t k ≡ χ k (ν t , ∆ t ) and χ t,l ν ≡ χ ν (ν l t , 0) for for k ∈ {ν, ∆} and t = 1. Realization-independent mechanisms For brevity, we now define χ t k ≡ χ k (ν t , ∆ t ) and χ t,l ν ≡ χ ν (ν l t , 0) for for k ∈ {ν, ∆} and t = 1, . . . , T . We also introduce notation
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[ "LEARNING TO PARALLELIZE WITH OPENMP BY AUGMENTED HETEROGENEOUS AST REPRESENTATION", "LEARNING TO PARALLELIZE WITH OPENMP BY AUGMENTED HETEROGENEOUS AST REPRESENTATION" ]
[ "Le Chen ", "Quazi Ishtiaque Mahmud ", "Hung Phan ", "Nesreen K Ahmed ", "Ali Jannesari " ]
[]
[]
Detecting parallelizable code regions is a challenging task, even for experienced developers. Numerous recent studies have explored the use of machine learning for code analysis and program synthesis, including parallelization, in light of the success of machine learning in natural language processing. However, applying machine learning techniques to parallelism detection presents several challenges, such as the lack of an adequate dataset for training, an effective code representation with rich information, and a suitable machine learning model to learn the latent features of code for diverse analyses. To address these challenges, we propose a novel graph-based learning approach called Graph2Par that utilizes a heterogeneous augmented abstract syntax tree (Augmented-AST) representation for code. The proposed approach primarily focused on loop-level parallelization with OpenMP. Moreover, we create an OMP Serial dataset with 18598 parallelizable and 13972 non-parallelizable loops to train the machine learning models. Our results show that our proposed approach achieves the accuracy of parallelizable code region detection with 85% accuracy and outperforms the state-of-the-art token-based machine learning approach. These results indicate that our approach is competitive with state-of-the-art tools and capable of handling loops with complex structures that other tools may overlook.
10.48550/arxiv.2305.05779
[ "https://export.arxiv.org/pdf/2305.05779v1.pdf" ]
258,588,061
2305.05779
39ad6be740f39dedf8d786e0364438c8f5ad2a9e
LEARNING TO PARALLELIZE WITH OPENMP BY AUGMENTED HETEROGENEOUS AST REPRESENTATION Le Chen Quazi Ishtiaque Mahmud Hung Phan Nesreen K Ahmed Ali Jannesari LEARNING TO PARALLELIZE WITH OPENMP BY AUGMENTED HETEROGENEOUS AST REPRESENTATION Detecting parallelizable code regions is a challenging task, even for experienced developers. Numerous recent studies have explored the use of machine learning for code analysis and program synthesis, including parallelization, in light of the success of machine learning in natural language processing. However, applying machine learning techniques to parallelism detection presents several challenges, such as the lack of an adequate dataset for training, an effective code representation with rich information, and a suitable machine learning model to learn the latent features of code for diverse analyses. To address these challenges, we propose a novel graph-based learning approach called Graph2Par that utilizes a heterogeneous augmented abstract syntax tree (Augmented-AST) representation for code. The proposed approach primarily focused on loop-level parallelization with OpenMP. Moreover, we create an OMP Serial dataset with 18598 parallelizable and 13972 non-parallelizable loops to train the machine learning models. Our results show that our proposed approach achieves the accuracy of parallelizable code region detection with 85% accuracy and outperforms the state-of-the-art token-based machine learning approach. These results indicate that our approach is competitive with state-of-the-art tools and capable of handling loops with complex structures that other tools may overlook. INTRODUCTION The growing demand and popularity for multi-core hardware systems over the past few decades require developing highly-parallel programs to maximize performance. Numerous parallel programming models and frameworks (Chandra et al., 2001;Gabriel et al., 2004;Pheatt, 2008;Bik et al., 2002) have been created to facilitate the development of parallel code, but the developer's expertise in using these frameworks and familiarity with the codes are crucial to achieving better performance. Loop-level auto-parallelism helps developers in carrying out parallel tasks within the loops to speed up the process. Modern compilers typically detect the loop-level parallelism during compile time statically. This process is conservative and overlooks parallelism to ensure the correctness of the detected parallelism opportunities. On the other hand, dynamic auto-parallelism tools detect loop-level parallelism at runtime. The dynamic information captured after executing the programs improves the accuracy but has overhead issues. Moreover, the application of current auto-parallelization tools is constrained by requiring either compilation or execution of the programs 1 Department of Computer Science, Iowa State University, Ames, USA 2 Intel Labs, Santa Clara, CA, USA. Correspondence to: Le Chen <[email protected]>, Ali Jannesari <[email protected]>. Proceedings of the 6 th MLSys Conference, Miami Beach, FL, USA, 2023. Copyright 2023 by the author(s). for analysis. Therefore, a more practical way to auto-detect parallelism is required. Machine learning (ML) techniques are usually more feasible and cost-effective by redefining conventional software engineering problems as prediction problems. Many attempts have been made recently to use machine learning and Natural Language Processing (NLP) techniques in software engineering, from performance optimization and passes in compilers to solving complex problems such as malicious code detection, code placement on CPU or GPU, and performance prediction. Auto-parallelization with ML techniques is also conducted in recent studies. Chen et al. (Chen et al., 2022) detect parallelism by training code static and dynamic information in a multi-view model. The code embedding in their work is an adaption of word2vec (Mikolov et al., 2013), a now classic NLP technique. Ben-nun et al. (Ben-Nun et al., 2018) introduce a Neural Code Comprehension (NCC) representation of code by using graph embeddings that are trained on unlabelled data before being used for simple code comprehension tasks. Brauckmann et al. (Brauckmann et al., 2020) show that graph-embedding methods applied to Abstract Syntax Tree (AST) or Control Data Flow Graph (CDFG) are more efficient at downstream tasks than the state-of-the-art (NLP-inspired) methods, with better ability to generalize to never-seen-before examples. Despite their success, previous studies have shown common challenges in applying ML and NLP techniques in code arXiv:2305.05779v1 [cs.LG] 9 May 2023 Figure 1. Proposed methodology. Data collection and generation: our dataset contains data from GitHub crawling, benchmark collection, and synthetic data generation. Data pre-processing: we extracted loops from codes with pre-processing steps, e.g., removing comments and blank lines. We also label the data according to the extracted pragma. Code representation: we generate the AST of each loop data and convert it to our proposed augmented heterogeneous AST. Training and Prediction: we feed our processed data and corresponding labels to the HGT model for 4 different downstream tasks. analysis. First, constructing relevant datasets is a major pain point when attempting to solve any problem using machine learning. Only a few public benchmarks for parallelization using OpenMP are applicable to the parallelism detection task. Second, code representation is crucial for machine learning models to comprehend programs. The intuitive solution is treating code as a natural language so NLP models can be applied directly (Dai et al., 2019). However, the context or token representation overlooks the code's structural information, which is crucial for parallelization analysis (Blume et al., 1994;Chen et al., 2022). Finally, the performance of ML models varies across different tasks. In this work, we propose to leverage state-of-the-art machine learning techniques to detect loop parallelism and suggest four possible OpenMP pragmas to assist developers in implementing parallelization with OpenMP. We tackle the above-mentioned challenges by (a) generating a dataset containing 18598 parallelizable and 13972 non-parallelizable loops from benchmarks, GitHub projects, and synthetic data, (b) introducing a heterogeneous augmented-AST (aug-AST) representation for loops that considers both textual and structural information of code, and (c) training the heterogeneous aug-AST of the loops in our dataset using a heterogeneous graph neural network. In particular, this paper makes the following contributions: • Dataset. OMP Serial: a C serial loop dataset with labels that can be used for parallelization or other code analysis purposes. • Method. Introducing a heterogeneous augmented AST code representation suitable for parallelism detection and other downstream tasks. • Evaluation. Comparing the proposed graph-based approach with AST and token-based code representation approach. • Application. Implementing a heterogeneous GNN on the proposed dataset and comparing the results with state-of-the-art parallelization tools. MOTIVATION EXAMPLES This section demonstrates and discusses the limitations of three widely used algorithm-based auto-parallelization tools: DiscoPoP (Li et al., 2016), Pluto (Bondhugula et al., 2008), and autoPar (Quinlan & Liao, 2011). These non-ML tools are generally classified into static and dynamic (hybrid) approaches. Dynamic or hybrid parallelization tools like DiscoPoP (Li et al., 2016) identify parallelism with runtime dynamic information generated by executing the programs. Profiling and executing programs are costly in terms of time and memory. In contrast, static analysis tools such as Pluto (Bondhugula et al., 2008) and autoPar (Quinlan & Liao, 2011) examine source codes statically without execution. However, these static analysis tools tend to be overly conservative, often overlooking parallelization opportunities. In addition to their inherent limitation, the use of non-ML tools is constrained due to their need for compilation or execution of the program. When applied to the OMP Serial dataset introduced in section 4, only 10.3% and 3.7% of the C loops can be processed with autoPar (static) and DiscoPoP (dynamic), respectively. There are four types of loops where tools mostly make mistakes in our observation: loops with reduction, loops with function calls, loops with reduction and function calls, and nested loops. Listings 1, 2, 3, 4 and 5 present the example of mistakes made by autoPar, Pluto, and DiscoPoP. Figure 2 illustrates the statistic of our findings regarding the number and type of the loops these tools fail to detect parallelism. f o r ( i = 0 ; i < 5 ; i ++) f o r ( k = 0 ; k < 6 ; k += 2 ) l ++; Listing 5. Nested parallel loop (outermost for) missed by Discopop and Pluto. We are motivated to explore cutting-edge machine learning techniques for a more feasible and precise solution. The evaluation in section 6 demonstrates that our proposed approach surpasses the tools we examined in detecting parallelism within complex-structure loops. BACKGROUND The field of source code analysis encompasses a broad spectrum of topics, including bug detection, optimization, and auto-parallelization. Specifically, the parallelization of sequential programs constitutes a sub-field that concentrates on tasks such as detecting parallelism, classifying parallelization patterns, and implementing parallelization. This section delves into the background of parallelization analysis and explores machine learning approaches pertinent to this task. Auto-parallelization and Algorithm-based Tools Sequential program parallelization poses considerable challenges, generally involving two phases: parallelism identification and parallelization implementation. Parallelism identification entails the analysis of sequential program fragments to identify opportunities for parallelism. Parallelization implementation or execution involves capitalizing on the detected parallelism to fully exploit the hardware capabilities. Parallelism can be expressed through two fundamental concepts: task-level parallelism and loop-level parallelism. Task-level parallelism demarcates regions within an application that can be executed simultaneously on multiple cores or threads. Task-level parallelism methods require predefined distinct regions in the program, which can limit fine-grained opportunities. Loop-level parallelism considers loop bodies parallel regions, where iterations can be distributed across threads (Wismüller, 2011). This work primarily focuses on loop-level parallelism. The identification of loops eligible for parallelism often relies on the program author, as modern compilers are unable to fully take advantage of parallel loop classification. However, This process imposes a significant burden on developers, particularly for extensive projects. Most dynamic approaches employ dependency analysis to record execution order constraints between instructions, enabling a more accurate automatic parallelizable loop identification. In contrast, static methods infer dependencies by conservatively analyzing the program during compilation. Different static, dynamic, and hybrid (i.e., combining static and dynamic) tools have been developed to automatically identify parallelization opportunities. Polly (Grosser et al., 2012), an automatic parallelism detection tool, is based on static analysis, LLVM (Lattner & Adve, 2004), and the polyhedral model. Kremlin (Garcia et al., 2012) determines the critical path length within the loops using dependency information and subsequently calculates a metric, namely selfparallelism, for parallelism detection. Alchemist (Zhang et al., 2009) identifies parallelization candidates by comparing the number of instructions with the read-after-write (RAW) dependencies, both of which are generated by Valgrind (Nethercote & Seward, 2007) during runtime. Dis-coPoP (Li et al., 2016;Huda et al., 2016) extracts dynamic profiling and instruction dependency data from instrumented sequential programs. Information like dependency type, the number of incoming and outgoing dependencies, and critical path length are extracted from a data dependency graph for parallelism detection. As a hybrid method tool, Dis-coPoP provides comprehensive dynamic analysis statistics that complement static analysis, yielding an improved understanding conducive to detecting parallel opportunities. Machine Learning-based Auto-Parallelization Machine learning, as defined by Alpaydin et al. (Alpaydin, 2020), involves programming computers to optimize a performance criterion using example data or past experience. Despite its potential, machine learning techniques have been under-explored and infrequently employed in parallelization analysis tasks. Fried et al. (Fried et al., 2013) investigated an automatic method for classifying regions of sequential programs that could be parallelized, using benchmarks with hand-annotated OpenMP directives for training. Tournavitis et al. (Tournavitis et al., 2009) applied SVM in conjunction with static and dynamic features extracted from source codes to identify parallel regions in programs. They used NAS parallel benchmarks (Jin et al., 1999) and SPEC OMP benchmarks (Aslot et al., 2001) to evaluate their model. Machine learning techniques have achieved significant progress since (Fried et al., 2013)'s and (Tournavitis et al., 2009)'s work, with the recent advancements demonstrating the capabilities of deep neural networks in code representation (Cummins et al., 2021;Ma et al., 2021) and parallelization analysis (Shen et al., 2021;Chen et al., 2022). Code Representations The representation of code is crucial for applying machine learning techniques in the area of code analysis. This subsection discusses commonly used code representations and their corresponding machine learning approaches. Token. Programming tokens are fundamental elements that comprise the source code of a program. A token is a string of characters that can be classified as constants, identifiers, operators, reserved words, or separators according to the syntax of the programming language. Inspired by word embedding in natural language processing (NLP), various studies have focused on generating token-based embedding that can serve as input for machine learning approaches. The state-of-the-art token embedding method, code2vec (Alon et al., 2019), is trained on the task of predicting method names. AST. The abstract syntax tree (AST) is one of the most viable representations for code. Every programming language has an explicit context-free grammar, allowing source code to be parsed into an abstract syntax tree (AST) that represents the source code's abstract syntactic structure. Each non-leaf node in an AST corresponds to a non-terminal in the context-free grammar that conveys structural information, while each leaf node corresponds to a terminal in the context-free grammar encoding program text. Figure 3 illustrates an example of AST for listing 1. An AST can be easily converted back to source code. As our work focuses on parallelism at the loop level, we concentrate on partial ASTs that represent the desired loop. CFG. The control flow graph (CFG) delineates the sequence in which code statements are executed and the requirements that must be satisfied for a specific path of execution. Nodes represent statements and predicates, while directed edges connect them and indicate the flow of control. Although edges of CFGs need not follow any specific order, as in abstract syntax trees, it is still necessary to identify each edge as true, false, or otherwise CFG has been employed for various purposes, such as detecting versions of wellknown malicious apps and guidng fuzz testing tools. They are also now a common code representation in reverse engineering to aid in program comprehension. However, control flow graphs do not reveal data flow, making them unsuitable for detecting statements that process data modified by an attacker, a limitation particularly relevant to tasks like vulnerability analysis. Heterogeneous Graph Neural Networks (HGNN) Graph Neural Networks (GNN) models have gained success in various research domains, including biology (Zhang et al., 2021;Kim et al., 2022), natural language processing (Yao et al., 2018;Huang et al., 2019), image processing (Vasudevan et al., 2022;Shi et al., 2019), and software engineering (Allamanis et al., 2017;Kammoun et al., 2022;Huda et al., 2016;TehraniJamsaz et al., 2022). The application of GNNs relies on the ability to represent sequential data or databases as a complex structure with large-scale nodes and edges with structural information (Kipf & Welling, 2016). However, the homogeneous representation of these GNN models hindered their ability to represent meaningful information for prediction. Heterogeneous Graph Neural Network (HGNN) models are proposed to overcome this challenge . Compared to original GNNs, HGNN has the following advantages. First, HGNNs allow nodes to connect to all types of neighborhood nodes. In HGNNs, we can define the connection between any type of node without any restriction, which overcomes the drawback of several graph datasets that restrict the type of source node and target node for each edge, such as in . Second, HGNNs can accept not only different types of nodes but also nodes with different attributes. For example, with an academic graph, HGNN allows embedding information of profile picture and description of the author, as well as embedding information of textual content of Paper node, since Paper has no information like "profile picture". HGNNs propose a new mechanism for concatenating information and linear transformation between nodes to handle this. Third, HGNNs provide a solution for aggregating neighborhood information between neighbor nodes of different types to a more meaningful embedding per each iteration of training/ inference. To achieve this, HGNN allows representing the learning thanks to different types and weights of edges beside the nodes. The first complete HGNN model was proposed by Zhang et al. , called HetGNN. Hu et al. (Hu et al., 2020) proposed HGT, a transformer-based HGNN model that utilizes the graphs' properties more efficiently than HetGNN by decomposing interaction and transformation matrices to capture common and specific patterns of relationships between nodes and edges' types. Moreover, HGT allows embedding dynamic features such as the timeline of nodes and edges. From the work of Hu et al. (Hu et al., 2020), we justify the original HGT model to be trained and inference on parallelism detection. DATASET SELECTION AND ANALYSIS In this study, we propose a dataset, OMP Serial, from two distinct sources: open-source projects containing OpenMP pragmas and synthetic codes with specific parallelization patterns generated by template programming. In this section, we will discuss both approaches in detail. Open-source code data Our primary source of data is GitHub, where we crawled around 16000 source files from over 6000 repositories. We focused on C source files containing loops with and without OpenMP pragmas (pragmas can be either "#pragma omp parallel for" or "#pragma omp for"), ensuring that developers have intentionally used OpenMP directives in their code. To validate the data, we attempted to compile all the source codes using Clang to verify their correctness. Out of the 16000 source files, we were able to compile and retain 5731 source files for further analysis and experiments. Finally, we examined the label of the collected data using parallelization tools: Pluto, autoPar, and DiscoPoP and observed a small number of parallel loops missed by developers. Data Processing Data processing is necessary for the crawled source codes. The source codes are parsed to extract loops with comments removed and pragmas extracted. The loops are initially labeled as either parallel or non-parallel based on the presence of OpenMP pragmas. Loops without a pragma are classified as non-parallel. Parallel loops with OpenMP pragmas are further divided into four categories, namely private, reduction, simd, and target based on the extracted pragma and verified with various parallelization tools. Consequently, the OMP Serial dataset comprises labeled loops with their corresponding pragma clause, if present. Synthetic data To ensure pattern diversity for the OMP Serial dataset, we complemented the filtered crawled data with synthetic data. Both the crawled and synthetic data will be processed as described in section 4.2. We utilized Jinja2 (Ronacher, 2008) to generate complete C programs. For the do-all and reduction patterns, we created ten templates for each pattern and generated 20 variations of C source files from each template. We sourced the templates mainly from well-known parallel benchmarks such as the NAS Parallel Benchmark (Jin et al., 1999), PolyBench (Pouchet & Yuki, 2017), the BOTS benchmark (Duran et al., 2009), and the Starbench benchmark (Andersch et al., 2013). To create complete C programs, we inserted randomly generated variables, constants, and operators into the templates. The variable names were generated using a combination of English language alphabets (a-z, A-Z), digits (0-9), and underscores ( ). For do-all loops, we considered the operators: +, −, * , /. For reduction loops, we considered only + and * operators since reduction operations need to be associative and commutative for parallelization. We used DiscoPoP to verify the generated reduction and doall templates. Loops not identified as do-all or reduction by DiscoPoP were manually checked for inter-iteration dependencies or data-race conditions. If such conditions existed in the loop body, they were labeled as non-parallel loops. More details and examples on the generation of synthetic data can be found in Appendix ??. Finally, the OMP Serial dataset, comprising both open-source and synthetic data, is summarized in Table 1. APPROACH The representation of code is crucial for any analysis task. We propose an augmented heterogeneous AST representation for comprehending code in semantic and structural views. We first introduce the augmented AST (aug-AST) representation based on the control flow graph (CFG) and token distance in text format. Next, we append the types of nodes and edges in the aug-AST and build the augmented heterogeneous AST graph for each data point in our OMP serial dataset. We use the heterogeneous graph transformer (HGT) model (Hu et al., 2020) as our base model, taking the augmented heterogeneous AST graph as input. Code representation Code representations like AST and CFG provide crucial data for code analysis. However, a single representation is often insufficient to capture all the dependencies and parallelism. To address this issue, we propose an augmented AST that merges edges and nodes from the CFG, creating a single graph that incorporates the benefits of each distinct representation. Additionally, we address long-dependence problems by incorporating texture edges that follow the token distance map. Transforming the Abstract Syntax Tree To build a joint representation, we propose an augmented AST that incorporates both AST and CFG. We express the AST as a heterogeneous graph HA = (V A , E A , λ A , µ A ), where nodes V A represent AST tree nodes and edges E A represent corresponding tree edges labeled as AST edges by the labeling function λ A . Each node is assigned an attribute using µ A that corresponds to the operator or operand the node represents. Furthermore, we assign an attribute to each node to reflect the tree's ordered structure (left or right). The color blocks in Figure 3 represent the heterogeneous node attributes, while the black edges represent edges from the AST. Merging the Control Flow Graph To include the CFG in the joint representation, we express it as a heterogeneous graph GC = (V C , E C , λ C , ·). The nodes V C represent statements and predicates in the loop AST. We also introduce edges from nodes shared by the AST and CFG to nodes in the AST graph. These edges are represented by yellow dash lines in figure 3, where node f 1 is a function call node shared by both AST and CFG. These edges enable the machine learning model to identify potential data races within the function call and explore parallelization opportunities. Texture token relations In the work of (Zügner et al., 2021) Figure 3. An example of the proposed heterogeneous augmented AST (Heterogeneous aug-AST) representation of code in Listing 1 is shown. The colored blocks indicate the heterogeneous attributes assigned to the AST nodes. The red and yellow lines represent the control flow graph (CFG) and token representation, respectively. formation, leading to difficulties in capturing long-distance dependence relations. To address this issue, we add extra edges to link each leaf with its neighbors in the token representation as shown in figure 3. The added lexical edges (represented by red dashes) help aug-AST track the token distance. Heterogeneous Graph Transformer In this study, the input for the Heterogeneous Graph Transformer (HGT) model is the aug-AST graph generated from the original AST plus augmented nodes and edges. An aug-AST graph is represented as a heterogeneous graph, denoted by G = (V, E, A, R). Here, V denotes the set of nodes, E denotes the set of edges, A represents the possible types of nodes in V , and R represents the possible types of edges in E. For a given edge e = (s, t) with source node s and target node t, a meta-relation of the edge e is defined by the type of s, the type of t, and the type of edge e. In our work, three types of edges are considered: parent-child edges generated by the original AST and augmented CFG and lexical edges added to capture the control flow information and the relationship between neighbor leaf nodes. In the original GNN model, information is updated from the (l − 1)-th layer to the l-th layer by the formula 1. H l [t] = Aggregate(Extract(H l−1 [s]; H l−1 [t]; e)) (1) Here, h (l) v is the feature representation of node v at the lth layer, σ is the activation function, N out r (v) is the set of nodes that have an outgoing edge of type r from v, W (l) r is the trainable weight matrix for edge type r at layer l, and d (l−1) v is the degree of node v in the (l − 1)-th layer. In formula 1, the Extract operator extracts information from neighbor nodes s to target node t and the Aggregate() combines information from all the source that has the target node as t. In HGT, the mechanism of passing information between layers is split into three components: Heterogeneous Mutual Attention, Heterogeneous Message Passing, and Target Specific Aggregation. Mutual Attention. The input of this step is the node t and a set of N (t), which represents all the source nodes of the relation r. The heterogeneous mutual attention mechanism is calculated by taking the dot product between the source node s (Key vector) and the node t (Query vector). Next, the Key vector is projected using a linear projection to h attention heads, where each head is represented by a vector with d h dimension. Similarly, the Query vector is also projected into h Query vectors. For each head h, the Query vector is compared with the projection of the Key vector using a distinct edge-based matrix W ATT . Finally, the attention vector for each pair of nodes is produced by concatenating the h attention heads. The gathering of all attention vectors from the set of neighbor nodes N (t) to the target node t is shown in the formula 2. Attention HGT (s, e, t) = Softmax ∀s∈N (t) ( || i∈[1,h] AT T − head i (s, e, t)) (2) Message Parsing. While the Mutual Attention compares between Key vector and Query vector as target node and source node, the Message Passing mechanism operates in parallel. The input of Message Passing is not only the edge but also its meta relations. The formula of the Message operator is shown in the formula 3, where the MSG-head function is calculated by a number of components. Message HGT (s, e, t) = || i∈[1,h] MSG − head i (s, e, t)(3) The amount of components in the equation 3 is equal to the number of hidden layers. Similar to Formula 2, the Message Passing step also needs a matrix W MSG that embeds information of the edge dependency. In the final step, the output of each head calculated by the formula 4 is combined with a type-specific distribution of target node t through a linear projection: H (l) [t] = A − Linear (type(t)) (σ(H (l) [t])) + H (l−1) [t] (5) In Graph2Par, the distribution of type(t) is the set of different node types in the aug-AST. In the work of Hu et al. (Hu et al., 2020), they provide Inductive Timestamp Assignment and Relative Temporal Encoding to represent the dynamic heterogeneous graphs. However, since Graph2Par works with static and structural information of AST, we set the same temporal encoding mechanism and deactivated the inductive timestamp assignment in our HGT model. RESULTS In this section, we present the results of our experiments aimed at answering two research questions: 1. evaluating the performance of the proposed Heterogeneous augmented AST code representation, and 2. assessing the effectiveness of the proposed Graph2Par method for OpenMP pragma suggestion. Additional training results are provided in the appendix (see Appendix ??). Performance of the Heterogeneous aug-AST We demonstrate that our proposed Heterogeneous aug-AST representation outperforms both token-based and original AST representations by evaluating its performance in predicting parallelism. We compare the vanilla AST and the Heterogeneous aug-AST by using them as inputs to the same HGT model. Additionally, we reproduce PragFormer, the work of Harel et. al (Harel et al., 2022), to compare the performance of token representation and Heterogeneous aug-AST representation. PragFormer uses token-based representation as input to a transformer model for parallelism detection. Table 2 shows that our Heterogeneous aug-AST outperforms PragFormer in parallelism detection. The results of the above experiments demonstrate that the proposed Heterogeneous aug-AST representation outperforms both original AST and token-based representations in parallelism detection. In this subsection, we continue the evaluation of the aug-AST presentation by comparing it with well-known algorithm-based parallelism assistant tools: PLUTO, autoPar, and DiscoPoP. PLUTO and autoPar are algorithm-based static analysis tools, whereas DiscoPoP is an algorithm-based dynamic analysis tool. All three autoparallelization tools can detect parallelism in codes they can handle. However, parallelization pattern classification is not supported by all the tools. For example, simd and target clause predictions are not supported by any tools at present. Therefore, we conduct a performance comparison for the task of parallelism detection. As mentioned in section 4, loops in the OMP serial dataset are labeled 1 when the OpenMP clauses are present and labeled 0 otherwise. Graph2Par predicts the parallelism within a loop by a binary classification. PLUTO directly reports the parallelism detection results within a loop. autoPar injects OpenMP clauses like "#pragma omp parallel for" including "private" clause and "reduction" clause to the programs. We mark the detection results as parallel when the injected clauses are present. DiscoPoP can detect reduction and do-all patterns within a loop, and we considered the loops detected as either do-all or reduction by DiscoPoP as parallel loops. Different tools usually work with different sizes of data because they may require different information about the codes. DiscoPoP, for example, requires execution information for analysis, making it works with a much smaller dataset compared with static tools like PLUTO. Therefore, we divided our test dataset into three subsets for a fair comparison between Graph2Par and different tools. The results are presented in Table 4. Our Graph2Par model achieves superior performance compared to the other tools, indicating its effectiveness in detecting parallelism in sequential programs. • Subset PLUTO: This subset contains the loops that are in our testing set and can also be successfully processed by PLUTO. This set contains 4032 loops. • Subset autoPar: This subset contains the loops that are in our testing set and can also be successfully processed by autoPar. This set contains 3356 loops. • Subset DiscoPoP: This subset contains the source files that are in our testing set and can also be successfully processed by DiscoPoP. This set contains 1226 loops. We train our Graph2Par approach the three subset described above separately for comparison. In each training, one of the subsets was excluded to ensure that the model had not seen the samples before. The results are presented in tables 3 and 4. For all three subsets, our Graph2Par model achieved better precision, recall, F1 score, and accuracy than all the other tools. OpenMP Clause Classification The above results demonstrate that Graph2Par has the ability to learn the latent features of code for parallelism detection. In this subsection, we evaluate the extensibility of our Graph2Par model for predicting OpenMP pragmas, including "private", "reduction", "simd", and "target". We apply the same labeling strategy as the parallelism detection task, where the presence of the corresponding pragma determines the label of the loop. We train Graph2Par on the entire OMP serial dataset and evaluate on a separate test set. The results are presented in Table 5. We observe that our Graph2Par model performs well for the "private" and "reduction" pragma prediction tasks but struggles with the "simd" and "target" pragma prediction tasks. This is due to the limited representation of the aug-AST for certain pragma patterns, as some patterns may require additional information beyond the control flow graph and lexical edges represented by the aug-AST. It is worth noting that algorithm-based tools are not able to predict all of these pragmas or process every data point in our dataset. As the state-of-the-art model, PragFormer is used as a baseline for comparing the results of Graph2Par. Table 5 shows that our Graph2Par approach outperforms the SOTA token-based approach in both "private" and "reduction" pragma prediction tasks. Overall, the results demonstrate that our Graph2Par model has the potential to be extended to other OpenMP pragma prediction tasks, but additional features and representations may be required to handle more complex patterns. Dealing with False Positives From Table 4, it can be observed that our proposed Graph2Par has some false positives, meaning that it predicted some loops that are not parallel as parallel loops. In contrast, traditional tools like PLUTO, autoPar, and Dis-coPoP have zero false positives. However, Graph2Par is able to detect 1.8x, 5.2x, and 1.2x more parallel loops (true positives) in the Subset PLUTO, Subset autoPar, and Subset DiscoPoP datasets, respectively. This suggests that although Graph2Par may wrongly predict some loops as parallel, it can discover more parallelization opportunities than traditional approaches that are often conservative and may miss out on such opportunities. False positives are inevitable when embracing machine learning techniques since no model is perfect and can make mistakes. Parallelizing serial programs is complex, which makes it hard to do endto-end auto-parallelization, even with algorithm-based tools. There is more to consider for end-to-end approaches other than the parallelization pattern within the code, such as the characteristics of the platform on which the code executes, as well as the input data size and data dependencies. These factors can significantly impact the performance of the paral-lelized code, and their consideration is crucial for achieving optimal speedup. Therefore, it is important to carefully analyze and tune these factors in addition to identifying the parallelism opportunities within the code. Therefore, Graph2Par handles the false positives by only providing suggestions instead of generating end-to-end parallel code. The suggestion provided by Graph2Par includes whether parallelism exists within a loop and whether the loop inhibits any parallel patterns when parallelism is present. Developers can then use this information to parallelize the loops using any framework they prefer. For example, if a developer finds that a loop is parallel and has a reduction pattern, they can easily parallelize the loop using the "#pragma omp parallel for reduction" clause of OpenMP. However, there may be scenarios where the false positives are significant and need to be reduced to avoid confusion and save developers time. In such cases, developers may use additional tools to manually verify the suggested parallelism by Graph2Par. 6.5 Overhead. When generating the proposed aug-AST representation for a loop, the steps mentioned in section 5 are followed. The overhead of creating an aug-AST comes from two steps: code compilation with Clang and AST traversal with treesitter (Brunsfeld, 2018). However, both steps introduce minimal overhead. It is important to note that the overhead of creating an aug-AST may increase for larger size codes. However, for the loops in the OMP serial dataset, which have an average size of 6.9 lines, the overhead is minimal and in the order of milliseconds. Case Study In the evaluation, it is observed that our proposed model can successfully identify 48 parallel loops missed by all three algorithm-based tools. An example of one such loop is presented in Listing 6, and other examples can be found in Listings 1, 2, 3, 4, and 5 in the motivation examples. These results demonstrate the effectiveness of our Graph2Par approach in detecting parallelism opportunities that are missed by traditional algorithm-based tools. f o r ( i = 0 ; i < 1 0 0 0 ; i ++){ a [ i ] = i * 2 ; sum += i ; } Listing 6. Parallel loop missed by DiscoPoP, PLUTO and autoPar with array and reduction Another example is shown in Listing 7. We believe that the conservative nature of non-AI-based parallelism assistant tools may be the reason for missing such opportunities. In this specific example, although there is a reduction operation on the variable sum and memory access to the 2D array a, only the j index is changing, and there are no interiteration dependencies. Therefore, this loop can be executed in parallel, and it is successfully detected by our Graph2Par model. Furthermore, our proposed Graph2Par model can handle parallelism detection in nested loops effectively, which is a challenging problem due to the complex dependencies between the loops. As an example, in Listing 8, the outer parallel loop has been missed by all traditional parallelism assistant tools due to its nested structure. However, our model successfully detects that the outer-most f or loop can be parallelized. By observing that each cell of the 3-d array a will eventually have the same value and that m is just a constant, we can verify that there are no loop-carried dependencies, and the loop can be safely parallelized. RELATED WORK Recent research has shown an increasing trend in employing machine learning techniques for parallelization analysis. These studies can be broadly classified into two categories based on their code representations. Token-based code analysis studies (Fried et al., 2013;Harel et al., 2022) used natural language processing (NLP) models trained on raw code text data. In contrast, recent studies such as (Shen et al., 2021;Chen et al., 2022) have leveraged structured graphical models with the structural representation of code, such as the Abstract Syntax Tree (AST). Compared to these works, our proposed Heterogeneous augment-AST representation is easy to process and contains rich information on nodes and edges, enabling more accurate and efficient parallelization analysis. CONCLUSION In this paper, we propose a static approach to discover parallelism in sequential programs using an augmented AST representation. To address the issue of data insufficiency, we created the OMP Serial dataset, which can be used for other parallelization tasks as well. We evaluate the aug-AST representation using a GNN-based model, and it outperforms traditional parallelization tools as well as token-based machine learning approaches. However, there is still room for improvement in our model. Currently, Graph2Par can only detect whether a pragma is applicable for a loop or not, but future research directions could focus on developing a model that can generate complete OpenMP pragmas for sequential loops. Figure 2 .Listing 1 .. 21Category-wise loops missed by renowned parallelization assistant tools. The results are generated using the OMP Serial dataset introduced in section 4. Parallel loop with reduction and function call. DiscoPoP, Pluto, and autoPar fail to detect the parallelism due to the f abs function call.f o r ( i n t i = 0 ; i < n u m p i x e l s ; i ++) Parallel loop with reduction and function call missed by Pluto because of the abs function call. . Parallel loop with a function call missed by autoPar because of the square function call. Listing 4 . 4Parallel loop with reduction missed by Discopop because of the reduction operation on variable v. . Parallel loop missed by DiscoPoP, PLUTO and autoPar with array and reduction . Parallel loop missed by DiscoPoP, PLUTO and autoPar with nested loop Comprehensive graph representations. Recent works with code representation have focused on comprehensive graph representations to incorporate more information about programs. Ben-Nun et al. (Ben-Nun et al., 2018) aimed to create an embedded representation of code based on LLVM IR, introducing an intermediate representation of programs by combining NLP techniques with code dependencies.Cummins et al.(Cummins et al., 2021) expanded upon the work of Ben-Nun et al. to propose an IR graph representation called PrograML, which is both comprehensive and rich in code information. The downstream task experiments set a new state-of-the-art standard. However, the requirements for using PrograML are stringent due to LLVM compilation, and only 31.2% of the data in our dataset can be processed with PrograML. Consequently, we adopt AST as our base representation of code to utilize all the data for training. Table 1 . 1Statistic Summary of the proposed OMP Serial dataset comprises synthetic code and code collected from GitHub. Each data in the OMP Serial represents a loop with labels indicating whether it is parallelizable or not. Parallelizable loops also include parallel pattern labels. The Loops column displays the number of loops for each type of pragmas. The Function Call and Nested Loops columns represent the number of loops with functions and nested loops for each type of pragmas, respectively. The Avg. LOC stands for the average length of code.Source Type Total Loops Pragma Type Loops Function Call Nested Loops Avg. LOC GitHub Parallel 18598 reduction 3705 279 887 6.35 private 6278 680 2589 8.51 simd 3574 42 201 2.65 target 2155 99 191 3.04 Non-parallel 13972 - - 3043 5931 8.59 Synthetic Parallel 400 reduction 200 200 100 31.59 private (do-all) 200 200 100 28.26 Non-parallel 700 - - 0 0 6.43 Table 2 . 2Result of pragma existence prediction. PragFormer uses token representations.Precision Recall F1 Accuracy AST 0.74 0.73 0.74 0.74 PragFormer 0.81 0.81 0.80 0.80 Graph2Par 0.92 0.82 0.87 0.85 Table 3 . 3Number of detected parallel loops comparing with algorithm-based approaches.Approach # of detected parallel loops Graph2Par 17563 HGT-AST 16236 DiscoPoP 953 PLUTO 1759 autoPar 6391 6.2 Parallelism Discovery: Comparing with other tools Table 4 . 4Comparing Graph2Par model with PLUTO, autoPar and DiscoPoP for the task of parallelism detection (Detecting the presence of "#pragma omp for" or "#pragma omp parallel for")TP TN FP FN Precision Recall F1 Accuracy(%) Subset PLUTO PLUTO 1593 0 0 2439 100.00 39.51 56.64 39.51 Graph2Par 2860 617 356 199 88.93 93.49 91.16 86.24 Subset autoPar autoPar 345 952 0 2059 100.00 14.35 25.10 38.65 Graph2Par 1800 897 187 472 90.59 79.23 84.53 80.36 Subset DiscoPoP DiscoPoP 541 240 0 445 100.00 54.87 70.86 63.70 Graph2Par 635 366 64 161 90.84 79.77 84.95 81.65 Table 5 . 5Performance of Graph2Par for four pragma prediction.Pragma Approach Precision Recall F1-score Accuracy private Graph2Par 0.88 0.87 0.87 0.89 PragFormer 0.86 0.85 0.86 0.85 reduction Graph2Par 0.9 0.89 0.91 0.91 PragFormer 0.89 0.87 0.87 0.87 SIMD Graph2Par 0.79 0.76 0.77 0.77 PragFormer N/A N/A N/A N/A target Graph2Par 0.75 0.74 0.74 0.74 PragFormer N/A N/A N/A N/A Learning to represent programs with graphs. 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Efficient estimation of word representations in vector space, 2013. Valgrind: a framework for heavyweight dynamic binary instrumentation. N Nethercote, J Seward, ACM Sigplan notices. 426Nethercote, N. and Seward, J. Valgrind: a framework for heavyweight dynamic binary instrumentation. ACM Sig- plan notices, 42(6):89-100, 2007. Intel® threading building blocks. C Pheatt, Journal of Computing Sciences in Colleges. 234Pheatt, C. Intel® threading building blocks. Journal of Computing Sciences in Colleges, 23(4):298-298, 2008. L.-N Pouchet, T Yuki, Polybench, The polyhedral benchmark suite. 2017version 4.2Pouchet, L.-N. and Yuki, T. Polybench: The polyhedral benchmark suite (version 4.2), 2017. The ROSE source-to-source compiler infrastructure. D Quinlan, C Liao, Cetus users and compiler infrastructure workshop, in conjunction with PACT. Citeseer2011Quinlan, D. and Liao, C. The ROSE source-to-source com- piler infrastructure. In Cetus users and compiler infras- tructure workshop, in conjunction with PACT, volume 2011, pp. 1. Citeseer, 2011. . A Ronacher, Jinja2 documentation. Welcome to Jinja2-Jinja2 Documentation (2.8-devRonacher, A. Jinja2 documentation. Welcome to Jinja2-Jinja2 Documentation (2.8-dev), 2008. Towards parallelism detection of sequential programs with graph neural network. Y Shen, M Peng, S Wang, Q Wu, Future Generation Computer Systems. 125Shen, Y., Peng, M., Wang, S., and Wu, Q. Towards paral- lelism detection of sequential programs with graph neural network. Future Generation Computer Systems, 125: 515-525, 2021. Skeleton-based action recognition with directed graph neural networks. L Shi, Y Zhang, J Cheng, H Lu, 10.1109/CVPR.2019.008102019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). Shi, L., Zhang, Y., Cheng, J., and Lu, H. Skeleton-based action recognition with directed graph neural networks. In 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 7904-7913, 2019. doi: 10.1109/CVPR.2019.00810. Learning intermediate representations using graph neural networks for numa and prefetchers optimization. A Tehranijamsaz, M Popov, A Dutta, E Saillard, Jannesari , A , 2022 IEEE International Parallel and Distributed Processing Symposium (IPDPS). IEEETehraniJamsaz, A., Popov, M., Dutta, A., Saillard, E., and Jannesari, A. Learning intermediate representations using graph neural networks for numa and prefetchers optimiza- tion. In 2022 IEEE International Parallel and Distributed Processing Symposium (IPDPS), pp. 1206-1216. IEEE, 2022. Towards a holistic approach to auto-parallelization: integrating profile-driven parallelism detection and machinelearning based mapping. G Tournavitis, Z Wang, B Franke, M F Boyle, ACM Sigplan notices. 446Tournavitis, G., Wang, Z., Franke, B., and O'Boyle, M. F. Towards a holistic approach to auto-parallelization: inte- grating profile-driven parallelism detection and machine- learning based mapping. ACM Sigplan notices, 44(6): 177-187, 2009. Image classification using graph neural network and multiscale wavelet superpixels. CoRR, abs/2201.12633. V Vasudevan, M Bassenne, M T Islam, L Xing, Vasudevan, V., Bassenne, M., Islam, M. T., and Xing, L. Image classification using graph neural network and mul- tiscale wavelet superpixels. CoRR, abs/2201.12633, 2022. URL https://arxiv.org/abs/2201.12633. . R Wismüller, Loops, 10.1007/978-0-387-09766-4_27Springer USBoston, MAWismüller, R. Loops, Parallel, pp. 1079-1087. Springer US, Boston, MA, 2011. ISBN 978-0-387-09766-4. doi: 10.1007/978-0-387-09766-4 27. URL https://doi. org/10.1007/978-0-387-09766-4_27. Graph convolutional networks for text classification. L Yao, C Mao, Y Luo, CoRR, abs/1809.05679Yao, L., Mao, C., and Luo, Y. Graph convolutional networks for text classification. CoRR, abs/1809.05679, 2018. URL http://arxiv.org/abs/1809.05679. Heterogeneous graph neural network. C Zhang, D Song, C Huang, A Swami, N V Chawla, Proceedings of the 25th ACM SIGKDD international conference on knowledge discovery & data mining. the 25th ACM SIGKDD international conference on knowledge discovery & data miningZhang, C., Song, D., Huang, C., Swami, A., and Chawla, N. V. Heterogeneous graph neural network. In Proceed- ings of the 25th ACM SIGKDD international conference on knowledge discovery & data mining, pp. 793-803, 2019. Alchemist: A transparent dependence distance profiling infrastructure. X Zhang, A Navabi, Jagannathan , S , 2009 International Symposium on Code Generation and Optimization. IEEEZhang, X., Navabi, A., and Jagannathan, S. Alchemist: A transparent dependence distance profiling infrastructure. In 2009 International Symposium on Code Generation and Optimization, pp. 47-58. IEEE, 2009. Graph neural networks and their current applications in bioinformatics. X.-M Zhang, L Liang, L Liu, M.-J Tang, Frontiers in genetics. 122021Zhang, X.-M., Liang, L., Liu, L., and Tang, M.-J. Graph neural networks and their current applications in bioinfor- matics. Frontiers in genetics, 12, 2021. Language-agnostic representation learning of source code from structure and context. D Zügner, T Kirschstein, M Catasta, J Leskovec, S Günnemann, arXiv:2103.11318arXiv preprintZügner, D., Kirschstein, T., Catasta, M., Leskovec, J., and Günnemann, S. Language-agnostic representation learn- ing of source code from structure and context. arXiv preprint arXiv:2103.11318, 2021.
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[ "Implementation and analysis of Ryze Tello drone vision-based positioning using AprilTags", "Implementation and analysis of Ryze Tello drone vision-based positioning using AprilTags" ]
[ "Kacper Hulek \nWarsaw University of Technology Św. A. Boboli 8\nWarsawPoland\n", "Mariusz Pawlicki \nWarsaw University of Technology\nŚw. A. Boboli 8WarsawPoland\n", "Adrian Ostrowski \nWarsaw University of Technology\nŚw. A. Boboli 8WarsawPoland\n", "Jakub Możaryn \nWarsaw University of Technology\nŚw. A. Boboli 8WarsawPoland\n" ]
[ "Warsaw University of Technology Św. A. Boboli 8\nWarsawPoland", "Warsaw University of Technology\nŚw. A. Boboli 8WarsawPoland", "Warsaw University of Technology\nŚw. A. Boboli 8WarsawPoland", "Warsaw University of Technology\nŚw. A. Boboli 8WarsawPoland" ]
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The paper describes of the Ryze Tello drone to move autonomously using a basic vision system. The drone's position is determined by identifying AprilTags' position relative to the drone's built-in camera. The accuracy of the drone's position readings and distance calculations was tested under controlled conditions, and errors were analysed. The study showed a decrease in absolute error with decreasing drone distance from the marker, a little change in the relative error for large distances, and a sharp decrease in the relative error for small distances. The method is satisfactory for determining the drone's position relative to a marker.
10.48550/arxiv.2305.05673
[ "https://export.arxiv.org/pdf/2305.05673v1.pdf" ]
258,588,173
2305.05673
245c58264b357184ea319130ce4de4f0c3c9739e
Implementation and analysis of Ryze Tello drone vision-based positioning using AprilTags Kacper Hulek Warsaw University of Technology Św. A. Boboli 8 WarsawPoland Mariusz Pawlicki Warsaw University of Technology Św. A. Boboli 8WarsawPoland Adrian Ostrowski Warsaw University of Technology Św. A. Boboli 8WarsawPoland Jakub Możaryn Warsaw University of Technology Św. A. Boboli 8WarsawPoland Implementation and analysis of Ryze Tello drone vision-based positioning using AprilTags Drone TelloComputer VisionMATLABPositioning System The paper describes of the Ryze Tello drone to move autonomously using a basic vision system. The drone's position is determined by identifying AprilTags' position relative to the drone's built-in camera. The accuracy of the drone's position readings and distance calculations was tested under controlled conditions, and errors were analysed. The study showed a decrease in absolute error with decreasing drone distance from the marker, a little change in the relative error for large distances, and a sharp decrease in the relative error for small distances. The method is satisfactory for determining the drone's position relative to a marker. I. INTRODUCTION Drones are becoming increasingly popular in various fields, such as photography, surveying, transportation, and military systems. The precise positioning of drones is essential for their safe and accurate movement. The GPS is one of the primary positioning systems used for drone flight and is widely used in the field. However, GPS accuracy is significantly reduced in closed spaces. Our project aimed to develop a solution that only requires a simple vision system to achieve accurate positioning in closed spaces. Our solution eliminates the need for additional hardware and involves providing good lighting and using standardized AprilTags to mark the places where the drone needs to move. We used the "tag36h11" family tag to ensure real-time performance while minimizing false-positive detections. In the following sections, we discussed the technologies we used and the preparation of the drone workspace required for the program to function correctly. We also presented the operating procedure of our application, including a discussion of the graphical user interface available to the user. Finally, we included an error analysis of the drone's accuracy in determining its position relative to the marker. II. IMPLEMENTATION DETAILS A. Required hardware The Tello drone is a small quadcopter with a vision positioning system and an onboard camera. It can hover in place using a vision positioning system and an advanced flight controller. It has advanced features such as Bounce mode, 8D Flip and EZ Shots. The drone takes 5 MPx photos and streams live video in 720 p resolution. Maximum flight length reaches up to 100 meters. Fig. 1). In addition, the library also has functions to collect data about its location, speed, and battery status. The capabilities of the library, and the functions we used, are described in more detail in the next section. B. Required software The first part of the project involved developing a functionality to detect AprilTags and determine their location. To accomplish this, we used the ReadAprilTag library [3], part of the Computer Vision Toolbox [4] in the MATLAB environment. Initially, we needed to establish the camera parameters to determine the actual displacement of the tags relative to the captured image. To obtain high-quality images, we chose to use the snapshot() function from the Support Package for Ryze Tello Drone library, which enables the capture of a single image from the drone's camera. Although streaming video using the preview() function is possible, the video quality in streaming mode is noticeably worse. It might need to be improved to determine the location of markers accurately. A Wi-Fi module is required to connect to the drone and run the program remotely. III. RYZE TELLO DRONE POSITIONING PROCEDURE A. Determining the camera parameters of the Ryze Tello drone To read the position of AprilTag tags correctly, it is necessary to define the actual size of the tag and then specify the camera parameters. In our tests, tags with a side size of 184 mm were used, but the created program allows to work with tags of any size. In this case, the user must specify a custom size in the application window. To enable the tag detection algorithm to work, the camera parameters required are the resolution of the camera image (ImageSize[x, y]), specified in two-axis parameters in pixels, the center point of the sensor (PrincipalPoint[x, y]), also given in two-axis parameters in pixels, and the axial focal length (FocalLength[x, y]), also specified in two-axis parameters in pixels. For the Ryze Tello drone, the ImageSize value documented equals 960 by 720 pixels. Assuming ideal sensor performance, we can assume that the PrincipalPoint parameter specifies the point exactly in the center of the image. Therefore, we can take half of the ImageSize parameter's value as its value. Regarding the FocalLength parameter, the readAprilTag [5] library documentation states that it is the product of the value of the primary focal length of the camera and the quotient of the resolution of the maximum working area of the sensor and the physical size of this area in a given axis (Fig. 2). We can determine this value using the following formula (1) Where fx,y is the desired axial focal length, f is the normal focal length value of 4mm for this camera, rx,y is the resolution along a given axis in pixels, and sx,y is the sensor dimension along a given axis in mm. Images from the drone's camera are captured using the snapshot() function, and then to get rid of image distortion, including mainly the fisheye effect causing angular warping at the edges of the image, the undistortimage() function, a part of the Computer Vision Toolbox library, is called (Fig. 3). B. Preparation of the flight environment It is crucial to ensure proper flight conditions to fly a Ryze Tello drone based on positioning relative to AprilTag tags. The drone comes equipped with a position stabilization system with a vision system consisting of cameras built into the device aimed at the ground. This system is designed to keep the drone in a fixed position relative to the ground (Fig. 4). To ensure that the system works properly, it is essential to ensure that the ground pattern is not uniform and is as contrasting as possible. A soft substrate can also minimize the airflow created by the drone's rotors, which can cause unwanted movement of the device. Additionally, the flight environment should be well-illuminated to enable the correct identification of the markers. These measures are crucial to ensure the proper functioning of the drone's vision system and accurate positioning relative to the AprilTag markers. C. The process of identifying the location of AprilTags To implement AprilTag detection, the readAprilTag() function from the readAprilTag library [3] is used. This function returns a pose object containing the detected tags' position in the dimensionless units used in the library. An empty pose object is returned if no marker is detected in the image. The Translation() method is used on this pose object to obtain the actual offset of the markers relative to the camera in meters. This method returns vectors that represent the position of the markers relative to the point [0, 0, 0] that defines the camera. These vectors contain [x, y, z] coordinates corresponding to rightward, downward, and forward displacement. Using this method, the precise location of the markers relative to the camera can be determined and used for accurate drone positioning. D. The process for autonomous movement of the Ryze Tello drone To enable control and measurement of the Ryze Tello drone, we utilized the Support Package for the Ryze Tello Drone library. This library provided the means to control the drone's movement and obtain real-time altitude and battery status parameters through a graphical interface. To establish a connection between the drone and the program, we used the ryze() function after connecting the drone to the computer via wifi. We also created a camera object using the camera() function to facilitate image capture. We employed the move() and turn() functions for drone movement. In the move() function, the [x, y, z] coordinates correspond to forward, right, and downward movement. However, this coordinate system differs from the one returned by the readAprilTag() function (Fig. 5). To ensure proper drone control, we transformed the readAprilTag() function vector as follows The turn() function performs rotation by a given angular value expressed in radians clockwise. When writing the algorithm, attention was paid to ensuring that the value of the calculated needed rotation angle followed the convention described for this function. This angle was obtained by dropping the normal vectors of the camera and the marker into the horizontal plane by replacing the "z" coordinates with "0" values and then calculating the angle between the two vectors using the following formula (3) Where θ is the angle between the vectors in radians, a and b are the normal vectors of the camera and a marker Knowing that after making the drone rotate by this angle in the opposite direction to the one obtained, it would be perpendicular to the marker, it was now still necessary to transform the movement vector to the marker by the rotation angle made. To do this, the coordinate system of the movement vector was converted to the polar system, and then the rotation angle was subtracted. The result of this operation was then converted back into a Cartesian vector, which became the new movement vector. In the flight process, a takeoff() function was also used to realize the ascent of the drone to an initial height of about 80cm and a land() function to allow the drone to land safely from whatever height it is at the given moment. The sequence of operations performed during the simple algorithm for positioning oneself relative to the marker was, therefore as follows: 1) Establishing a connection with the drone and creating a camera object. 2) Specifying the camera parameters and the size of the marker. 3) Launch the drone using the takeoff() function. In the case of this project, the drone performed actions 4 through 9 in a loop, returning to the starting position in the horizontal plane after performing a certain action. After failing to detect a marker or returning to the initial position after performing a certain action, the drone would perform a 45-degree turn to find the next marker. After performing a full turn, it would fly up by the height specified by the user in the program and start the operations again until it exceeded the maximum height. In our case, the values were 0.5 and 2 meters, respectively. The other functions used in the project referred to the abovementioned analysis of the drone's positioning with the readHeight() function reading the height from takeoff, readOrientation()reading the drone's position relative to the takeoff in the form of Euler angles, readSpeed() reading the drone's speed, and the BatteryLevel() method for reading the drone battery status value. IV. USE OF GRAPHICAL INTERFACE TO CONTROL THE FLIGHT PROCEDURE The graphical user interface was created using the App Designer tool provided by MATLAB (Fig. 6). With the interface, one can establish designated parameters, initiate or cease the program, and retrieve crucial flight analysis information (Fig. 6). V. ERROR ANALYSIS OF THE VISION-BASED POSITIONING METHOD A. Method of determining the error A series of tests were conducted indoors with consistent lighting to verify the accuracy of the drone's position readings and distance calculations. External factors that could affect the algorithm's accuracy were excluded. The measurements were performed using a Bosch DLE 40 laser rangefinder with a precision of +/-0.0015 meters. The test involved flying the drone in a straight line towards a marker at a fixed height, taking pictures using the drone's camera, and then measuring the distance to the marker's centre from the camera's location using the rangefinder. We then used a previously written function in MATLAB to read the marker's position in subsequent photos, which returned the position as a vector [x, y, z], where x represents the left/right shift, y represents the up/down shift, and z represents the forward shift (into the photo). The distance to the marker was defined as the length of the resultant vector [x, y, z]. Using the value measured with the rangefinder as a reference reading, we calculated the absolute and relative errors of reading the marker's offset relative to the camera using the method employed in this project. B. Analysis of positioning errors After conducting a series of measurements, moving the drone away from the marker position, we obtained the following characteristics of the absolute error in the vector line drawn from the camera position to the centre of the marker. The graph above shows a decrease in absolute error with decreasing drone distance from the marker (Fig. 7). Ideally, the characteristic would be linear. Still, many points appear that reject this hypothesis. Nevertheless, the error trend is downward, which allows the drone's position to improve as it approaches the marker. Analysing the graph of relative error (Fig 8.), there is little change for large distances, that is, in the range of 7 to 3 meters of distance from the marker. The error fluctuates between 5.5 and 3 per cent. For small distances, i.e., less than 2 meters, a sharp decrease occurs, below 3 per cent error, and at distances of less than 1 meter, the error comes as low as 0.5 per cent. Over its entire range, the relative error does not exceed 7%, which is a very satisfactory result. To analyse the origin of the errors in more depth, we have marked the points for which increases are visible on the graphs concerning the previous measurement (marked with red diamonds). Our considerations are conducted assuming the drone is getting closer to the markers. We can observe that the increases appear mainly for measurements taken at a distance of more than 3 meters. They may be due to the measurement inaccuracy of the laser meter operator, the drone's camera's low resolution, the drone's camera, or the angular offset of the drone relative to the marker. For the relative error, changes at large distances do not matter, while the closer the marker is to the camera, the greater the impact of the disturbance of the absolute error; that is, despite small increases in the absolute error, the relative error visibly increases. The final element of this analysis is to examine the effect of the angular offset of the drone, relative to the marker, on distance reading errors (Fig. 9). To do this, a line representing the change in angular displacement was plotted on the absolute error graph. The angular values are placed on the right axis at a scale of 1 : 100. As can be seen from the graph, the error values cannot be related to the angular displacement values. The data appear to be independent. VI. SUMMARY The goal of positioning the drone relative to the markers has been achieved. Based on the analysis of the measurement error of the method used in the project, it can be seen that the angular displacement of the marker does not affect the accuracy of determining its position relative to the camera. In addition, the data presented show that when determining the position of the marker at distances of less than 3 meters, the relative error decreases significantly and reaches values of about 0.5%, which allows for very accurate motion correction in the final stage of positioning. On the other hand, at distances of more than 3 meters, the relative error reading remains constant at between 3 and 7 percent. These types of readings allow for satisfactory positioning accuracy of the drone in this type of application. The examined method shows promising results in exemplary testing although further development is necessary to achieve consistent outcomes. In the case of continuous marker detection through video transmission, it would be possible to obtain more accurate positioning results due to continuous position correction. However, this solution requires a more accurate video capture quality. Fig. 2 . 2Designation of the quantities needed to calculate the focal values. Fig. 3 . 3Example comparisons of the photo before distortion correction (left), and after (right). Fig. 4 . 4Ryze Tello drone in correctly prepared flight environment. ( 2 ) 2Where x,y,z are the values sent to the drone's move() function, and x',y',z' are the values received from the readAprilTag() function. Fig. 5 . 5Comparison of coordinate systems from the perspective of the drone's move() function (left) and from the perspective of the readAprilTag() function after returning the tag move vector (right). 4 ) 4Capturing the photo using the snapshot() function. 5) Removing the distortion of the photo using the undistortimage() function. 6) Performing a tag position calculation using the readAprilTag() function. 7) Rotating the drone using the turn() function by the opposite value of the camera and tag normal angles in order to position the drone perpendicular to the tag 8) Converting the measured displacement vector by the calculated rotation angle. 9) Executing a move using the move() function by the transformed move vector with the forward movement distance value subtracted by the user-specified value of the final distance of the drone in front of the marker. 10) Performing a landing using the land() function. Fig. 6 . 6Graphical interface with the following elements: (1.) a box for entering the size of the side of the marker, (2.) a box for entering the maximum flight altitude, (3.) a button to start the execution of the marker search program and display the camera preview and the flight trajectory, (4.) a button to stop the program, forcing the drone to land, (5.) a box in which messages are displayed when certain actions are not performed, (6.) a battery status indicator expressed in percent, (7.) a drone altitude indicator expressed in meters. Fig. 6 . 6An example of a drone's flight trajectory and the detected markers. Distances are given in meters. Fig. 7 . 7Changes of the absolute error during the drone flight. Fig. 8 . 8Changes of relative error during the drone flight. Fig 9 . 9The effect of the angular offset of the drone. The Ryze Tello drone has a dedicated Support Package for Ryze Tello Drone [2] library in the MATLAB environment allowing for easy control, image capture and video upload(Table 1, TABLE 1 . 1RYZE TELLO DRONE PARAMETERSFlight time Maximum flight speed Battery capacity Weight Camera resolution 13 min 8 m/s 1100 mAh 80 g 960 / 720 px MATLAB readAprilTag R2022b The MathWorks, Inc. Natick, Massachusetts, United StatesMATLAB readAprilTag R2022b The MathWorks, Inc., Natick, Massachusetts, United States https://www.mathworks.com/help/vision/ref/readapriltag/ MATLAB Computer Vision Toolbox R2022b The MathWorks, Inc. Natick, Massachusetts, United StatesMATLAB Computer Vision Toolbox R2022b The MathWorks, Inc., Natick, Massachusetts, United States https://www.mathworks.com/products/computer-vision.html Camera Calibration Using AprilTag Markers R2022b The MathWorks. Inc., Natick, Massachusetts, United StatesCamera Calibration Using AprilTag Markers R2022b The MathWorks, Inc., Natick, Massachusetts, United States https://www.mathworks.com/help/vision/ug/camera-calibration- using-apriltag-markers.html Tag-based visual-inertial localization of unmanned aerial vehicles in indoor construction environments using an on-manifold extended Kalman filter. Navid Kayhani, Wenda Zhao, Brenda Mccabe, Angela P Schoellig, 0926-5805Automation in Construction. 135Navid Kayhani, Wenda Zhao, Brenda McCabe, Angela P. Schoellig, Tag-based visual-inertial localization of unmanned aerial vehicles in indoor construction environments using an on-manifold extended Kalman filter, Automation in Construction, Volume 135, 2022, 104112, ISSN 0926-5805 UAV Autonomous Landing Technology Based on AprilTags Vision Positioning Algorithm. Zhou &amp; Li, Yang &amp; Chen, Lu, &amp; Hao, Wu, Lei Cheng, 8148-8153.10.23919/ChiCC.2019.8865757Li, Zhou & Chen, Yang & Lu, Hao & Wu, Huaiyu & Cheng, Lei. (2019). UAV Autonomous Landing Technology Based on AprilTags Vision Positioning Algorithm. 8148-8153. 10.23919/ChiCC.2019.8865757. AprilTag: A robust and flexible visual fiducial system. Edwin Olson, 3400-3407.10.1109/ICRA.2011.5979561Proceedings -IEEE International Conference on Robotics and Automation. -IEEE International Conference on Robotics and AutomationOlson, Edwin. (2011). AprilTag: A robust and flexible visual fiducial system. Proceedings -IEEE International Conference on Robotics and Automation. 3400 -3407. 10.1109/ICRA.2011.5979561.
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[ "TARGET: Traffic Rule-based Test Generation for Autonomous Driving Systems", "TARGET: Traffic Rule-based Test Generation for Autonomous Driving Systems" ]
[ "Yao Deng [email protected] \nMacquarie University Sydney\nNSWAustralia\n", "Jiaohong Yao [email protected] \nMacquarie University Sydney\nNSWAustralia\n", "Zhi Tu \nPurdue University West Lafayette\nINUSA\n", "Xi Zheng [email protected] \nMacquarie University Sydney\nNSWAustralia\n", "Mengshi Zhang [email protected] \nMeta\nMenlo ParkCAUSA\n", "Tianyi Zhang [email protected] \nPurdue University West Lafayette\nINUSA\n" ]
[ "Macquarie University Sydney\nNSWAustralia", "Macquarie University Sydney\nNSWAustralia", "Purdue University West Lafayette\nINUSA", "Macquarie University Sydney\nNSWAustralia", "Meta\nMenlo ParkCAUSA", "Purdue University West Lafayette\nINUSA" ]
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Recent accidents involving self-driving cars call for extensive testing efforts to improve the safety and robustness of autonomous driving. However, constructing test scenarios for autonomous driving is tedious and time-consuming. In this work, we develop an end-to-end test generation framework called TARGET, which automatically constructs test scenarios from human-written traffic rules in an autonomous driving simulator. To handle the ambiguity and sophistication of natural language, TARGET uses GPT-3 to extract key information related to the test scenario from a traffic rule and represents the extracted information in a test scenario schema. Then, TARGET synthesizes the corresponding scenario scripts to construct the test scenario based on the scenario representation. We have evaluated TARGET on four autonomous driving systems, 18 traffic rules, and 8 road maps. TARGET can successfully generate 75 test scenarios and detect 247 traffic rule violations. Based on the violation logs (e.g., waypoints of ego vehicles), we are able to identify three underlying issues in these autonomous driving systems, which are either confirmed by the developers or the existing bug reports.
10.48550/arxiv.2305.06018
[ "https://export.arxiv.org/pdf/2305.06018v1.pdf" ]
258,588,387
2305.06018
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TARGET: Traffic Rule-based Test Generation for Autonomous Driving Systems Yao Deng [email protected] Macquarie University Sydney NSWAustralia Jiaohong Yao [email protected] Macquarie University Sydney NSWAustralia Zhi Tu Purdue University West Lafayette INUSA Xi Zheng [email protected] Macquarie University Sydney NSWAustralia Mengshi Zhang [email protected] Meta Menlo ParkCAUSA Tianyi Zhang [email protected] Purdue University West Lafayette INUSA TARGET: Traffic Rule-based Test Generation for Autonomous Driving Systems Recent accidents involving self-driving cars call for extensive testing efforts to improve the safety and robustness of autonomous driving. However, constructing test scenarios for autonomous driving is tedious and time-consuming. In this work, we develop an end-to-end test generation framework called TARGET, which automatically constructs test scenarios from human-written traffic rules in an autonomous driving simulator. To handle the ambiguity and sophistication of natural language, TARGET uses GPT-3 to extract key information related to the test scenario from a traffic rule and represents the extracted information in a test scenario schema. Then, TARGET synthesizes the corresponding scenario scripts to construct the test scenario based on the scenario representation. We have evaluated TARGET on four autonomous driving systems, 18 traffic rules, and 8 road maps. TARGET can successfully generate 75 test scenarios and detect 247 traffic rule violations. Based on the violation logs (e.g., waypoints of ego vehicles), we are able to identify three underlying issues in these autonomous driving systems, which are either confirmed by the developers or the existing bug reports. INTRODUCTION For the huge investments and efforts from academia and industry, autonomous driving has been shining the great potential of revamping our transportation style. As safety-critical systems, during the development and deployment of autonomous driving systems (ADSs), rigorous testing is a must-have to ensure ADSs' safety and reliability under diverse driving scenarios. For example, in China, autonomous vehicles are required to pass an assessment that includes hundreds of driving scenarios derived from traffic rules and human driver experiences [1]. To evaluate the performance of ADSs before on-road testing, simulation testing has been widely applied as a preliminary matter. Specifically, simulators such as Carla [13] and LGSVL [34] can provide safe and highly customized environments for ADS testing, which pave a promising way towards developing more reliable and robust ADSs. According to a recent research work [27], how to automatically construct test scenarios from traffic rules in the simulation environment is an urgent problem to solve in the industry. Generally, to construct such scenarios in simulators, ADS testers need to explicitly specify the driving scenarios and define the metrics of how to evaluate the performance of the ADS under test. Traffic rules are valuable external resources that can provide both scenario descriptions and driving regulations. However, such information cannot be directly leveraged by simulators to create driving scenarios. Specifically, one of the main challenges of traffic-rule-based test scenario generation is about effectively extracting key information from the documents in natural language. Some research efforts have been made to address this problem. For instance, AC3R [17] is a pioneer work for reconstructing crash scenarios using police reports. It involves traditional natural language processing (NLP) methods to extract knowledge of driving scenarios from well-structured crash description reports. However, as a shallow parsing [35] method, the traditional NLP techniques can only identify specific keywords but not understand the input text. Therefore, such methods cannot process unstructured descriptions of more complex driving scenarios. Moreover, LawBreaker [37] is a recent work that encodes traffic rules as evaluation metrics using a formal specification language, for the performances of ADSs in driving scenarios. However, the parsing process from traffic rules to evaluation metrics is manually implemented. In addition, even though the aforementioned key information can be effectively extracted from traffic rules, it is still very challenging to automatically leverage such information to generate driving scenarios in simulators. The current practice is, testers have to manually describe driving scenarios derived from traffic rules using Domain Specific Languages (DSLs) like OpenScenario [3]. However, the learning curve of mastering such DSLs is very steep. Also, the DSLs' cumbersome syntax hurts developement efficiency. For example, a tester needs to use about 260 lines to describe a lane-cutting scenario [27]. Such manual work of specifying all the scenario details (e.g., the position of occurred vehicles) is very timeconsuming and frustrating. In this paper, we propose a TrAffic Rule-based test GEneraTion framework (TARGET) to solve the above problems. TARGET can automatically generate test scenarios based on traffic rules written in natural language. To fully extract knowledge about testing scenarios in traffic rules, TARGET adopts a large-scale language model (i.e., GPT-3 [6]) and converts the knowledge extraction problem to a question-answering problem. By asking questions to the GPT-3 model, GPT-3 can analyze the traffic rule sentences and answer them. Based on the model responses, the needed information about test scenarios is extracted. Compared with traditional NLP techniques, the GPT-3 model and question-answering paradigm are more powerful in extracting information from natural languages. In addition, a scenario script generator is implemented in TARGET to automatically construct corresponding test scenarios. Based on the extracted information and the descriptions of driving maps in a simulator, the scenario script generator applies a proposed scenario match algorithm to rapidly locate possible positions where the driving scenario can occur and calculate all actors' behaviors in the driving scenario. Then, the scenario generator encodes the driving scenario to a script (e.g., a Python file) using the simulator APIs. Generated scenarios are used for ADS testing and test reports are generated to disclose whether ADSs violate traffic rules or have other issues. To evaluate the effectiveness of TARGET, we generate 75 test scenarios and use them to test 4 ADSs. The experimental results show that scenarios accurately match descriptions of corresponding traffic rules. Moreover, we in total detected 247 rule violations and issues (e.g., collisions). Specifically, we observed that most of the tested ADSs cannot always conform to traffic rules to yield to other vehicles, stop before stop signs, and keep a safe distance from the front vehicle. There are three main contributions of this work: • We propose an end-to-end test generation method to automatically generate test scenarios based on traffic rules for evaluating ADSs. • To our best knowledge, we are the first to leverage a largescale language model (e.g., GPT-3) to automatically parse traffic rules in a QA manner. • We implement a test generation tool on the simulation platform Carla and generate 75 test scenarios to test 4 ADSs. The rest of the paper is organized as follows: Section 2 introduces and compares related works. Section 3 describes the proposed test generation methods based on traffic rules. Section 4 introduces research questions and experiment settings. Section 5 demonstrates the experiment results. Section 6 describes threats to the validity of the work.Section 7 concludes the paper and introduces the future research directions. RELATED WORK To generate driving scenarios for ADS testing, researchers have explored and proposed several methods. One of the most related works to ours is AC3R (Automatic Crash Constructor from Crash Reports) [17], which constructs corner cases of car crash accident scenarios [28] for ADS testing. It uses domain-specific ontology [4,25] and applies NLP (Natural Language Processing) techniques [23] to extract information from police reports. However, AC3R requires restricted domain-specific descriptions of driving scenarios, and only uses collision as the evaluation criterion, which cannot guarantee the safety and robustness of ADSs. In contrast, our work evaluates ADSs more comprehensively and thoroughly, by implementing a QA model to automatically parse the existing traffic rules under the guidance of our proposed driving scenario schema (Section 3.2). Another related work is LawBreaker [37], a framework to test whether ADSs violate traffic laws. LawBreaker describes real-world traffic laws and test scenarios by domain-specific languages (DSLs) and then searches for various scenarios that lead ADSs to violate traffic laws by using a fuzzing engine. LawBreaker focuses on using a search-based method to generate test scenarios where the ADS under test may violate specific traffic rules. The generated scenarios may not match the descriptions in traffic rules. In contrast, our work aims to accurately generate the driving scenarios described in traffic rules. Furthermore, traffic rules are manually converted to DSL descriptions in LawBreaker while in our work traffic rules are automatically parsed. In addition, some DSLs are designed and proposed for ADS test generation. Fremont et al. [15] propose Scenic, a domain-specific language to describe the distribution of complex scenarios (e.g., driving environment), to synthesize corner-case testing data. Queiroz et al. [33] propose GeoScenario for the representation and evaluation of ADS test scenarios, by identifying relevant elements. However, it is not a trivial effort to design, understand, and master DSLs. Compared with the DSL-based methods, our approach focuses on test auto-generation, which aims to reduce testers' learning and operation costs. Moreover, some ADS testing studies focus on generating driving scenes by applying affine transformations [41], generative adversarial networks [43], and adversarial perturbations [22,46]. The generated driving images can only be used to test simple imagebased driving models on steering angle or speed prediction [10,11] in static scenarios. In our work, the generated test scenarios are dynamic, and they are used to evaluate the performances of complicated ADSs (e.g., multi-modility model). Figure 1 shows the architecture of TARGET. It contains four components: Scenario schema, Rule parser, Scenario generator, and Scenario monitor. Scenario schema defines an intermediate representation to describe test scenarios derived from traffic rules (Section 3.2). Rule parser extracts information from the input traffic rule and then generates the test scenario representation following the syntax of the proposed driving scenario schema (Section 3.3). Scenario generator outputs a scenario script, which the simulator can execute, based on the test scenario representation (Section 3.4). Scenario monitor records the running status of the ADS in the test scenario and reports whether its driving behavior violates the test oracle described in the traffic rule (Section 3.5). METHODOLOGY 3.1 Overview Specifically, given the pre-defined scenario schema, a list of rule parsing question templates is created. Then, for each input traffic rule, the rule parser iteratively generates questions based on the rule parsing question template and calls the QA model to extract corresponding information from the traffic rule. A driving scenario representation in YAML format is then generated. Based on the driving scenario representation and existing simulation map information, the scenario generator identifies the position where the driving scenario occurs in the map, orchestrates all actors' behaviors in the driving scenario, and generates the corresponding scenario script. The scenario script is fed to the simulator to render the test scenario. During the running of the test scenario, the scenario monitor records the status of the ADS, checks whether the ADS violates test oracles, and finally outputs a test report to summarize the performance of the ADS under test. Scenario schema In this work, we first define a new driving scenario schema to describe driving scenarios. Compared with the state-of-the-art scenario description standard such as OpenScenario [3], the proposed schema has a simpler syntax and does not require precise definitions for all parameters such as ego vehicle's coordinates that are not provided in traffic rules. As shown in Figure 2, in our schema, a driving scenario is composed of elements in four categories: Environment, Road network, Actor, and Oracle. Environment: This category contains elements about the current driving time (e.g., night) and weather (e.g., rainy) to describe the driving environment of the test scenario. Road network: This category contains three types of elements including traffic sign, road type, and road marker to describe the location where the driving scenario occurs. Traffic sign contains common road signs (e.g., stop signs). Road type describes the shape and properties of the road network, such as intersections, crosswalks, and roundabouts. Road marker is the type of lane dividing lines, including solid lines (unbroken lines) and broken lines. A road network element can be explicitly mentioned in a traffic rule such as "You must stop at a stop sign". In addition, a road network element could be implicitly mentioned. For example, in the traffic rule "When turning left, always yield to any vehicle coming straight through from the other direction.", turning left and vehicle coming straight through from the other direction indicates an intersection in the driving scenario. Actor: This category describes all the participants, including the ego vehicle and other actors (e.g. other vehicles or pedestrians) in the test scenario. For each actor, its type, behavior, and position need to be further specified as sub-categories. Type: The subcategory defines the type of actor, its elements can be various vehicles (e.g., car, fire truck, school bus), or other actors (e.g., pedestrian and cyclist). Behavior: This subcategory defines the action or movement of an actor. For example, go forward means traveling straight along the current lane, static means the actor does not move, and turn left means turning to the road on the left. Position: This subcategory describes the initial position of the actor in the driving scenario. It comprises position reference to define the reference object, and position relation to define the relative location of the actor to the reference object. The elements for position reference can be road network elements (e.g., traffic signs) mentioned in the traffic rule, or be the ego vehicle when describing the position of other actors. The elements for position relation can be position prepositions such as front, in, and right. In addition, it can be a composite of two position prepositions such as left behind, which means the actor is behind of the ego vehicle and at the left lane relative to the ego vehicle. Oracle: Elements in this category describe the required actions of the ego vehicle in traffic rules, such as yield and decelerate. These elements in the category can be further used for creating testing oracles to evaluate erroneous behaviors of ADS, which is introduced in Section 3.4.4 in detail. Rule parser Given a traffic rule, the rule parser first extracts key information from the rule, creates the corresponding test scenario representation (Section 3.2), and saves the representation into a YAML file. Unlike prior works [11,18] that apply rule-based NLP techniques to extract information, we propose to use a pre-trained large-scale language model, GPT-3 [6], to handle the inherent ambiguity and complexity of traffic rules. We reformulate the knowledge extraction problem as a Question Answering (QA) task [12,20]. To extract a piece of information (e.g., road type) from a traffic rule, we generate a prompt that asks a targeted question about the piece of information (e.g., what is the road type described in the traffic rule?) to GPT-3. Specifically, We use text-DaVinci-003, a version of GPT-3 model offered by OpenAI [5]. We directly use the original GPT-3 model, without further fine-tuning the model on traffic rules. The Rule parser pipeline is shown in Figure 3. Given a traffic rule, the question selector chooses question topics from possible elements of the proposed scenario schema (e.g., stop sign) and passes it to the prompt generator to generate multiple parallel questions with different wording and narratives but the same semantic meaning. We do this because, during the pilot study on GPT-3, we found it sensitive to the syntax of input questions. When adding or deleting some phrases in the question, the output answer may change accordingly. Therefore, we use multiple questions with the same semantic meaning to improve the robustness of the model outputs. For question answering, we use GPT-3 as a QA model via prompt engineering [26,32,36,42,42]. An example of our designed prompt is as the following: Based on the description, "You must stop at a 'Stop' sign. ", is stop sign mentioned? Answer yes or no. In the example above, the bold text is the template; the text in quotation marks is the traffic rule for parsing, which is filled in by the prompt generator; the italicized text is an instantiated question based on a question template selected by the prompt generator. Column Question Template in Table 1 shows more question templates. The full list of question templates is in the supplementary material. The prompt generator iteratively generates questions by using all question templates. The placeholders (<> ) in the question templates are filled with the selected element by the question selector. To efficiently ask questions and parse information, the question selector first asks questions about Environment and Road network and parses corresponding elements. Then, when parsing elements of Actor category, the question selector selects elements from the parsing result of Road network to fill placeholders in question templates of the subcategory Ego vehicle position and Other actor position. Using these questions as prompts, GPT-3 generates the answer either "yes" or "no" with a confidence score. We ensemble the outputs of these parallel questions to obtain a single concrete answer using majority voting and calculate the average confidence score of the voted answer. If the answer of majority voting is "yes" with average confidence higher than a threshold, which is 0.8 in this work, the corresponding element is recorded in the test scenario YAML file. The setting is based on our pilot study that GPT-3 tends to output answers with relatively low confidence (lower than 0.8) if it does not confirm the correctness of the answer. Scenario generator The scenario generator aims to create a scenario script (e.g., a Python file) and configuration files (e.g. scenario initialization configuration required by the target simulation platform). The generated Python script uses the primitive APIs provided by the simulator (Carla [13] in this case) to retrieve simulation data and control the simulation environment. The scenario generator implements four components to construct test scenarios: a) Control the environment in the driving scenario (Algorithm 1 Line 1); b) Identify possible locations where the driving scenario occurs in the simulator map (Algorithm 1 Lines 3-6); c) Define all actors' initial positions and destinations in the driving scenario (Algorithm 1 Lines 7-9); d) Create test oracles to check whether the ego vehicle violates the given traffic rule (Algorithm 1 Line 10). Finally, the configuration files and the scenario script are generated ((Algorithm 1 Lines 11-12)). All functions mentioned in Algorithm 1 are implemented in TARGET based on simulation APIs. corresponding simulator APIs to change the weather in the driving environment (e.g., change the degree of fog density) using a default constant parameter (e.g., fog density equals to 0.5). In addition, the generator also controls traffic signals. If traffic light is mentioned in the YAML file, the scenario generator ensures that the traffic light is in the corresponding status (e.g., red) when the ego vehicle approaches it. Environment Creation. Based on elements in the category Scenario Location Identification. To accurately identify the location where the driving scenario occurs, the scenario generator parses map information from the map description file in the target simulation platform and then automatically selects roads that meet the requirements of the traffic rule. For example, if a rule describes that the ego vehicle is driving on the road with an intersection, all roads containing intersections will be selected for further use. As With the extracted lane-level map information, the scenario generator further constructs route-level map information (Algorithm 1 Line 4). First, the generator samples waypoints on all lanes, as black dots shown in Intersection (a) in Figure 4. By connecting two adjacent waypoints, a route is then formed, as blue lanes with arrows in Intersection (a) in Figure 4, from the start waypoint to the end waypoint. For each route, based on the lane-level information of the start and end waypoints, the scenario generator calculates the route's direction (e.g., turning left), and other information such as whether there are traffic signs on the route and whether the route crosses an intersection. Based on the route-level map information, the scenario generator selects routes that match the elements in the category Road network of the YAML file (Algorithm 1 Line 6). Actor Creation. With selected routes, the scenario generator tries to identify spawn points (i.e., the positions to initialize actors) and the destinations of the ego vehicle and other actors. For each route in the selected routes, the generator checks whether it satisfies the requirements for the ego vehicle's behavior (e.g., turning left) as shown in Algorithm 2 Lines 3-4. If so, the route is marked as the current ego route (e.g., the red route in Intersection (b) in Figure 4). Then, in Algorithm 2 Lines 5-9, the generator checks nearby routes that satisfy the requirements of other actors (e.g., another route for going straight and crossing with the ego route). The process is iteratively executed until all actors find their corresponding routes or all selected routes are checked. To make sure that the ADS under test can be evaluated by the desired test scenario, the generator does not directly use start points and ending points of identified routes as spawn points and destinations of the ego vehicle and other actors. Specifically, for the ego vehicle, the spawn point is assigned as the start point of the predecessor route of the ego route (i.e., the route ending at the start point of the ego route), as point A shown in Intersection (b) in Figure 4. The reason is that the ADS needs this route to initialize and increase its speed to a steady level. The destination of the ego vehicle is set as the ending point of the ego route. For other actors, When the ego vehicle reaches the start point of the ego route, the other actors' behaviors are triggered and move along their driving routes. For example, in Intersection (b) in Figure 4, the NPC vehicle starts to move when the ego vehicle reaches point E. The reason for setting the trigger point is that if the NPC vehicle starts to move too early or too late, the ego vehicle may already reach its destination. In this condition, a possible situation that the ego vehicle and the NPC vehicle collide in the intersection cannot be evaluated. When spawn points and destinations of all actors are identified, they are stored in the scenario initialization file for later rendering the initial driving scenario. Test Oracle Creation. Several oracle options that specify the expected behavior of the ego vehicle in traffic rules are implemented in the scenario generator. By checking elements in category Oracle in the YAML file, the scenario generator integrates the corresponding test oracle in the scenario script. The implemented oracle options contain yield, stop, decelerate, keep distance, and keep lane. These options are summarized from traffic rule handbooks, and they cover the expected behaviors of drivers in most driving scenarios. Specifically, yield measures whether the ego vehicle yields to other actors that have the rightof-way on the road. Stop evaluates whether the ego vehicle stops before traffic signs and signals. Decelerate checks whether the ego vehicle reduces its speed to the required speed limit. Keep distance measures whether the ego vehicle maintains a safe distance from other actors in the front. Keep lane measures whether the ego vehicle keeps driving in the current lane. For decelerate, the degree of deceleration is decided by speed limit signs on the road in the simulation environment. For example, if the speed limit sign is "40 km/h" in a generated test scenario, the decelerate oracle will check whether the ego vehicle's speed decelerate to less than 40 km/h when the ego vehicle passes through the speed limit sign. If there are no speed limit signs in the driving scenario, decelerate checks whether the average speed of the ego vehicle in the last half driving route is smaller than the average speed in the first half driving route. For keep distance, the scenario generator adopts the predefined parameter in the driver handbook that the ego vehicle should keep 3 seconds safe driving distance. Scenario monitor When the scenario script and the initialization file are generated, they are fed into the simulator to construct the driving scenario. The ADS under test is loaded into the driving scenario to control the ego vehicle moving to its destination. The other actors are controlled by the implemented behaviors in the scenario script. During the running of the test scenario, the scenario monitor records the statuses of the ego vehicle including its speed and location as well as the distance between the ego vehicle and other actors. Each scenario is running for a pre-defined time limit. Once the scenario ends, the scenario monitor analyzes recorded data and generates a test report to show the performance of the ADS. The test report contains four metrics to evaluate the performance of the ADSs in the driving scenario. The first one is rule violation, which shows whether the ADS violates the test oracles in the scenario YAML file of the traffic rule. The second metric collision measures whether the ADS under test collides with other actors on the road or objects on the roadside. The third metric timeout measures whether the ADS reaches the destination within the specified time limits. The fourth metric other problems evaluates whether the ADS violates other test oracles not mentioned in the corresponding traffic rules but implemented in the driving scenario (e.g., additional traffic signs on the road). EXPERIMENT We proposed four research questions (RQs) and conducted corresponding experiments to evaluate the effectiveness of TARGET. The conducted experiments were used to evaluate TARGET in an end-to-end manner and then we further analyzed two key components: Rule parser and Scenario generator. For the end-to-end evaluation, we proposed the first two research questions: • . The driver handbook has 14 chapters in total, in which Chapters 4-9 introduce traffic rules for the safe driving of vehicles. To create the benchmark, we selected sentences from Chapters 4-9 based on the following criteria. First, we only selected traffic rules that describe driving scenarios in one sentence. The reason is that currently the state-of-the-art NLP and QA models only analyze words statistically but have difficulty understanding the semantic meaning of natural language with complex grammar and context [14]. Second, we filtered out traffic rules that require additional figures (e.g Figure 4) to explain corresponding driving scenarios. Such traffic rules are also beyond the capability of the QA model. Third, we filtered out sentences that describe non-driving scenarios such as parking and signaling because existing ADSs do not support these functionalities. After the filter process, we obtained a list of traffic rules containing 98 rules, provided in the supplementary material. We randomly sampled 30 rules from them as the benchmark. In these 30 rules, 18 rules can be used to generate driving scenarios while other rules cannot because they describe scenarios related to unsupportable actors and behaviors (e.g., ambulance vehicles and school buses flashing lights) by the state-of-the-art simulators [13,34], including Carla, which we used for the paper. The 18 rules are used for RQ1 and RQ2 to evaluate TARGET's ability end-to-end. For each traffic rule, TARGET tries to generate driving scenarios on eight maps in Carla (Town01-Town07, and Town10). These maps cover diverse driving environments in rural, urban, and highway areas. Different maps contain different traffic signs and road markers. Therefore, the test scenario of a specific traffic rule may not be generated on all maps. For each traffic rule, we generated the corresponding test scenarios on all supported maps (75 test scenarios in total). The traffic rules and numbers of generated scenarios for each rule are shown in Table 2 ("# S" means the number of generated scenarios). For RQ1, to evaluate the consistency between the generated test scenarios and the traffic rules, we randomly split 75 test scenarios into three groups for three online surveys. The surveys were distributed to six researchers without conflict of interest in the university. We asked participants to evaluate whether generated test scenarios match the description of the corresponding traffic rules in five options including Totally match, Mostly match, Partially match, Mostly not match, and Totally not match. Specifically, we show the traffic rule and the corresponding generated driving scenario in Carla to each participant. By watching the driving scenario video, the participant needs to watch the driving scenario video and to check whether the scenario match the traffic rule description by considering the fields such as weather, actor positions, and actor behaviors in forementioned scenario schema. For RQ2, we tested four ADSs including Auto [44], MMFN [45], LAV [8], and Autoware [21]. Auto [44] is an ADS modified based on the NPC agent in Carla. MMFN and LAV are two ADSs that achieved good performances on Carla leaderboard challenge [7]. These three ADSs were obtained from the Github repositories 1 2 . We used their pre-trained models with default settings. Autoware [21] is a ROSbased industrial-level ADS. We summarized the performances of these ADSs based on testing reports generated by the scenario monitor in Section 3.5, which contains four metrics rule violation, collision, timeout, and other problems. The time limit of timeout was set as 60 seconds, which is enough for reaching the destination in generated test scenarios if the ADS drives normally. For RQ3, we first manually constructed all the ground truth YAML files of the 30 traffic rules following the proposed driving scenario schema. The YAML files were initially created by the first and second authors separately. Then, the two authors checked each other's work and discussed the differences. Finally, they made agreements for all YAML files and ensured that each one contains all information that is explicitly or implicitly mentioned in the traffic rule. For each traffic rule, Rule parser outputs the corresponding YAML file. Then, we compared the generated one and the ground truth. For each category, if the parsed elements in the YAML file are the same as the ones in the ground truth, we think the category is matched successfully by the rule parser. We finally used Accuracy to measure the performance of the rule parser, which is calculated by the number of matched categories versus the total categories in the scenario representation (10 in our paper). We tried to compare our rule parser with prior works like AC3R [17] and LawBreaker [37]. However, we found that we cannot compare with them because of the following reasons: Comparison with AC3R: We failed to use AC3R to parse sampled traffic rules. The reason is that AC3R requires a rigidly formatted XML file as input, specifying details of each vehicle and pedestrian individually. Such information can be manually defined in crash police reports but not provided in traffic rules. Therefore, we cannot compare TARGET with this work. Comparison with LawBreaker: The traffic laws they target are mostly paragraphs with complicated descriptions, and these rules are manually parsed by the authors. These rules are out of the scope of our work because TARGET only supports traffic rules described in one sentence, due to a lack of precision by the used language model. Therefore, we cannot conduct a comparison with LawBreaker. For RQ4, we evaluated the performance of Scenario generator when the inputs are ground truth YAML files, by a relatively largescale human study. The generated 75 scenarios were split into five surveys. The surveys were randomly distributed to students majoring in software engineering in five practical classes of software testing. Due to the different numbers of students in classes and other factors (e.g., some students were not willing to participate in the survey), the numbers of participants for five surveys vary from 10 to 35. Every participant was asked the same question as in RQ1. Figure 5 reveals the overall votings of scenarios on all rules. Each bar shows the distribution of votings on five options for generated scenarios for a traffic rule. The blue part plus the orange part can be regarded as a match between the generated driving scenario and the corresponding traffic rule. For scenarios generated on most rules, votings are either Totally match or Mostly match, which means that overall generated scenarios match descriptions of traffic rules. In the 75 test scenarios, 50 scenarios are thought to at least partially match the rules (no voting on options Most not match or Totally not match), in which 23 scenarios are thought totally match their corresponding traffic rules. RESULTS RQ1: Test Scenario Generation Only for Rules 3 and 6, the number of votings on Totally not match is much greater than other options. Specifically, four scenarios generated on Rule 3 and six scenarios generated on Rule 6 are thought Totally not match by more than 67% participants. The reason is the ego vehicle and the other vehicle's positions and behaviors are wrongly parsed. The detailed analysis for these rules is in Section 5.3. In addition, another driving scenario that was thought as totally not match was generated on Rule 8 for evaluating Rule # S 1 when approaching intersections not controlled by signs, signals, multi-Lanes, or pavement, yield the right-of-way to any vehicle that has entered or is approaching the intersection on your right. 4 2 when turning left, always yield the right-of-way to any vehicle coming straight through from the other direction. 4 3 when approaching an intersection of a through street traveling from a street that ends at the intersection, you must stop and yield the rightof-way to vehicles on the through street. 6 4 drivers must give the right-of-way to pedestrians at an uncontrolled crosswalk. you must stop at a 'Stop' sign. 3 10 on roads where there's a speed limit sign, you must not drive faster than that speed limit. 15 slow down and increase the following distance when the road is wet. 6 16 slow down and increase the following distance in the dark. 6 17 slow down and increase the following distance in bad weather. 6 18 slow down as you approach the roundabout intersection. 1 Table 2: Traffic rules for scenario generation and testing results on them red traffic lights. We found the reason for the mismatch was that the ego vehicle was initialized at the wrong spawn point because of a road overlapping problem in the map, which caused the ADS to fail to reach the target lane in the time limit. Table 3: Detected violations for all ADSs is much fewer, and Auto and LAV are less likely to cause collisions. However, LAV leads to more timeout problems. RQ2: Rule Violation Detection on ADS Systems To be more specific, we found LAV quite conservative. It often slows down or even stops when it reaches an intersection without traffic signals, even though there are no vehicles or pedestrians in the front, as shown in Figure 6d. This issue is reported to the authors and get confirmed. With this driving strategy, LAV reduces the risk of collisions and rule violations such as giving way or keeping a safe distance from the front vehicle (e.g., . However, such conservative behaviors lead to more timeout problems for LAV since it takes a long time to give way or stop and then fails to reach the destination within the time limit. We found MMFN and Autoware cause more collisions. For MMFN, the problem is possibly in the localization module, which mistakenly thinks the ego vehicle is at an intersection. This problem has been confirmed by the authors of MMFN. Then the ego vehicle starts to turn and collide on the roadside, as shown in Figure 6b. For Autoware, the problem is possibly in the perception system, which has difficulty detecting static or slow-moving vehicles. Then the ego vehicle collides with the front vehicle, as shown in Figure 6c, which was similar to a reported issue in Github 3 . Except for LAV, the other three ADSs have many violations of Rules that require the ego vehicle to give way as shown in Figure 6a (Rules 1, 2, and 4), stop at the stop sign (Rule 9), and increase following distances in different weather conditions (Rules 14-17). 3 Autoware issue: https://github.com/carla-simulator/carla-autoware/issues/66 Table 4 shows the accuracy of parsing each category in the YAML files of 30 rules. The second column in the table shows the results of majority voting (ensemble), and the following columns are the results of using five question templates independently. RQ3: Performance of Rule Parser The parsing accuracies of the ensemble on all categories are between 0.70 to 1, with an average value of 0.85. The highest accuracy is for parsing Oracle. This is because all oracle behaviors of the ego vehicle are explicitly described in traffic rules. The lowest accuracy is for parsing Other actor position relation (0.70), which means that GPT-3 model is not good at analyzing the relative position between actors and objects in traffic rules. The accuracy of parsing Road network is 0.77, mainly because the GPT-3 model tends to believe some traffic signs like speed limit signs exist in the driving scenario even if the traffic rule does not mention the sign. Though this problem causes relatively low accuracy of parsing Road network, the generated driving scenarios still match the description of traffic rules in general, as shown in RQ1. Furthermore, the ensemble operation has two advantages. First, it increases the parsing accuracy. For the first 30 rules, the ensemble accuracies in all the categories except for Other actor type are higher or equal to that of the best individual template. And even for the subcategory Other actor type, the ensemble result 0.86 is almost the same as the highest parsing result 0.87 (template 2). Second, the ensemble inhibits large variations in the parsing results when using different templates. As for different categories, the highest parsing accuracy is achieved by different templates. We cannot identify which template can achieve the best performance consistently. Therefore, ensembling multiple templates is an ideal approach to ensure the good performance of rule parsing. Among all 18 rules in Table 2, for Rules 2,4,8,9,13,15,17, and 18, all categories elements in the YAML files can be correctly parsed. For rules 1, 7, 10 − 12, and 14, the accuracy achieves 88.9% (one element mismatch). For Rules 3 and 6, the performances are the lowest, with 56% and 33% accuracy respectively. The reason is Rule 3 is written in a relatively confusing sentence structure, which might be too complicated for the QA model to analyze and extract correct information. For Rule 6, the parsing model mistakenly thinks that the other vehicle is changing lane in front of the ego vehicle. RQ4: Performance of Scenario Generator When using the ground truth YAML files as inputs of the Scenario generator, the Totally match voting ratio achieves 79.87% on average among 5 surveys in the relatively large-scale human study, which is better than in RQ1 where 70% votings in total are Totally match. In 75 scenarios, 30 scenarios achieve 90% -100% Totally match voting ratio and another 20 achieve 80% to 90% Totally match voting ratio. The performance is slightly better than RQ1. To further analyze why participants thought some generated driving scenarios do not match descriptions in the corresponding traffic rules, we manually checked driving scenarios with less than 60% Totally match ratio and summarized problems in them. In these scenarios, four of them are from Rule 11 that drivers cannot overtake other vehicles by crossing unbroken lines. For these four driving scenarios, the lines at the left or right side of the ego vehicle are not solid lines, as the example shown in Figure 6e. Therefore, participants thought generated scenarios are not consistent with traffic rule descriptions. By analyzing our code and Carla API, we found the problem is that Carla API cannot return the correct labels of lines on the road. The API can only show whether the part of lane lines near the ego vehicle is solid. For example, in Figure 6e, even though the API perceives the segment of the line at the left side of the ego vehicle as unbroken, the lines at the left side of the road overall are broken. Another three driving scenarios with low Totally match ratios are from Rule 15 about driving in wet weather. The reason is that the scenarios rendered by Carla do not look wet, more like with shadows, as shown in Figure 6f. Interestingly, scenarios for the two rules in RQ1 are not thought as mismatched. For example, in Rule 11 the other vehicle's behavior in the ground truth YAML file is defined as static. In the generated scenario, the other vehicle is spawned near a broken line and does not move. Therefore, participants thought the scenario does not match the description in the traffic rule. However, in the parsed YAML file, the other vehicle's behavior is going forward. In the generated scenario the other vehicle and the ego vehicle keep moving and pass through many roads with unbroken lines. The survey participants thus thought the scenario matches the traffic rule. This phenomenon indicates that when the traffic rule does not clearly describe all key information such as vehicle's behavior, there are not one, but many driving scenarios that match the description of the traffic rule. A possible future research direction can be mutating possible elements in a YAML file to create large amounts of driving scenarios for a rule and then search scenarios that most likely cause rule violation of ADSs. THREAT TO VALIDITY The external validity is about the generalizability of the proposed test generation method. For traffic rule parsing, GPT-3 cannot correctly extract semantic information from paragraphs or complicated sentences. This is the limitation of the language model. With the emergence of new language models (e.g., ChatGPT [30]), the problem might be mitigated. We tried to use ChatGPT in our work but currently met some problems. One was that ChatGPT did not provide the official API. The web interface was usually unusable due to the large volume of visits. This made us difficult to evaluate and improve our method on ChatGPT. In addition, in our pilot study, the output from ChatGPT still contained many phrases or sentences that were not precise and might need further post-processing using NLP techniques. We will explore the use of ChatGPT in future research. For scenario generation, our method mainly leveraged simulator (Carla) APIs to retrieve simulation information and set simulation control functions (e.g., changing the weather and adding actors). These APIs are common designs for the most of existing simulators [13,34]. Therefore, our method can be implemented on these simulation platforms based on their APIs. The internal validity is that GPT-3 does not output deterministic answers by default and it is sensitive to question formats and syntax. To solve these problems, we set the temperature (a parameter controlling the model's randomness) of GPT-3 as 0 and adopted majority voting to decrease the model's variance. The construct validity is about the selection of ADSs under test and the simulator. In this paper, we tested four ADSs. Two of them (MMFN [45] and LAV [8]) were published at top conferences and achieved good performances in Carla Challenge [7]. Auto [44] was implemented based on the source code in the simulator Carla. Autoware [21] is one of the largest open-source industry-level ADSs. Therefore, test results on these ADSs are representative of current ADSs. We did not test another widely researched [9,19,31,39] ADS Baidu Apollo [2] because currently there was no useable bridge to integrate Apollo into Carla environment. Specifically, we tried some existing bridges 4 but it cannot launch normally as described in the GitHub issue 5 . We did not conduct experiments on all 98 filtered traffic rules. One reason is that it is not cost-effective considering one driving scenario might be needed to be generated on each map for each rule. In addition, some rules have similar structures or express similar meanings. In our pilot study of randomly choosing a few other rules, the results were almost the same. We selected Carla as the simulation platform because it is the most active open-source simulation project and it provides several different maps with high fidelity. We did not evaluate our method on another simulator LGSVL [34] that is also widely used in ADS testing research [16,24,38,40]. The reason is that LGSVL has been terminated and it was no longer officially developed and maintained 6 . CONCLUSION This paper proposes TARGET to automatically generate ADS test scenarios according to real-world traffic rules written in natural language. To bridge the traffic rules and ADS simulators, we propose a scenario schema to describe test scenarios described by traffic rules. Then, we implement a QA model based on the language model GPT-3 to parse traffic rules as test scenario representations. We Table 4: Parsing results of rules for scenario generation also implement a scenario generator and monitor to generate test scenarios and evaluate ADSs. The experiments show that TARGET is able to detect traffic rule violations of state-of-the-art ADSs, which leads to the detection of underlying issues. These issues are either confirmed by the developers or existing bug reports. TARGET is the first work to leverage large-scale language models to automate the driving scenario generation directly from human rules. As future research directions, search-based algorithms can be developed to mutate our YAML files to generate more rule-violating test scenarios. Also, ChatGPT can be further studied to parse more complex traffic rules. rainy | foggy | snowy | ... := daytime | nighttime Road network := | | := stop sign | speed limit sign | ... := one-way | one-lane | intersection | ... := solid line | broken line | ... Actor := < , ℎ , > := ego vehicle | pedestrian | car | school bus | ... ℎ := go forward | turn left | change lane left| static| ... := < , > := | lane | ego vehicle := front | behind | left | in | left behind | ... Oracle := yield | decelerate | keep lane | ... Figure 2 : 2An example of Scenario Schema containing a subset of elements in different categories. Figure 3 : 3Pipeline of Rule parser. shown in Algorithm 1 Line 3, the scenario generator first extracts map information at the lane level. For each lane, its length, relative position on the road (e.g., left lane or right lane), predecessor lanes (i.e., lanes ending at the start point of the current lane), and successor lanes (i.e., lanes starting from the end of the current lane) are identified. In addition, other information, including the type of lines (e.g, solid or broken), whether the lane contains traffic signs like a stop sign, and the positions of contained signs are extracted. Figure 4 : 4The demonstration of waypoints, routes and a driving scenario the spawn points and destinations are directly assigned by their routes' start points and endpoints. In addition, other actors are set to be static when they are spawned. Figure 5 : 5Overall votings of scenarios on all rules left at an intersection where 'no turn left sign' is posted is prohibited. 1 13 you must not drive in that direction when you see 'do not enter' sign. ' 1 14 you should keep a safe distance between your car and the one in front of you.5 Figure 6 : 6Traffic rule violation: Not Give way to the other vehicle at T-intersection (b) MMFN model collides on the roadside (c) Autoware collides with the front vehicle (d) LAV does not move even if the pedestrian has crossed the road (e) Generated scenario for unbroken line (f) Generated scenario for wet weather Examples Table 1 : 1A subset of QA model question templates. Environment of the YAML file, the scenario script generator calls Q1: Based on...is it a stop sign, ...? Q2: Based on...is there a stop sign, ....? Q2: Based on...is stop sign mentioned, ...? ... You must stop at a 'Stop' sign. Influence the choice of the following questionsGPT -3 Ensemble Question Answering "Stop Sign" Answer YAML Traffic Rule Prompt Generator Prompts "yes" "no" "yes" ... Question Selector ... road_network: -stop_sign participant: ego_vehicle: behavior: -travel position_target: stop_sign position_relation: behind ... Algorithm 1 : 1Scenario Generation Input : map: a map file; scenarioRep: the test scenario YAML file, which is loaded as a dictionary containing four categories road_network, actor, environment, and oracle as introduced before; Output : initConfig: the scenario initialization configuration file; scenarioScript: the scenario script; 1 envFunc ← ConstructEnvFunc (scenarioRep ['environment']); 2 /* Parse map information from the map file. */ 3 laneMapInfo ← ParseMapLanes (map); 4 routeMapInfo ← ParseMapRoutes (map, laneMapInfo); 5 /* Find routes that match road network elements in the test scenario presentation. */ 6 routes ← FilterRoutes (routeMapInfo, scenarioRep ['road_network']) 7 /* Identify spawn points, destinations, and behaviors of actors in the scenario. */ 8 actorInit, actorBehaviors ← CreateActor (routes, scenarioRep ['road_network'], scenarioRep ['actor']); 9 behaviorFunc ← ConstructBehaviorFunc (actorInit, actorBehaviors); 10 criteriaFunc ← ConstructCriteriaFunc (scenarioRep ['oracle']); 11 initConfig ← GenerateInitConf (actorInit); 12 scenarioScript ← GenerateScenario (behaviorFunc, envFunc, criteriaFunc); 13 return initConfig, scenarioScript; Algorithm 2 : 2Create Actor Input : filteredRoutes: a list of routes that meet the requirements in the test scenario YAML file. roadNetworkInfo: a list containing elements of the category Road network in the YAML file. actorInfo: a dictionary containing elements of the category Actor in the YAML file. Output : actorConfig: a configuration file for rendering the driving scenario. 1 for route in filteredRoutes do2 // check the route that meets the requirements for spawning ego vehicle 3 if MatchBehavior (route, actorInfo ['ego vehicle']) then 4 egoRoute ← route; 5 for otherRoute in routes do 6 // check the route that meets the requirements for spawning the other actor 7 if MatchBehavior (route, actorInfo ['other_actor']) then 8 otherActorRoute ← otherRoute; 9 end 10 end 11 end 12 if egoRoute and otherActorRoute then 13 break; 14 end 15 end 16 // identify start points, destinations, and the trigger point for all actors 17 startPointEgo, destinationEgo ← IdentifyEgoBehavior (egoRoute, otherRoute, waypoints, actorInfo); 18 startPointOA, destinationOA, triggerPointOA ← IdentifyOABehavior (egoRoute, otherRoute, waypoints, actorInfo); 19 actorConfig ← genActorConfig (startPointOA, startPointEgo, destinationOA, destinationEgo, triggerPointOA); 20 return actorConfig; RQ1: Can TARGET accurately generate test scenarios from traffic rules? • RQ2: Can TARGET detect erroneous behaviors of ADSs on generated test scenarios? For the component evaluation, we proposed the last two research questions: • RQ3: How accurately does the Rule parser create test scenario YAML files from traffic rules? • RQ4: How accurately does the Scenario generator construct test scenarios from the YAML files? To evaluate these RQs, we first created a traffic rule benchmark based on Texas Driver Handbook [29] Table 3 3shows the testing results of four ADSs in the 64 generated test scenarios as 11 scenarios reported in RQ1 are not matched with traffic rule descriptions via human voting. The generated test scenarios can find many issues of these ADSs based on four metrics including rule violation, timeout, collision, and other problems as introduced in Section 3.5. 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[ "https://github.com/carla-simulator/carla-autoware/issues/66", "https://github.com/Kin-Zhang/carla-expert", "https://github.com/dotchen/LAV", "https://github.com/guardstrikelab/carla_apollo_bridge", "https://github.com/guardstrikelab/carla_apollo_bridge/issues/30" ]
[ "DeepCatra: Learning Flow-and Graph-based Behaviors for Android Malware Detection", "DeepCatra: Learning Flow-and Graph-based Behaviors for Android Malware Detection" ]
[ "Yafei Wu ", "Jian Shi ", "Peicheng Wang ", "Dongrui Zeng ", "Cong Sun " ]
[]
[]
As Android malware grows and evolves, deep learning has been introduced into malware detection, resulting in great effectiveness. Recent work is considering hybrid models and multi-view learning. However, they use only simple features, limiting the accuracy of these approaches in practice. This paper proposes DeepCatra, a multi-view learning approach for Android malware detection, whose model consists of a bidirectional LSTM (BiLSTM) and a graph neural network (GNN) as subnets. The two subnets rely on features extracted from statically computed call traces leading to critical APIs derived from public vulnerabilities. For each Android app, DeepCatra first constructs its call graph and computes call traces reaching critical APIs. Then, temporal opcode features used by the BiLSTM subnet are extracted from the call traces, while flow graph features used by the GNN subnet are constructed from all the call traces and inter-component communications. We evaluate the effectiveness of DeepCatra by comparing it with several state-ofthe-art detection approaches. Experimental results on over 18,000 real-world apps and prevalent malware show that DeepCatra achieves considerable improvement, e.g., 2.7% to 14.6% on the F1 measure, which demonstrates the feasibility of DeepCatra in practice.
10.1049/ise2.12082
[ "https://arxiv.org/pdf/2201.12876v2.pdf" ]
246,430,408
2201.12876
65237c1af447c6819e8dc1302a9b06923998fa57
DeepCatra: Learning Flow-and Graph-based Behaviors for Android Malware Detection Yafei Wu Jian Shi Peicheng Wang Dongrui Zeng Cong Sun DeepCatra: Learning Flow-and Graph-based Behaviors for Android Malware Detection 1Index Terms-Androidmalware detectionstatic analysisdeep learninggraph neural network As Android malware grows and evolves, deep learning has been introduced into malware detection, resulting in great effectiveness. Recent work is considering hybrid models and multi-view learning. However, they use only simple features, limiting the accuracy of these approaches in practice. This paper proposes DeepCatra, a multi-view learning approach for Android malware detection, whose model consists of a bidirectional LSTM (BiLSTM) and a graph neural network (GNN) as subnets. The two subnets rely on features extracted from statically computed call traces leading to critical APIs derived from public vulnerabilities. For each Android app, DeepCatra first constructs its call graph and computes call traces reaching critical APIs. Then, temporal opcode features used by the BiLSTM subnet are extracted from the call traces, while flow graph features used by the GNN subnet are constructed from all the call traces and inter-component communications. We evaluate the effectiveness of DeepCatra by comparing it with several state-ofthe-art detection approaches. Experimental results on over 18,000 real-world apps and prevalent malware show that DeepCatra achieves considerable improvement, e.g., 2.7% to 14.6% on the F1 measure, which demonstrates the feasibility of DeepCatra in practice. I. INTRODUCTION Android system dominates the smartphone market with around 84% share in 2021 [1]. Due to the high occupancy rate and the open-source development ecosystem, Android suffers drastic malware dissemination. Indeed, smartphone malware on Android has become a significant and persistent security threat, such as the recent boost in exploiting automated messaging functionality and Banking Trojans [2]. Therefore, effective identification of malware behaviors is in urgent demand to detect malware and protect Android users' assets. The most effective malware detection approaches for Android apps rely on machine learning and deep learning-based classification [3], [4], which classify a given app as benign or malicious according to various potential malicious features. To accommodate various characteristics of malicious app behaviors, different deep neural network structures have been adopted, including Convolutional Neural Networks (CNN) [5]- [8], Recurrent Neural Networks (RNN) [9], [10], Deep Belief Networks (DBN) [11]- [13], Multi-Layer Perceptron (MLP) [14], auto-encoder [15], heterogeneous information network [16], and Graph Neural Networks (GNN) [17]- [20]. Features considered in these approaches include request/used permissions, API call sequences, system call sequences, opcode sequences, and graph structures (e.g., abstract syntax trees, control-flow graphs, and data-flow graphs). Although various features and feature selection approaches have been proposed and high accuracies have been reported on largescale benchmarks, the recent complex malicious behaviors, e.g., cooperative data flows or even inter-component collusion, may lead to the learning model's detection performance depression and raise the requirement to use more complicated features and learning models. The advance in static and dynamic analysis provides new knowledge about malicious behaviors and promotes the efficiency and interpretability of the model by avoiding wild characteristics being used. Recent research has shown that combining the temporal features of actions and the high-order graph knowledge of system/API call sequences as the representation of malicious behaviors is adequate [18], [20], [21]. However, none of the related work has utilized the existing vulnerability knowledge, e.g., whether the calls are sensitive or critical to any public CVE. Meanwhile, although current approaches based on GNNs can capture structural knowledge (i.e., function call graph [20] and system call graph [18]) of the app code and generalize to different but structurally similar apps, the homogeneous graph structures are coarse-grained. They did not take diverse flow types, e.g., inter-component communications (ICC), into the embedding. The flows in the app have different categories of sources and sinks, which decide the flow types. The flow types refine the knowledge of app behaviors. Inspired by the observation that the benign and malicious apps differ in flow types [22], we infer that the flow types can be valuable knowledge of potential malicious behaviors. Therefore, we take heterogeneous edges into the graph embedding to improve malicious behavior identification. In this paper, we present a critical call trace guided multiview learning approach that uses the sampled opcode sequences along the critical call traces and a global abstract flow graph bridging the critical call traces and inter-component communications. In detail, we first extract a critical API set from the known vulnerability repositories with a text mining approach. Then, we traverse the static call graph with this API set and figure out the call traces ending with a call to a critical API. Based on the call traces of each app, we build the data embedding for each view of learning. We sample and take the nearest opcode sequences leading to the critical API calls for the embedding of bidirectional LSTM. We extend the critical edges with the ICC-related edges to build the global abstract flow graph for the embedding of GNN. The sampled opcode sequences and the abstract flow graph derived from the same set of call traces exhibit two different modalities, making multi-view learning feasible for our goal. In the end, we use an unweighted view combination to determine an app's benign/malicious verdict. The contributions of this paper are summarized as follows: 1) We propose a multi-view deep learning approach to detect Android malware. The approach is guided by the call traces reaching critical APIs derived from the existing vulnerability reports. The deep neural network model takes temporal features leading to critical actions and the graph structure inferring different flow types to achieve fine-grained feature extraction. 2) We design a practical flow graph abstraction to represent the relations between the critical call traces and the ICCrelated flows that are potentially diverse in benign and malicious apps. The abstraction facilitates the efficient training of the GNN-involved hybrid model. 3) We evaluate the effectiveness of our approach by comparing it with the popular deep learning-based detection techniques using CNN, LSTM, and GCN. The results on real-world benign and malicious datasets demonstrate the accuracy of our approach. The remainder of the paper is organized as follows. Section II provides the design of our approach. We describe the implementation and evaluations of our approach in Section III. Section IV discusses the threats to validity, and Section V presents related work. We conclude our paper in Section VI. II. DESIGN OF DEEPCATRA DeepCatra is a deep-learning-based embedding approach to statically detect malicious behaviors for Android Applications. We present the overall workflow of DeepCatra in Fig. 1. DeepCatra first identifies the critical APIs with the NLP technique (Section II-A). Then, DeepCatra analyzes the sensitive call traces and inter-component communications over the call graph and derives the abstract flow graph (Section II-B). Our deep-learning-based detection procedure uses a multi-view neural network (Section II-C). A graph neural network embeds the abstract flow graph derived from various sensitive traces of the app. A decision-level fusion is applied to combine the graph neural network model with a bidirectional Long Short-Term Memory (BiLSTM) model that preserves the local temporal features of executed code. Our hybrid model (Fig. 4) can efficiently realize and predict malicious apps. A. Critical APIs Identification We bridge the real-world Android vulnerabilities and popular codebases to build a more comprehensive list of critical APIs. We use the text mining technique to identify the critical APIs for the flow-based behavior modeling [23]. To derive a complete list of critical APIs, we first collect the literal descriptions related to the potential malicious or sensitive behaviors from known vulnerability repositories, e.g. [24], [25]. We also collect vulnerable Android app code samples crawled from Stack Overflow. Then, we use the term frequency-inverse document frequency (TF-IDF) to rank and select a set of keywords. This procedure excludes Java keywords, built-in types, and variable names as stopwords. To make the keyword ranking more informative, we also introduce new weighting metrics, i.e., verified / unverified status of Exploit DB entries [25], over the keywords. As a result, we collected and ranked 10,782 keywords. Thirdly, we select the top 150 keywords and search the official online document of Android platform APIs [26] for these top-ranking keywords. If more than one keyword is present in the signature text and the description of a specific API, we identify this API as a critical API. On the other hand, we collect the configurations of off-theshelf tools [27]- [31], including source/sink lists, callback lists, and taint wrapper lists. We use the top-ranking keywords to filter these APIs and merge the result with the above critical APIs. Finally, we identify 632 critical APIs. These APIs serve as the knowledge base of the malicious behaviors inducing Android app vulnerabilities. Without loss of generality, our approach is extensible to identify more critical APIs when more literal vulnerability descriptions, code samples, and tool configurations are involved or more top-ranking keywords are considered. B. Call Trace based Graph Modeling In this section, we capture the call traces used in our neural network embeddings. We sample and derive opcode sequences and build the flow graph structure for the GNN embedding based on these call traces. The runtime behaviors of Android apps are event-driven, and specific user events may trigger the exploits frequently. Therefore, we define the call trace as a static directed path in the call graph from some app entry-point to a call of some critical API identified in Section II-A. We use static analysis to capture the call traces to avoid the incompleteness of dynamic profiling or the unfolding loop events. To achieve the static analysis, we firstly generate a precise call graph for each app. With multiple entry-point methods in each app, we construct the call graph by bridging a set of subgraphs with the edges of Intent-based inter-component communications (ICC). Algorithm 1 presents the procedure to generate the call graph. All the call relations are reserved in E in a one-to-many form, i.e., callee mtd is an ordered list of user-defined methods that appeared in sequence Search Vi, Ei , add callback listeners to 15: end for 16: Add ICC-edges for SubCGs and to the ICC-edge set 17: return CG ≡ { Vj, Ej | Vj, Ej ∈ SubCGs ∨ Vj, Ej merged from subgraphs in SubCGs using ICC-edges} 18: end procedure in mtd. It could be empty for mtd if it calls no user-defined methods or only performs Intent-based ICC. H is the class hierarchy of the app. We traverse the class hierarchy for each component located at n c with type τ c to find the class object c and collect all the life-cycle methods and event listeners into the set of entry points. We use breadth-first traversal over E from each entry point in to build the subgraphs. Then we iterate on the callback listeners found in the subgraphs, add them to , and update the set of subgraphs until no new entry method is added to . The callback listeners we use include 3,390 callbacks derived by EdgeMiner [31]. Finally, we add the implicit ICC edges to derive the complete call graph CG. To identify the call traces over CG, we apply a depth-first traversal from each entry point in to see if any call to some critical API is on the forwarding control flows. The identified call trace should be in the form of ω = e 1 m 2 m 3 . . . m k−1 s k such that e 1 ∈ and s k is a call of a critical API. To build the graph model for GNN, we define an abstract flow graph G = (V, E), which captures the interrelation between the critical traces and the sensitive inter-component behaviors of apps. To define the nodes in V, we treat the app's code as many code chunks connected by different types of edges in E. For realworld apps, we disassemble the app's bytecode into smali Fig. 2 code. We separate each method's smali code into several chunks by the call sites of 1) user-defined method and 2) intent sending method. As a node of G, each code chunk may end with one of these call sites or end with the exit point of userdefined methods. Then we define different types of edges in E ≡ (ε ct , ε is , ε nb , ε ic , ε in ), as illustrated in Fig. 2. Firstly, we define the critical edge, e.g., (e 1 , s k ) ∈ ε ct , to abstract the call trace e 1 m 2 m 3 . . . m k−1 s k . The intent-sending edges in ε is represent the traces from the entry point to some sender method through Intents. We define the neighbor edge in ε nb as the edges connecting the call-ended node and its subsequent node to bridge the contexts before and after the call. We hold the ICC-edge set ε ic collected by Algorithm 1. For the method that issues at least one critical edge, ICC edge, or intentsending edge, we analyze the nodes of the method. For the ICC-ended node, we capture all the possible returning ICC edges to some intent receiver method at the same component into ε ic . We also define an implicit neighbor edge in ε in as the edge between this ICC-ended node and the beginning node of the intent receiver method. A typical example of an implicit neighbor edge in a parent component can be from a node calling startActivityForResult to the node calling onActivityResult, which receives the data from the child component. Fig. 3 presents the abstract flow graph for the call graph in Fig. 2. This abstract flow graph is afforded for the embedding of GNN. For simplicity, we do not define edge type for the returns from the callees in the call graph. The effects of edges in the abstract flow graph are twofold. The critical edges, ICC edges, or Intent-sending edges deliver information to the critical API calls or the Intents between components. On the other hand, the neighbor edges deliver context information of the current call or activity to the node of the subsequent call or activity. Because the neighbor edges are mainly for delivering information, if a neighbor edge is unconnected with any other types of edges, this isolated neighbor edge will be omitted by ε nb . Moreover, for all the edges in E, we define their respective backward edges, doubling the number and types of edges. The backward edges, represented asÊ ≡ (ε ct ,ε is ,ε nb ,ε ic ,ε in ), make the graph model more expressive and help to propagate information faster across the GNN. C. Network Structure The hybrid structure of our deep neural network mainly combines a graph neural network (GNN) [32], [33] and a bidirectional LSTM network, as depicted in Fig. 4. The BiL-STM network model is trained to capture the temporal features and sequential constraints of the potential malicious behaviors. The GNN focuses on the more complicated graph-based semantics, i.e., inter-component data flow behaviors exploited by malicious apps. The output vectors of GNN and BiLSTM layers are merged with a hidden fully connected layer. The softmax activation function then maps the output of multiple neurons to the interval of (0, 1) to produce the classification results, i.e., the probability of being malicious or benign. Without loss of generality, the softmax-based classification can be replaced by an MLP-based multi-class classification, as in [20], to categorize the malicious apps further. 1) BiLSTM Network: For each call trace identified in the call graph, we extract the opcode sequence of the call trace. We follow each invoke operation on the call trace into the userdefined callee method to accumulate the opcode sequence. The value of each opcode is normalized [34]. To avoid computing resource exhaustion, we conduct a sampling procedure over the opcode sequences to reduce the input size to the network. We specify an upper bound L of opcode samples for the apps. We retain the original opcodes for the app whose opcodes on all its call traces count less than or equal to L. For the app whose call traces have more than L opcodes, if it has y call traces, we define the upper bound of each call trace as L/y. The opcode sequences shorter than L/y are retained. For the opcode sequence longer than L/y, we truncate and hold the last L/y opcodes and drop the preceding opcodes in this sequence. Since the malicious feature tends to be reflected by the critical API call, this backward sampling policy ensures that the samples always take the ending critical call, and each app is sampled at most L opcodes. After this step, neither any opcode sequence nor the app has more than L opcodes. Then, we split the opcode sequence of each call trace into a set of fixed-length sequences for embedding. Assuming this length is , the call trace ω = m 1 m 2 m 3 . . . m k−1 m k , (k ≤ L) will be split into ω 0 = m 1 . . . m k− · k ω i = m k− ( k −i+1)+1 . . . m k− ( k −i) , s.t. i = 1.. k (1) Generally, the length of ω 1 , ω 2 , . . . , and ω k is , respectively. ω 0 may be empty or shorter than . For all the sampled call traces of an app, we drop all the ω 0 of each call trace and collect all the size-opcode sequences into an n × matrix, where n is the number of size-opcode sequences of each app. This backward splitting outperforms a forward splitting with null paddings after the critical API call to fill up a sizesequence. The input matrix of each app fits the embedding of BiLSTM. The outputs of BiLSTM are delivered to the fully connected layer for merging with the results of GNN. 2) Graph Neural Network: The GNN for malware detection is built upon the abstract flow graph G = (V, E ∪Ê). The node label l v for each node v is an opcode vector of v. Let the dimension of the opcode vector be L v . The vector is constructed with the first L v opcodes of v. If v has less than L v opcodes, we pad the vector with 0. The edge label l e ∈ {ct, is, nb, ic, in,ĉt,îs,nb,îc,în}. Let the local transition function f be a linear function, and the local output function g be an aggregation function. Then the state vector h v and the output o v are computed iteratively over timestep t as follows until convergence. h v (t) = f * (l v , l co[v] , h ne[v] (t − 1), l ne[v] ) (2) o v (t) = g(h v (t), l v ) (3) co[v] returns the set of edges incoming to v, while ne[v] returns the set of nodes with an outgoing edge to v. h v (1) is randomly initialized. Because we are dealing with a graph classification problem, the final output o v = g(h v (t), l v ) for the node is inadequate to label the graph. We take a graph-level representation to convert node score to a graph vector, h G = tanh( v∈V (i(h v (t)) h v (t)))(4) where i is the network that outputs real-valued vectors. The representation vector h G is then delivered to the hidden fully connected layer for merging the results with BiLSTM. III. IMPLEMENTATION AND EVALUATION In this section, we elaborate on the implementation issues of DeepCatra, demonstrate the parameter tuning, and evaluate the efficiency of our model. A. Implementation Issues Our call graph generation algorithm is developed with the WALA framework [35]. We use Androguard [36] to derive the opcode sequences for the critical traces and the nodes of the abstract flow graph. Each node of the abstract flow graph is persisted as a quadruple id, offset, opcode seq, invoke mtd . With the derived nodes, we identify the edges of the abstract flow graph in different types. Each edge is persisted as a triple source, target, type . To identify the ICC edges precisely, we use IC3 [37], [38] to capture the forward ICC edges of the abstract flow graph. To build the ICC edge for the explicit ICC, we bridge the node that ends with the intent-sending invoke operation with the first node of the onCreate/onReceive method of the explicit target component. For the implicit ICC, the intent-sending invoke operation is mapped by IC3 to a specific intent type of intent filter. We analyze the AndroidManifest.xml of the app and find all the components that hold an intent filter with this intent type. For each of these components, we set the first node of its onCreate/onReceive method as the target node to build the ICC edge. The intent-sending edges help indicate the malicious behaviors triggered through some inter-component communications from the major components of the app. Using a breadth-first search on the control-flow graph starting from each entry node of the app, we identify a complete set of intent-sending edges for each app. Intuitively, we focus on the user actions to launch a malicious component through ICC. In several apps, when the initial node of some entry point method ends with an intent-sending action, this node has an intent-sending edge pointing to itself. We ignore such intentsending edge in the abstract flow graph for simplicity. We implement the neural network of DeepCatra in Python 3 with PyTorch 1.7.0 [39]. For the upper bound of opcode samples used by the BiLSTM, we set L = 8, 000. B. Experimental Setup and Metrics 1) Dataset: The benign dataset consists of 9,185 realworld apps. These apps were released from 2012 to 2021 on Google Play (over 88% were released between 2016 and 2021), and we got them randomly through AndroZoo [40]. To establish ground truth, we first get a much bigger real-world app dataset. We exclude potential malware from this dataset by uploading each app in the dataset to VirusTotal [41] and retaining the apps that cannot raise any alarm by the anti-virus scanners of VirusTotal in the dataset. The malicious dataset consists of 9,443 malware from VirusShare [42], Drebin [43], DroidAnalytics [44], and CICInvesAndMal2019/2000 [45]. We also submit the malware to VirusTotal to confirm that at least one alarm is raised for each malware. Duplicated apps have been removed if they share the same hash values. Overall, our dataset is balance with 18,628 Android applications. We have further analyzed that 54.1% of the apps (5,727 benign and 4,350 malicious apps) in our dataset are obfuscated by renaming. Our approach is resilient to obfuscation because the graph features we address are robust to the common obfuscation options. We do not distinguish between obfuscated and unobfuscated apps in the following evaluations. To validate the experimental results, we divided the training/validation/testing set into around 8:1:1. To justify that our model can be generalized to evolutional apps over time, we hold the newest 10% benign and malicious apps as the testing set, which consists of 929 benign apps and 944 malicious apps. For the rest of the apps in the dataset, we randomly divide both the benign and malicious apps into 8:1. Specifically, there are 7,348 benign and 7,555 malicious apps in the training set. There are 908 benign apps and 944 malicious apps in the validation set. 2) Experimental Environment: We conduct the experiments for the classifications of these approaches on an elastic compute service with Nvidia V100 (32GB NVLink) GPU, assisted by a 2.5GHz×12 Intel Xeon (Skylake) Platinum 8163 CPU and 92GB RAM. The operating system is Linux 4.15.0-135-generic kernel (Ubuntu 18.04). To compare with other approaches, we deploy torch [46], PyG [47], and TensorFlow [48] in our environment to reproduce related approaches. 3) Metrics for measurement: We take standard metrics for the decision system to evaluate the performance of DeepCatra. accuracy = T P + T N T P + F P + T N + F N (5) precision = T P T P + F P (6) recall = T P T P + F N (7) F 1 = 2 · T P 2 · T P + F P + F N (8) F P R = F P T N + F P (9) F N R = 1 − recall(10) In these definitions, the true positives (T P ) refers to the number of malware correctly classified as the malicious app. It is more dangerous if we take a malicious app as trusted. Therefore the recall is usually more concerned. On the other hand, in some situations, e.g., exploit construction, high precision (less FPs) is more desired than high recall (less FNs). Thus F1 score measures the overall efficacy of our approach by treating precision and recall with equal importance. The area under the ROC curve represents the probability that a classifier will rank a randomly chosen malicious instance higher than a randomly chosen benign one. An area of 1.0 means a perfect classifier, while 0.5 indicates a worthless classifier. Another effective metric for measuring classifier performance is the PRC (precision-recall curve) [10], [49]. The higher the area under the PRC curve, the better is the classifier. C. Hyperparameters Tuning We use cross-entropy as the loss function to guide the training process of the model. We use the Adam algorithm [50], i.e., an algorithm for first-order gradient-based optimization of stochastic objective functions, as the optimization algorithm. The initial learning rate is 0.001. The hyperparameters of the deep neural network affect the performance of the classifier of DeepCatra. To confirm the optimal combination of the hyperparameters, we use the grid search approach in our tuning procedure. We list the related hyperparameters, their ranges, and the step intervals in Table I to specify the search space. To speed up the grid search, we use subsets of the training set and validation set for the hyperparameter tuning. We randomly chose 1/8 of the training set (including 920 benign and 944 malicious apps) and 1/8 of the validation set (including 115 benign and 118 malicious apps) to perform the grid search. This choice makes the ratio of benign apps to malicious apps on the subsets the same as that of benign apps to malicious apps on the original training/validation set. We apply a validating procedure on the complete training set to ensure the optimal hyperparameters are also optimal on the complete training and validation set. We use the optimal and suboptimal hyperparameters to train classifiers over the complete training set. We get the metrics of the classifiers on these grid points using the complete validation set. Then we ensure the classifier on optimal hyperparameters outperforms the classifiers on the suboptimal hyperparameters. Table I also lists the optimal value of hyperparameters whose tuning procedures are described below. For the local hyperparameters of BiLSTM, we split the opcode sequence sampled on each call trace into a set of length-opcode sequences. The length of these sequences affects the performance of classifiers. We set from 50 to 200 with an increment of 25. All other hyperparameters are set to their optimal value. Through grid searching, the metrics on different lengths of opcode sequences are presented in Table II. When is set to 100, the hyperparameter-tuning classifier reaches its highest F1 score. Shorter opcode sequences will retain less representative information about benign and malicious behaviors. Longer opcode sequences will introduce more interference from irrelevant information to degrade the effectiveness of hidden knowledge extraction. On the complete training and validation set, we obtain a similar trend that the optimal = 100 results in a classifier better than the classifiers on the suboptimal = 50 or 175. This trend validates the tuning procedure. Similarly, we investigate and validate the optimal number of hidden layers and neurons in each hidden layer of the BiLSTM. Their optimal values in Table I are decided by the results in Table III and Table IV. The dimension of opcode vector L v for the GNN node is tunable. The optimal L v = 13 is validated by the results in Table V. A smaller dimension value causes a significant loss of node features. In contrast, a more considerable dimension value introduces more null padding and noise. Besides, the GNN sub-model relies on the proper iteration times to effectively update the node states and propagate information between the nodes. Table VI shows that the classifiers reach optimal under ten iterations. The node state usually fails to get a fixed point when iterating less than ten times, while more iterations may cause the overfitting of the model. We observe that the above local hyperparameters are generally more sensitive to the performance of the classifiers. On the other hand, the global hyperparameters under tuning (i.e., number of epochs and batch size) reach optimal when setting the number of epochs to 25 and the batch size to 16. D. Effectiveness Evaluation We compare the effectiveness of our approach with several related works [6], [10], [18] of malware detection. In these works, the CNN-based approach [6] has released the implementation of feature extraction and model construction [51]. The LSTM-based approach [10] has released its model construction [52]. The hybrid classifier of [52] combines two LSTM models with a decision-level fusion to capture the features of both static API calls and dynamic system calls. We use androguard [36] to decompile and collect the userdefined methods in the apk. For each method, we capture all the library method calls in this method to derive a library call sequence. We use the Android 7.0 instance of Genymotion emulator to derive the dynamic feature. We parse the package name and main activity of the apk, launch the app, and capture its pid. We use Strace to track the app's events as the dynamic feature. To emulate the user actions, we automatically use the Monkey tool [53] to issue 500 random UI events. These random events contain 30% motion events, 55% touch events, and 15% other events. Due to the difference in datasets, we tune the API/system call sequence length as a hyperparameter. The optimal length is 7,000 for our dataset. Because of the call sequence length increase, we need more epochs to learn from the features. We set the number of epochs to 10 for the static model and 14 for the dynamic model. To facilitate the comparison, we reimplemented the approach based on the graph convolutional network [18]. The feature extraction was to build a system call graph for the app. The first step is similar to the dynamic feature extraction of [10]. For the system call traces generated by Strace, we only retain the 26 types of system calls claimed in [18]. We use these system call types as the nodes of the system call digraph. If a system call s 2 follows s 1 immediately in a system-call trace, we add an edge s 1 → s 2 to the digraph. We use the same centrality measures (i.e., Katz, Betweenness, Closeness, and Pagerank) as the label of each node of the digraph. We use the system call graph as the input of the GCN to train the model. Based on our dataset, we tune the hyperparameters. The optimal number of epochs is 100, the learning rate is 0.001, and the batch size is 64. For our approach, we use the classifier trained under the optimal hyperparameters in Table I. This classifier has been validated on the complete validation set with an F1 score of 0.9470. Because the comparisons are conducted on the testing set for prediction performance, the metrics values are different from the values in Table II∼Table VI. The results of comparisons are presented in Table VII. Our approach is more effective than the related works. For example, our approach reaches a 2.86%∼10.69% improvement on the recall and a 2.75%∼14.63% improvement on the F1 measure. We also sketch the ROC curves of each approach in Fig. 5. We obtain 96.59% area under the ROC curve on our dataset. The precision-recall curves (PRC) [49] show how precision varies with recall when the discrimination threshold is varied. The closer the value of the area under the PRC curve gets to 1, the better the classifier's performance. The PRC curves of each approach are given in Fig. 6. Our classifier achieves an area of 97.69% under the PRC curve on our dataset. The abrupt part of the GCN's PRC curve indicates that the GCN classifier predicts around 14.5% of samples in the testing set to have the same probability. Stepping over one specific discrimination threshold causes these samples to become negative, and the precision and recall vary drastically. We also find some factors other than the difference in the datasets that may impact the comparison results, discussed in Section IV. IV. DISCUSSION Incomparability of critical API lists. Most of the malicious behaviors of Android apps are conducted by specific APIs or based anomalies is to identify the critical APIs used by malicious behaviors. The sensitive APIs have been potentially used as the sources and sinks of data-flow analysis [27], [28], [30], [54], the target APIs for dynamic analysis [55], or the features differentiating malicious behaviors by the usage frequency [56]. We summarize the information of sensitive APIs addressed by these works in Table VIII. The sensitive APIs of these works are manually crafted with experts' domain knowledge or dynamically profiled in sample apps by sandboxing, which may have significant bias. Although some learning-based approaches, e.g., [29], automatically identify and categorize sensitive data sources and sinks of the Android framework or apps, the derived APIs indicate restricted correlations with real-world vulnerabilities. In contrast, our attempt to use NLP techniques to identify the critical APIs concretizes the correlation with real-world vulnerabilities. Due to the diverse usage of the API lists, quantifying the advantage of our API list against other works' API lists is infeasible. [27] 85(sources)/198(sinks) Argus-SAF 3.2.0 [30] 32(sources)/42(sinks) DroidSafe (commit 1eab2fc) [54] 4,051(sources)/2,116(sinks) IntelliDroid (commit fde1cae) [55] 300 DroidAPIMiner [56] 169 this work 632 Applying different API lists into our framework requires high analysis costs in our call traces generation. Some API lists tightly integrated into the analysis framework, e.g., [54], are undeployable in our static analysis. Pros and cons of the static analysis. The scalability and robustness of program analysis significantly impact the applicability of our detection model. The feature extraction of many other works, e.g., [57], [58], rely on straightforward static analysis to abstract the sketchy features like permissions, API calls, components, opcodes, and strings. Compared with these features, the high-order call trace and ICC features are hard to capture, and the analysis is more likely to be confronted with failures. When analyzing the inter-component communications on our dataset, we investigated that the IC3 tool reports failure on 1,607 apps (i.e., 642 benign and 965 malicious apps), which account for 8.6% of our dataset. The abstract flow graphs of these apps then miss this type of edge, limiting the accuracy of our approach. On the other hand, we only use static analysis to decide the features. Our feature extraction is more stable than the dynamic analysis approaches, e.g., [10], [18]. We do not depend on pseudo-random inputs generated for emulation, which cannot efficiently induce high feature coverage. For example, the events triggered by the Monkey tool in the compared approaches are random to miss the button press leading to malicious behavior potentially. Also, dynamic feature profiling is time-consuming. We infer that one reason for the relatively low performance of [18] in Table VII is because we inject 500 random events before terminating the dynamic analysis of each sample, while in [18], they inject more than one thousand, which costs unrealistic time on our extensive dataset. The relatively shorter system call sequences may miss certain connections and introduce isolated nodes in the system call graph. When it comes to the static analysis of [10], another threat is the obfuscation of APIs. Considering an obfuscated API call to lcom/noveo/pdf/e/e;.a:(iljava/lang/string;)v, the definition of method a is in the parent class of e. Such method call should be ignored as a call to user-defined API because such a method is unlikely to appear in this form in other apps. However, [10] failed to capture such inheritance relation. Lack of sustainability consideration. The evolution of malware poses another threat to the applicability of our approach. From an over-time perspective, recent works on this issue depict specific features of malware about callbacks, component behaviors, inter-component communications, data flows, and framework usages, differently from the benign apps [59]- [61]. Specifically, the longitudinal study [59] investigates apps' code and runtime behavior evolutions on diversified metrics in complementary dimensions and makes valuable recommendations about app analysis and defense. The evolution of method calls and ICCs in apps indicates malware and benign apps' diverse behaviors to different components and the callback usage [61]. On the repackaging malware, the rider behaviors are analyzed with differential analysis on top of annotated CFGs [60]. Although these works identify different characteristics of ICCs and method calls from our observation of flow types in Section I, we believe these characteristics can bring us new knowledge to develop new abstract flow graphs for our multiview learning. To sustain the malware detection models effectively, DroidEvolver [62] takes the API usage as detection features and updates old models with the detected app and the classification result of the model pool. The feature set is also updated to adapt to the feature changes. The malware detector can rely on the dynamic features and metrics sustainable to the emerging malware [63], [64]. A potential adaption for our evaluation is to use the aged samples in our dataset to train the model and the relative new samples to test, which may require more aged samples in the dataset. Attaining sustainability for our approach is even more challenging than these works [62]- [64] because the abstract flow graphs used by GNN have higher dimensionality to complicate differentiating sustainable features. We may resort to the evolving structure of GNN [65] to mitigate the sustainability challenge. V. RELATED WORK A. API features-related malware detection API-related features are critical for Android malware detection. DroidAPIMiner [56] addressed the frequency of API calls, the package information, and the parameters of APIs. The data flows are analyzed to estimate the value of the critical API parameters. MalPat [66] uses the app's permissions to decide the sensitivity of APIs and their coarse-grained correlations. Build-in data-flow analysis can also derive abnormal data dependence paths and generate modalities bridged by specific source-sink API pairs [22], [67]. Some multi-level and behavior-based approaches, e.g. [68], [69], detect the anomaly based on the system calls, critical API calls, Binder communication, user-level activities, and package-level metadata. DroidCat [70] profiles method calls and inter-component communication dynamically and uses these features to classify malware accurately. Higher-dimensional program features, e.g., graph-level structures, are crucial to malware analysis. For example, the similarity between API dependency graphs has been featured to detect anomalies in apps [71]. The behavioral graphs derived with relations of either lifecycle methods or permission-related APIs are used to mine the patterns of malicious behaviors [72]. B. Deep learning-based malware detection GNN [17]- [20], [73], [74] is a practical approach to capturing malware's structural and complicated semantics features. Yan et al. [17] used graph convolutional neural network (GCN) to classify CFG-represented binary malware. John et al. [18] proposed to use GCN to classify whether the system call graphs constructed by the control-dependent file management and network access syscalls exhibit malicious behavior. Busch et al. [74] extracted network flow graphs based on the network traffic data generated during the execution of the apps. They proposed to use GNN and its variants to learn the representations of the network flow graphs. GDroid [19] proposed a heterogeneous graph fed into the graph convolutional network. The heterogeneous graph consists of edges representing patterns of the API invocations by the apps and the API occurrence in the methods. Compared with our abstract flow graph in Section II-B, such heterogeneous graphs are coarsegrained. Xu et al. [20] generated the graph embedding from the function call graph for the detection model, and the NLP technique inspired their node embedding. CGDroid [73] also relies on a precise call graph and the NLP technique to learn the graph representation for malware detection. Several approaches have concatenated or combined different neural network models for the efficiency and effectiveness of malware detection. The LSTM-based hierarchical denoise network (HDN) model [75] learns features from raw opcode sequences. The HDN has a method block denoise module to filter out opcode segments irrelevant to the malicious behaviors. DeepRefiner [76] is a two-layer architecture for malware detection. After capturing potential malicious features on required system resources in XML files with MLP-based prediction, the uncertain apps are fed into a second detection layer. This layer uses LSTM on variable-length bytecode vector sequences representing method-level and app-level bytecode semantics. Wang et al. [77] used deep autoencoder as a pre-training method for the CNNs to reduce the training time cost to learn the malicious features. Pektas et al. [21] combined CNN and LSTM to derive the latent features from opcode sequences. Lu et al. [78] combined DBN with gated recurrent units (GRU) to accelerate learning of both static features and longer-time operation sequences. The multi-view and multi-modal approaches also integrate neural network models into a hybrid structure for learning effectiveness [57], [58], [79]. DANdroid [57] proposed a multi-view discriminative adversarial network (DAN) that adapts obfuscation-resilient feature sets to remove bias to obfuscation. Kim et al. [58] proposed the first multi-modal deep learning framework to detect Android malware. The framework extracts different features to reflect the properties of apps from various aspects. It refines the features with the existence-based and similaritybased feature extraction methods to achieve effective feature representation. Zhu et al. [79] also addressed multi-modal detection on different features and with submodels based on CNN. To the best of our knowledge, the state-of-the-art approaches have never integrated graphic neural networks into a multi-view approach to classify malware. VI. CONCLUSION We proposed DeepCatra, a multi-view learning-based detection of Android malware. We built the hybrid learning model based on the public knowledge of vulnerabilities and the fine-grained features we used. The call traces leading to the critical actions are sampled into opcode sequences and embedded into the BiLSTM component of the hybrid model. An abstract flow graph inferring the relations between different flow types is built for the app as the fine-grained features of the graph neural network component. By taking both the temporal characteristics and graph features into different views of learning, our detection model can outperform several state-of-the-art detection approaches using CNN, LSTM, and GCN. In future work, we expect to extend our abstract flow graph model to accommodate more flow types, which may further benefit the effectiveness of our malware detection. More scalable static analyses are also expected to reduce the feature missing in very complex applications. The code and models of DeepCatra have been made publicly available at https://github.com/shijiansj/DeepCatra. APPENDIX A NETWORK STRUCTURE IN DETAIL The structure of our deep neural network is in Fig. 7. For the GNN, the node label l v ∈ R L v and L v is a tunable parameter. The edge label l (u,v) ∈ R 10 is the one-hot encoding of the edge types. The state vector h v (t) ∈ R s and s = 32 in the implementation. linear 1 takes the concatenation of l u , l v , and l (u,v) as input and outputs the size-32 2 tensor, which is then resized to 32 × 32. linear 2 takes l v as input and outputs a size-32 tensor. For the BiLSTM, we suppose there are n size-opcode sequences for an app. The embedding takes these fix-length sequences as input, and outputs the threedimensional tensor with the size n × × 128. There are two hidden layers in our BiLSTM, with 256 neurons in each hidden layer. The output of the two hidden layers is with the size n × × 512. After the dimensionality reduction, linear 3 and linear 4 respectively output n × 64 and n × 32 tensors. The output of average has a size of 32. As seen in Section III-C, the optimal parameters for our neural network are = 100 and L v = 13. Fig. 2 . 2Network Edge Types Fig. 3 . 3Abstract Flow Graph w.r.t. the Call Graph in Fig. 4 . 4Structure of Neural Network Model Fig. 5 . 5ROC curves and the comparison on AUC Fig. 7 . 7Detailed Deep Neural Network Structure Algorithm 1 Call Graph Generation for Android App α E ← { mtd, callees mtd | ∀mtd ∈ Classes(α)} C ← { nc, τc | path name nc and category τc of component c extracted from AndroidManifest.xml} SubCGs ← SubCGs ∪ Vi, Ei1: procedure CALLGRAPHGEN(α) 2: 3: H ← ClassHierarchy(α) 4: 5: ← ∅ 6: for all nc, τc ∈ C do 7: c ← traverse(H, nc) 8: ← ∪ getEntryM td(c, τc) 9: end for 10: SubCGs ← ∅ 11: for all entryi ∈ do 12: Vi, Ei ← BF S(entryi, E, H) 13: 14: TABLE I HYPERPARAMETERS IOF DEEPCATRA WITH SEARCH SPACES AND OPTIMAL VALUESHyperparameter Scope Network type Sampling space Optimal Length of splitted opcode sequence ( ) Local BiLSTM min:50; max:200; step:25 100 Number of hidden layers Local BiLSTM min:1; max:4; step:1 2 LSTM unit size Local BiLSTM 64, 128, 256, 512 256 Dimension of opcode vector (L v ) Local GNN min:9; max:15; step:2 13 Iteration times of node state Local GNN min:6; max:12; step:2 10 Number of epochs Global All min:15; max:30; step:5 25 Batch size Global All 4, 8, 16, 32 16 TABLE II TUNING AND VALIDATING THE LENGTH OF SPLITTED OPCODE SEQUENCES Tuning on sub-datasets Validating 50 75 100 125 150 175 200 50 100 175 accuracy 0.8969 0.8841 0.9141 0.8884 0.8884 0.9055 0.8798 0.9157 0.9476 0.9113 precision 0.9284 0.9115 0.9415 0.9179 0.9208 0.9308 0.9105 0.9508 0.9774 0.9512 recall 0.8641 0.8584 0.8882 0.8505 0.8569 0.8790 0.8484 0.8802 0.9184 0.8686 F1 0.8950 0.8841 0.9140 0.8829 0.8877 0.9041 0.8783 0.9141 0.9470 0.9080 TABLE III TUING AND VALIDATING NUMBERS OF HIDDEN LAYERS IN BILSTM #hidden Tuning on sub-datasets Validating layer 1 2 3 4 1 2 3 accuracy 0.9055 0.9141 0.8583 0.8154 0.9193 0.9476 0.8659 precision 0.9378 0.9415 0.8603 0.8090 0.9497 0.9774 0.8778 recall 0.8708 0.8882 0.8583 0.8208 0.8891 0.9184 0.8527 F1 0.9030 0.9140 0.8580 0.8148 0.9184 0.9470 0.8651 TABLE IV TUNING IVAND VALIDATING NUMBERS OF NEURONS IN HIDDEN LAYER OF BILSTMTuning on sub-datasets Validating #neuron 64 128 256 512 128 256 512 accuracy 0.8712 0.9012 0.9141 0.8602 0.9065 0.9476 0.8905 precision 0.9058 0.9345 0.9415 0.8614 0.9436 0.9774 0.9404 recall 0.8398 0.8700 0.8882 0.8583 0.8686 0.9184 0.8358 F1 0.8706 0.9010 0.9140 0.8579 0.9045 0.9470 0.8850 TABLE V TUNING VAND VALIDATING DIMENSION OF OPCODE VECTOR FOR GNNTuning on sub-datasets Validating L v 9 11 13 15 9 13 15 accuracy 0.8969 0.8755 0.9141 0.9012 0.9044 0.9476 0.9249 precision 0.9250 0.9088 0.9415 0.9417 0.9432 0.9774 0.9579 recall 0.8667 0.8435 0.8882 0.8638 0.8622 0.9184 0.8919 F1 0.8949 0.8749 0.9140 0.9010 0.9009 0.9470 0.9237 TABLE VI TUNING AND VALIDATING ITERATION TIMES OF NODE STATE FOR GNN Tuning on sub-datasets Validating #iterations 6 8 10 12 8 10 12 accuracy 0.8669 0.8927 0.9141 0.8798 0.9087 0.9476 0.8957 precision 0.8712 0.9057 0.9415 0.9100 0.9457 0.9774 0.9204 recall 0.8630 0.8802 0.8882 0.8476 0.8686 0.9184 0.8707 F1 0.8670 0.8927 0.9140 0.8776 0.9055 0.9470 0.8949 a sequence of sensitive API calls, which are usually triggered by user inputs. Therefore, a prerequisite of learning the flow- TABLE VII COMPARISON VIIWITH OTHER WORKS OF ANDROID MALWARE DETECTIONApproach accuracy precision recall FPR FNR F1 CNN-based [6], [51] 0.9215 0.9473 0.8941 0.0506 0.1059 0.9199 LSTM-based [10], [52] 0.9327 0.9669 0.8972 0.0312 0.1028 0.9308 GCN-based [18] 0.8089 0.8052 0.8189 0.2013 0.1811 0.8120 this work 0.9594 0.9932 0.9258 0.0065 0.0742 0.9583 Fig. 6. PRC curves and the comparison on AUC TABLE VIII SENSITIVE APIS USED BY EXISTING TOOLS Tool #Sensitive APIs TaintDroid [28] 62 FlowDroid 2.8 Smartphone Market Share. IDC, "Smartphone Market Share," Available at https://www.idc.com/ promo/smartphone-market-share, 2021. . Mcafee, McAfee Mobile Threat ReportMcAfee, "McAfee Mobile Threat Report," Available at https://www.mcafee.com/content/dam/global/infographics/ McAfeeMobileThreatReport2021.pdf, 2021. A survey of android malware detection with deep neural models. J Qiu, J Zhang, W Luo, L Pan, S Nepal, Y Xiang, ACM Comput. Surv. 53636J. 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[ "https://github.com/shijiansj/DeepCatra.", "https://github.com/androguard.", "https://github.com/siis/ic3.", "https://github.com/sankardasroy/" ]
[ "THE MEAN-SQUARE DICHOTOMY SPECTRUM AND A BIFURCATION TO A MEAN-SQUARE ATTRACTOR", "THE MEAN-SQUARE DICHOTOMY SPECTRUM AND A BIFURCATION TO A MEAN-SQUARE ATTRACTOR" ]
[ "Thai Son Doan \nDepartment of Mathematics Imperial College London 180 Queen's Gate\nInstitut für Mathematik\nGoethe Universität\nSW7 2AZ, D-60054London, Frankfurt am MainUnited Kingdom, Germany\n", "Martin Rasmussen \nDepartment of Mathematics Imperial College London 180 Queen's Gate\nInstitut für Mathematik\nGoethe Universität\nSW7 2AZ, D-60054London, Frankfurt am MainUnited Kingdom, Germany\n", "Peter E Kloeden \nDepartment of Mathematics Imperial College London 180 Queen's Gate\nInstitut für Mathematik\nGoethe Universität\nSW7 2AZ, D-60054London, Frankfurt am MainUnited Kingdom, Germany\n" ]
[ "Department of Mathematics Imperial College London 180 Queen's Gate\nInstitut für Mathematik\nGoethe Universität\nSW7 2AZ, D-60054London, Frankfurt am MainUnited Kingdom, Germany", "Department of Mathematics Imperial College London 180 Queen's Gate\nInstitut für Mathematik\nGoethe Universität\nSW7 2AZ, D-60054London, Frankfurt am MainUnited Kingdom, Germany", "Department of Mathematics Imperial College London 180 Queen's Gate\nInstitut für Mathematik\nGoethe Universität\nSW7 2AZ, D-60054London, Frankfurt am MainUnited Kingdom, Germany" ]
[]
The dichotomy spectrum is introduced for linear mean-square random dynamical systems, and it is shown that for finite-dimensional mean-field stochastic differential equations, the dichotomy spectrum consists of finitely many compact intervals. It is then demonstrated that a change in the sign of the dichotomy spectrum is associated with a bifurcation from a trivial to a non-trivial mean-square random attractor.
10.3934/dcdsb.2015.20.875
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119,132,532
1403.1068
7204dbf41d7a916757018c56898b61e18a2e12a1
THE MEAN-SQUARE DICHOTOMY SPECTRUM AND A BIFURCATION TO A MEAN-SQUARE ATTRACTOR Thai Son Doan Department of Mathematics Imperial College London 180 Queen's Gate Institut für Mathematik Goethe Universität SW7 2AZ, D-60054London, Frankfurt am MainUnited Kingdom, Germany Martin Rasmussen Department of Mathematics Imperial College London 180 Queen's Gate Institut für Mathematik Goethe Universität SW7 2AZ, D-60054London, Frankfurt am MainUnited Kingdom, Germany Peter E Kloeden Department of Mathematics Imperial College London 180 Queen's Gate Institut für Mathematik Goethe Universität SW7 2AZ, D-60054London, Frankfurt am MainUnited Kingdom, Germany THE MEAN-SQUARE DICHOTOMY SPECTRUM AND A BIFURCATION TO A MEAN-SQUARE ATTRACTOR The dichotomy spectrum is introduced for linear mean-square random dynamical systems, and it is shown that for finite-dimensional mean-field stochastic differential equations, the dichotomy spectrum consists of finitely many compact intervals. It is then demonstrated that a change in the sign of the dichotomy spectrum is associated with a bifurcation from a trivial to a non-trivial mean-square random attractor. 1. Introduction. Mean-square properties are of traditional interest in the investigation of stochastic systems in engineering and physics. This is quite natural since the Ito stochastic calculus is a mean-square calculus. At first sight, it is thus somewhat surprising that the classical theory of random dynamical systems and their spectra is a pathwise theory, although this can be justified by Doss-Sussman-like transformations between stochastic differential equations and path-wise random ordinary differential equations [1]. Such transformations, however, do not apply to mean-field stochastic differential equations, which include expectations of the solution in their coefficient functions [6]. Mean-square random dynamical systems based on deterministic two-parameter semi-groups from the theory of nonautonomous dynamical systems acting on a state space of random variables or random sets with the mean-square topology were introduced in [7]. These act like deterministic systems with the stochasticity built into the state spaces of mean-square random variables. A mean-square random attractor was defined as a nonautonomous pullback attractor for such systems from the theory of nonautonomous dynamical systems [9]. The main difficulty in applying the theory is the lack of useful characterisations of compact sets of such spaces of mean-square random variables. In this paper, a theory of mean-square exponential dichotomies is presented for linear mean-field stochastic differential equations. (It also applies to classical linear stochastic differential equations). Although the corresponding mean-square random dynamical systems are essentially infinite-dimensional their dichotomy spectrum is given by the union of finitely many intervals. This is applied to analyse a nonlinear mean-field stochastic differential equation, for which it is shown that the trivial solution undergoes a mean-square bifurcation leading to a nontrivial mean-square attractor. The paper is structured as follows. Section 2 contains the definition of a meansquare random dynamical system, and the notions of mean-square exponential dichotomy and mean-square dichotomy spectrum are introduced. Section 3 explains under which conditions, a mean-field stochastic differential equation generates a mean-square random dynamical system. In Section 4, the spectral theorem is established, which says that the mean-square spectrum of a linear mean-field stochastic differential equation consists of finitely many compact intervals. Finally, in the last section, it is shown that for a one-dimensional mean-field SDE of pitchfork-type, a stability change in the mean-square spectrum is associated with a bifurcation from a trivial to a non-trivial mean-square random attractor. 2. Mean-square random dynamical systems. Consider the time set R, and define R 2 ≥ := (t, s) ∈ R 2 : t ≥ s . Let (Ω, F, {F t } t∈R , P) be a complete filtered probability space satisfying the usual hypothesis, i.e., {F t } t∈R is an increasing and right-continuous family of sub-σ-algebras of F, which contain all P-null sets. Essentially, F t represents the information about the randomness at time t ∈ R. Finally, define X := L 2 (Ω, F; R d ) and X t := L 2 (Ω, F t ; R d ) for t ∈ R with the norm X ms := E|X| 2 , where | · | is the Euclidean norm on R d . Definition 1. A mean-square random dynamical system (MS-RDS for short) ϕ on the underlying phase space R d with the filtered probability space (Ω, F, {F t } t∈R , P) is a family of mappings ϕ(t, s, ·) : X s → X t , for (t, s) ∈ R 2 ≥ , which satisfies: (1) Initial value condition. ϕ(s, s, X s ) = X s for all X s ∈ X s and s ∈ R. (2) Two-parameter semigroup property. For all X ∈ X w and all (t, s), (s, w) ∈ R 2 ≥ ϕ(t, w, X) = ϕ(t, s, ϕ(s, w, X)). (3) Continuity. ϕ is continuous. Mean-square random dynamical systems are essentially deterministic with the stochasticity built into or hidden in the time-dependent state spaces. A MS-RDS ϕ is called linear if for each (t, s) ∈ R 2 ≥ , the map ϕ t,s (·) := ϕ(t, s, ·) is a bounded linear operator. It will be denoted by Φ t,s , and Φ t,s (X) will conventionally be written Φ t,s X. A spectral theory for linear mean-square random dynamical systems can be established based on exponential dichotomies. Definition 2 (Mean-square exponential dichotomy). Let γ ∈ R. A linear meansquare random dynamical system Φ t,s : X s → X t is said to admit an exponential dichotomy with growth rate γ if there exist positive constants K, α and a timedependent decomposition X t = U γ (t) ⊕ S γ (t) for t ∈ R such that Φ t,s X s ms ≤ Ke (γ−α)(t−s) X s ms for X s ∈ S γ (s) and t ≥ s , Φ t,s X s ms ≥ 1 K e (γ+α)(t−s) X s ms for X s ∈ U γ (s) and t ≥ s . A special case of exponential dichotomy, when the growth rate is equal to zero and the space of initial condition consists of the deterministic vectors in R d , is also investigated in [2,12], where a Perron-type condition for existence of this exponential dichotomy is established. Definition 3 (Mean-square dichotomy spectrum). The mean-square dichotomy spectrum for a linear MS-RDS Φ is defined as Σ := γ ∈ R : Φ has no exponential dichotomy with growth rate γ . The set ρ := R \ Σ is called the resolvent set of Φ. The dichotomy spectrum was first introduced in [11] for nonautonomous differential equations. Dichotomy spectra for random dynamical systems have been discussed recently in [3,4,13]. 3. Mean-field stochastic differential equations. Mean-field stochastic differential equations of the form dX t = f (t, X t , EX t ) dt + g(t, X t , EX t ) dW t(1) were introduced in [6]. Here {W t } t∈R is a two-sided scalar Wiener process defined on a probability space (Ω, F, P), and F := (f, g) : R × R d × R d → R d × R d . Let Lip(R d ) denote the set of Lipschitz continuous functions f : R d → R d , and for each f ∈ Lip(R d ), set f (·) lg := sup x∈R d |f (x)| 1 + |x| . Suppose that (A1) Γ := sup (t,x)∈R×R d LipF (t, x, ·) + F (t, x, ·) lg < ∞. (A2) For each R > 0, there exist a constant L R and a modulus continuity ω R such that F (t 1 , x 1 , ·) − F (t 2 , x 2 , ·) 2 lg ≤ L R |x 1 − x 2 | 2 + ω R (|t 1 − t 2 |) for all (t k , x k ) ∈ R × R d with t k + |x k | 2 ≤ R, k ∈ {1, 2}. Let {F t } t∈R be the natural filtration generated by {W t } t∈R , and define X := L 2 (Ω, F; R d ), X t := L 2 (Ω, F t ; R d ) for t ∈ R. Given any initial condition X s ∈ X s , s ∈ R, a solution of (1) is a stochastic process {X t } t≥s with X t ∈ X t for t ≥ s, satisfying the stochastic integral equation X t = X s + t s f (u, X u , EX u ) du + t s g(u, X u , EX u ) dW u . It was shown in [6] that the SDE (1) has a unique solution and generates a MS-RDS {ϕ t,s } t≥s on the underlying phase space R d with a probability set-up (Ω, F, {F t } t∈R , P), defined by ϕ t,s : X s → X t with ϕ t,s (X s ) = X t for X s ∈ X s . 4. Mean-square dichotomy spectrum for linear mean-field stochastic differential equations. Consider a linear mean-field stochastic differential equation dX t = A(t)X t + B(t)EX t dt + C(t)X t + D(t)EX t dW t ,(2) where A, B, C, D : R → R d×d are continuous bounded functions, which generates a linear mean-square random dynamical system Φ t,s . Proposition 4 (Equations for the first and second moments). Let X s ∈ X s , and define X t := Φ t,s X s for t ≥ s. Then d dt EX t = A(t) + B(t) E(X t ) and for all i, j ∈ {1, . . . , d}, d dt EX i t X j t = d k=1 a ik (t)EX k t X j where 1 ≤ i ≤ j ≤ d. Then for any X s ∈ X s , Φ t,s X s ms = d i=1 V i,i (t, s)(π s X s ) 1 2 ,(4) where the map π s = π 1 s × π 2 s : X s → R d(d+1) 2 × R d(d+1) 2 is defined by (π 1 s X s ) i,j = EX i s EX j s and (π 2 s X s ) i,j = EX i s X j s . Remark 6. It is interesting to compare the above equations with the ordinary differential equations for the first moment and second moments of linear stochastic differential equations, see e.g. [5, Section 6.2]. The proofs of the following preparatory results are straightforward. Lemma 7. Let γ ∈ R be such that that the linear MS-RDS Φ t,s : X s → X t generated by (1) admits an exponential dichotomy with the growth rate γ and a decomposition X t = U γ (t) ⊕ S γ (t). Then the subspace S γ (t) is uniquely determined, i.e., if the linear MS-RDS Φ t,s also admits an exponential dichotomy with the growth rate γ and another decomposition X t = U γ (t) ⊕ S γ (t), then S γ (t) = S γ (t) for all t ∈ R. The subspaces S γ (t) of an exponential dichotomy with the growth rate γ are its stable subspaces. The following lemma provides an inclusion relation between these stable subspaces. Its proof follows directly from the definition of an exponential dichotomy. Lemma 8. Let γ 1 < γ 2 be such that the linear MS-RDS admits an exponential dichotomy with the growth rates γ 1 and γ 2 . Then S γ1 (t) ⊂ S γ2 (t) for all t ∈ R. One of the main results of this paper is the following characterisation of the dichotomy spectrum. Theorem 9 (Spectral Theorem). Suppose that the coefficient functions in the linear mean-field stochastic differential equation (2) satisfy max |a ij (t)|, |b ij (t)|, |c ij (t)|, |d ij (t)| ≤ m for i, j ∈ {1, . . . , d} and t ∈ R (5) with some m > 0. Then the dichotomy spectrum Σ is the disjoint union of at most d(d + 1) compact intervals [a 1 , b 1 ], . . . , [a n , b n ] with a 1 ≤ b 1 < a 2 ≤ b 2 ≤ · · · < a n ≤ b n . Furthermore, for each s ∈ R, there exists a filtration of subspaces {0} V 1 (s) V 2 (s) · · · V n (s) = X s which satisfies that for any i ∈ {1, . . . , n}, a random variable X s ∈ V i (s) if and only if for any ε > 0, there exists K(ε) > 0 such that Φ t,s X s ms ≤ K(ε)e (bi+ε)(t−s) for t ≥ s . Proof. The proof is divided into several steps. Step 1. First it will be shown that (−∞, −Γ) ⊂ ρ and (Γ, ∞) ⊂ ρ, where Γ := 2dm + 2d 2 m 2 . Let X s ∈ X s be arbitrary, and define α(t) := max i,j∈{1,...,d} By the inequalities EXY ≤ √ EX 2 EY 2 and (EX) 2 ≤ EX 2 , it follows that α(t) = max 1≤i≤d (EX i t ) 2 ≤ Φ(t, s)X s 2 ms . Then by Corollary 5, α(t) ≤ α(s) + 2Γ t s α(u) du , and Gronwall's inequality then yields Φ(t, s)X s 2 ms ≤ dα(t) ≤ de 2Γ(t−s) α(s) ≤ de 2Γ(t−s) X s 2 ms . This proves that (Γ, ∞) ⊂ ρ. Time reversal of the equations in Corollary 5 leads to Φ(t, s)X s 2 ms ≥ 1 d e −2Γ(t−s) X s 2 ms for (t, s) ∈ R 2 ≥ , which proves (−∞, Γ) ⊂ ρ. Step 2. It will be shown that for any t ∈ R, the set S γ (t) : γ ∈ ρ ∩ (−Γ − 1, Γ + 1) consists of at most d(d + 1) + 1 elements. Suppose the contrary, i.e., there exist n + 1 numbers γ 0 < γ 1 < · · · < γ n in ρ ∩ (−Γ − 1, Γ + 1), where n > d(d + 1), such that S γi (t) = S γj (t) for i = j. Then by Lemma 8, S γ0 (t) S γ1 (t) · · · S γn (t). Thus, there exist X 1 t , . . . , X n t such that X i t ∈ S γi (t), X i t ∈ S γi−1 (t) for i ∈ {1, . . . , n} . By definition of the γ i , there exist K, α > 0 and complementary subspaces U γi (t) such that X t = U γi (t) ⊕ S γi (t) and Φt ,t X t ms ≤ Ke (γi−α)(t−t) X t ms for X t ∈ S γi (t) andt ≥ t(6) and Φt ,t X t ms ≥ 1 K e (γi+α)(t−t) X t ms for X t ∈ U γi (t) and t ≥ t . Since R d(d+1) 2 × R d(d+1) 2 is d(d + 1)-dimensional, it follows that there exist k ≤ n and α 1 , . . . , α k−1 with α 2 1 + · · · + α 2 k−1 = 0 and π t X k t = α 1 π t X 1 t + · · · + α k−1 π t X k−1 t .(8) Consequently, by Corollary 5, Φt ,t X k t 2 ms = d i=1 V i,i (t, t)(π t X k t ) = d i=1 k−1 j=1 α j V i,i (t, t)(π t X j t ) ≤ k−1 j=1 |α j | d i=1 k−1 j=1 V i,i (t, t)(π t X j t ) = k−1 j=1 |α j | Φt ,t X 1 t 2 ms + · · · + Φt ,t X k−1 t 2 ms . By definition of γ i and (6) Φt ,t X k t 2 ms ≤ (k − 1)K   k−1 j=1 |α j |     k−1 j=1 X j t 2 ms   e (γ k−1 −α)(t−t) . Hence, it follows by (6) and (7) that X k t ∈ S γ k−1 (t), which leads to a contradiction. Step 3. As proved in Step 2, for t ∈ R, let S 0 (t) S 1 (t), . . . , S n (t) . S 1 (t) · · · S n (t) with n ≤ d(d + 1) satisfy S γ (t) : γ ∈ ρ ∩ (−Γ − 1, Γ + 1) = {S 0 (t), By Step 1, it follows that S 0 (t) = {0} and S n (t) = X t . For each i ∈ {0, . . . , n}, define I i := γ ∈ ρ ∩ (−Γ − 1, Γ + 1) : S γ (t) = S i (t) . It will be shown that I i = (b i , a i+1 ), where b i = inf{γ : γ ∈ I i } and a i+1 = sup{γ : γ ∈ I i } for i ∈ {0, . . . , n} . First, let γ ∈ I i be arbitrary. By the definition of I i , there exist K, α > 0 and a decomposition X t = U (t) ⊕ S i (t) and Φt ,t X t ms ≤ Ke (γ−α)(t−t) X t ms for X t ∈ S i (t) andt ≥ t and Φt ,t X t ms ≥ 1 K e (γ+α)(t−t) X t ms for X t ∈ U (t) andt ≥ t . This implies that (γ − α, γ + α) ⊂ I i , so I i is open. It can be shown similarly that I i is connected. Hence I i = (a i , b i ). Combining this result and Step 1 gives ρ = (−∞, a 1 ) ∪ (b 1 , a 2 ) ∪ · · · ∪ (b n−1 , a n ) ∪ (b n , ∞), which implies that Σ = [a 1 , b 1 ] ∪ · · · ∪ [a n , b n ]. To conclude the proof, the filtration corresponding to the spectral intervals is constructed as follows: for t ∈ R, V 0 (t) := {0}, V n (t) := X t , and V i (t) := S γ (t), where γ ∈ (b i , a i+1 ), i ∈ {1, . . . , n − 1} . Due to Lemma 7, the definition of V i is independent of γ ∈ (b i , a i+1 ) for i ∈ {1, . . . , n − 1}. The strict inclusion V i V i+1 for i ∈ {0, . . . , n − 1} follows from the construction of the open interval (b i , a i+1 ) above. Finally, the dynamical characterisation of V i follows from the definition of (b i , a i+1 ) and the definition of exponential dichotomy. This completes the proof. 5. Bifurcation of a mean-square random attractor. A mean-square random attractor was defined in [7] as the pullback attractor of the nonautonomous dynamical system formulated as a mean-square random dynamical system. Specifically, a family A = {A t } t∈R of nonempty compact subsets of X with A t ⊂ X t for each t ∈ R is called a pullback attractor if it pullback attracts all uniformly bounded families D = {D t } t∈R of subsets of {X t } t∈R , i.e., lim s→−∞ dist(ϕ(t, s, D s ), A t ) = 0. Uniformly bounded here means that there is an R > 0 such that X ms ≤ R for all X ∈ D t and t ∈ R. The existence of pullback attractors follows from that of an absorbing family. A uniformly bounded family B = {B t } t∈R of nonempty closed subsets of {X t } t∈R is called a pullback absorbing family for a MS-RDS ϕ if for each t ∈ R and every uniformly bounded family D = {D t } t∈R of nonempty subsets of {X t } t∈R , there exists some T = T (t, D) ∈ R + such that ϕ(t, s, D s ) ⊆ B t for s ∈ R with s ≤ t − T. Theorem 10. Suppose that a MS-RDS ϕ has a positively invariant pullback absorbing uniformly bounded family B = {B t } t∈R of nonempty closed subsets of {X t } t∈R and that the mappings ϕ(t, s, ·) : X s → X t are pullback compact (respectively, eventually or asymptotically compact) for all (t, s) ∈ R 2 ≥ . Then, ϕ has a unique global pullback attractor A = {A t } t∈R with its component sets determined by A t = s≤t ϕ(t, s, B s ) for t ∈ R. Consider the nonlinear mean-field SDE dX t = αX t + βEX t − X t EX 2 t dt + X t dW t(9) with real-valued parameters α, β. Note that the theory in Section 3 can be easily extended to include the second moment of the solution in the equation. This SDE has the steady state solutionX(t) ≡ 0. Linearising along this solution gives the bi-linear mean-field SDE dZ t = (αZ t + βEZ t ) dt + Z t dW t .(10) Theorem 11. The dichotomy spectrum of the linear MS-RDS Φ generated by (10) is given by Σ = {α + 1/2} ∪ {α + β} if β > 1/2 , {α + 1/2} if β ≤ 1/2 . Proof. Taking the expectation of two sides of (10) yields that d dt EZ t = (α + β)EZ t , which implies that EΦ(t, s)Z s = e (α+β)(t−s) EZ s for (t, s) ∈ R 2 ≥ and Z s ∈ X s . Ito's formula for the function U (x) = x 2 then gives dZ 2 t = (2α + 1)Z 2 t + 2βZ t EZ t dt + 2Z 2 t dW t . Consequently, d dt EZ 2 t = (2α + 1)EZ 2 t + 2β(EZ t ) 2 . Thus, using (11) for (t, s) ∈ R 2 ≥ and Z s ∈ X s , it follows that Φ(t, s)Z s The assertions of the lemma will be shown for the three cases β < 1/2, β = 1/2 and β > 1/2. Case 1 (β < 1/2). By (12), Φ(t, s)Z s 2 ms = e (2α+1)(t−s) Z s 2 ms + 2β 1 − 2β 1 − e (2β−1)(t−s) (EZ s ) 2 , which together with the inequality (EX) 2 ≤ EX 2 yields Φ(t, s)Z s 2 ms ≤ e (2α+1)(t−s) 1 + 2|β| (1 − 2β) Z s 2 , and Φ(t, s)Z s 2 ms ≥ e (2α+1)(t−s) Z s 2 ms if β ≥ 0 , 1 1−2β e (2α+1)(t−s) Z s 2 ms if β < 0 . This implies that Σ = {α + 1/2}. Case 2 (β = 1/2). By (12), Φ(t, s)Z s 2 ms = e (2α+1)(t−s) Z s 2 ms + (t − s)(EZ s ) 2 , which implies that Φ(t, s)Z s 2 ms ≥ e (2α+1)(t−s) Z s 2 ms for (t, s) ∈ R 2 ≥ . Let ε > 0 be arbitrary. Since (t − s) ≤ 1 ε e ε(t−s) for t ≥ s and (EX) 2 ≤ EX 2 , it follows that Φ(t, s)Z s 2 ms ≤ 1 + 1 ε e (2α+1+ε)(t−s) Z s 2 ms . Consequently, Σ ⊂ α + 1 2 , α + 1 2 + ε . The limit ε → 0 leads to Σ = {α + 1/2}. Case 3 (β > 1/2). By (12), Φ(t, s)Z s 2 ms = e (2α+1)(t−s) Z s 2 ms + 2β 2β − 1 e (2α+2β)(t−s) − e (2α+1)(t−s) (EZ s ) 2 . (13) Together with the inequality (EX) 2 ≤ EX 2 , this implies that e (2α+1)(t−s) Z s 2 ms ≤ Φ(t, s)Z s 2 ms ≤ 1 + 2β 2β − 1 e (2α+2β)(t−s) Z s 2 ms . Consequently, Σ ⊂ α + 1 2 , α + β . Let γ ∈ α + 1 2 , α + β be arbitrary. Choose and fix ε > 0 such that (γ − ε, γ + ε) ⊂ α + 1 2 , α + β . The aim is to show that Φ admits an exponential dichotomy with the growth rate γ for the decomposition X s = U s ⊕ S s , where S s := f ∈ X s : Ef = 0 , U s := f ∈ X s : f is independent of noise . Obviously, any X ∈ X s can be written as X = (X − EX) + EX, with X − EX ∈ S s and EX ∈ U s . By (13), for any Z s ∈ S s , Φ(t, s)Z s 2 ms = e (2α+1)(t−s) Z s 2 ms . Now (EZ s ) 2 = EZ 2 s for any Z s ∈ U s , so by (13), for t − s ≥ 1, Φ(t, s)Z s 2 ms ≥ 2β 2β − 1 e (2γ+2ε)(t−s) − e (2γ−2ε)(t−s) Z s 2 ms ≥ 4εβ 2β − 1 e 2γ(t−s) Z s 2 ms . Here the inequality e x ≥ 1 + x for x ≥ 0 has been used. Thus, Φ admits an exponential dichotomy with the growth rate γ, which means that Σ ⊂ {α + 1/2} ∪ {α + β}. Considering the decomposition S s ⊕ U s , it follows that α + 1 2 , α + β ∈ Σ. Thus, Σ = {α + 1/2} ∪ {α + β}. This completes the proof. A globally bifurcation of pullback attractor of (9) with β = 1, i.e., dX t = αX t + EX t − X t EX 2 t dt + X t dW t ,(14) as α varies, will be investigated in a series of theorems. The first and second moment equations of the mean-field SDE (14) are given by d dt EX t = (α + 1)EX t − EX t EX 2 t ,(15)d dt EX 2 t = (2α + 1)EX 2 t + 2(EX t ) 2 − 2(EX 2 t ) 2 ,(16) where Ito's formula with y = x 2 was used to derive (16). These can be rewritten as the system of ODEs dx dt = x(α + 1 − y) and dy dt = (2α + 1)y + 2x 2 − 2y 2 , where x 2 ≤ y , which has a steady state solutionx =ȳ = 0 for all α corresponding to the zero solution X t ≡ 0 of the mean-field SDE (14). There also exist valid (i.e., with y ≥ 0) steady state solutionsx = ± (α + 1)/2 ,ȳ = α+1 for α > −1 andx = 0,ȳ = α+ 1 2 for α ≥ − 1 2 . It needs to be shown if there are solutions of the SDE (14) with these moments. Theorem 12. The MS-RDS ϕ generated by (14) has a uniformly bounded positively invariant pullback absorbing family. Proof. Let α be arbitrary and define B t = X ∈ X t : X ms ≤ |α| + 2 for t ∈ R . Using (2α + 1)EX 2 Let D = {D t } t∈T be a uniformly bounded family of nonempty subsets of {X t } t∈R , i.e., D t ⊂ X t and there exists R > 0 such that X ms ≤ R for all X ∈ D t . Specifically, it will be shown that ϕ(t, s, D s ) ⊂ B t for t − s ≥ T , where T is defined by T := log R 2 |α| + 2 .(18) Pick X s ∈ D s arbitrarily and (t, s) ∈ R 2 ≥ with t − s ≥ T . Motivated by the differential inequality (17), consider the scalar systeṁ y = (2α + 3)y − 2y 2 , where y(s) ≤ R 2 .(19) A direct computation yields y(t) = y(s) exp t s (2α + 3) − 2y(u) du ≤ R 2 exp t s (2α + 3) − 2y(u) du . From the definition of T in (18), it follows that min s≤u≤t y(u) ≤ |α| + 2. Furthermore, y = 0 and y = α + 3 2 are stationary points of the ODE (19). For this reason, min s≤u≤t y(u) ≤ |α| + 2 implies that y(t) ≤ |α| + 2. Then from (17), it follows that y(t) ≥ ϕ(t, s)X s 2 ms . This means that ϕ t,s X s 2 ms ≤ y(t) ≤ |α| + 2 , i.e., ϕ t,s X s ∈ B t for t − s ≥ T . Hence, {B t } t∈R is a pullback absorbing family for the MS-RDS ϕ. It is clear that this family is uniformly bounded and positively invariant for the MS-RDS ϕ. Theorem 13. The MS-RDS ϕ generated by (14) has a pullback attractor with component sets {0} when α < −1. Proof. Let α < −1 be arbitrary. Let D = {D t } t∈R be a uniformly bounded family of nonempty subsets of {X t } t∈R with X ms ≤ R for all X ∈ D t where R > 0. Let (X s ) s∈R be an arbitrary sequence with X s ∈ D s . The moment equations (15)-(16) can be written as which implies that lim s→−∞ ϕ(t, s, X s ) ms = 0. Thus {0} is the pullback attractor of (14) in this case. d dt E[ϕ(t, s)X s ] = E[ϕ(t, s)X s ] α + 1 − E[ϕ(t, s)X s ] 2 d dt E[ϕ(t, s)X s ] 2 = (2α + 1)E[ϕ(t, s)X s ] 2 + 2(E[ϕ(t, s)X s ]) 2 − 2(E[ϕ(t, s)X s ] 2 ) 2 . Theorem 14. The MS-RDS ϕ generated by (14) has a nontrivial pullback attractor when −1 < α < − 1 2 . Remark 15. The idea for the proof is taken from [7, Subsection 4.1], but their result cannot be applied directly, since the Lipschitz constant of the nonlinear terms is at least 1, and α is not less than −4. Instead the uniform equicontinuity of the mapping t → EX t , and then the positivity of the second moment to obtain a better estimate are used. Proof. In order to apply Theorem 10, it needs to be shown that ϕ is pullback asymptotically compact, i.e., given a uniformly bounded family D = {D t } t∈R of nonempty subsets of X t and sequences {t k } k∈N in (−∞, t) with t k → −∞ as k → ∞ and {X k } k∈N with X k ∈ D t k ⊂ X t k for each k ∈ N, then the subset {ϕ(t, t k , X k )} k∈N ⊂ X t is relatively compact. For this purpose, let ε > 0 be arbitrary. A finite cover of {ϕ(t, t k , X k )} k∈N with diameter less than ε will be constructed. Choose and fix s ∈ R with s < t such that 4e (2α+1)(t−s) (|α| + 2) 2 < ε 2 2 , and define Y k s := ϕ(s, t k , X k ). Using (20) it can be assumed without loss of generality that E(Y k s ) 2 ≤ |α| + 2 for k ∈ N. Y k t = e α(t−s) Y k s + t s e α(t−u) EY k u − Y k u E(Y k u ) 2 du + 2α+1)(t−u) (EZ u ) 2 du = e (2α+1)(t−s) Z s 2 ms + 2β(EZ s ) 2 t s e (2α+1)(t−u) e (2α+2β)(u−s) du = e (2α+1)(t−s) Z s 2 ms + 2β(EZ s ) 2 t s e (2β−1)(u−s) du . + 1 − 1EX 2 u du , which implies that (E[ϕ(t, s)X s ]) 2 ≤ e (α+1)(t−s) R 2 . Moreover, by the variation of constants formula, 2α+1)(t−u) E[ϕ(u, s)X s ] 2 − (E[ϕ(u, s)X s ] 2 ) 2 du ≤ e (2α+1)(t−s) R 2 + 2R 2 t s e (2α+1)(t−u) e (α+1)(u−s) du ≤ e (2α+1)(t−s) R 2 + 2e (α+1)(t−s) (t − s)R 2 , ( 22 ) 22For u ∈ [s, t], let Y k u := ϕ(u, s)Y k s and consider a family of functions f k : [s, t] → R defined by f k (u) := EY k u . By Ito's formula, t + a jk (t)EX k t X i t + d m,n=1 c im (t)c jn (t)EX m t X n t + d k=1 b ik (t)EX k t EX j t + b jk (t)EX k t EX i t + d m,n=1 (c im (t)d jn (t) + c jn (t)d im (t) + d im (t)d jn (t)) EX m t EX n t .Proof. From (2),dX i t = d k=1 a ik (t)X k t + b ik (t)EX k t dt + d k=1 c ik (t)X k t + d ik (t)EX k t dW tholds for t ∈ R. Taking the expectation of variables in both sides givesEX t = EX s + t s A(u) + B(u) EX u du for t ≥ s,which proves the first statement. Ito's product formula [8, Example 3.4.1] and the expectation then yield the second statement.Corollary 5. Let (U i,j (t, s), V i,j (t,s)) be the evolution operator of the linear nonautonomous differential equation in R d(d+1)u i,j = d k=1 a ik (t) + b ik (t) u k,j + a jk (t) + b jk (t) u k,i v i,j = d k=1 (a ik (t)v k,j + a jk (t)v k,i ) + d m,n=1 c im (t)c jn (t)v m,n + d k=1 (b ik (t)u k,j + b jk (t)u k,i )(3)+ d m,n=1(c im (t)d jn (t) + c jn (t)d im (t) + d im (t)d jn (t)) u m,n , t X j t , EX i t EX j t for t ≥ s. t + 2(EX t ) 2 − 2(EX 2 t ) 2 ≤ (2α + 3)EX 2 t − 2(EX 2 t ) 2 , which holds since (EX t ) 2 ≤ EX 2t , the second moment equation(16)gives the differential inequality d dtEX 2 t ≤ (2α + 3)EX 2 t − 2(EX 2 t ) 2(17) t s e α(t−u) Y k u dW u , from which it can be shown that that (f k ) k∈N is a uniformly equicontinuous sequence of functions. By the Arzelà-Ascoli theorem, for δ := ε α 2 + 1 4 , there exists a finite index set J(δ) ⊂ N such that for any k ∈ N there exists n k ∈ J(δ) for whichTo conclude the proof of asymptotic compactness, it is sufficient to show the in-which together with(22)and(23)implies thatHere (21) was used to obtain the preceding inequality. Thus the set {ϕ(t, t k , X k )} k∈N is covered by the finite union of open balls with radius ε centered at ϕ(t, t k , X k ), where k ∈ J(λ), i.e., is totally bounded. Hence the MS-RDS is asymptotically compact and by Theorem 10 has a pullback attractor with component subsets {A t } t∈R that contain the zero solution. It remains to show that the sets A t also contain other points. Consider a uniformly bounded family of bounded sets defined by D t := X ∈ X t : EX = (α + 1)/2 , EX 2 = α + 1 .Recall that these values are a steady state solution of the moment equations (15)-(16). Hence ϕ(t, s, X s ) ∈ D t for all t > s when X s ∈ D s . Then by pullback attraction dist(ϕ(t, s, X s ), A t ) ≤ dist(ϕ(t, s, D s ), A t ) → 0 as s → −∞ .The convergence here is mean-square convergence, and Eϕ(t, s, X s ) 2 ≡ α + 1 for all t > s. Thus, there exists a random variable X * t ∈ A t ∩ D t for each t ∈ R, i.e., the pullback attractor is nontrivial. Note that by arguments in[10](see also[9]), it follows that there is in fact an entire solutionX t ∈ A t ∩ D t for all t ∈ R. Random Dynamical Systems. L Arnold, SpringerBerlin, Heidelberg, New YorkL. Arnold. Random Dynamical Systems. Springer, Berlin, Heidelberg, New York, 1998. About bounded solutions of linear stochastic Ito systems. A M Ateiwi, Miskolc Math. Notes. 31A.M. Ateiwi. About bounded solutions of linear stochastic Ito systems. Miskolc Math. Notes, 3(1):3-12, 2002. The dichotomy spectrum for random dynamical systems and pitchfork bifurcations with bounded noise. M Callaway, T S Doan, J S W Lamb, M Rasmussen, submittedM. Callaway, T.S. Doan, J.S.W. Lamb, and M. Rasmussen. 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Negatively invariant sets and entire trajectories of set- valued dynamical systems. Set-Valued and Variational Analysis, 19(1):43-57, 2011. A spectral theory for linear differential systems. R J Sacker, G R Sell, Journal of Differential Equations. 27R.J. Sacker and G.R. Sell. A spectral theory for linear differential systems. Journal of Differ- ential Equations, 27:320-358, 1978. Uniform exponential dichotomy of stochastic cocycles. D Stoica, Stochastic Processes and their Applications. 120D. Stoica. Uniform exponential dichotomy of stochastic cocycles. Stochastic Processes and their Applications, 120(10):1920-1928, 2010. Dynamical spectrum in random dynamical systems. G Wang, Y Cao, Journal of Dynamics and Differential Equations. to appear inG. Wang and Y. Cao. Dynamical spectrum in random dynamical systems. to appear in: Journal of Dynamics and Differential Equations. E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]. 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[ "A Theorem of the Alternative for Personalized Federated Learning", "A Theorem of the Alternative for Personalized Federated Learning" ]
[ "Shuxiao Chen \nUniversity of Pennsylvania\n\n", "Qinqing Zheng \nUniversity of Pennsylvania\n\n", "Qi Long \nUniversity of Pennsylvania\n\n", "Weijie J Su \nUniversity of Pennsylvania\n\n" ]
[ "University of Pennsylvania\n", "University of Pennsylvania\n", "University of Pennsylvania\n", "University of Pennsylvania\n" ]
[]
A widely recognized difficulty in federated learning arises from the statistical heterogeneity among clients: local datasets often come from different but not entirely unrelated distributions, and personalization is, therefore, necessary to achieve optimal results from each individual's perspective. In this paper, we show how the excess risks of personalized federated learning with a smooth, strongly convex loss depend on data heterogeneity from a minimax point of view. Our analysis reveals a surprising theorem of the alternative for personalized federated learning: there exists a threshold such that (a) if a certain measure of data heterogeneity is below this threshold, the FedAvg algorithm[McMahan et al., 2017]is minimax optimal; (b) when the measure of heterogeneity is above this threshold, then doing pure local training (i.e., clients solve empirical risk minimization problems on their local datasets without any communication) is minimax optimal. As an implication, our results show that the presumably difficult (infinitedimensional) problem of adapting to client-wise heterogeneity can be reduced to a simple binary decision problem of choosing between the two baseline algorithms. Our analysis relies on a new notion of algorithmic stability that takes into account the nature of federated learning. * 2. A closer look at the above-mentioned bounds reveals a surprising theorem of the alternative, which states that given a problem instance with a specified level of heterogeneity, either Fe-dAvg is minimax optimal, or PureLocalTraining is minimax optimal. Such a statement is reminiscent of the celebrated Fredholm alternative in functional analysis [Fredholm, 1903] and Farkas' lemma in linear programming [Farkas, 1902], both of which give two assertions and state that exactly one of them must hold 1 .3. With the established theorem of the alternative, the originally infinite-dimensional problem of adapting to client-wise heterogeneity is reduced to a binary decision problem of making a choice 1 Technically, our result is slightly weaker than a theorem of the alternative, as there are scenarios (i.e., when R 2 ≍ m/N ) where the two algorithms are simultaneously minimax optimal. See Remark 3.1 for more details.
null
[ "https://arxiv.org/pdf/2103.01901v1.pdf" ]
232,092,376
2103.01901
5e8b511102120ea831bfb16ca164dafc31e5e602
A Theorem of the Alternative for Personalized Federated Learning 2 Mar 2021 March 3, 2021 Shuxiao Chen University of Pennsylvania Qinqing Zheng University of Pennsylvania Qi Long University of Pennsylvania Weijie J Su University of Pennsylvania A Theorem of the Alternative for Personalized Federated Learning 2 Mar 2021 March 3, 2021 A widely recognized difficulty in federated learning arises from the statistical heterogeneity among clients: local datasets often come from different but not entirely unrelated distributions, and personalization is, therefore, necessary to achieve optimal results from each individual's perspective. In this paper, we show how the excess risks of personalized federated learning with a smooth, strongly convex loss depend on data heterogeneity from a minimax point of view. Our analysis reveals a surprising theorem of the alternative for personalized federated learning: there exists a threshold such that (a) if a certain measure of data heterogeneity is below this threshold, the FedAvg algorithm[McMahan et al., 2017]is minimax optimal; (b) when the measure of heterogeneity is above this threshold, then doing pure local training (i.e., clients solve empirical risk minimization problems on their local datasets without any communication) is minimax optimal. As an implication, our results show that the presumably difficult (infinitedimensional) problem of adapting to client-wise heterogeneity can be reduced to a simple binary decision problem of choosing between the two baseline algorithms. Our analysis relies on a new notion of algorithmic stability that takes into account the nature of federated learning. * 2. A closer look at the above-mentioned bounds reveals a surprising theorem of the alternative, which states that given a problem instance with a specified level of heterogeneity, either Fe-dAvg is minimax optimal, or PureLocalTraining is minimax optimal. Such a statement is reminiscent of the celebrated Fredholm alternative in functional analysis [Fredholm, 1903] and Farkas' lemma in linear programming [Farkas, 1902], both of which give two assertions and state that exactly one of them must hold 1 .3. With the established theorem of the alternative, the originally infinite-dimensional problem of adapting to client-wise heterogeneity is reduced to a binary decision problem of making a choice 1 Technically, our result is slightly weaker than a theorem of the alternative, as there are scenarios (i.e., when R 2 ≍ m/N ) where the two algorithms are simultaneously minimax optimal. See Remark 3.1 for more details. Introduction As one of the most important ingredients driving the success of machine learning, data have been being generated and subsequently stored in an increasingly decentralized fashion in many realworld applications. For example, mobile devices will in a single day collect an unprecedented amount of data from users. These data commonly contain sensitive information such as web search histories, online shopping records, and health information, and thus are often not available to service providers [Poushter, 2016]. This decentralized nature of (sensitive) data poses substantial challenges to many machine learning tasks. To address this issue, McMahan et al. [2017] proposed a new learning paradigm, which they termed federated learning, for collaboratively training machine learning models on data that are locally possessed by multiple clients with the coordination of the central server (e.g., service provider), without having direct access to the local datasets. In its simplest form, federated learning considers a pool of m clients, where the i-th client has a local dataset S i of size n i , consisting of i.i.d. samples w (global) t+1 ← w (global) t − mηt N |Ct| i∈Ct n i (w (global) t − w (i) t+1 ) Output: w (i) = w (global) T , i ∈ [m] {z (i) j : j ∈ [n i ]} (denote [n] := {1, 2, . . . , n}) from some unknown distribution D i . Letting ℓ(w, z) be a loss function, where w denotes the model parameter, the optimal local model for the i-th client is given by w (i) ⋆ ∈ argmin w E Z i ∼D i ℓ(w, Z i ). (1.1) From the client-wise perspective, any data-dependent estimator w (i) (S), with S = {S i } m i=1 denoting the collection of all samples, can be evaluated based on its individualized excess risk: IER i := E Z i ∼D i [ℓ( w (i) , Z i ) − ℓ(w (i) ⋆ , Z i )], where the expectation is taken over a fresh sample Z i ∼ D i . At a high level, this learning paradigm of federated learning aims to obtain possibly different trained models for each client such that the individualized excess risks are low (see, e.g., Kairouz et al. 2019). From a statistical viewpoint, perhaps the most crucial factor in determining the effectiveness of federated learning is data heterogeneity. When the data distribution D i is (approximately) homogeneous across different clients, presumably a single global model would lead to small IER i for all i. In this regime, indeed, McMahan et al. [2017] proposed the federated averaging algorithm (FedAvg, see Algorithm 1), which can be regarded as an instance of local stochastic gradient descent (SGD) for solving [Mangasarian andSolodov, 1993, Stich, 2019] min w 1 N i∈ [m] n i L i (w, S i ), ( 1.2) where L i (w, S i ) := j∈[n i ] ℓ(w, z (i) j )/n i is the empirical risk minimization (ERM) objective of the i-th client and N = n 1 + · · · + n m denotes the total number of training samples. Translating Algorithm 1 into words, FedAvg in effect learns a shared global model using gradients from each client and outputs a single model as an estimate of w (i) ⋆ for all clients. When the distributions {D i } coincide with each other, FedAvg with a strongly convex loss achieves a weighted average excess risk of O(1/N ), which is minimax optimal up to a constant factor [Shalev-Shwartz et al., 2009, Agarwal et al., 2012, see the formal statement in Theorem 3.2. However, it is an entirely different story in the presence of data heterogeneity. FedAvg has been recognized to give inferior performance when there is a significant departure from complete homogeneity (see, e.g., Bonawitz et al. 2019). To better understand this point, consider the extreme case where the data distributions {D i } are entirely unrelated. This roughly amounts to saying that the model parameters {w (i) ⋆ } can be arbitrarily different from each other. In such a "completely heterogeneous" scenario, the objective function (1.2) simply has no clear interpretation, and any single global model-for example, the output of FedAvg-would lead to unbounded risks for most, if not all, clients. As a matter of fact, it is not difficult to see that the optimal training strategy for federated learning in this regime is arguably PureLocalTraining, which lets each client separately run SGD to minimize its own local ERM objective min w (i) L i (w (i) , S i ) (1.3) without any communication. Indeed, PureLocalTraining is minimax rate optimal in the completely heterogeneous regime, just as FedAvg in the completely homogeneous regime (see Theorem 3.2). The level of data heterogeneity in practical federated learning problems is apparently neither complete homogeneity nor complete heterogeneity. Thus, the foregoing discussion raises a pressing question of what would happen if we are in the wide middle ground of the two extremes. This underlines the essence of personalized federated learning, which seeks to develop algorithms that perform well over a wide spectrum of data heterogeneity. Despite a venerable line of work on personalized federated learning (see, e.g., Kulkarni et al. 2020), the literature remains relatively silent on how the fundamental limits of personalized federated learning depend on data heterogeneity, as opposed to two extreme cases where both the minimax optimal rates and algorithms are known. Main Contributions The present paper takes a step toward understanding the statistical limits of personalized federated learning by establishing the minimax rates of convergence for both individualized excess risks and their weighted average with smooth strongly convex losses. We briefly summarize our main contributions below. 1. We prove that if the client-wise sample sizes are relatively balanced, then there exists a problem instance on which the IER i 's of any algorithm are lower bounded by Ω(1/N + R 2 ) if R 2 = O(m/N ) Ω(m/N ) if R 2 = Ω(m/N ),(1.4) where R 2 := min w∈W i∈[m] n i w (i) ⋆ − w 2 /N measures the level of heterogeneity among clients (here · throughout the paper denotes the Euclidean distance). Meanwhile, we show that the IER i 's of FedAvg are upper bounded by O(1/N + R 2 ), whereas the guarantee for PureLocalTraining is O(m/N ), regardless of the specific value of R. Moreover, we also establish similar upper and lower bounds for a weighted average of the IER i 's under a weaker condition. between the two algorithms. Indeed, the foregoing results suggest that the naïve dichotomous strategy of (1) running FedAvg when R 2 = O(m/N ), and (2) running PureLocalTraining when R 2 = Ω(m/N ), attains the lower bound (1.4). Moreover, for supervised problems, this dichotomous strategy can be implemented without knowing R by (1) running both Fe-dAvg and PureLocalTraining, (2) evaluating the test errors of the two algorithms, and (3) deploying the algorithm with a lower test error. We emphasize that the notion of optimality under our consideration overlooks constant factors. In practice, a better personalization result could be achieved by more sophisticated algorithms. 4. As a side product, we provide a novel analysis of FedProx, a popular algorithm for personalized federated learning that constrains the learned local models to be close via ℓ 2 regularization [Li et al., 2018]. In particular, we show that its IER 5. On the technical side, our upper bound analysis is based on a generalized notion of algorithmic stability [Bousquet and Elisseeff, 2002], which we term federated stability and can be of independent interest. Briefly speaking, an algorithm A(S) = { w (i) (S)} has federated stability {γ i } if for any i ∈ [m], the loss function evaluated at w (i) (S) can only change by an additive term of O(γ i ), if we perturb S i a little bit, while keeping the rest of datasets {S i ′ : i ′ = i} fixed. Similar ideas have appeared in Maurer [2005] and have been recently applied to multi-task learning [Wang et al., 2018]. However, their notion of perturbation is based on the deletion of the whole client-wise dataset, whereas our notion of federated stability operates at the "record-level" and is more fine-grained. On the other hand, our construction of the lower bound is based on a generalization of Assound's lemma [Assouad, 1983] (see also Yu 1997), which enables us to handle multiple heterogeneous datasets. Related Work Ever since the proposal of federated learning by McMahan et al. [2017], recent years have witnessed a rapidly growing line of work that is concerned with various aspects of FedAvg and its variants (see, e.g., Khaled et al. 2019, Haddadpour and Mahdavi 2019, Li et al. 2020b, Bayoumi et al. 2020, Malinovsky et al. 2020, Li and Richtárik 2020, Woodworth et al. 2020, Yuan and Ma 2020, Zheng et al. 2021. In the context of personalized federated learning, there have been significant algorithmic developments in recent years. While the idea of using ℓ 2 regularization to constrain the learned models to be similar has appeared in early works on multi-task learning [Evgeniou and Pontil, 2004], its applicability to personalized federated learning was only recently demonstrated by Li et al. [2018], where the FedProx algorithm was introduced. Similar regularization-based methods have been proposed and analyzed from the scope of convex optimization in Hanzely and Richtárik ], Dinh et al. [2020], and Hanzely et al. [2020. In particular, Hanzely et al. [2020] showed that an accelerated variant of FedProx is optimal in terms of communication complexity and the local oracle complexity. There is also a line of work using model-agnostic meta learning [Finn et al., 2017] to achieve personalization [Jiang et al., 2019, Fallah et al., 2020. Other strategies have been proposed (see, e.g., Arivazhagan et al. 2019, Li and Wang 2019, Mansour et al. 2020, Yu et al. 2020), and we refer readers to Kulkarni et al. [2020] for a comprehensive survey. We briefly remark here that all the papers mentioned above only consider the optimization properties of their proposed algorithms, while we focus on statistical properties of personalized federated learning. Compared to the optimization understanding, our statistical understanding (in terms of sample complexity) of federated learning is still limited. Deng et al. [2020] proposed an algorithm for personalized federated learning with learning-theoretic guarantees. However, it is unclear how their bound scales with the heterogeneity among clients. More generally, exploiting the information "shared among multiple learners" is a theme that constantly appears in other fields of machine learning such as multi-task learning [Caruana, 1997], meta learning [Baxter, 2000], and transfer learning [Pan and Yang, 2009], from which we borrow a lot of intuitions (see, e.g., Ben-David et al. 2006, Ben-David and Borbely 2008, Ben-David et al. 2010, Maurer et al. 2016, Cai and Wei 2019, Hanneke and Kpotufe 2019, Du et al. 2020, Tripuraneni et al. 2020a,b, Kalan et al. 2020, Shui et al. 2020, Li et al. 2020a, Jose and Simeone 2021. More related to our work, a series work by Denevi et al. [2018Denevi et al. [ , 2019, , and assumes the optimal local models lie in a small sub-parameter-space, and establishes "heterogeneity-aware" bounds on a weighted average of individualized excess risks. However, we would like to point out that they operate under the online learning setup, where the datasets are assumed to come in streams, and this is in sharp contrast to the federated learning setup, where the datasets are decentralized. Our notion of heterogeneity is also related to the hierarchical Bayesian model considered in Bai et al. [2020], Lucas et al. [2020], Konobeev et al. [2020], and Chen et al. [2020]. Paper Organization The rest of this paper is organized as follows. In Section 2, we give an exposition of the problem setup and main assumptions. Section 3 presents our main results. In Section 4, we present the general statements of our main results with relaxed assumptions and give an overview of our proof strategies. We conclude this paper with a discussion of open problems in Section 5. For brevity, detailed proofs are deferred to the appendix. Problem Setup In this section, we detail some preliminaries to prepare the readers for our main results. Notation. We introduce the notation we are going to use throughout this paper. For two real numbers a, b, we let a ∨ b = max{a, b} and a ∧ b = min{a, b}. For two non-negative sequences a n , b n , we denote a n b n (resp. a n b n ) if a n ≤ Cb n (resp. a n ≥ Cb n ) for some constant C > 0 when n is sufficiently large. We use a n ≍ b n to indicate that a n b n , a b b n hold simultaneously. We also use a n = O(b n ), whose meaning is the same as a n b n , and a n = Ω(b n ), whose meaning is the same as a n b n . We use W to denote the parameter space and Z to denote the sample space. Finally, we let P W (x) := argmin y∈W x − y denote the operator that projects x onto W in Euclidean distance. Evaluation Metrics. The presentation of our main results relies on how to evaluate the performance of a federated learning algorithm. To this end, we consider the following two evaluation metrics. Definition 2.1. Consider an algorithm A that outputs A(S) = { w (i) (S)} m i=1 for the m clients. For the i-the client, its individualized excess risk (IER) is defined as IER i (A) := E Z i ∼D i [ℓ( w (i) (S), Z i ) − ℓ(w (i) ⋆ , Z i )],(2.1) where Z i ∼ D i is a fresh data point independent of S. In addition, the average excess risk (AER) of A is defined as AER(A) := 1 N i∈[m] n i · IER i (A). (2.2) Recall that N = i∈[m] n i is the total sample size. In words, the IER measures the performance of the algorithm from the client-wise perspective, whereas the AER evaluates the performance of the algorithm from the system-wide perspective. Notably, while a uniform upper bound on all the IER i 's can be carried over to the same bound on the AER, a bound on the AER alone in general does not imply a tight bound on each IER i , other than the trivial bound IER i ≤ AER · N/n i . Such a subtlety is a distinguishing feature of personalized federated learning in the following sense: under homogeneity, it suffices to estimate a single shared global model, and thus the AER and all of the IER i 's are mathematically equivalent. In the definition of the AER (2.2), the weight we put on each client is n i /N . In some situations, one may want to use other weighting schemes (e.g., use the uniform weight 1/m to ensure "fairness" among clients). While our main results in Section 3 are stated for this choice of weights, our upper bounds can be readily generalized to other weighting schemes. See Section 4 for the general statements. Regularity Conditions. In this paper, we restrict ourselves to bounded, smooth, and strongly convex loss functions. Such assumptions are common in the federated learning literature (see, e.g., Li et al. 2020b, Hanzely et al. 2020) and cover many unsupervised learning problems such as mean estimation in exponential families and supervised learning problems such as generalized linear models. Assumption A (Regularity conditions). Suppose the following conditions hold: (a) Compact and convex domain. The parameter space W is a compact convex subset of R d with diameter D := sup w,w ′ ∈D w − w ′ < ∞; (b) Smoothness and strong convexity. For any i ∈ [m], the loss function ℓ(·, z) is β-smooth for almost every z in the support of D i , and the i-th ERM objective L i (·, S) is almost surely µstrongly convex on the convex domain W ⊆ R d . We also assume that there exists a universal constant ℓ ∞ such that 0 ≤ ℓ(·, z) ≤ ℓ ∞ for almost every z in the support of D i ; (c) Bounded gradient variance at optimum. There exists a positive constant σ such that for any i ∈ [m], we have E Z i ∼D i ∇ℓ(w (i) ⋆ , Z i ) 2 ≤ σ 2 . Heterogeneity Conditions. To quantify the level of heterogeneity among clients, we start by introducing the notion of an average global model. Assuming a strongly convex loss, the optimal local models (1.1) are uniquely defined. Thus, we can define the average global model as w (global) avg = 1 N i∈[m] n i w (i) ⋆ . We remark that the average global model w (global) avg defined above should not be interpreted as the "optimal global model". Rather, it is more suitable to think of w (global) avg as a point in the parameter space, from which every local model is close to. Indeed, a bit of analysis shows that the average global model is the minimizer of 1 N i∈[m] n i w (i) ⋆ − w (global) avg 2 . We are now ready to quantify the level of client-wise heterogeneity as follows. Assumption B (Level of heterogeneity). There exists a positive constant R such that (a) either 1 N i∈[m] n i w (i) ⋆ − w (global) avg 2 ≤ R 2 , (b) or w (i) ⋆ − w (global) avg 2 ≤ R 2 for all i ∈ [m]. Our study of the AER and IER will be based on Part (a) and (b) of Assumption B, respectively. Note that this assumption is slightly more general than what we have imposed in Section 1.1, where we considered exact equalities. Main Results This section presents our main results. For ease of exposition, many results are stated in their special forms, and we refer readers to Section 4 for the general forms. Fundamental Limits and Costs of Heterogeneity The main result in this subsection is the following minimax lower bound, which characterizes the information-theoretic limits of personalized federated learning. Theorem 3.1 (Minimax lower bound). Assume there exist constants C, C ′ > 0, c ≥ 0 such that n i ≥ Cβ ∀i ∈ [m] and m ≤ C ′ (N/m) c . Moreover, assume n i ≍ n i ′ for any i = i ′ ∈ [m]. Then there exists an absolute constant c ′ such that the following two statements hold: 1. There exists a problem instance such that Assumptions A and B(a) are satisfied with probability at least 1 − e −c ′ √ N/m . Call this high probability event E. On this problem instance, any randomized algorithm A must necessarily suffer E A,S [AER(A) · 1 E ] µ · β N/m ∧ R 2 + β N ; (3.1) For any i ∈ [m] , there exists a problem instance such that Assumptions A and B(b) are satisfied with probability at least 1 − e −c ′ √ N/m . Call this high probability event E i . On this problem instance, any randomized algorithm A must necessarily suffer E A,S [IER i (A) · 1 E i ] µ · β n i ∧ R 2 + β N . (3.2) In the two displays above, the expectation is taken over the randomness in both the algorithm A and the sample S. Proof. See Section 4.1. Focusing on the dependence on the sample sizes, the above theorem tells that the heterogeneity measure R enters the lower bounds in a dichotomous fashion: • If R 2 m/N , then both lower bounds become Ω(1/N ), which agree with the minimax rate as if we were under complete homogeneity; • If R 2 m/N , then both lower bounds become Ω(m/N ). When sample sizes are balanced, they agree with the minimax rate as if we were under complete heterogeneity. The proof is based on a generalization of Assouad's lemma [Assouad, 1983], which enables us to handle multiple heterogeneous datasets. Curious readers may wonder why our working assumptions are only satisfied with high probability in the statement of the above theorem. This is because we construct this problem instance as a logistic regression problem with random design, for which Assumption A(b) holds with high probability. Analysis of the Two Baseline Algorithms In this subsection, we characterize the performance of two baseline algorithms, namely FedAvg and PureLocalTraining, under the heterogeneity conditions imposed by Assumption B. Under our working assumptions, FedAvg is guaranteed to converge to the global optimum under a proper hyperparameter choice (see, e.g., Khaled et al. 2019, Li et al. 2020b, Bayoumi et al. 2020, and so does PureLocalTraining (see, e.g., Rakhlin et al. 2011). Hence, our analyses are to be conducted for the exact minimizers of (1.2) and (1.3), respectively. This is without loss of generality, as the bounds for the approximate minimizers only involve an extra additive term representing the optimization error, and this term will be negligible since our focus is sample complexity. Theorem 3.2 (Performance of two baseline algorithms). Let assumptions A hold, and assume n i ≥ 4β/µ for any i ∈ [m]. Suppose the FedAvg algorithm A FA and the PureLocalTraining algorithm A PLT output the exact minimizers of (1.2) and (1.3), respectively. Then, under Assumption B(a), we have E S [AER(A FA )] β ℓ ∞ µN + βR 2 , (3.3) E S [AER(A PLT )] β ℓ ∞ µN/m . (3.4) Meanwhile, under Assumption B(b), for any i ∈ [m] we have E S [IER i (A FA )] βσ 2 µ 2 N + β 3 R 2 µ 2 , (3.5) E S [IER i (A PLT )] β ℓ ∞ µn i . (3.6) Proof. This is a special case of Theorems 4.2 and 4.3 in Section 4.1. We remark that while the lower bounds in Theorem 3.1 require an extra assumption of relatively balanced sample sizes, such an assumption is not needed for the upper bounds in (3.5) and (3.6). This is favorable, as in practice, client-wise sample sizes can be extremely imbalanced. While the bounds for PureLocalTraining follows from standard stability arguments, the bounds for FedAvg rely on the notion of federated stability, which we will introduce in Section 4.2. We also remark that for FedAvg, the bound on its IER is slightly worse than the corresponding bound on the AER in terms of the dependence on β and µ. This is expected, as the IER is a more stringent criterion. A Theorem of the Alternative Inspecting the lower bounds in Theorem 3.1 and the upper bounds in Theorem 3.2, we see an interesting phase transition phenomenon. If R 2 m/N , then the upper bound on the AER of FedAvg (3.3) matches the corresponding lower bound (3.1); if R 2 m/N , then the upper bound on the AER of PureLocalTraining matches the corresponding lower bound (3.1), provided the client-wise sample sizes are relatively balanced and (β, µ, ℓ ∞ ) are all of the constant order. The same phase transition phenomenon also holds for the IER i 's. The foregoing observation allows us to establish a theorem of the alternative for personalized federated learning. If FedAvg is not minimax optimal in terms of its AER, then we are certain that R 2 ≫ m/N , from which we can conclude that PureLocalTraining is minimax optimal in terms of its AER. Again, such a statement holds for all the IER i 's. This theorem of the alternative is formally stated as follows. Theorem 3.3 (Theorem of the alternative for personalized federated learning). Under the setup of Theorems 3.1 and 3.2, if (β, µ, ℓ ∞ ) are all of constant order, then one of the following two assertions must hold: 1. either FedAvg is minimax optimal in terms of its AER; 2. or PureLocalTraining is minimax optimal in terms of its AER. Moreover, the same conclusion holds for the IER i 's under the additional assumption that σ ≍ 1. Remark 3.1. Strictly speaking, a theorem of the alternative would give two assertions and state that exactly one of them must hold. In this sense, the above result is slightly weaker than a theorem of the alternative, as there are scenarios (i.e., when R ≍ m/N ) where FedAvg and PureLocalTraining are simultaneously minimax optimal. The implications of this theorem are two-fold. From the technical side, it effectively reduces the problem of adapting to client-wise heterogeneity to a binary decision problem of making a choice between the two baseline algorithms, as detailed in the following corollary. E S [AER(A)] β ℓ ∞ µN/m ∧ R 2 + β ℓ ∞ µN . If in addition, n i ≍ n i ′ for any i = i ′ ∈ [m], then under Assumption B(b), for any i ∈ [m], we have E S [IER i (A)] β 3 µ 2 ℓ ∞ µn i ∧ R 2 + βσ 2 µ 2 N . Proof. This is a direct consequence of Theorem 3.2. Comparing the above result to the lower bounds in Theorem 3.1, we see that such a dichotomous strategy is minimax optimal, provided the problem-dependent parameters (µ, β, ℓ ∞ , σ) are all of constant order. From the practical side, we see that for supervised learning problems, such a dichotomous strategy can be implemented without prior knowledge of R. Indeed, given a problem instance, we can first run both FedAvg and PureLocalTraining separately, evaluate their test errors, and deploy the one with a lower test error. Due to the theorem of the alternative, such a strategy is guaranteed to be minimax optimal. As a caveat, however, one should refrain from interpreting our results as saying either of the two baseline algorithms is sufficient for practical problems. From a practical viewpoint, constants that are omitted in the minimax analysis are crucial. Nevertheless, our results suggest that the two baseline algorithms can at least serve as a good starting point in the search for efficient personalized algorithms. For unsupervised problems where the quality of a model is hard to evaluate, implementing the dichotomous strategy in Corollary 3.1 requires estimating the level of heterogeneity R. This is an important open problem, which we leave for future work. Meanwhile, we would like to emphasize that the notion of optimality under our consideration is worst-case in nature and overlooks constant factors. Thus, even for supervised problems, a better personalization result could be achieved by more sophisticated algorithms in practice. Algorithm 2: FedProx [Li et al., 2018] Initialize w (global) 0 , {w (i) 0 }, number of communication rounds T , step sizes {η t } T −1 t=0 for t = 0, 1, . . . , T − 1 do Randomly sample a batch C t ⊆ [m] of clients; for i ∈ [m] do if i ∈ C t then w (i) t+1 ← w (i) t else Obtain w (i) t+1 by running several steps of SGD on S i with the regularized loss function L i (·, S i ) + λ 2 w (global) t − · 2 w (global) t+1 ← w (global) t − mηt N |Ct| i∈Ct n i (w (global) t − w (i) t+1 ) return w (i) = w (i) T , i ∈ [m] Performance of FedProx In this subsection, we provide an analysis of the FedProx algorithm Li et al. [2018]. FedProx considers the following optimization problem: min w (global) {w (i) } m i=1 1 N i∈[m] n i L i (w (i) , S i ) + λ 2 w (global) − w (i) 2 , (3.7) where we recall that L i (w, S i ) := j∈[n i ] ℓ(w, z (i) j )/n i is the ERM objective for the i-th client. Compared to (1.2), which imposes a "hard" constraint w (i) ≡ w (global) , and compared to (1.3), where there is no constraint at all, the above formulation imposes a "soft" constraint that the norm of (w (global) − w (i) ) should be small, with a hyperparameter λ controlling the strength of this constraint. The rationale behind the optimization formulation (3.7) of FedProx is clear: by setting λ = 0, the optimization formulation of PureLocalTraining (1.3) is recovered, and as λ → ∞, the optimization formulation of FedAvg (1.2) is recovered. The hope is that by varying λ ∈ (0, ∞), one can interpolate between the two extremes. The idea of local SGD can be seamlessly applied to solve (3.7). Note that on the one hand, if the local models {w (i) } are fixed, then solving for the global model w (global) of (3.7) reduces to taking the weighted average of local models. On the other hand, if a global model is fixed, then solving for the local models of (3.7) "decouples" into m sub-problems, and each one can be solved by running SGD on its local data with an ℓ 2 regularized objective function. Thus, one naturally obtains a "soft constraint" version of FedAvg, which is exactly the FedProx algorithm proposed by Li et al. [2018] (see also Hanzely and Richtárik 2020, Dinh et al. 2020, Hanzely et al. 2020 for related ideas). A detailed description of FedProx is given in Algorithm 2. Note that similar to FedAvg described in Algorithm 1, we introduce an "aggregation step size" η t and use a generalized notion of weighted average, sometimes known as the "elastic average" in the literature [Zhang et al., 2015]. As is the case with FedAvg, in the aggregation step, only intermediate local models (namely w (i) t 's), but not the raw data (namely S i 's), are synchronized with the central server, thus satisfying the computational constraint of federated learning. While the optimization convergence of FedProx has been established in the literature (see, e.g., Li et al. 2018, Dinh et al. 2020, Hanzely et al. 2020, we find that a direct application of existing results is not sufficient to obtain meaningful excess risk guarantees in our case. Hence, in contrast to Theorem 3.2 where the analysis is done for the global minimizers, we will do an algorithm-dependent analysis. Next, we present the performance guarantees for FedProx. Theorem 3.4 (Performance of FedProx). Let Assumption A hold. Moreover, assume that n i ≍ n i ′ for any i = i ′ ∈ [m] and n i ≥ 4β/µ for any i ∈ [m] . Consider the FedProx algorithm defined in Algorithm 2, denoted as A FP . With a proper hyperparameter choice (see Theorem 4.4 and 4.5 for details), under Assumption B(a), we have E A FP ,S [AER(A FP )] µ + β ℓ ∞ µ · 1 N/m ∧ R N/m + 1 N , (3.8) and under Assumption B(b), for any i ∈ [m] we have E A FP ,S [IER i (A FP )] (µ + µ −1 ) β ℓ ∞ + σ 2 β 2 + β 2 + σ 2 µ 2 + µD 2 1 n i ∧ R √ n i + √ m N . (3.9) Proof. This is a special case of Theorem 4.4 and 4.5 in Section 4.3. Both the AER and IER guarantees in the above theorem need the assumption that client-wise sample sizes are relatively balanced. This assumption arises because of the technical subtleties in choosing the regularization parameter λ. The bounds in Theorem 3.4 in general do not attain the lower bounds in Theorem 3.1. • For the AER, we have the following three cases. If R 2 m/N , then (3.8) becomes O(m/N ), which matches the lower bound. Meanwhile, if 1/mN R 2 m/N , then (3.8) becomes O(m/N ), whereas the lower bound reads Ω(R 2 + 1/N ), and thus (3.8) is suboptimal unless R 2 ≍ m/N . Moreover, if R 2 1/mN , then (3.8) becomes O(1/N ), and is minimax optimal again. • For the IER, we still have three cases as follows. If R 2 m/N , then (3.9) is O(1/n i ), which agrees with the lower bound. Meanwhile, if 1/N R 2 m/N , then (3.9) is O(R/ √ n i ), and is suboptimal compared to the Ω(R 2 + 1/N ) lower bound unless R 2 ≍ m/N . Moreover, if R 2 1/N , then (3.9) is O( √ m/N ), and is off by a factor of order √ m compared to the Ω(1/N ) lower bound. Despite their suboptimality, the bounds in Theorem 3.4 are still non-trivial in the sense that they scale with the heterogeneity measure R. While there are some recent works establishing the AER guarantees for an objective similar to (3.7) under the online learning setup (see, e.g., Denevi et al. 2019, to the best of our knowledge, this is the first result that establishes both the AER and IER guarantees for (3.7) under the federated learning setup. The proof is based on federated stability, which we will introduce in Section 4.2 and detail in Section 4.3. The suboptimality arises partly because in our current proof, we cannot choose λ to be arbitrarily large: if we were able to do so, then the objective function for FedAvg (1.2) can be recovered, and we would have been able to get an O(1/N + R 2 ) bound whenever R 2 m/N . We suspect that this is an artifact of our technical approaches, and we conjecture that FedProx is indeed minimax optimal, especially in view of its empirical success reported in the literature. Proofs Construction of Lower Bounds In this subsection, we present our construction of the lower bounds in Theorem 3.1, which characterizes the information-theoretic limit of personalized federated learning. Our construction starts by considering a special class of problem instances: logistic regression. In logistic regression, given the collection of regression coefficients {w (i) ⋆ } ⊆ W where W has a diameter D, the data distributions D i 's are supported on R d × {±1} and specified by a two-step procedure as follows: 1. Generate a feature vector x, whose coordinates are i.i.d. copies from some distribution P X on R, which is assumed to have mean zero and is almost surely bounded by some absolute constant c X ; 2. Generate the binary label y ∈ {±1}, which is a biased Rademacher random variable with head probability 1 + exp{−x ⊤ w (i) ⋆ } −1 . The loss function is naturally chosen to be the negative log-likelihood function, which takes the following form: ℓ(w, z) = ℓ(w, x, y) = log(1 + e −yx ⊤ w ). The following lemma says that Assumption A holds for the aforementioned logistic regression models. µ ≍ µ 0 = exp{c X D √ d/2} + exp{−c X D √ d} −2 . (4.1) Proof. The compactness of the domain and the boundedness of the loss function hold by construction. To verify the rest parts of Assumption A, with some algebra one finds that ∇ 2 ℓ(w, x, y) = xx ⊤ exp{yx ⊤ w} 1 + exp{yx ⊤ w} 2 1 4 xx ⊤ , (4.2) where is the Loewner order and the inequality holds because x/(1+x) 2 = 1/(x −1/2 +x 1/2 ) 2 ≤ 1/4 for x > 0. Since the population gradient has mean zero at optimum, the gradient variance at optimum can be upper bounded by the trace of the expected Hessian matrix, which, by the above display, is further upper bounded by c 2 X d/4. Thus, we can take σ 2 = c 2 X d/4 in Part (c). Another message of the above display is that we can set the smoothness constant in Part (b) to be β = c 2 X d/4. The only subtlety that remains is to ensure each local loss function is µ-strongly convex. Note that since x/(1 + x) 2 is decreasing from (0, 1) and is increasing from (1, ∞), the right-hand side of (4.2) dominates µ 0 xx ⊤ in Loewner order, where µ 0 is the right-hand side of (4.1). Thus, the local population losses E (x,y)∼D i [ℓ(·, x, y)] are all µ 0 -strongly convex. Now, note that ∇ 2 L i (w (i) , S i ) = 1 n i j∈[n i ] x (i) j (x (i) j ) ⊤ exp{y (i) j x (i) j , w (i) } 1 + exp{y (i) j x (i) j , w (i) } 2 µ 0 · 1 n i j∈[n i ] x (i) j (x (i) j ) ⊤ . Invoking Theorem 5.39 of Vershynin [2010] along with a union bound over all clients, we conclude that for any i ∈ [m], the minimum eigenvalue of j∈[n i ] x (i) j (x (i) j ) ⊤ is lower bounded by a constant multiple of n i − p n i (this is the definition of the event E) with probability at least 1 − me −O(n i ) 1 − e −O( √ N/m) , and the proof is concluded. Note that in the proof of the above lemma, we have established the µ 0 ≍ µ-strong convexity of the client-wise population losses. Hence, lower bounding the excess risks reduces to lower bounding the ℓ 2 estimation errors w (i) − w (i) ⋆ 2 of the estimators w (i) for w (i) ⋆ . Such a reduction allows us to use powerful tools from information theory. Before we rigorously state the main result of this subsection, from which Theorem 3.1 follows, we pause to introduce two parameter spaces, corresponding to Part (a) and (b) of Assumption B. Recalling that w (global) avg = 1 N i∈[m] n i w (i) ⋆ , we define P 1 := {w (i) ⋆ } m i=1 ⊆ W : 1 N i∈[m] n i w (i) ⋆ − w (global) avg 2 ≤ R 2 , P 2 := {w (i) ⋆ } m i=1 ⊆ W : w (i) ⋆ − w (global) avg 2 ≤ R 2 ∀i ∈ [m] . Note that P 1 and P 2 index all possible values of {w With the notations introduced so far, we are ready to state the main result of this subsection. Theorem 4.1 (Minimax lower bounds for estimation errors). Consider the logistic regression model described above. Suppose n i ≍ n i ′ for any i = i ′ ∈ [m]. Then we have inf { w (i) } sup {w (i) ⋆ }∈P 1 1 N i∈[m] n i E S w (i) − w (i) ⋆ 2 d N/m ∧ R 2 + d N , (4.3) inf w (i) sup {w (i) ⋆ }∈P 2 E S w (i) − w (i) ⋆ 2 d n i ∧ R 2 + d N (4.4) for all i ∈ [m], where the infimum is taken over all possible w (i) 's that are measurable functions of the data S. Proof. See Appendix A. From Lemma 4.1 and Theorem 4.1, Theorem 3.1 follows by the fact that the smoothness constant β is of the same order as d and the population losses are all µ 0 ≍ µ-strongly convex. Note that both lower bounds in Theorem 4.1 are a superposition of two terms, and they correspond to two distinct steps in the proof. The first step in our proof is to argue that the lower bound under complete homogeneity is in fact a valid lower bound under our working assumptions, which gives the Ω(d/N ) term. This is reasonable, since estimation under complete homogeneity is, in many senses, an "easier" problem. The proof of the Ω(d/N ) term is based on the classical Assouad's method [Assouad, 1983]. The second step is to use a generalized version of Assouad's method that allows us to deal with multiple heterogeneous datasets. In particular, we need to carefully choose the prior distributions over the parameter space based on the level of heterogeneity, which ultimately leads to the Ω( d N/m ∧ R 2 ) term. Recall that in the vanilla version of Assouad's method where there is only one parameter, say w ⋆ , one can lower bounds the minimax risk by the Bayes risk, and the prior distribution is usually chosen to be w ⋆ = δv, where v follows a uniform distribution over all d-dimensional binary vectors and δ is chosen so that the resulting hypothesis testing problem has large type-I plus type-II error. In our case where there are m parameters {w (i) ⋆ }, we need to consider a different prior of the following form: w (i) ⋆ = δ i v (i) , where v (i) are i.i.d. samples from the uniform distribution over all d-dimensional binary vectors, and δ i 's are scalers that need to be carefully chosen to make the resulting hypothesis testing problem hard. Analysis of Baseline Algorithms In this subsection, we present our analyses of the two baseline algorithms, FedAvg and PureLo-calTraining, as well as the naïve dichotomous strategy that combines the two baseline algorithms. Analysis of PureLocalTraining. The analysis of PureLocalTrainingis based on the classical notion of uniform stability, proposed by Bousquet and Elisseeff [2002]. Definition 4.1 (Uniform stability). Consider an algorithm A that takes a single dataset S = {z j } n j=1 of size n as input and outputs a single model: A(S) =ŵ (S). We say A is γ-uniformly stable if for any dataset S, any j ∈ [n], and any z ′ j ∈ Z, we have ℓ(ŵ(S), ·) − ℓ(ŵ(S \j ), ·) ∞ ≤ γ, where S \j is the dataset formed by replacing z j with z ′ j : S \j = {z 1 , . . . , z j−1 , z ′ j , z j , . . . , z n }. The main implication of uniformly stable algorithms is that "stable algorithms do not overfit": if A is γ-uniformly stable, then its generalization error is upper bounded by a constant multiple of γ. Thus, one can dissect the analysis of A into two separate parts: (1) bounding its optimization error; (2) bounding its stability term. As pointed out in Section 3.2, under our working assumptions, SGD with properly chosen step sizes is guaranteed to converge to the global minimum of (1.3) (see, e.g., Rakhlin et al. [2011]). Thus, we consider without loss of generality the global minimizer of (1.3), whose performance is given by the following theorem. Theorem 4.2 (Performance of PureLocalTraining). Let Assumption A(b) hold and assume n i ≥ 4β/µ ∀i ∈ [m]. Then the algorithm A PLT which outputs the minimizer of (1.3) satisfies E S [IER i (A PLT )] β ℓ ∞ µn i for all i = 1, . . . , m. Proof. The proof is a direct consequence of standard results on uniform stability of strongly convex ERM (see, e.g., Section 5 of Shalev-Shwartz et al. [2009] and Section 13 of Shalev-Shwartz and Ben-David [2014]), which assert that under the current assumptions, the minimizer of (1.3) is O β ℓ ∞ µn i uniformly stable. We omit the details. Analysis of FedAvg. As mentioned in Section 2, sometimes we are interested in the following more general weighted version of the AER. Definition 4.2 (p-average excess risk). Consider an algorithm A that outputs A(S) = { w (i) (S)}. For a vector p = (p 1 , . . . , p m ) lying in the m-dimensional probability simplex (i.e., all p i 's are non-negative and they sum to one), we define the p-average excess risk (AER p ) of A to be AER p (A) := i∈[m] p i · IER i (A). Intuitively speaking, the weight vector p can be regarded as the importance weight on each client and controls "how many resources are allocated to each client". For example, setting p i = 1/m enforces "fair allocation", so that each client is treated uniformly, regardless of sample sizes. As another example, setting p i = n i /N means that the central server pays more attention to clients with larger sample sizes, which, to a certain extend, incentivize the clients to contribute more data. In view of Definition 4.2, it is natural to consider the following generalization of Assumption B. Assumption C (p-heterogeneity). For a vector p = (p 1 , . . . , p m ) lying in the m-dimensional probability simplex, define the p-average global model as w (global,p) avg = i∈[m] p i w (i) ⋆ . We assume that there exists a positive constant R such that (a) either i∈[m] p i w (global,p) avg − w (i) ⋆ 2 ≤ R 2 , (b) or w (i) ⋆ − w (global,p) avg 2 ≤ R 2 ∀i ∈ [m]. As is the case with Assumption B, we assume Part (a) when we deal with AER p , and we assume Part (b) when we deal with the IER i 's. We consider the following weighted version of (1.2): min w∈W i∈[m] p i L i (w, S i ),(4.5) and the FedAvg algorithm also seamlessly generalizes. The above optimization formulation is in fact covered by the general theory of Li et al. [2020b], where they showed that FedAvg is guaranteed to converge to the global optimum under a suitable hyperparameter choice, even in the presence of heterogeneity (but the convergence is slower). Thus, in the following discussion, we only consider the global minimizer of (4.5). It turns out that a tight analysis of FedAvg requires a more "fine-grained" notion of uniform stability, which we present below. Definition 4.3 (Federated stability). An algorithm A that outputs A(S) = { w (i) (S)} has federated stability {γ i } m i=1 if for every S ∼ i D ⊗n i i and for any i ∈ [m], j i ∈ [n i ], z ′ i,j i ∈ Z, we have ℓ( w (i) (S), ·) − ℓ( w (i) (S \(i,j i ) ), ·) ∞ ≤ γ i . Above, S \(i,j i ) is the dataset formed by replacing z (i) j i in the i-the dataset with z ′ i,j i : S \(i,j i ) = {S 1 , . . . , S i−1 , S \j i i , S i+1 , . . . , S m }, S \j i i = {z (i) 1 , . . . , z (i) j i −1 , z ′ i,j i , z (i) j i +1 , . . . , z (i) n i }. Compared to the conventional uniform stability in Definition 4.1, federated stability provides a finer control by allowing distinct stability measures {γ i } for different clients. Moreover, the classical statement that "stable algorithms do not overfit" still holds, in the sense that the average (resp. individualized) generalization error can be upper bounded by O( i∈[m] n i γ i /N ) (resp. O(γ i )), plus a term scaling with the level of heterogeneity R. And this again enables us to separate the analysis of A into two parts (namely bounding the optimization error and bounding the stability), as is the case with the conventional uniform stability. The notion of federated stability has other implications when restricted to the FedProx algorithm, and we refer the readers to Section 4.3 for details. We are now ready to state the theorem that characterizes the performance of FedAvg. Theorem 4.3 (Performance of FedAvg). Let Assumption A(b, c) hold and assume n i ≥ 4βp i /µ ∀i ∈ [m] . Suppose the FedAvg algorithm A FA outputs the minimizer of (4.5). Then under Assumption C(a), we have E S [AER p (A FA )] β ℓ ∞ µ i∈[m] p 2 i n i + βR 2 , (4.6) and under Assumption C(b), we have E S [IER i (A FA )] βσ 2 µ 2 i ′ ∈[m] p 2 i ′ n i ′ + β 3 µ 2 R 2 . (4.7) Proof. The proof of (4.6) is, roughly speaking, based on the fact that the global minimizer of (4.5) has federated stability γ i β ℓ ∞pi µn i , and thus the first term in the right-hand side of (4.6) corresponds to the average federated stability i∈[m] p i γ i . The second term βR 2 in the right-hand side of (4.6) reflects the presence of heterogeneity. For Equation (4.7), we were not able to obtain a federated stability based proof, and our current proof is based on an adaptation of the arguments in Theorem 7 of Foster et al. [2019], which explains why the dependence on (σ, β, µ) are different (and slightly worse) compared to Equation (4.6). We refer the readers to Appendix C.1 for details. Note that both bounds in the above theorem are minimized by choosing p i = n i /N . This makes sense, since this choice of weight corresponds to the ERM objective under complete homogeneity. This observation also suggests that ensuring "fair resource allocation" (i.e., setting p i = 1/m) can lead to statistical inefficiency, especially when the sample sizes are imbalanced. Analysis of the dichotomous strategy. With Theorem 4.2 and 4.3 at hand, we are ready to state the following result, from which Corollary 3.1 follows as a special case. Corollary 4.1 (Performance of a dichotomous strategy). Let Assumptions A(b, c) hold and assume n i ≥ 4βp i /µ ∀i ∈ [m] . Consider the following naïve dichotomous strategy: if output R 2 ≤ ℓ ∞ µN/m , then output A = A FA ; otherwise, output A = A PLT . Then under Assumption C(a), we have E S [AER p (A)] β ℓ ∞ µN/m ∧ R 2 + β ℓ ∞ µ i∈[m] p 2 i n i If in addition, n i ≍ n i ′ for any i = i ′ ∈ [m], then under Assumption C(b), for any i ∈ [m], we have E S [IER i (A)] β 3 µ 2 ℓ ∞ µn i ∧ R 2 + βσ 2 µ 2 i ′ ∈[m] p 2 i ′ n i ′ . Proof. This is a direct consequence of Theorem 4.2 and 4.3. We conclude this subsection by noting that though the compactness assumption (Assumption A(a)) is not needed in the above result, it is usually needed in the analysis of the optimization error of FedAvg and PureLocalTraining(see, e.g., Rakhlin et al. [2011], Li et al. [2020b]). Analysis of FedProx In this subsection, we are concerned with the performance guarantees for FedProx. As with our earlier analysis of FedAvg, we will consider a p-weighted version of FedProx, and Algorithm 2 is a special case with p i = n i /N . Specifically, we consider the following p-weighted generalization of (3.7): min w (global) ∈W {w (i) } m i=1 ⊆W i∈[m] p i L i (w (i) , S i ) + λ 2 w (global) − w (i) 2 . (4.8) In this subsection, we let (w (global) , {w (i) }) be the global minimizer of the above problem. The FedProx algorithm generalizes in a straightforward fashion (see Algorithm 3 for a detailed description), where the only change to Algorithm 2 is the aggregation step: at the t-th communication round, we aggregate by w (global) t+1 = w (global) t − λmη (global) |C t | i∈[Ct] p i (w (global) t − w (i) t+1 ). Curious readers may wonder why we deliberately separate the training into two stages in the algorithm description. This is because, in order for FedProx to generalize well, we need both global and local models output by this algorithm to approach the optimum: w (global) T ≈w (global) , w (i) T +1 ≈w (i) ∀i ∈ [m] . Thus, the two stages have distinct interpretations: in Stage I, the central server aims to learn a good global model with the help of local clients, whereas in Stage II, each local client takes advantage of the global model to personalize. Alternatively, one can also interpret FedProx as an instance of the general framework of model-agnostic meta learning [Finn et al., 2017], where Stage I learns a good initialization, and Stage II trains the local models starting from this initialization. In contrast to our analyses for FedAvg and PureLocalTraining, where we largely focus on global minimizers, the analysis for FedProx will be carried out for the approximate minimizer output by Algorithm 3. The reason for this, again, is rooted in the fact that a rigorous statement on (i) 0 } m i=1 ≡ {w (i) 0,0 } m i=1 , global rounds T , global batch size B (global) , global step sizes {η (global) t } T −1 t=0 , local rounds {K t } T t=0 , local batch sizes {B (i) } m i=0 local step sizes {η (i) t,k : 0 ≤ t ≤ T, 0 ≤ k ≤ K t − 1}. Output: Local models {w (i) T +1 } m i=1 . # Stage I: joint training for t = 0, 1, . . . , T − 1 do Randomly sample a batch C t ⊆ [m] of size B (global) for i ∈ [m] do if i ∈ C t then w (i) t+1 ← w (i) t else Pull w (global) t from the server w (i) t+1 ← SoftLocalSGD(i, w (i) t , w (global) t , K t , B (i) , {η (i) t,k } Kt−1 k=0 ) Push w (i) t+1 to the server w (global) t+1 ← w (global) t − λmη (global) t B (global) i∈Ct p i (w (global) t − w (i) t+1 ) # Stage II: final training before deployment for i ∈ [m] do Pull w (global) T from the server w (i) T +1 ← SoftLocalSGD(i, w (i) T , w (global) T , K T , B (i) , {η (i) T,k } KT −1 k=0 ) return {w (i) T +1 } m i=1 # Local SGD subroutine Function SoftLocalSGD(i, w (i) , w (global) , K, B, {η k } K−1 k=0 ) for k = 0, 1, . . . , K − 1 do Randomly sample a batch I ⊆ [n i ] of size |I| = B w (i) ← P W w (i) − η k B j∈I ∇ℓ(w (i) , z (i) j ) + λ(w (i) − w (global) ) return w (i) the generalization ability of FedProx calls for both global and local models to approach the optimum. While the convergence of w Implications of federated stability for FedProx. We have briefly mentioned the main implications of federated stability in Section 4.2: for an algorithm A = { w (i) } with federated stability {γ i }, its average generalization error (resp. individualized) generalization error can be upper bounded by O( i∈[m] p i γ i ) (resp. O(γ i )), plus a term scaling with the level of heterogeneity R. We make such a statement precise here. Let us first define the optimization error of a generic algorithm A = ( w (global) , { w (i) }) (which tries to solve (4.8)) as E OPT := i∈[m] p i L i ( w (i) , S i ) + λ 2 w (global) − w (i) 2 − i∈[m] p i L i (w (i) , S i ) + λ 2 w (global) −w (i) 2 . The main implications of federated stability, when applied to the specifics of FedProx, can then be summarized in the following proposition. Proposition 4.1 (Implications of federated stability restricted to FedProx). Consider an algo- rithm A = ( w (global) , { w (i) }) with federated uniform stability {γ i } m 1 . Then we have E A,S [AER p (A)] ≤ E A,S [E OPT ] + 2 i∈[m] p i E A,S [γ i ] + λ 2 i∈[m] p i w (global) avg − w (i) ⋆ 2 , (4.9) E A,S [IER i (A)] ≤ E A,S [E OPT ] p i + 2E A,S [γ i ] + λ 2 E A,S w (global) − w (i) ⋆ 2 ∀i ∈ [m]. (4.10) Proof. The proof of (4.9) is based on the following basic inequality for the AER: i∈[m] p i L i (w (i) , S i ) + λ 2 w (global) −w (i) 2 ≤ i∈[m] p i L i (w (i) ⋆ , S i ) + λ 2 w (global) avg − w (i) ⋆ 2 , (4.11) whereas the proof of (4.10) is based on the following basic inequality for the IER: for any s ∈ [m], we have i∈[m] p i L i (w (i) , S i ) + λ 2 w (global) −w (i) 2 ≤ p s L s (w (s) ⋆ , S s ) + λ 2 w (global) − w (s) ⋆ 2 + i =s p i L i ( w (i) , S i ) + λ 2 w (global) − w (i) 2 . (4.12) We refer the readers to Appendix C.2 for details. Note that both bounds in Proposition 4.1 involve a term that scales linearly with both λ and the heterogeneity measure. In general, we expect the stability measures to scale inversely with λ, and thus opening the possibility of carefully choosing λ to balance the stability term and the heterogeneity term. Let us observe that the heterogeneity term of (4.10) is slightly different than that of (4.9), in that it involves the estimated global model w (global) . This suggests that achieving the IER guarantees might be intrinsically more difficult than achieving the AER guarantees, which partially explains why the IER guarantees are largely missing in the literature, whereas the AER guarantees are already available in recent works such as Denevi et al. . It also explains why our bound for the IER in Theorem 3.4 is worse compared to the bound for the AER. In view of Proposition 4.1, we are left to bound the optimization error and the federated stability of FedProx. As discussed above, achieving the AER and IER guarantees requires somewhat different assumptions, as the latter involves characterizing the performance of the global model. So we split our discussion into two parts below. Bounding the average excess error. The following theorem characterize the performance of Fed-Prox in terms of the AER. Theorem 4.4 (AER guarantees for FedProx). Let Assumptions A and C(a) hold, and assume n i ≥ 4β/µ for all i ∈ [m]. Choose the weight vector p such that p max i∈[m] p i /n i i∈[m] p 2 i /n i ≤ C p (4.13) for some constant C p , where p max = max i p i . Consider the FedProx algorithm, A FP , with the following hyperparameter configuration: 1. In the joint training stage (i.e., 0 ≤ t ≤ T − 1), set η (i) t,k = 1 (µ + λ)(k + 1) , η (global) t = 2(µ + λ) λµ(t + 1) , K t + 1 ≥ C 1 (λ 2 ∨ 1)t, (4.14) T ≥ C 2 λ(λ ∨ 1)m p 2 · i∈[m] p i /n i −1 ∨ λ(λ ∨ 1)n 2 max ; 2. In the final training stage (i.e., t = T ), set η (i) T,k = 1 (µ + λ)(k + 1) , (4.15) K T ≥ C 3 (λ + 1) 2 · i∈[m] p i /n i −1 ∨ λ 2 max i∈[m] (p i n i ) 2 , where C 1 , C 2 , C 3 are constants depending only on (µ, β, ℓ ∞ , D). Then, there exists a choice of λ such that E A FP ,S [AER p (A FP )] µ 1 ∧ C p + (1 ∨ C p )β ℓ ∞ µ R i∈[m] p i n i ∧ i∈[m] p i n i + i∈[m] p 2 i n i . (4.16) Proof. See Appendix C.3. A few remarks are in order. First, (4.13) essentially says that the weight p cannot be too imbalanced, and too much imbalance in p can hurt the performance in view of the multiplicative factor of C p in our bound (4.16). If we set p i = 1/m, then C p is naturally of constant order; whereas if we set p i = n i /N , we have C p ≍ mn max /N , where n max = max i n i , which calls for relative balance of the sample sizes. We then briefly comment on the hyperparameter choice in the above theorem. The step sizes are of the form 1/(strongly convex constant × iteration counter), and such a choice is common in strongly convex stochastic optimization problems (see, e.g., Rakhlin et al. 2011, Shamir andZhang 2013). Such a choice, along with the smoothness of the problem, is also the key for us to by-pass the need of doing any time-averaging operation, as is done by, for example, Dinh et al. [2020]. In Theorem 4.4, the choice of the communication rounds T and the final local training round K T both scale polynomially with λ, which means that the optimization convergence of FedProx is slower when the data are less heterogeneous. This phenomenon happens more generally. For example, in Hanzely and Richtárik [2020], they proposed a variant of SGD that optimizes (4.8) with p i = 1/m in O L+λ µ log 1/ε -many iterations, where L is the Lipschitz constant of the loss function and ε is the desired accuracy level. The constants C 1 , C 2 , C 3 in the statement of Theorem 4.4 can be explicitly traced in our proof. We remark that the dependence on problem-specific constants (µ, β, ℓ ∞ , D) in our hyperparameter choice and on λ may not be tight. A tight analysis of the optimization error is interesting, but less relevant for our purpose of understanding the sample complexity. So we defer such an analysis to future work 2 . Bounding individualized excess errors. The following theorem gives the IER guarantees for Fed-Prox. Theorem 4.5 (IER guarantees for FedProx). Let Assumptions A and C(b) hold. Moreover, assume that n i ≍ n i ′ for any i = i ′ ∈ [m] and n i ≥ 4β/µ ∀i ∈ [m]. Let the weight vector be chosen as p i ≍ 1/m ∀i ∈ [m]. Consider the FedProx algorithm, A FP , with the following hyperparameter configuration: 1. In the joint training stage (i.e., 0 ≤ t ≤ T − 1), set η (i) t,k , η (global) t , K t as in (4.14), and set T ≥ C ′ 2 λ(λ ∨ 1) max i∈[m] n i · p −1 i ∨ [λ(λ ∨ 1)n i ] ; 2. In the final training stage (i.e., t = T ), set η (i) T,k as in (4.15), and set K T ≥ C ′ 3 (λ + 1) 2 max i∈[m] n i p −1 i ∨ λ 2 p 2 i n i , where C ′ 2 , C ′ 3 are constants only depending on (µ, β, ℓ ∞ , D). Then, there exists a choice of λ such that for any i ∈ [m], we have E A FP ,S [IER i (A FP )] (µ + µ −1 ) β ℓ ∞ + σ 2 β 2 + β 2 + σ 2 µ 2 + µD 2 · R √ n i ∧ 1 n i + √ m N . Proof. See Appendix C.4. Compared to Theorem 4.4, the above theorem imposes extra assumptions that the sample sizes are relative balanced and that p i ≍ 1/m, both of which are due to the fact that we need to additionally take care of the estimation error of the global model. The hyperparameter choice slightly differs from that in Theorem 4.4 for the same reason. Note that Theorem 3.4 is then a direct consequence of Theorem 4.4 and 4.5. We conclude this subsection by a remark on practical implementations of FedProx. In practice, when one is to use FedProx to optimize highly non-convex functions like the loss function of deep neural networks, instead of sticking to the choices made in Theorems 4.4 and 4.5, the hyperparameters are usually tuned by trial-and-error for best test performance. Discussion This paper studies the statistical properties of personalized federated learning. Focusing on stronglyconvex, smooth, and bounded empirical risk minimization problems, we have established a theorem of the alternative, stating that given a specific level of heterogeneity, either FedAvg is minimax optimal, or PureLocalTraining is minimax optimal. In the course of proving this theorem of the alternative for personalized federated learning, we obtained a novel analysis of FedProx and introduced a new notion of algorithmic stability termed federated stability, which is possibly of independent interest for analyzing generalization properties in the context of federated learning. We close this paper by mentioning several open problems. • Dependence on problem-specific parameters. This paper focuses on the dependence on the sample sizes, and in our bounds, the dependence on problem-specific parameters (e.g., the smoothness and strong convexity constants) may not be optimal. This can be problematic if those parameters are not of constant order, and it will be interesting to give a refined analysis that gives optimal dependence on those parameters. • A refined analysis of FedProx. The upper bounds we develop for FedProx, as we have mentioned, do not match our minimax lower bounds. We suspect that this is an artifact of our analysis and a refined analysis of FedProx would be a welcome advance. • Estimation of the level of heterogeneity. For unsupervised problems where evaluation of a model is difficult, implementation of the dichotomous strategy described in Corollaries 3.1 and 4.1 would require estimating the level of heterogeneity R. Even for supervised problems, estimation of R would be interesting, as it allows one to decide which algorithm to choose without model training. • Beyond convexity. Our analysis is heavily contingent upon the strong convexity of the loss function, which, to the best of our knowledge, is not easily generalizable to the non-convex case. Meanwhile, our notion of heterogeneity, which is based on the distance of optimal local models to the convex combination of them, may not be natural for non-convex problems. It is of interest, albeit difficult, to have a theoretical investigation of personalized federated learning for non-convex problems. E S w (global) − w (global) avg 2 d N . Proof. This is a classical result. See, e.g., Example 8.4 of Duchi [2019]. Proof of (4.3). We first give a lower bound based on the observation that the homogeneous case is in fact included in the parameter space P 1 . More explicitly, let us define P 0 = {{w (i) ⋆ } ∈ P 1 : w (i) ⋆ = w (global) avg ∀i ∈ [m]}. By Lemma A.1, we have inf { w (i) } sup {w (i) ⋆ }∈P1 i∈[m] p i E S w (i) − w (i) ⋆ 2 ≥ inf { w (i) } sup {w (i) ⋆ }∈P0 i∈[m] p i E S w (i) − w (i) ⋆ 2 = inf w (global) sup w (global) avg E S w (global) − w (global) avg 2 d N . (A.1) We now use a variant of Assouad's method [Assouad, 1983] that allows us to tackle multiple datasets. Consider the following data generating process: nature generates V = {v (i) : i ∈ [m]} i.i.d. from the uniform distribution on V = {±1} d and sets w (i) ⋆ = δ i v (i) for some δ i such that the following constraint is satisfied: i∈[m] p i w (i) ⋆ − w (global) avg 2 = i∈[m] p i δ i v (i) − s∈[m] p s δ s v (s) 2 ≤ R 2 . (A.2) We will specify the choice of δ i 's later. Denoting E X as the marginal expectation operator with respect to all the features {x (i) j } and E Y |X as the conditional expectation operator with respect to {y (i) j }|{x (i) j }, we can lower bound the minimax risk by the Bayes risk as follows: inf { w (i) } sup {w (i) ⋆ }∈P i∈[m] p i E S w (i) − w (i) ⋆ 2 ≥ inf { w (i) } E {v (i) } i∈[m] p i E S w (i) − δ i v (i) 2 = inf { v (i) }⊆V i∈[m] p i E V ,S δ i v (i) − δ i v (i) 2 ≥ E X i∈[m] p i δ 2 i inf v (i) ∈V E V ,Y |X v (i) − v (i) 2 ≥ E X i∈[m] p i δ 2 i k∈[d] inf v (i) k ∈{±1} E V ,Y |X ( v (i) k − v (i) k ) 2 E X i∈[m] p i δ 2 i k∈[d] inf v (i) k ∈{±1} P V ,Y |X ( v (i) k = v (i) k ) = 1 2 E X i∈[m] p i δ 2 i k∈[d] inf v (i) k ∈{±1} P i,+k ( v (i) k = −1) + P i,−k ( v (i) k = +1) , where in the last line, we have let P i,±k (·) = P V ,Y |X (·|v More explicitly, we can write P i,±k = s =i P v (s) ⊗ P {y (s) } ns j=1 |v (s) ,{x (s) j } ns j=1 ⊗ P v (i) |v (i) k =±1 ⊗ P {y (i) j } n i j=1 |v (i) ,v (i) k =±1,{x (i) j } n i j=1 = 1 2 (m−1)d+d−1 V \{v (i) k } P V ,i,±k , where the ⊗ symbol stands for taking the product of two measures and P V ,i,±k corresponds to the law of all the labels Y conditional on a specific realization of {V : v (i) k = ±1} and the features X. With the current notations and letting P − Q TV be the total variation distance between two probability measures P and Q, we can invoke Neyman-Pearson lemma to get inf { w (i) } sup {w (i) ⋆ }∈P i∈[m] p i E S w (i) − w (i) ⋆ 2 E X i∈[m] p i δ 2 i k∈[d] 1 − P i,+k − P i,−k TV = d i∈[m] p i δ 2 i − E X i∈[m] p i δ 2 i k∈[d] P i,+k − P i,−k TV . (A.3) We then proceed by i∈[m] p i δ 2 i k∈[d] P i,+k − P i,−k TV ≤ i∈[m] p i δ 2 i √ d k∈[d] P i,+k − P i,−k 2 TV 1/2 = i∈[m] p i δ 2 i √ d k∈[d] 1 2 (m−1)d+d−1 V \{v (i) k } P V ,i,+k − P V ,i,−k 2 TV 1/2 = i∈[m] p i δ 2 i √ d k∈[d] 1 2 (m−1)d+d−1 V \{v (i) k } P V ,i,+k − P V ,i,−k 2 TV 1/2 , where the last inequality is by convexity of the total variation distance. Note that P V ,i,±k is the product of biased Rademacher random variables: if we let Rad(p) be the ±1-valued random variable with positive probability p, we can write P V ,i,±k = s∈[m] j∈[ns] Rad 1 1 + exp{−δ s v (s) , x (s) j } , v (i) k = ±1. Thus, by Pinsker's inequality, we have P V ,i,+k − P V ,i,−k 2 TV ≤ 1 2 D JS (P V ,i,+k P V ,i,−k ) = 1 2 s =i j∈[ns] 0 + 1 2 j∈[ni] D JS Rad 1 1 + exp{−δ i v (i) , x (i) j } Rad 1 1 + exp{−δ i ṽ (i) , x (i) j } , where D JS (P Q) = D KL (P Q)+D KL (Q P) 2 is the Jensen-Shannon divergence between P and Q, and v (s) ,ṽ (s) are two V-valued vectors that only differs in the k-th coordinate. By a standard calculation, one finds that D JS Rad 1 1 + exp{−δ i v (i) , x (i) j } Rad 1 1 + exp{−δ i ṽ (i) , x (i) j } ≤ δ 2 i (v (i) k −ṽ (i) k ) 2 (x (i) j,k ) 2 = 4δ 2 i (x (i) j,k ) 2 . This gives P V ,i,+k − P V ,i,−k 2 TV ≤ 2δ 2 i j∈[ni] (x (i) j,k ) 2 ≤ 2δ 2 i c 2 X n i . and hence i∈[m] p i δ 2 i k∈[d] P i,+k − P i,−k TV ≤ √ 2c X i∈[m] p i δ 3 i dn 1/2 i . Plugging the above display to (A.3) gives inf { w (i) } sup {w (i) ⋆ }∈P i∈[m] p i E S w (i) − w (i) ⋆ 2 d i∈[m] p i δ 2 i − √ 2c X i∈[m] p i δ 3 i √ n i . (A.4) To this end, all that is left is to choose δ i approriately so that (1) the above display is as tight as possible; (2) (A.2) is satisfied. We consider the following two cases: 1. Assume R 2 ≥ d i∈[m] p i /n i = dm/N .p i δ 2 i − i∈[m] p i δ i v (i) 2 ≤ R 2 . Under the current assumption, this requirement will be satisfied if we choose δ i = c/ √ n i for any c ≤ 1. Under such a choice, the right-hand side of (A.4) becomes c 2 dm N (c − √ 2c X ). Thus, by setting c = 2 √ 2c X , we get the following lower bound: inf { w (i) } sup {w (i) ⋆ }∈P i∈[m] p i E S w (i) − w (i) ⋆ 2 d N/m . 2. Assume R 2 ≤ d i∈[m] p i /n i = dm/N . Note that if we set δ i ≡ δ = cR/ √ d where c ≤ 1, (A.2) reads c 2 R 2 − i∈[m] p i δ i v (i) 2 ≤ R 2 , which trivially holds. Now, the right-hand side of (A.4) becomes c 2 R 2 (1 − √ 2cc X i∈[m] p i R √ n i / √ d). Since p i = n i /N and n i ≍ N/m, our assumption on R gives √ 2cc X i∈[m] p i R √ n i / √ d i n i N · mn i N = 1. This means that we can choose c to be a small constant such that the following lower bound holds: inf { w (i) } sup {w (i) ⋆ }∈P i∈[m] p i E S w (i) − w (i) ⋆ 2 R 2 . Summarizing the above two cases, we arrive at inf { w (i) } sup {w (i) ⋆ }∈P i∈[m] p i E S w (i) − w (i) ⋆ 2 d N/m ∧ R 2 . Combining the above bound with (A.1), we get inf { w (i) } sup {w (i) ⋆ }∈P i∈[m] p i E S w (i) − w (i) ⋆ 2 d N/m ∧ R 2 + d N , which is the desired result. Proof of (4.4). The proof is similar to the proof of (3.1), and we only provide a sketch here. Without loss of generality we consider the first client. By the same arguments as in the proof of (3.1), the left-hand side of (4.4) is lower bounded by a constant multiple of d/N . Now, by considering the same prior distribution on P as in the proof of (3.1), we get inf w (1) sup {w (i) }∈P E S w (1) − w (1) ⋆ 2 dδ 2 1 (1 − δ 1 √ n 1 ), where the δ i 's should obey the following inequality: δ i v (i) − s∈[m] p s δ s v (s) 2 ≤ R 2 . Choosing δ i ≍ 1/ √ n i when R ≥ dm/N and δ i ≍ R/ √ d otherwise, we arrive at inf w (1) sup {w (i) }∈P E S w (1) − w (1) ⋆ 2 d n 1 ∧ R 2 , and the proof is concluded. Appendix B Optimization Convergence of FedProx This section concerns the optimization convergence of FedProx. We first introduce some notations. Let w (i) t,k be the output of k-th step of Algorithm 3 when the initial local model is given by w (i) t ≡ w (i) t,0 ≡ w (i) t−1,K , let I (t) t,k be the corresponding minibatch taken, and denote the initial global model by w (global) t . Let F t,k be the sigma algebra generated by the randomness by Algorithm 3 up to w (i) t,k , namely the randomness in C τ , {I (i) τ,l : i ∈ C τ , 0 ≤ l ≤ K τ − 1} t−1 τ =0 , C t , and {I (i) t,l : i ∈ C t , 0 ≤ l ≤ k − 1}. For notational convenience we let C T = [m] (i.e., all clients are involved in local training in Stage II of Algorithm 3). Then the sequence {w (i) t,k } is adapted to the following filtration: F 0,0 ⊆ F 0,1 ⊆ · · · ⊆ F 0,K ⊆ F 1,0 ⊆ F 1,1 ⊆ · · · ⊆ F 1,K ⊆ · · · ⊆ F T,K . We write the optimization problem (4.8) as min w (global) ∈W i∈[m] p i F i (w (global) , S i ), (B.1) where F i (w (global) , S i ) = min w (i) ∈W L i (w (i) , S i ) + λ 2 w (global) − w (i) 2 . (B.2) To simplify notations, we introduce the proximal opertor Prox Li/λ (w (global) ) = Prox Li/λ (w (global) , S i ) = argmin w (i) ∈W L i (w (i) , S i ) + λ 2 w (global) − w (i) 2 . (B.3) The high-level idea of this proof is to regard λ i∈Ct (w w (i) t,k − Prox Li/λ (w (global) t ) 2 F t,0 , i ∈ C t ≤ 8β 2 D 2 µ 2 (k + 1) . Proof. See Appendix B.1. λµ(t+1) and assume K τ + 1 ≥ (4τ + 20)λ 2 β 2 D 2 µ 2 (β 2 D 2 ∧ 2λ ℓ ∞ ∧ λ 2 D 2 ) ∀0 ≤ τ ≤ t − 1. (B.4) Then for any t ≥ 0, we have E A FP w (global) t −w (global) 2 ≤ 12(λ + µ) 2 m p 2 (β 2 D 2 ∧ 2λ ℓ ∞ ∧ λ 2 D 2 ) λ 2 µ 2 (t + 1) , (B.5) where the expectation is taken over the randomness in Algorithm 3. E A FP [E OPT ] ≤ 4(β + λ)β 2 D 2 µ 2 (K T + 1) + 6(λ + µ) 2 m p 2 (β 2 D 2 ∧ 2λ ℓ ∞ ∧ λ 2 D 2 ) λµ 2 (t + 1) Proof. By definition we have E A FP [E OPT ] := E A FP i∈[m] p i L i (w (i) T +1 , S i ) + λ 2 w (global) T − w (i) T +1 2 − i∈[m] p i F i (w (global) , S i ) (a) ≤ i∈[m] p i (β + λ) 2 · E A FP w (i) T +1 − Prox Li/λ (w (global) T ) 2 + E A FP i∈[m] p i F i (w (global) T , S i ) − i∈[m] p i F i (w (global) , S i ) (b) ≤ 4(β + λ)β 2 D 2 µ 2 (K T + 1) + λ 2 E A FP w (global) T −w (global) 2 (c) ≤ 4(β + λ)β 2 D 2 µ 2 (K T + 1) + 6(λ + µ) 2 m p 2 (β 2 D 2 ∧ 2λ ℓ ∞ ∧ λ 2 D 2 ) λµ 2 (t + 1) . where ( B.1 Proof of Lemma B.1: Convergence of the Inner Loop The proof is an adaptation of the proof of Lemma 1 in Rakhlin et al. [2011]. However, we need to deal with the extra complication that the hyperparameter λ can in principle be arbitrarily large. We start by noting that w (i) t,k+1 − Prox Li/λ (w (global) t ) 2 = P W w (i) t,k − η (i) t,k B (i) j∈I (i) t,k ∇ℓ(w (i) t,k , z (i) j ) + λ(w (i) t,k − w (global) t ) − Prox Li/λ (w (global) t ) 2 ≤ w (i) t,k − η (i) t,k B (i) j∈I (i) t,k ∇ℓ(w (i) t,k , z (i) j ) + λ(w (i) t,k − w (global) t ) − Prox Li/λ (w (global) t ) 2 = w (i) t,k − Prox Li/λ (w (global) t )) 2 + η (i) t,k B (i) j∈I (i) t,k ∇ℓ(w (i) t,k , z (i) j ) + λ(w (i) t,k − w (global) t ) 2 − 2 w (i) t,k − Prox Li/λ (w (global) t ), η (i) t,k B (i) j∈I (i) t,k ∇ℓ(w (i) t,k , z (i) j ) + λ(w (i) t,k − w (global) t ) , where the inequality is because Prox Li/λ (w (global) t ) ∈ W and P W is non-expansive. Now by strong convexity and unbiasedness of the stochastic gradients, we have E w (i) t,k − Prox Li/λ (w (global) t ), 1 B (i) j∈I (i) t,k ∇ℓ(w (i) t,k , z (i) j ) + λ(w (i) t,k − w (global) t ) F t,k , i ∈ C t ≥ L i (w (i) t,k , S i ) + λ 2 w (i) t,k − w (global) 2 − L i (Prox Li/λ (w (global) t ), S i ) + λ 2 Prox Li/λ (w (global) t ) − w (global) t 2 + 1 2 µ i + λn mn i w (i) t,k − Prox Li/λ (w (global) t ) 2 ≥ (µ + λ) w (i) t,k − Prox Li/λ (w (global) t ) 2 . On the other hand, applying Lemma B.5 gives E η (i) t,k B (i) j∈I (i) t,k ∇ℓ(w (i) t,k , z (i) j ) + λ(w (i) t,k − w (global) t ) 2 F t,k , i ∈ C t = (η (i) t,k ) 2 · n i /B (i) − 1 n i (n i − 1) j∈[ni] ∇ℓ(w (i) t,k , z (i) j ) − ∇ℓ(w (i) t,k , z (i) • ) 2 + 1 n i j∈[ni] ∇ℓ(w (i) t,k , z (i) j ) + λ(w (i) t,k − w (global) t ) 2 ≤ 2(η (i) t,k ) 2 β 2 D 2 · n i /B (i) − 1 (n i − 1) + β + λn mn i 2 w (i) t,k − Prox Li/λ (w (global) t ) 2 , where in the second line we let ∇ℓ(w (i) t,k , z (i) • ) := j∈[ni] ∇ℓ(w (i) t,k , z (i) j )/n i , and in the last line is by the β-smoothness of ℓ(·, z). Thus, we get E w (i) t,k+1 − Prox Li/λ (w (global) t ) 2 F t,k , i ∈ C t ≤ 1 − 2η (i) t,k (µ + λ) + (η (i) t,k ) 2 (β + λ) 2 w (i) t,k − Prox Li/λ (w (global) t ) 2 + 2(η (i) t,k ) 2 β 2 D 2 · n i /B (i) − 1 (n i − 1) . (B.6) We then proceed by induction. Note that if k + 1 ≤ 8β 2 µ 2 , then we have the following trivial bound: E w (i) t,k − Prox Li/λ (w (global) t ) 2 F t,0 , i ∈ C t ≤ D 2 ≤ 8β 2 D 2 µ 2 (k + 1) , (B.7) where the first inequality is by w (i) t,k , Prox Li/λ (w (global) t ) ∈ W and the second inequality is by our assumption on k. Thus, it suffices to show E w (i) t,k+1 − Prox Li/λ (w (global) t ) 2 F t,0 , i ∈ C t ≤ 8β 2 D 2 µ 2 (k + 2) (B.8) based on the inductive hypothesis (B.7) and k + 1 ≥ 8β 2 /µ 2 . By the recursive relationship (B.6) and taking expectation, we have E w (i) t,k+1 − Prox Li/λ (w (global) t ) 2 F t,0 , i ∈ C t ≤ 1 − 2η (i) t,k (µ + λ) + (η (i) t,k ) 2 β + λ 2 8β 2 D 2 k + 1 + 2(η (i) t,k ) 2 β 2 D 2 · n i /B (i) − 1 (n i − 1) . Hence (B.8) is satisfied if 8β 2 D 2 · 1 k + 2 − 1 k + 1 + 2η (i) t,k k + 1 (µ + λ) − (η (i) t,k ) 2 k + 1 (β + λ) 2 ≥ 2(η (i) t,k ) 2 β 2 D 2 · n i /B (i) − 1 (n i − 1) . By our choice of η (i) t,k , the above display is equivalent to 8β 2 D 2 · − 1 (k + 1)(k + 2) + 2 (k + 1) 2 − 1 (k + 1) 3 β + λ µ + λ 2 ≥ 2β 2 D 2 (µ + λ) 2 (k + 1) 2 · n i /B (i) − 1 n i − 1 , which is further equivalent to 8β 2 D 2 · − k + 1 k + 2 + 2 − 1 k + 1 β + λ µ + λ 2 ≥ 2β 2 D 2 (µ + λ) 2 · n i /B (i) − 1 n i − 1 . We now claim that 1 k + 1 β + λ µ + λ 2 ≤ 1 2 . Indeed, since k + 1 ≥ 8β 2 /µ 2 , (1) if λ ≤ β, then the left-hand side above is less than 4β 2 µ 2 (k+1) ≤ 1 2 ; and (2) if λ ≥ β, the left-hand side above is less than 4 k+1 ≤ µ 2 2β 2 ≤ 1 2 . By the above claim, (B.8) would hold if 4β 2 D 2 ≥ 2β 2 D 2 (µ + λ) 2 · n i /B (i) − 1 n i − 1 . We finish the proof by noting that the right-hand side above is bounded above by 2β 2 D 2 µ 2 . B.2 Proof of Lemma B.2: Convergence of the Outer Loop By construction we have w (global) t −w (global) 2 = λmη (global) t B (global) i∈Ct p i (w (global) t − w (i) t+1 ) 2 = w (global) t −w (global) 2 + λmη (global) t B (global) i∈Ct p i (w (global) t − w (i) t+1 ) 2 − 2 w (global) t −w (global) , λmη (global) t B (global) i∈Ct p i (w (global) t − w (i) t+1 ) ≤ w (global) t −w (global) 2 − 2 w (global) t −w (global) , λmη (global) t B (global) i∈Ct p i w (global) t − Prox Li/λ (w (global) t ) I + 2 λmη (global) t B (global) i∈Ct p i w (global) t − Prox Li/λ (w (global) t ) 2 II + 2 λmη (global) t B (global) i∈Ct p i Prox Li/λ (w (global) t ) − w (i) t+1 2 III − 2 w (global) t −w (global) , λmη (global) t B (global) i∈Ct p i Prox Li/λ (w (global) t ) − w (i) t+1 IV . We first consider Term I. Note that λm B (global) i∈Ct p i (w (global) t −Prox Li/λ (w (global) t )) is an unbiased stochastic gradient of i p i F i , which is µ F = λµ/(λ + µ)-strongly convex. Thus, we have E[I | F t−1,Kt−1 ] = 2η (global) t w (global) t −w (global) , i∈[m] p i ∇F i (w (global) t , S i ) ≥ 2η (global) t µ F w (global) t −w (global) 2 . Now for Term II, we have E[II | F t−1,Kt−1 ] ≤ 2(η (global) t ) 2 · E 1 B (global) i∈Ct mp i 2 | F t−1,Kt−1 · max i∈[m] ∇F i (w (global) t , S i ) 2 ≤ 2(η (global) t ) 2 · E 1 B (global) i∈Ct mp i 2 | F t−1,Kt−1 · max i∈[m] ·(β 2 D 2 ∧ 2λ ℓ ∞ ∧ λ 2 D 2 ) ≤ 2(η (global) t ) 2 · 1 m i∈[m] (mp i − 1) 2 + 1 · (β 2 D 2 ∧ 2λ ℓ ∞ ∧ λ 2 D 2 ) = 2(η (global) t ) 2 m p 2 (β 2 D 2 ∧ 2λ ℓ ∞ ∧ λ 2 D 2 ), where the second line is by Lemma B.4 and the third line is by Lemma B.5. For Term III, we invoke Lemma B.1 to get E[III | F t−1,Kt−1 ] ≤ 2λ 2 (η (global) t ) 2 · 8β 2 D 2 µ 2 (K t + 1) · E 1 B (global) i∈Ct mp i 2 | F t−1,Kt−1 ≤ 16λ 2 (η (global) t ) 2 β 2 D 2 m p 2 µ 2 (K t + 1) , where the last line is again by Lemma B.5. For Term IV, we invoke Young's inequality for products to get E[−IV | F t−1,Kt−1 ] ≤ (η (global) t µ F ) w (global) t −w (global) 2 + (η (global) t µ F ) −1 · E III 2 F t−1,Kt−1 ≤ (η (global) t µ F ) w (global) t −w (global) 2 + (η (global) t µ F ) −1 · 8λ 2 (η (global) t ) 2 β 2 D 2 m p 2 µ 2 (K t + 1) . Summarizing the above bounds on the four terms, we arrive at E w (global) t+1 −w (global) 2 F t−1,Kt−1 ≤ (1 − η (global) t µ F ) w (global) t −w (global) 2 + 2 (η (global) t ) 2 m p 2 (β 2 D 2 ∧ 2λ ℓ ∞ ∧ λ 2 D 2 ) V + λ 2 (η (global) t ) 2 β 2 D 2 m p 2 µ 2 (K t + 1) · 16 + 8 δ (global) t µ F VI . We claim that VI ≤ V. Indeed, with our choice of η (global) t = 2 µF (t+1) , with some algebra, one recognizes that this claim is equivalent to 20 + 4t µ 2 (K t + 1) ≤ 1 λ 2 ∧ 2 ℓ ∞ λβ 2 D 2 ∧ 1 β 2 , which is exactly (B.4). Thus, we have E w (global) t+1 −w (global) 2 F t−1,Kt−1 ≤ (1 − η (global) t µ F ) w (global) t −w (global) 2 + 3 · V = 1 − 2 t + 1 w (global) t −w (global) 2 + 12m p 2 (β 2 D 2 ∧ 2λ ℓ ∞ ∧ λ 2 D 2 ) µ 2 F (t + 1) 2 . (B.9) We then proceed by induction. For the base case, we invoke the strong convexity of i p i F i and Lemma B.4 to get µ 2 F 4 w (global) 0 −w (global) 2 ≤ i∈[m] p i ∇F i (w (global) 0 , S i ) 2 ≤ β 2 D 2 ∧ 2λ ℓ ∞ ∧ λ 2 D 2 . Along with the fact that 1 = ( i∈[m] p i ) 2 ≤ m p 2 , we conclude that (B.2) is true for t = 0. Now assume (B.5) hold for any 0 ≤ t ≤ τ . For t = τ + 1, using (B.9) and the inductive hypothesis, we have E A FP w (global) τ +1 −w (global) 2 ≤ 1 − 2 τ + 1 12m p 2 (β 2 D 2 ∧ 2λ ℓ ∞ ∧ λ 2 D 2 ) (τ + 1)µ 2 F + 12m p 2 (β 2 D 2 ∧ 2λ ℓ ∞ ∧ λ 2 D 2 ) (τ + 1) 2 µ 2 F = 1 τ + 1 − 1 (τ + 1) 2 · 12m p 2 (β 2 D 2 ∧ 2λ ℓ ∞ ∧ λ 2 D 2 ) µ 2 F ≤ 12m p 2 (β 2 D 2 ∧ 2λ ℓ ∞ ∧ λ 2 D 2 ) (τ + 2)µ 2 F , which is the desired result. Lemma B.3 (Convexity and smoothness F i ). Under Assumption A(b), each F i is λ-smooth and µλ µ+λ -strongly convex. B.3 Auxiliary lemmas Proof. The smoothness is a standard fact about the Moreau envelope. The strongly convex constant of F i follows from Theorem 2.2 of Lemaréchal and Sagastizábal [1997]. ∇F i (w, S i ) 2 ≤ β 2 D 2 ∧ 2λ ℓ ∞ ∧ λ 2 D 2 . Proof. Since ∇F i (w, S i ) = λ(w −Prox Li/λ (w)), its norm is trivially bounded by λD. Now, since Prox Li/λ (w) achieves a lower objective value than w for the objective function L i (·, S i ) + λ 2 w − · 2 , we have λ 2 w − Prox Li/λ (w) 2 ≤ L i (w, S i ) − L i (Prox Li/λ (w), S i ) ≤ ℓ ∞ , and hence ∇F i (w, S i ) 2 ≤ 2λ ℓ ∞ . Finally, by the first-order condition, we have ∇L i (Prox Li/λ (w), S i ) + λ(Prox Li/λ (w) − w) = 0. Hence, we get ∇F i (w, S i ) = ∇L i (Prox Li/λ (w), S i ) ≤ βD.E B 1 B i∈B x i 2 = n/B − 1 n(n − 1) i∈[n] x i −x 2 + x 2 ≤ 1 n i∈[n] x i −x 2 + x 2 , wherex := i∈[n] x i /n. Proof. Since E B i∈B x i /B =x, we have E B 1 B i∈B x i 2 = E B 1 B i∈B x i −x 2 + x 2 = 1 B 2 i∈[n] 1{i ∈ B} x i −x 2 + 2 i<j 1{i, j ∈ B} x i −x, x j −x + x 2 = 1 B 2 B n i∈[n] x i −x 2 + 2B(B − 1) n(n − 1) i<j x i −x, x j −x + x 2 , where the last line is by P B (i ∈ B) = B/n and P B (i, j ∈ B) = B(B − 1)n −1 (n − 1) −1 for any i = j. Now, since i∈[n] x i −x 2 + 2 i<j x i −x, x j −x = 0, we arrive at E B 1 B i∈B x i 2 = 1 B 2 B n − B(B − 1) n(n − 1) i x i −x 2 + x 2 = n/B − 1 n(n − 1) i∈[n] x i −x 2 + x 2 , which is the desired result. Appendix C Proofs of Upper Bounds C.1 Proof of Theorem 4.3 In this proof, we let w (global) be the global minimizer of (4.5) and we write w (global,p) avg ≡ w (global) avg when there is no ambiguity. Proof of (4.6). We have 0 = − i∈[m] p i L i ( w (global) (S), S i ) + i∈[m] p i L i ( w (global) (S), S i ) ≤ − i∈[m] p i L i ( w (global) (S), S i ) + i∈[m] p i L i (w (1) ⋆ , S i ) = − i∈[m] p i n i j∈ [ni] ℓ( w (global) (S \(i,j) ), z (i) j ) − ℓ(w (1) ⋆ , z (i) j ) + i∈[m] p i n i j∈[ni] ℓ( w (global) (S \(i,j) ), z (i) j ) − ℓ( w (global) (S), z (i) j ) , where S \(i,j) stands for the dataset formed by replacing z (i) j by another z ′ i,j ∼ D i , which is independent of everything else. Taking expectation in both sides, we get 0 ≤ − i∈[m] p i · E S,Zi∼Di [ℓ( w (global) (S), Z i ) − ℓ(w (1) ⋆ , Z i )] + i∈[m] p i n i j∈[ni] E S,z ′ i,j [ℓ( w (global) (S \(i,j) ), z (i) j ) − ℓ( w (global) (S), z (i) j )] = − i∈[m] p i · E S,Zi∼Di [ℓ( w (global) (S), Z i ) − ℓ(w (i) ⋆ , Z i )] − i∈[m] p i · E Zi∼Di [ℓ(w (i) ⋆ , Z i ) − ℓ(w (1) ⋆ , Z i )] + i∈[m] p i n i j∈[ni] E S,z ′ i,j [ℓ( w (global) (S \(i,j) ), z (i) j ) − ℓ( w (global) (S), z (i) j )]. Noting that w (i) ⋆ is the argmin of E Zi∼Di [ℓ(·, Z i )] and invoking the β-smoothness assumption, we get E S [AER p ( w (global) )] ≤ β i∈[m] p i w (1) ⋆ − w (i) ⋆ 2 + i∈[m] p i n i j∈[ni] E S,z ′ i,j [ℓ( w (global) (S \(i,j) ), z (i) j ) − ℓ( w (global) (S), z (i) j )] ≤ 2β w (1) ⋆ − w (global) avg 2 + 2β i∈[m] p i w (i) ⋆ − w (global) avg + i∈[m] p i n i j∈[ni] E S,z ′ i,j [ℓ( w (global) (S \(i,j) ), z (i) j ) − ℓ( w (global) (S), z (i) j )]. Taking a weighted average, we arrive at E S [AER p ( w (global) )] ≤ 4βR 2 + i∈[m] p i n i j∈[ni] E S,z ′ i,j [ℓ( w (global) (S \(i,j) ), z (i) j ) − ℓ( w (global) (S), z (i) j )] (C.1) To bound the second term in the right-hand side above, we bound the federated stability of w (global) . Without loss of generality we consider the first client. By µ-strongly convexity of L 1 , for any j 1 ∈ [n 1 ] we have µ 2 w (global) (S) − w (global) (S \(1,j1) ) 2 ≤ i∈[m] p i L i ( w (global) (S \(1,j1) ), S i ) − L i ( w (global) (S), S i ) = i =1 p i L i ( w (global) (S \(1,j1) ), S i ) + p 1 L 1 ( w (global) (S \(1,j1) ), S \j1 1 ) − i =1 p i L i ( w (global) (S), S i ) + p 1 L 1 ( w (global) (S), S \j1 1 ) + p 1 L 1 ( w (global) (S \(1,j1) ), S 1 ) − L 1 ( w (global) (S \(1,j1) ), S \j1 1 ) + p 1 L 1 ( w (global) (S), S \j1 1 ) − L 1 ( w (global) (S), S 1 ) ≤ p 1 L 1 ( w (global) (S \(1,j1) ), S 1 ) − L 1 ( w (global) (S \(1,j1) ), S \j1 1 ) + p 1 L 1 ( w (global) (S), S \j1 1 ) − L 1 ( w (global) (S), S 1 ) = p 1 n 1 ℓ( w (global) (S \(1,j1) ), z (1) j1 ) − ℓ( w (global) (S), z (1) j1 ) + p 1 n 1 ℓ( w (global) (S), z ′ 1,j1 ) − ℓ( w (global) (S \(1,j1) ), z ′ 1,j1 ) , (C.2) where the second inequality is because w (global) (S \(1,j1) ) minimizes L 1 (·, S \j1 1 ) + i =1 n i L i (·, S i ). By an identical argument as in the proof of Lemma C.3, we have ℓ( w (global) (S \(1,j1) ), z (1) j1 ) − ℓ( w (global) (S), z (1) j1 ) ≤ 2β ℓ ∞ · w (global) (S) − w (global) (S \(1,j1) ) + β 2 w (global) (S) − w (global) (S \(1,j1) ) 2 (C.3) The same bound also holds for ℓ( w (global) (S), z ′ 1,j1 ) − ℓ( w (global) (S \(1,j1) ), z ′ 1,j1 ). Plugging these two bounds to (C.2) and rearranging terms, we get (S \(1,j1) ) ≤ 2 2β ℓ ∞ · p 1 n 1 . µ 2 − βp 1 n 1 w (global) (S) − w (global) Since n 1 ≥ 4βp 1 /µ, we in fact have (S \(1,j1) ) ≤ 2 2β ℓ ∞ · p 1 n 1 . µ 4 w (global) (S) − w (global) Plugging the above display back to (C.3), we arrive at ℓ( w (global) (S \(1,j1) ), z (1) j1 ) − ℓ( w (global) (S), z (1) j1 ) ≤ 16β ℓ ∞ p 1 µn 1 1 + 4βp 1 µn 1 ≤ 32β ℓ ∞ p 1 µn 1 , where the last inequality is again by n 1 ≥ 4βp 1 /µ. The desired result follows by plugging the above inequality back to (C.1) Proof of (4.7). Without loss of generality we consider the first client. Since w (1) ⋆ is the minimizer of E Z1∼D1 ℓ(·, Z 1 ), by β-smoothness we have E Z1∼D1 [ℓ( w (global) , Z 1 ) − ℓ(w (1) ⋆ , Z 1 )] β · E Z1∼D1 w (global) − w (1) ⋆ 2 β · E Z1∼D1 w (global) − w (global) avg 2 + βR 2 , (C.4) where the last inequality is by Part (b) of Assumption C. By optimality of w (global) and the strong convexity of L i 's, we have i∈[m] p i ∇L i (w (global) avg , S i ), w (global) − w (global) avg + µ 2 w (global) − w (global) avg 2 ≤ 0. If w (global) − w (global) avg = 0 then we are done. Otherwise, the above display gives w (global) − w (global) avg ≤ 2 µ i∈[m] p i ∇L i (w (global) avg , S i ) ≤ 2 µ i∈[m] p i ∇L i (w (i) ⋆ , S i ) + i∈[m] p i ∇L i (w (global) avg , S i ) − ∇L i (w (i) ⋆ , S i ) ≤ 2 µ i∈[m] p i ∇L i (w (i) ⋆ , S i ) + β i∈[m] p i w (global) avg − w (i) ⋆ ≤ 2 µ i∈[m] p i ∇L i (w (i) ⋆ , S i ) + βR . Thus, we get w (global) − w (global) avg 2 ≤ 8 µ 2 i∈[m] p i ∇L i (w (i) ⋆ , S i ) 2 + β 2 R 2 . Taking expectation with respect to the sample S at both sides, we have E S w (global) − w (global) avg 2 1 µ 2 E S i∈[m] p i ∇L i (w (i) ⋆ , S i ) − E S [∇L i (w (i) ⋆ , S i )] 2 + β 2 R 2 µ 2 ≤ 1 µ 2 · i∈[m] p 2 i σ 2 n i + β 2 R 2 µ 2 . Plugging the above inequality to (C.4) gives the desired result. C.2 Proof of Proposition 4.1 Proof of (4.9). By the definitions of the AER and E OPT , we have AER p = E OPT + λ 2 i∈[m] p i w (global) (S) −w (i) (S) 2 − w (global) (S) − w (i) (S) 2 + i∈[m] p i E Zi∼Di [ℓ( w (i) (S), Z i )] − L i ( w (i) (S), S i ) + i∈[m] p i L i (w (i) (S), S i ) − E Zi∼Di [ℓ(w (i) ⋆ , Z i )] . By the basic inequality (4.11), we can bound the AER by AER p ≤ E OPT + λ 2 i∈[m] p i w (global) avg − w (i) ⋆ 2 + i∈[m] p i E Zi∼Di [ℓ( w (i) (S), Z i )] − L i ( w (i) (S), S i ) + i∈[m] p i L i (w (i) ⋆ , S i ) − E Zi∼Di [ℓ(w (i) ⋆ , Z i )] . Now, invoking federated stability, we can further bound the AER by AER p ≤ E OPT + λ 2 i∈[m] p i w (global) avg − w (i) ⋆ 2 + 2 i∈[m] p i γ i + i∈[m] p i · 1 n i j∈[ni] E z ′ i,j ∼Di E Zi∼Di [ℓ( w (i) (S \(i,j) ), Z i )] − ℓ( w (i) (S \(i,j) ), z (i) j ) + i∈[m] p i L i (w (i) ⋆ , S i ) − E Zi∼Di [ℓ(w (i) ⋆ , Z i )] , where S \(i,j) is the dataset formed by replacing z to be an independent sample from D i . Note that the last two terms of the above display have mean zero under the randomness of the algorithm A, the dataset S, and {z ′ i,j : i ∈ [m], j ∈ [n i ]}. Thus, the desired result follows by taking expectation in both sides. Proof of (4.10). Without loss of generality we consider the first client. By definitions of IER 1 and E OPT , we have p 1 · IER 1 = E OPT + i∈[m] p i L i (w (i) (S), S i ) + λ 2 w (global) (S) −w (i) (S) 2 − i∈[m] p i L i ( w (i) (S), S i ) + λ 2 w (global) (S) − w (i) (S) 2 + p 1 E Z1∼D1 [ℓ( w (1) (S), Z 1 ) − ℓ(w (1) ⋆ , Z 1 )]. Invoking the basic inequality (4.12), with some algebra, we arrive at p 1 · IER 1 ≤ E OPT + p 1 λ 2 w (global) (S) − w (1) ⋆ 2 + p 1 E Z1∼D1 [ℓ( w (1) (S), Z 1 )] − L 1 ( w (1) (S), S 1 ) + p 1 L 1 (w (1) ⋆ , S 1 ) − E Z1∼D1 [ℓ(w (1) ⋆ , Z 1 )] . Now, invoking federated stability for the first client, we can bound its IER by p 1 · IER 1 ≤ E OPT + p 1 λ 2 w (global) (S) − w (1) ⋆ 2 + 2p 1 γ 1 + p 1 n 1 j∈[n1] E z ′ 1,j ∼D1 E Z1∼D1 [ℓ( w (1) (S \(1,j) ), Z 1 )] − ℓ( w (1) (S \(1,j) ), z (1) j ) + p 1 L 1 (w (1) ⋆ , S 1 ) − E Z1∼D1 [ℓ(w (1) ⋆ , Z 1 )] , where we recall that S \(1,j) is the dataset formed by replacing z (1) j with a new sample z ′ 1,j , and here we are choosing z ′ 1,j to be an independent sample from D 1 . We finish the proof by taking the expectation with respect to A, S, {z ′ 1,j : j ∈ [n 1 ]} at both sides. C.3 Proof of Theorem 4.4 In this proof, we let A = ( w (global) , { w (i) } be a generic algorithm that tries to minimize (4.8). For notiontional simplicity, we use a n β b n (resp. a n β b n ) to denote that a n ≤ C β b n (resp. a n ≥ C β b n ) for large n, where C β has explicit dependence on a parameter β. Recall that (w (global) , {w (i) }) is the global minimizer of (4.8), and recall the notations in (B.1)- (B.3). We start by bounding the federated stability of approximate minimizers of (4.8). We need the following definition. Definition C.1 (Approximate minimizers). We say an algorithm A = ( w (global) , { w (i) } m 1 ) produces an (ε (global) , {ε (i) } m 1 )-minimizer of the objective function (4.8) on the dataset S if the following two conditions hold: 1. there exist a positive constant ε (global) such that w (global) −w (global) ≤ ε (global) ; 2. for any i ∈ [m], there exist a positive constant ε (i) such that w (i) − Prox Li/λ ( w (global) ) ≤ ε (i) . The stability bound is as follows. Proposition C.1 (Federated stability of approximate minimizers). Let Assumption A(b) holds, and consider an algorithm A = ( w (global) , { w (i) } m 1 ) that produces an (ε (global) , {ε (i) } m 1 )-minimizer of the objective function (4.8) on the dataset S. Assume in addition that n i ≥ 4β µ , p i λ ≤ µ 16 ∀i ∈ [m]. (C.5) Then A has federated stability γ i ≤ 160β ℓ ∞ n i (µ + λ) + Err i , where Err i := 2 2β ℓ ∞ 4ε (global) β + λ µ + λ + 3λ µ + ε (i) β + λ µ + λ + 16p i λ µ + 8β 2 16(ε (global) ) 2 β + λ µ + λ + 3λ µ 2 + (ε (i) ) 2 β + λ µ + λ + 16p i λ µ 2 is the error term due to not exactly minimizing the soft weight sharing objective (3.7). Proof. See Appendix C.3.1. Taking the optimization error into account, we have the following result. Proposition C.2 (Federated stability of A FP ). Let Assumption A(a, b) and (C.5) hold. Run A FP with hyperparameters chosen as in Lemma B.1 and B.2. Then, as long as T ≥ C 1 · λ 2 (λ ∨ 1) 2 m p 2 n 2 i , K T ≥ C 2 · λ 2 (λ ∨ 1) 2 p 2 i n 2 i ∀i ∈ [m], (C.6) the algorithm A FP have expected federated stability E A FP [γ i ] ≤ C · β ℓ ∞ n i (µ + λ) , where C 1 , C 2 are two constants only depending on (µ, β, ℓ ∞ , D), and C is an absolute constant. Proof. By Proposition C.1, it suffices to upper bound the error term Err i by a constant multiple of β ℓ ∞ ni(µ+λ) . Invoking Lemma B.2, we have E A FP [ε (global) ] 2 ≤ E A FP [(ε (global) ) 2 ] ≤ 12(λ + µ) 2 m p 2 (β 2 D 2 ∧ 2λ ℓ ∞ ∧ λ 2 D 2 ) λ 2 µ 2 (T + 1) . for any i ∈ [m], the algorithm A FP satisfies E A FP ,S [AER p (A FP )] ≤ C · β ℓ ∞ µ + λ i∈[m] p i n i + λ 2 i∈[m] p i w (global) avg − w (i) ⋆ 2 , where C 1 , C 2 are two constants only depending on (µ, β, ℓ ∞ , D), and C is an absolute constant. Proof. In view of Propositions 4.1 and C.2, it suffices to set T, K T such that (1) (C.6) is satisfied; and (2) E A FP [E OPT ] is upper bounded by a constant multiple of β ℓ ∞ µ+λ i∈[m] pi ni . To achive the second goal, note that by Proposition B.1, the optimization error is bounded by E A FP [E OPT ] (µ,β, µ ∞,D) λ ∨ 1 K T + 1 + λm p 2 T + 1 . (C.9) Thus, it suffices to require T (µ,β, µ ∞,D) λ(λ∨1)m p 2 i∈[m] pi/ni and K T (µ,β, µ ∞,D) (λ∨1) 2 i∈[m] pi/ni . This requirement, combined with (C.6), is exactly (C.8). With the above proposition at hand, we are ready to give our proof of Theorem 4.4. Proof of Theorem 4.4. We first define the following three events: A := R ≥ i∈[m] p i n i , B := i∈[m] p 2 i /n i i∈[m] p i /n i ≤ R ≤ i∈[m] p i n i , C := R ≤ i∈[m] p 2 i /n i i∈[m] p i /n i . We then choose λ to be λ = µ 16R 2 i∈[m] p i n i · 1 A + µ 16C p R i∈[m] p i n i · 1 B + µ 16C p i∈[m] p 2 i /n i i∈[m] p i n i · 1 C . We now consider the three events separately. E A FP ,S [AER p ] ≤ 16CC p β ℓ ∞ µ + µ 32C p R i∈[m] p i n i right-hand side of (4.16). 3. If C holds, then p i λ = piµ 16Cp i∈[m] p 2 i /ni i∈[m] pi ni ≤ µ 16 , and thus Proposition C.3 gives E A FP ,S [AER p ] ≤ 16CC p β ℓ ∞ µ + µ 32C p i∈[m] p 2 i n i right-hand side of (4.16). The desired result follows by combining the above three cases together. C.3.1 Proof of Proposition C.1: Stability of Approximate Minimizers We first present two lemmas, from which Proposition C.1 will follow. p i F i ( w (global) , S i ) ≤ δ (global) + i∈[m] p i F i (w (global) , S i ), (C.10) i∈[m] p i ∇F i ( w (global) , S i ) ≤ ζ (global) . (C.11) 2. for any i ∈ [m], there exist positive constants {δ (i) , ζ (i) , ε (i) } m i=1 such that L i ( w (i) , S i ) + λ 2 w (global) − w (i) 2 ≤ δ (i) + F i ( w (global) , S i ), (C.12) ∇L i ( w (i) , S i ) + λ( w (i) − w (global) ) ≤ ζ (i) , (C.13) w (i) − Prox Li/λ ( w (global) ) ≤ ε (i) . (C.14) Assume in addition that (C.5) holds. Then A has federated stability γ i ≤ 160β ℓ ∞ n i (µ + λ) + 2β ℓ ∞ · E λ,i + βE 2 λ,i , (C.15) where E λ,i := 8ζ (i) µ + λ + 8δ (i) µ + λ + 8µ −1 2ζ (global) + 4p i λε (i) + 2µλδ (global) µ + λ (C.16) is the error term due to not exactly minimizing (4.8). Lemma C.2 (Federated stability of approximate minimizers, Part II). Let Assumption A(b) holds and consider an algorithm A = ( w (global) , { w (i) } m 1 ) that produces an (ε (global) , {ε (i) } m 1 )-minimizer in the sense of Definition C.1. Then A also satisfies Equations (C.10)-(C.14) with δ (global) = λ 2 ε (global) , ζ (global) = λε (global) , δ (i) = β + λ 2 ε (i) , ζ (i) = (β + λ)ε (i) . Proof. These correspondences are consequences of λ-smoothness of F i and (β + λ)-smoothness of L i (·, S i ) + λ 2 w (global) − · 2 . We omit the details. With the above two lemmas at hand, the proof of Proposition C.1 is purely computational: Proof of Proposition C.1 given Lemma C.1 and C.2. Invoking C.2, the error term E λ,i defined in Equation C.16 can be bounded above by E λ,i ≤ 8(β + λ) µ + λ · ε (global) + 4(β + λ) µ + λ · ε (i) + 8 µ 2λε (global) + 4p i λε (i) + µλ 2 µ + λ = 8ε (global) β + λ µ + λ + 2λ µ + λ µ(µ + λ) + 2ε (i) β + λ µ + λ + 16p i λ µ ≤ 8ε (global) β + λ µ + λ + 3λ µ + 2ε (i) β + λ µ + λ + 16p i λ µ . + ζ (global) w (global) (S \(1,j1) ) − w (global) (S) = p 1 F 1 ( w (global) (S \(1,j1) (1,j1) ), S 1 ) − F 1 ( w (global) (S \(1,j1) ), S ≤ δ (global) + ζ (global) w (global) (S \(1,j1) ) − w (global) (S) + p 1 F 1 ( w (global) (S \(1,j1) ), S 1 ) − F 1 ( w (global) (S), S 1 ) ), S \j1 1 ) + i =1 p i F i ( w (global) (S \(1,j1) ), S i ) − p 1 F 1 ( w (global) (S, S \j1 1 ) + i =1 p i F i ( w (global) (S, S i ) + p 1 F 1 ( w (global) (S \+ F 1 ( w (global) (S), S \j1 1 ) − F 1 ( w (global) (S \(1,j1) ), S \j1 1 ) . Since F 1 is λ-smooth by Lemma B.3, we can proceed by µ F 2 w (global) (S) − w (global) (S \(1,j1) ) 2 ≤ δ (global) + ζ (global) w (global) (S \(1,j1) ) − w (global) (S) + p 1 λ w (global) (S \(1,j1) ) − w (global) (S) 2 + p 1 ∇F 1 ( w (global) (S), S 1 ) − ∇F 1 ( w (global) (S \(1,j1) ), S \j1 1 ), w (global) (S \(1,j1) ) − w (global) (S) . Since ∇F 1 (w (global) , S 1 ) = λ w (global) − Prox L1/λ (w (global) , S 1 ) , with some algebra, the right-hand side above is in fact equal to δ (global) + ζ (global) w (global) (S \(1,j1) ) − w (global) (S) + p 1 λ w (1) (S) − Prox L1/λ ( w (global) (S), S 1 ), w (global) (S \(1,j1) ) − w (global) (S) + p 1 λ Prox L1/λ ( w (global) (S \(1,j1) ), S \j1 1 ) − w (1) (S \(1,j1) , w (global) (S \(1,j1) ) − w (global) (S) + p 1 λ w (1) (S \(1,j1) ) − w (1) (S), w (global) (S \(1,j1) ) − w (global) (S) (C.14) ≤ δ (global) + (ζ (global) + 2p 1 λ) w (global) (S \(1,j1) ) − w (global) (S) + p 1 λ w (i) (S \(1,j1) ) − w (i) (S) w (global) (S \(1,j1) ) − w (global) (S) . The above bound gives a quadratic inequality: if we let s G := w (global) (S \(1,j1) ) − w (global) (S) and s 1 := w (i) (S \(1,j1) ) − w (i) (S) , then the above bound can be written as µ F 2 · s 2 G − (ζ (global) + 2p 1 λε (1) + p 1 λs 1 ) · s G − δ (global) ≤ 0. Solving this inequality gives s G ≤ 1 µ F · ζ (global) + 2p 1 λε (1) + p 1 λs 1 + (ζ (global) + 2p 1 λε (1) + p 1 λs 1 ) 2 + 2µ F δ (global) ≤ 1 µ F 2ζ (global) + 4p 1 λε (1) + 2p 1 λs 1 + 2µ F δ (global) , which is exactly (C.17). Lemma C.5 (Parameter stability). Under the same assumptions as Proposition C.1, for any i ∈ [m], j i ∈ [n i ], we have w (i) (S \(i,ji) ) − w (i) (S) ≤ 16 2β ℓ ∞ n i (µ + λ) + E λ,i . Proof. Without loss of generality we consider the first client. Since L 1 (·, S 1 ) + λ 2 w (global) (S) − · 2 is (µ + λ)strongly convex, we have 1 2 (µ + λ) w (1) (S) − w (1) (S \(1,j1) ) 2 ≤ L 1 ( w (1) (S \(1,j1) ), S 1 ) + λ 2 w (global) (S) − w (1) (S \(1,j1) (S \(1,j1) ) − w (1) (S) 2 − λ 2 w (global) (S) − w (1) (S) 2 + ζ (1) w (1) (S \(1,j1) ) − w (1) (S) (C.12) ≤ δ (1) + ζ (1) w (1) (S \(1,j1) ) − w (1) (S) + λ w (global) (S) − w (global) (S \(1,j1) ), w (S) − w (1) (S \(1,j1) ) + 1 n 1 ℓ( w (1) (S), z ′ 1,j1 ) − ℓ( w (1) (S \(1,j1) ), z ′ 1,j1 ) + ℓ( w (1) (S \(1,j1) ), z ) 2 − L 1 ( w (1) (S), S 1 ) + λ 2 w (global) (S) − w (1) (S) 2 + ∇L 1 ( w (1) (S), S 1 ) + λ( w (1) (S) − w (global) (S)), w (1) (S \(1,j1) ) − w (1) (S) (C.13) ≤ L 1 ( w (1) (S \(1,j1) ), S \j1 1 ) + λ 2 w (global) (S \(1,j1) ) − w (1) (S \(1,j1) ) 2 − L 1 ( w (1) (S), S \j1 1 ) + λ 2 w (global) (S \(1,j1) ) − w (1) (S) 2 − 1 n 1 ℓ( w (1) (S \(1,j1) ), z ′ 1,j1 ) + 1 n 1 ℓ( w (1) (S \(1,j1) ), z (1) j1 ) + 1 n ℓ( w (1) (S), z ′ 1,j1 ) − 1 n 1 ℓ( w (S) , z (1) j1 ) − λ 2 w (global) (S \(1,j1) ) − w (1) (S \(1,j1) ) 2 + λ 2 w (global) (S) − w (1) (S \(1,j1) ) 2 + λ 2 w (global) (1) j1 ) − ℓ( w (1) (S), z (1) j1 ) ≤δ (1) + ζ (1) w (1) (S \(1,j1) ) − w (1) (S) + 2 n 1 2β ℓ ∞ w (1) (S) − w (1) (S \(1,j1) ) + β 2 w (1) (S) − w (1) (S \(1,j1) ) 2 + λ + µ µ 2ζ (global) + 2λµδ (global) λ + µ + 4p i λε (i) + 2p i λ w (i) (S \(i,j1) ) − w (i) (S) × w (1) (S) − w (1) (S \(1,j1) ) , where the last inequality is by Lemma C.3 and Lemma C.4. Denoting s 1 := w (1) (S) − w (1) (S \(1,j1) ) , the above inequality can be written as C λ,1 s 2 1 − 2 2β ℓ ∞ n 1 + ζ (1) + λ + µ µ 2ζ (global) + 4p 1 λε (1) + 2λµδ (global) λ + µ s 1 − δ (1) ≤ 0, (C.18) where C λ,1 := 1 2 (µ + λ) − β n 1 − 2p 1 λ(λ + µ) µ . By (C.5), we have C λ,1 ≥ µ + λ 2 − µ 4 − 2p 1 λ(λ + µ) µ ≥ λ + µ 4 − 2p 1 λ(λ + µ) µ = λ + µ 4 · 1 − 8p 1 λ µ ≥ λ + µ 8 . In particular, C λ,1 > 0, and thus we can solve the quadratic inequality (C.18) (similar to the proof of Lemma C.4) to get s 1 ≤ 2 2β ℓ ∞ C λ,1 n 1 + ζ (1) + λ+µ µ 2ζ (global) + 4p 1 λε (1) + 2λµδ (global) λ+µ C λ,1 + δ (1) C λ,1 . Plugging in C λ,1 ≥ (λ + µ)/8 to the above inequality gives the desired result. We are finally ready to present a proof of Lemma C.1: Proof of Lemma C.1. Invoking Lemma C.3, we have γ i ≤ 2β ℓ ∞ · 16 2β ℓ ∞ n i (µ + λ) + E λ,i + β 2 16 2β ℓ ∞ n i (µ + λ) + E λ,i 2 ≤ 32β ℓ ∞ n i (µ + λ) · 1 + β n i (µ + λ) + 2β ℓ ∞ · E λ,i + βE 2 λ,i , where in the last line we have used (a+b) 2 ≤ 2a 2 +2b 2 . We finish the proof by noting that β ni(µ+λ) ≤ β niµ ≤ 4, where the last inequality is by (C.5). C.4 Proof of Theorem 4.5 Compared to the proof of Theorem 4.4, we need to additionally control the estimation error of the global model. p i √ n i 2 + 48β 2 R 2 µ 2 + 12(µ + λ) 2 σ 2 µ 2 λ 2 i∈ [m] p 2 i n i . Proof. See Appendix C.4.1 With the above proposition, the following result is a counterpart of Proposition C.3. Proposition C.5 (λ-dependent bound on the IER). Let Assumptions A(a, b), C(b) and Equation (C.5) hold. Run A FP with hyperparameters chosen as in Lemma B.1 and B.2. Then, for any i ∈ [m], as long as where C 1 , C 2 are two constants only depending on (µ, β, ℓ ∞ , D), and C is an absolute constant. T ≥ C 1 λ(λ ∨ 1)m p 2 n i · p −1 i ∨ [λ(λ ∨ 1)n i ] , K T ≥ C 2 (λ + 1) 2 n i p −1 i ∨ λ 2 p 2 i n i ,(C. β ℓ ∞ + σ 2 β 2 + β 2 + σ 2 µ 2 · 1 λN/m + λ(R 2 + N −1 ) = β ℓ ∞ + σ 2 β 2 + β 2 + σ 2 µ 2 · √ R 2 N + 1 c B N/ √ m + c B N m(R 2 N + 1) ≤ β ℓ ∞ + σ 2 β 2 + β 2 + σ 2 µ 2 · R c B N/m + 1 c B N/ √ m + c B √ mR √ N + c B √ m N = β ℓ ∞ + σ 2 β 2 + β 2 + σ 2 µ 2 · (c B + c −1 B ) R N/m + √ m N . Note that p max λ ≤ c B p max √ m ≍ c B / √ m ≤ c B . So to satisfy p max λ ≤ µ/16, we can choose c B ≍ µ. This gives E A FP ,S [IER 1 ] β ℓ ∞ + σ 2 β 2 + β 2 + σ 2 µ 2 (µ + µ −1 ) · R N/m + √ m N ≤ right-hand side of (C.22). The desired result follows by combining the above two cases together. C.4.1 Proof of Proposition C.4: Estimation Error of the Global Model We begin by proving a useful lemma. Lemma C.6 (Estimating w Proof. This follows from an adaptation of the arguments in Theorem 7 of Foster et al. [2019]. By strong convexity, we have ∇L i (w (i) ⋆ , S i ) + λ(w (i) ⋆ − w (global) avg ), Prox Li/λ (w (global) avg ) − w (i) ⋆ + µ + λ 2 w (i) ⋆ − Prox Li/λ (w (global) avg ) 2 ≤ L i (w (i) ⋆ , S i ) + λ 2 w (global) − w (i) ⋆ 2 − L i (Prox Li/λ (w (global) avg ), S i ) − λ 2 w (global) avg − Prox Li/λ (w (global) avg ) 2 ≤ 0. If w (i) − Prox Li/λ (w (global) avg ) = 0 we are done. Otherwise, Cauchy-Schwartz inequality applied to the above display gives the desired result. Now, since i∈[m] p i F i is µ F = µλ/(µ + λ)-strongly convex, we have i∈[m] p i ∇F i (w (global) avg ),w (global) − w (global) avg + µ F 2 w (global) − w (global) avg 2 ≤ i∈[m] p i F i (w (global) ) − i∈[m] p i F i (w (global) avg ) ≤ 0. of communication rounds T , step sizes {η t } T −1 t=0 for t = 0, 1, . . . , T − 1 do Randomly sample a batch of clients C t ⊆ [m] for client i ∈ C t do Obtain w (i) t+1 by running several steps of SGD on S i using w (global) t as the initialization bound, where a ∧ b = min{a, b} for two real numbers a and b. Corollary 3 . 1 ( 31Performance of a dichotomous strategy). Under the setup of Theorem 3.2, consider the following naïve dichotomous strategy: if R 2 ≤ ℓ ∞ µN/m , then output A = A FA ; otherwise, output A = A PLT . Then under Assumption B(a), we have Lemma 4 . 1 ( 41Logistic regressions are valid problem instances). The logistic regression problem described above is a class of problem instances that satisfies Assumption A with ℓ ∞ = c X D √ d and σ 2 = β = c 2 X d/4. Moreover, if m (N/m) c for some c ≥ 0 and N/m ≥ Cd for some C > 1, then there exists some event E which only depends on the features {x (i) j : i ∈ [m], j ∈ [n i ]} and happens with probability at least 1 − e −O( √ N/m) , such that on this event, the strongly convex constant in Assumption A satisfies ⋆ } that can arise in the logistic regression models under Assumption B (a) and (b), respectively. tow(global) has appeared inLi et al. [2018] and Dinh et al.[2020], to the best of our knowledge, the convergence of w (i) T tow (i) has been largely overlooked in the literature, and we need to address this additional technical challenge in order to have a fully rigorous statement on the performance of FedProx. [2019], Balcan et al. [2019], andKhodak et al. [2019] S. Ben-David and R. S. Borbely. 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Convergence of FedProx 29 B.1 Proof of Lemma B.1: Convergence of the Inner Loop . . . . . . . . . . . . . . . . . . . . . . . 31 B.2 Proof of Lemma B.2: Convergence of the Outer Loop . . . . . . . . . . . . . . . . . . . . . . . 33 B.3 Auxiliary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Appendix C Proofs of Upper Bounds 36 C.1 Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 C.2 Proof of Proposition 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 C.3 Proof of Theorem 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 C.3.1 Proof of Proposition C.1: Stability of Approximate Minimizers . . . . . . . . . . . . . 43 C.4 Proof of Theorem 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 C.4.1 Proof of Proposition C.4: Estimation Error of the Global Model . . . . . . . . . . . . 49 Appendix A Proof of Theorem 4.1: Lower Bounds We start by presenting a lower bound when all w (i) ⋆ 's are the same. Lemma A.1 (Lower bound under homogeneity). Consider the logistic regression model with w ±1) to denote the probability measure with respect to the randomness in (V , S) conditional on the features {x (i) j } as well as the realization of v (i) k = ±1. , S i ). This idea has appeared in various places (see, e.g., the proof of Proposition 5 inDenevi et al. [2019] and the proof of Theorem 1 in Dinh et al.[2020]). However, the implementation of this idea in our case is more complicated than the above mentioned works in that(1)we are not in an online learning setup (compared to Denevi et al. [2019]); (2) we don't need to assume all clients are training at every round (compared to Dinh et al. [2020]); and (3) we use local SGD for the inner loop (instead of assuming the inner loop can be solved with arbitrary precision as assumed in Dinh et al. [2020]), so the gradient norm depends on λ, and could in principle be arbitrarily large, which causes extra complications. Lemma B.1 (Convergence of the inner loop). Let Assumption A(a, b) holds. Choose η )(k+1) . Then for any k ≥ 0, we have E Lemma B. 2 ( 2Convergence of the outer loop). Let the assumptions in Lemma B.1 hold. Choose η a) is by smoothness of L i , (b) is by Lemma B.1 and λ-smoothness of i∈[m] p i F i (which holds by Lemma B.3), and (c) is by Lemma B.2. Lemma B.4 (A priori gradient norm bound). Under Assumption A(a, b), for any w ∈ W and i ∈ [m], we have Lemma B. 5 ( 5Variance of minibatch sampling). Let B ⊆ [n] be a randomly sampled batch with batch size B and let {x i } n i=1 ⊆ R d be an arbitrary set of vectors, then j with a new sample z ′ i,j , and here we are choosing z ′ i,j + F 1 ( w (global) (S), S \j1 1 ) − F 1 ( w (global) (S), S 1 ) + ζ (global) w (global) (S \(1,j1) ) − w (global) (S) (C.10) Proposition C. 4 ( 4Estimation error of the global model). Let Assumptions A(b) and C(b) hold. ThenE S w (global) − w (global) E A FP ,S [IER i (A FP )] ≤ C β ℓ ∞ µn i + λD 2 , (C.21) Algorithm 3 : 3FedProx, general versionInput: Initial global model w(global) 0 , initial local models {w 1 . 1If A holds, then p i λ = piµ 16R 2 i∈[m] pi µi ≤ piµ 16 ≤ µ 16 . Thus we can invoke Proposition C.3 to get E A FP ,S [AER p ] ≤ If B holds, then p i λ = piµ where the last inequality is by the definition of C p . Hence, by Proposition C.3, we haveCβ ℓ ∞ µ + µ 32 i∈[m] p i n i right-hand side of (4.16). 2. 16CpR i∈[m] pi ni ≤ pmaxµN 16Cp i∈[m] p 2 i /ni i∈[m] pi ni ≤ µ 16 , Lemma C.1 (Federated stability of approximate minimizers, Part I). Let Assumption A(b) holds, and consider an algorithm A = ( w (global) , { w (i) } m 1 ) that satisfies the following conditions: 1. there exist positive constants δ (global) , ζ (global) such that i∈[m] 19 ) 19the algorithm A FP satisfies both E A FP ,S [IER i (A FP )]≤ C λn i β ℓ ∞ + σ 2 β 2 n i µ 2 i∈[m] p i √ n i 2 + σ 2 n i i∈[m] p 2 i n i + Cλ 1 + β 2 µ 2 R 2 + σ 2 µ 2 i∈[m] The theories developed by Hanzely et al. [2020] can be useful for such an analysis. AcknowledgmentsThis work was supported in part by NIH through R01-GM124111 and RF1-AG063481, NSF through CAREER DMS-1847415, CCF-1763314, and CCF-1934876, and an Alfred P. Sloan Research Fellowship.This gives[(ε (global)) 2 ](µ,β, µ ∞,D)(λ ∨ 1) 2 m p 2 (1 ∧ λ ∧ λ 2 ) λ 2 (T + 1)where we recall that a n (µ,β, µ ∞,D) b n means |a n | ≤ Cb n for a constant C that only depending on (µ, β, µ ∞ , D), and the last inequality follows from (λ ∨ 1) 2 (1 ∧ λ ∧ λ 2 ) ≤ λ 2 regardless λ ≥ 1 or λ ≤ 1.Meanwhile, by Lemma B.1, we haveRecalling the definition of Err i , we havewhich is equivalent towhich is exactly (C.6).Combining the above proposition with Proposition 4.1. we get the following result.Proposition C.3 (λ-dependent bound on the AER). Let Assumption A(a, b) and (C.5) hold. Run A FP with hyperparameters chosen as inLemma B.1 and B.2. Then, as long asThis givesPlugging the above two displays to (C.15) gives the desired result.We now present our proof of Lemma C.1. We start by stating and proving several useful lemmas.Lemma C.3 (From loss stability to parameter stability). Let Assumption A(b) holds. Then the algorithmProof. This lemma has implicitly appeared in the proofs of many stability-based generalization bounds (see, e.g., Section 13.3.2 of Shalev-Shwartz and Ben-David [2014]), and we provide a proof for completeness. By β-smoothness, for an arbitrary z ∈ Z we havewhere the last inequality follows from boundedness of ℓ. By a nearly identical argument, the above upper bound also holds for −ℓ( w (i) (S), z) + ℓ( w (i)(S \(i,ji)), z), and the desired result follows.Lemma C.4 (Local stability implies global stability). Assume Assumption A(b) holds and consider an algorithm A = ( w (global) , { w (i) } m 1 ) that satisfies Equations (C.10), (C.11) and (C.14). Then for any i ∈ [m], j i ∈ [n i ], we haveProof. Without loss of generality we consider the first client. Let µ F be the strongly convex constant ofProof. Without loss of generality we consider the first client. Our assumptions allow us to invoke Propositions 4.1 and C.2 to getWe first show that the expected value of E OPT /p 1 and λ(ε (global) ) 2 are both bounded above by a constant multiple of β ℓ ∞ n1(µ+λ) . Indeed, by the estimates we have established in Equations (C.7) and (C.9), it suffices to require T (µ,β, µ ∞,D) λ(λ ∨ 1)m p 2 n 1 ,respectively. And the above two displays, combined with (C.6), is exactly (C.19). (C.21) then follows from the compactness of W.To prove (C.20), we invoke Proposition C.4 to getand (C.21) follows by rearranging terms.We now present our proof of Theorem 4.5.Proof of Theorem 4.5. Without loss of generality we consider the first client. Since all n i 's are of the same order, it suffices to showWe define the following two events:and we setwhere c A , c B are two constants to be specified later. We consider two cases:≤ right-hand side of (C.22).2. If B holds, and if λp max ≤ µ/16 holds, then from (C.20) we have= 0 we are done. 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[]
[ "Integral formulas for a foliated sub-Riemannian manifold", "Integral formulas for a foliated sub-Riemannian manifold" ]
[ "Vladimir Rovenski " ]
[]
[]
In this article, we deduce a series of integral formulas for a foliated sub-Riemannian manifold, which is a new geometric concept denoting a Riemannian manifold equipped with a distribution D and a foliation F , whose tangent bundle is a subbundle of D. Our integral formulas generalize some results for foliated Riemannian manifolds and involve the shape operators of F with respect to normals in D and the curvature tensor of induced connection on D. The formulas also include arbitrary functions f j (0 ≤ j < dim F ) depending on scalar invariants of the shape operators, and for a special choice of f j reduce to integral formulas with the Newton transformations of the shape operators. We apply our formulas to foliated sub-Riemannian manifolds with restrictions on the curvature and extrinsic geometry of F and to codimension-one foliations.
10.1007/s40879-023-00593-5
[ "https://export.arxiv.org/pdf/2208.13461v1.pdf" ]
232,105,209
2208.13461
6111087bfa64f5101e76644fe5393efd38eb48d4
Integral formulas for a foliated sub-Riemannian manifold Aug 2022 Vladimir Rovenski Integral formulas for a foliated sub-Riemannian manifold 29Aug 2022Distributionfoliationshape operatorNewton transformation Mathematics Subject Classifications (2010) 53C1253C17 In this article, we deduce a series of integral formulas for a foliated sub-Riemannian manifold, which is a new geometric concept denoting a Riemannian manifold equipped with a distribution D and a foliation F , whose tangent bundle is a subbundle of D. Our integral formulas generalize some results for foliated Riemannian manifolds and involve the shape operators of F with respect to normals in D and the curvature tensor of induced connection on D. The formulas also include arbitrary functions f j (0 ≤ j < dim F ) depending on scalar invariants of the shape operators, and for a special choice of f j reduce to integral formulas with the Newton transformations of the shape operators. We apply our formulas to foliated sub-Riemannian manifolds with restrictions on the curvature and extrinsic geometry of F and to codimension-one foliations. Introduction A non-holonomic manifold is a pair (M, D), where M is a smooth manifold and D is a distribution on M , i.e., a subbundle of the tangent bundle T M , see [4]. In terms of an almost product structure on M , i.e., a (1,1)-tensor field P of constant rank for which P 2 = P , see [9], we obtain D = P (T M ). A non-holonomic manifold (M, D) with a Riemannian metric g on the distribution D, e.g., the restriction of a metric on the manifold M , is the main object of sub-Riemannian geometry. In [6], they explore sub-Riemannian structures arising as the transversal distribution to a foliation. In [15], we introduced a new geometrical concept of a foliated sub-Riemannian manifold, i.e., (M, D, g) equipped with a foliation F, whose tangent bundle T F is a subbundle of any codimension of D. (Recall that a foliation F is a partition of M into equivalence classes that locally models the partition of R n+m by submanifolds parallel to R n ). We proved in [15] a series of integral formulas for a codimension-one foliated sub-Riemannian manifold, i.e., D = T F ⊕ span(N ), where N is a unit vector field orthogonal to F; these integral formulas generalize results in [2,7] for foliated Riemannian manifolds. On the other hand, in [14], we extended the above mentioned results in [2,7] for foliations of arbitrary codimension of Riemannian manifolds, as a series of integral formulas depending on the Newton or even more general transformations of the shape operators of the leaves. Analyzing history of extrinsic geometry of foliations, we see that from the origin it was related to some integral formulas containing the shape operator (or the second fundamental form) of leaves and its invariants (mean curvature, higher order mean curvatures σ r , etc.) and some expressions corresponding to geometry (curvature) of M . Integral formulas are useful for solving many problems in differential geometry, both manifolds (for example, the Gauss-Bonnet formula for closed surfaces) and foliations, see surveys [1,20,21]. The first known integral formula for a codimension-one foliation of a closed Riemannian manifold by G. Reeb [12] tells that the integral of the mean curvature H of the leaves is equal to zero. Its proof is based on the application of the Divergence theorem to the identity div N = −H with N a unit normal to the leaves. The second formula in the series of total higher mean curvatures σ r 's of a codimension-one foliation is, e.g., [14], M (2 σ 2 − Ric N,N ) d vol g = 0. (1) In [2], the Newton transformations T r (A) (of the shape operator A of the leaves) were applied to codimension one foliations, and a series of integral formulas for r ≥ 0 starting with (1) was obtained (with consequences for foliated space forms, see [7]): M (r + 2) σ r+2 − tr F (T r (A)R N ) − div F T r (A), ∇ N N d vol g = 0.(2) Here, the Jacobi operator R N : T F → T F is given by R N (Y ) = R(Y, N )N . There is also a series of integral formulas for two complementary orthogonal distributions D and D ⊥ (or foliations) of arbitrary dimension on a closed Riemannian manifold (M, g), see [14], starting with the the following generalization of (1), see [24]: M S mix + h 2 + h ⊥ 2 − H 2 − H ⊥ 2 − T 2 − T ⊥ 2 d vol g = 0.(3) Here, h, T : D × D → D ⊥ and H = tr g h are the second fundamental form, the integrability tensor and the mean curvature vector field of D, and similarly for D ⊥ , S mix is the mixed scalar curvature. The following question naturally arises: if the integral formulas of the type (2) and (3) can be proved for foliated sub-Riemannian manifolds? In this article, we answer this question in the affirmative and derive a series of integral formulas (in Theorems 1 and 2 and Corollaries 1-7) for a foliated sub-Riemannian manifold, which generalize some known integral formulas for foliated Riemannian manifolds, see (2) and [2,8], and extend the results in [15], where codim F = 1. Our integral formulas involve the shape operators of F with respect to unit normals in D, some components of the curvature tensor of the induced connection on D. The formulas also include arbitrary functions f j (0 ≤ j < dim F) depending on scalar invariants of the shape operators, and for a special choice f j = (−1) j σ r−j reduce to integral formulas with Newton transformations of the shape operators (and their scalar invariants -the rth mean curvatures) of F. We apply our formulas to foliated sub-Riemannian manifolds with restrictions on the curvature and extrinsic geometry of the leaves of F and to codimension-one foliations. Preliminaries Here, we define the shape operator A ξ and its Newton transformations T r (A ξ ), the rth mean curvatures σ r (ξ) and power sums symmetric functions τ k (ξ). Let D be an (n + p)-dimensional distribution on a smooth m-dimensional manifold M , i.e., a subbundle of T M of rank n + p (where n, p > 0 and n + p < m). In other words, to each point x ∈ M we assign an (n + p)-dimensional subspace D x of the tangent space T x M smoothly depending on x. An integrable distribution determines a foliation; in this case, the Lie bracket of any two vector fields from the distribution D also belongs to D. A pair (M, D), where D is a non-integrable distribution on a manifold M , is called a nonholonomic manifold, see [4]. The concept of a non-holonomic manifold was introduced for the geometric interpretation of constrained systems in classical mechanics. A sub-Riemannian manifold is a non-holonomic manifold (M, D), where D is equipped with a sub-Riemannian metric g = · , · , i.e., the scalar product on D x for all x ∈ M , see [4]. Usually, they assume that the sub-Riemannian metric on D (the horizontal bundle) is extended to a Riemannian metric on the whole M , also denoted by g. This allows us to define the orthogonal distribution D (the vertical subbundle) such that T M = D ⊕ D. Definition 1. A sub-Riemannian manifold (M, D, g) equipped with a foliation F such that the tangent bundle T F is a subbundle of D will be called a foliated sub-Riemannian manifold. Let F be a foliation of codimension p relative to D, i.e., dim F = n, and let N F be the orthogonal complement of T F in D; thus, the following decomposition holds: D = T F ⊕N F. Let ⊤ denotes the projection from T M on the vector subbundle T F. The shape operator A ξ : T F → T F of the foliation F with respect to the unit normal ξ ∈ N F is defined by A ξ (X) = −(∇ X ξ) ⊤ , X ∈ T F. The elementary symmetric functions σ j (A ξ ) of A ξ are given by the equality, e.g., [20], n r=0 σ r (A ξ ) t r = det( id T F + t A ξ ), t ∈ R. Note that σ r (A ξ ) = i 1 <···<ir λ i 1 (ξ) · · · λ ir (ξ), where λ 1 (ξ) ≤ . . . ≤ λ n (ξ) are the eigenvalues of A ξ . The power sums symmetric functions of A ξ are τ j (A ξ ) = tr (A j ξ ) = 1≤i≤n (λ i (ξ)) j , j ∈ N. For short, set σ r (ξ) = σ r (A ξ ), τ r (ξ) = τ r (A ξ ). For example, σ 0 (ξ) = 1, σ n (ξ) = det A ξ , σ 1 (ξ) = τ 1 (ξ) = tr A ξ and 2 σ 2 (ξ) = τ 2 1 (ξ) − τ 2 (ξ). Definition 2. The Newton transformations T r (A ξ ) of the shape operator A ξ of a foliated sub-Riemannian manifold (M, D, F, g) are defined recursively or explicitly by T 0 (A ξ ) = id T F , T r (A ξ ) = σ r (ξ) id T F −A ξ T r−1 (A ξ ) (0 < r ≤ n),(4)T r (A ξ ) = r j=0 (−1) j σ r−j (ξ) A j ξ = σ r (ξ) id T F −σ r−1 (ξ) A ξ + . . . + (−1) r A r ξ .(5) For example, T 1 (A ξ ) = σ 1 (ξ) id T F −A ξ and T n (A ξ ) = 0. Define a (1,1)-tensor field A ξ := n−1 j=0 f j (τ 1 (ξ), . . . , τ n (ξ)) A j ξ ,(6) where f j : R n → R (0 ≤ j < n) are given functions. We write f j = f j (ξ) shortly. The choice of the RHS for A ξ in (6) is natural for the following reasons, [20, Section 1.2.2]: • the powers A j ξ are the only (1,1)-tensors, obtained algebraically from A ξ , while τ 1 (ξ), . . . , τ n (ξ), or, equivalently, σ 1 (ξ), . . . , σ n (ξ), generate all scalar invariants of A ξ . • the Newton transformation T r (A ξ ) (0 ≤ r < n) of (5) depends on all A j ξ (0 ≤ j ≤ r). In this article, for illustrate the results for A ξ by the particular case of A ξ = T r (A ξ ). Let ∇ F : T M × T F → T F be the induced connection on the vector subbundle T F. Since the (1,1)-tensors A ξ and T r (A ξ ) are self-adjoint, we have (∇ F X T r (A ξ ))Y, V = (∇ F X T r (A ξ ))V, Y , X, Y, V ∈ T F. The following properties of T r (A) are proved similarly as for codimension-one foliations of a Riemannian manifold, e.g., [2] or [20, Lemma 1.3]. Lemma 1. For the shape operator A ξ of an n-dimensional foliation F we have tr F T r (A ξ ) = (n − r) σ r (ξ), tr F (A ξ · T r (A ξ )) = (r + 1) σ r+1 (ξ), tr F (A 2 · T r (A ξ )) = σ 1 (ξ) σ r+1 (ξ) − (r + 2) σ r+2 (ξ), tr F (T r−1 (A ξ )(∇ F X A ξ )) = X(σ r (ξ)), X ∈ T F, k tr F (A k−1 ξ ∇ F X A ξ ) = X(τ k (ξ)), X ∈ T F, k > 0. 2 The induced connection and curvature on D Here, we define the induced linear connection ∇ P and the curvature tensor R P related to a foliated sub-Riemannian manifold, we also prove Codazzi type equation. The orthoprojector P : T M → D on the distribution D is characterized by the properties, e.g., [9], P = P * (self-adjoint), P 2 = P. The Levi-Civita connection ∇ on (M, g) induces a linear connection ∇ P on D: ∇ P X (P Y ) = P ∇ X (P Y ), X, Y ∈ Γ(T M ), which is compatible with the metric on D: X U, V = ∇ P X U, V + U, ∇ P X V for U, V ∈ D. Set R P (X, Y, U, V ) = R P (X, Y )U, V for U, V ∈ D, where R P : T M × T M → End(D) is the curvature tensor of ∇ P . As for any linear connection, we have the anti-symmetry for the first pair of vectors: R P (Y, X)U = −R P (X, Y )U . Since ∇ P is compatible with g, then the anti-symmetry for the last pair of vectors in D is valid, e.g., [10], R P (X, Y, U, V ) = −R P (X, Y, V, U ).(7) Let ⊥ denotes the projection on the vector subbundle (T F) ⊥ orthogonal to F in T M . The Codazzi equation for a foliation (or a submanifold) of (M, g) has the form, e.g., [13]: (∇ X h)(Y, U ) − (∇ Y h)(X, U ) = (R(X, Y )U ) ⊥ ,(8) where R : T M × T M → End(T M ) is the Riemannian curvature tensor of ∇, R(X, Y ) = ∇ X ∇ Y − ∇ Y ∇ X − ∇ [X, Y ] , and h : T F × T F → (T F) ⊥ is the second fundamental form of F in (M, g) defined by h(X, Y ) = (∇ X Y ) ⊥ . Hence, A ξ (X), Y = h(X, Y ), ξ for ξ ∈ N F. Lemma 2. The following Codazzi type equation is valid: (∇ F X A ξ )Y − (∇ F Y A ξ )X = −R P (X, Y )ξ, X, Y ∈ T F, ξ ∈ N F.(9) Proof. From (8), for all vectors X, Y, U ∈ T F we get (∇ F X A ξ )Y − (∇ F Y A ξ )X, U + h(X, U ), ∇ Y ξ − h(Y, U ), ∇ X ξ = − R(X, Y )ξ, U . (10) Applying the orthoprojector from T M on the vector subbundle (T F) ⊥ , we find R(X, Y )ξ, U = R P (X, Y )ξ, U + ∇ X ((∇ Y ξ) ⊥ ) − ∇ Y ((∇ X ξ) ⊥ ) − ∇ [X, Y ] ⊥ ξ, U . (11) Using the equalities [X, Y ] ⊥ = 0 (since T F is integrable), (11) and h(X, U ), ∇ Y ξ = ∇ Y ξ, (∇ X U ) ⊥ = − ∇ X ((∇ Y ξ) ⊥ ), U , h(Y, U ), ∇ X ξ = ∇ X ξ, (∇ Y U ) ⊥ = − ∇ Y ((∇ X ξ) ⊥ ), U , in (10) completes the proof. 3 The F -divergence of (1,1)-tensors on D An orthonormal frame {e i , e a } 1≤i≤n, 1≤a≤p on D built either of single vectors or of (local) vector fields is said to be adapted (to F), whenever e i ∈ T F and e a ∈ N F. Following [2,14], define the F-divergence of (1,1)-tensors A k ξ and T r (A ξ ) by div F A k ξ = n i=1 (∇ F e i A k ξ ) e i , div F T r (A ξ ) = n i=1 (∇ F e i T r (A ξ )) e i .(12) Note that div F T 0 (A ξ ) = div F id T F = ∇ F e i (id T F e i ) − id T F (∇ F e i e i ) = 0. Remark 1. Given a (1, 1)-tensor S, they usually define the 1-form div F S by (div F S)(X) = n i=1 (∇ i S)(X), e i , and the vector field ∇ * F S (called the adjoint of the covariant F-derivative) by (∇ * F S)(X) = − n i=1 (∇ i S)(e i , X). Using a metric g on a manifold, we identify (1, 0)-tensors with (0, 1)-tensors. For simplicity, in the article we use the definition (12) and do not mention the ∇ * F -notation. For any X ∈ D and ξ ∈ N 1 F, define a linear operator R P X, ξ : T F → T F by R P X, ξ : V → (R P (V, X) ξ) ⊤ , V ∈ T F.Proposition 1. (a) The following formula is valid for any ξ ∈ N 1 F and X ∈ T F: div F A ξ , X = k<n (A k ξ X)(f k ) + f k k j=1 1 k − j + 1 (A j−1 ξ X)(τ k−j+1 (ξ)) + tr F (A k−j ξ R P (−A ξ ) j−1 X, ξ ) .(13) (b) For f j = (−1) j σ r−j and 0 < r < n, (13) gives the following: div F T r (A ξ ), X = r j=1 tr F T r−j (A ξ ) R P (−A ξ ) j−1 X, ξ .(14) Proof. (a) We will calculate at a point x ∈ M . One may assume ∇ e i ξ| x ∈ T x F for all i. Decomposing A k ξ = A ξ A k−1 ξ for k ≥ 1, we get at a point x, div F A k ξ = A ξ div F A k−1 ξ + n i=1 (∇ F e i A ξ )A k−1 ξ e i .(15) Since (15) is tensorial, it is valid for any point of M . Using (9), we find for X ∈ T x F, n i=1 (∇ F e i A ξ )A k−1 ξ e i , X = n i=1 A k−1 ξ e i , (∇ F e i A ξ )X = n i=1 A k−1 ξ e i , (∇ F X A ξ ) e i + R P (X, e i ) ξ = tr F (A k−1 ξ ∇ F X A ξ ) + tr F (A k−1 ξ R P X, ξ ). For X ∈ T x F and for k ≥ 1, (15) gives us div F A k ξ , X = − A ξ div F A k−1 ξ , X + tr F (A k−1 ξ ∇ F X A ξ ) + tr F (A k−1 ξ R P X, ξ ). The above and the last identity of Lemma 1, yield the inductive formula div F A k ξ , X = div F A k−1 ξ , −A ξ X + 1 k X(τ k (ξ)) + tr F (A k−1 ξ R P X, ξ ).(16) By induction, from (16) we obtain the following for k ≥ 1: div F A k ξ , X = k j=1 1 k−j+1 (A j−1 ξ X)(τ k−j+1 (ξ)) + tr F (A k−j ξ R P (−A ξ ) j−1 X, ξ ) .(17) For the (1,1)-tensor A ξ , see (6), we get div F A ξ , X = k<n ∇ F f k , A k ξ X + f k div F A k ξ , X . Thus, (13) follows from (17). (b) By inductive definition (4), we obtain the following for r > 0, compare with (15): div F T r (A ξ ) = ∇ F σ r (ξ) − A ξ div F T r−1 (A ξ ) − n i=1 (∇ F e i A ξ )(T r−1 (A ξ )e i ). Note that the (1,1)-tensor ∇ F e i A ξ is self-adjoint. Then, using the third identity of Lemma 1, and Codazzi type equation (9), we get, compare with (16), div F T r (A ξ ), X = div F T r−1 (A ξ ), (−A ξ )X + tr F (T r−1 (A ξ ) R P X, ξ ).(18) From (18) by induction we obtain (14). Example 1. (a) For k = 1, (17) reads as div F A ξ , X = X(τ 1 (ξ)) + tr F R P X,ξ . (b) Let the distribution T F be P -curvature invariant, that is R P (X, Y )V ∈ T F (X, Y, V ∈ T F).(19) Then, in view of (7), equation (14) implies that div F T r (A ξ ) = 0 for every r and ξ. Note that (19) is satisfied, if T F is auto-parallel, i.e., ∇ X Y ∈ Γ(T F) for all X, Y ∈ Γ(T F). A sufficient condition for (19) is the following equality: R P (X, Y )Z = c ( Y, Z X − X, Z Y ), X, Y, Z ∈ D,(20) for some constant c ∈ R. Remark 2. Let F be a foliation of a Riemannian manifold (M, g) and D = T M ; thus, R P = R. Using the symmetry R(X, Y )U, V = R(U, V )X, Y of the curvature tensor, we simplify (18) to the following form: div F T r (A ξ ) = −A ξ div F T r−1 (A ξ ) + n i=1 (R(ξ, T r−1 (A ξ )e i )e i ) ⊤ , where ⊤ denotes the orthogonal projection on the vector subbundle T F, see [ 4 The F -divergence of vector fields on D For any local vector field ξ in N 1 F, put Z ξ = (∇ ξ ξ) ⊤ . The F-divergence of a vector field X on M is defined by div F X = n i=1 ∇ e i X, e i , where {e i } is a local orthonormal frame of T F. Let N 1 F ⊂ N F be a subbundle of unit vectors orthogonal to T F in D, (N 1 F) x its fiber (a unit sphere) at a point x ∈ M , d ω ⊥ x the volume form on (N 1 F) x with the induced metric. Applying Lemma 3 given below, and (13), we generalize [14, Proposition 2.8]. Proposition 2. The divergence of the vector field (N 1 F )x A ξ Z ξ d ω ⊥ x at a point x ∈ M is div F (N 1 F )x A ξ Z ξ d ω ⊥ x = (N 1 F )x div F A ξ , Z ξ + tr F (A ξ R P ξ,ξ ) + k<n f k τ k+2 (ξ) − f k k + 1 ξ(τ k+1 (ξ)) + p a=1 A ξ (∇ ea ξ) ⊤ , ∇ ξ e a d ω ⊥ x , where the underlined term is given in (13) with X = Z ξ . In particular, the divergence of the vector field ( N 1 F )x T r (A ξ )Z ξ d ω ⊥ x at a point x ∈ M is div F (N 1 F )x T r (A ξ )Z ξ d ω ⊥ x = (N 1 F )x div F T r (A ξ ), Z ξ − ξ(σ r+1 (ξ)) − (r + 2) σ r+2 (ξ) + σ 1 (ξ) σ r+1 (ξ) + tr F (T r (A ξ )R P ξ, ξ ) + p a=1 T r (A ξ )(∇ ea ξ) ⊤ , ∇ ξ e a d ω ⊥ x ,(21) where the underlined term is given by (14) with X = Z ξ . Proof. Assume that ∇ X ξ ∈ T x F for any X ∈ T x M at a point x ∈ M , and calculate div F (N 1 F )x A ξ Z ξ d ω ⊥ x = n i=1 (N 1 F )x ∇ e i (A ξ Z ξ ), e i d ω ⊥ x = (N 1 F )x div F A ξ , Z ξ d ω ⊥ x + n i=1 (N 1 F )x ∇ F e i Z ξ , A ξ e i d ω ⊥ x .(22) By Lemma 3 in what follows, we find the integrand (22), n i=1 ∇ F e i Z ξ , A ξ e i ink<n f k τ k+2 (ξ) + tr F (A ξ R P ξ, ξ ) − tr F (A ξ (∇ F ξ A ξ )) + p a=1 A ξ (∇ ea ξ) ⊤ , ∇ ξ e a .(23) We transform the term tr F (A ξ ∇ F ξ A ξ ) in (23), using the definition A ξ = k<n f k A k ξ , see (6), and the last identity of Lemma 1, as tr F (A ξ ∇ F ξ A ξ ) = k<n f k k + 1 ξ(τ k+1 (ξ)). Finally, we conclude that n i=1 ∇ F e i Z ξ , A ξ e i = tr F (A ξ R P ξ, ξ ) + k<n f k τ k+2 (ξ) − f k k + 1 ξ(τ k+1 (ξ)) + p a=1 A ξ (∇ ea ξ) ⊤ , ∇ ξ e a . Since the result is tensorial, it is valid for any x ∈ M . The proof of (21) is similar. Remark 3. The integrals over (N 1 F) x when p > 1 and x ∈ M can be found explicitly. To show this, denote λ = (λ 1 , . . . , λ p ) and y = (y 1 , . . . , y p ). Then, see e.g., [14], I λ := y =1 y λ d ω p−1 = 2 Γ p/2 + (1/2) 1≤a≤p λ a 1≤a≤p 1 2 (1 + (−1) λa ) Γ 1 + λ a 2 , where y λ = a≤p y λa a , and Γ is the Gamma function. For example, I 0,...,0 = 2 π p/2 Γ(p/2) = vol(S p−1 (1)), I 2λ 1 ,0,...,0 = 2 π p−1 2 Γ(1/2 + λ 1 ) Γ(p/2 + λ 1 ) . The following lemma generalizes [2, Lemma 3.1], see also [14,Lemma 2.7]. Lemma 3. Let {e i , e a } be an adapted orthonormal frame of D such that • ∇ F X e i = 0 (1 ≤ i ≤ n) and ∇ X e a , e b = 0 (1 ≤ a, b ≤ p) for any vector X ∈ T x M ; • ∇ P µ e i = 0 (1 ≤ i ≤ n) for any vector µ ∈ D x at a point x ∈ M . Then for any unit vector ξ ∈ (N 1 F) x we have ∇ F e i Z ξ , e j = A 2 ξ e i , e j + R P (e i , ξ)ξ, e j − (∇ F ξ A ξ )e i , e j + p a=1 ∇ ξ e a , e i ∇ ea ξ, e j . Proof. Taking covariant derivative of Z ξ , e j = − ξ, ∇ ξ e j with respect to e i , we find − Z ξ , ∇ e i e j = ∇ e i Z ξ , e j + ∇ e i ξ, P ∇ ξ e j + ξ, ∇ e i P ∇ ξ e j .(24) For a foliation F, we obtain A 2 ξ e i , e j = ∇ ξ ξ, ∇ e i e j = Z ξ , ∇ e i e j + ξ, ∇ ξ P ∇ e i e j . Therefore, we calculate at the point x ∈ M : A 2 ξ e i , e j + R P (e i , ξ) ξ, e j − (∇ F ξ A ξ )e i , e j = A 2 ξ e i , e j − R P (e i , ξ) e j , ξ + ξ ∇ e i ξ, e j = A 2 ξ e i , e j − Z ξ , ∇ e i e j − ∇ e i P ∇ ξ e j , ξ + ∇ [e i ,ξ] e j , ξ .(25) Using (24) ∇ ξ e k , e i ∇ e k ξ, e j . From the above, the claim follows. Main results Here, we prove integral formulas for a foliated sub-Riemannian manifold (M, D, F, g). The idea is to find the divergence of a suitable vector field and apply the Divergence Theorem and Remark 4 in what follows to (M, g) or to a compact leaf with the induced metric. We will integrate over the normal sphere bundle N 1 F ⊂ N F of F with the induced metric. N 1 F f (ξ) d ω ⊥ = M (N 1 F )x f (ξ) d ω ⊥ x d vol g , where d ω ⊥ is the volume form on N 1 F and d vol g is the volume form of (M, g). N 1 F | L div F A ξ , Z ξ + k<n f k τ k+2 (ξ) − f k k + 1 ξ(τ k+1 (ξ)) + tr F (A ξ R P ξ,ξ ) + p a=1 A ξ (∇ ea ξ) ⊤ , ∇ ξ e a d ω ⊥ L = 0,(26) where d ω ⊥ L is the volume form on N 1 F| L , and the underlined term is given by (13) with X = Z ξ . For f j = (−1) j σ r−j , (26) gives N 1 F | L div F T r (A ξ ), Z ξ − ξ(σ r+1 (ξ)) − (r + 2) σ r+2 (ξ) + σ 1 (ξ) σ r+1 (ξ) + tr F (T r (A ξ )R P ξ, ξ ) + p a=1 T r (A ξ )(∇ ea ξ) ⊤ , ∇ ξ e a d ω ⊥ L = 0,(27) where the underlined term is given by (14) with X = Z ξ . For any vector field X in the distribution T F, we have div X = div F X − X, H ⊥ − X, H ,(28) where H ⊥ is the mean curvature vector of N F, and H is the mean curvature vector field of D. Recall that D is a harmonic distribution if H = 0. There are topological restrictions for the existence of a Riemannian metric on closed manifold, for which a given distribution is harmonic, see [22]. Using Proposition 2, we get the following theorem with integral formulas along a closed foliated manifold, which generalizes [14,Theorem 3.3]. N 1 F div F A ξ , Z ξ + k<n f k τ k+2 (ξ) + τ k+1 (ξ) k + 1 (ξ(f k ) − f k τ 1 (ξ)) + tr F (A ξ R P ξ,ξ ) − A ξ Z ξ , H ⊥ + p a=1 A ξ (∇ ea ξ) ⊤ , ∇ ξ e a d ω ⊥ = 0,(29) where the underlined term is given by (13) with X = Z ξ . For f j = (−1) j σ r−j , (29) gives N 1 F div F T r (A ξ ), Z ξ − (r + 2) σ r+2 (ξ) − T r (A ξ )Z ξ , H ⊥ + tr F (T r (A ξ ) R P ξ, ξ ) + p a=1 T r (A ξ )(∇ ea ξ) ⊤ , ∇ ξ e a d ω ⊥ = 0,(30) where the underlined term is given by (14) with X = Z ξ . Proof. Using 1≤a≤p (∇ ea e a ) ⊤ = P (H ⊥ ) and (28) with X(x) = (N 1 F )x A ξ Z ξ d ω ⊥ x and our assumption H = 0, we get div (N 1 F )x A ξ Z ξ d ω ⊥ x = div F (N 1 F )x A ξ Z ξ d ω ⊥ x + p a=1 (N 1 F )x ∇ ea (A ξ Z ξ ), e a d ω ⊥ x = div F (N 1 F )x A ξ Z ξ d ω ⊥ x − (N 1 F )x A ξ Z ξ , H ⊥ d ω ⊥ x .(31) Assuming ∇ X ξ ⊥ N F for any X ∈ T x M and ξ ∈ (N 1 F) x at some point x ∈ M , we get div ξ = div F ξ = −τ 1 (ξ). Using this equality, we also find div f k k + 1 τ k+1 (ξ) ξ = f k k + 1 div τ k+1 (ξ) ξ + 1 k + 1 τ k+1 (ξ) ξ(f k ) = f k k + 1 τ k+1 (ξ) div ξ + ξ(τ k+1 (ξ)) + 1 k + 1 τ k+1 (ξ) ξ(f k ) = f k k + 1 ξ(τ k+1 (ξ)) + 1 k + 1 τ k+1 (ξ)(ξ(f k ) − f k τ 1 (ξ)). Then at the point x ∈ M we obtain the equality for tensors div (N 1 F )x A ξ Z ξ + k<n f k τ k+1 (ξ) k + 1 ξ d ω ⊥ x = (N 1 F )x div F A ξ , Z ξ + k<n f k τ k+2 (ξ) + 1 k + 1 τ k+1 (ξ) (ξ(f k ) − f k τ 1 (ξ)) + tr F (A ξ R P ξ, ξ ) − A ξ Z ξ , H ⊥ + p a=1 A ξ (∇ ea ξ) ⊤ , ∇ ξ e a d ω ⊥ x , see (13) for the underlined term. Applying the Divergence Theorem yields (29). For the Newton transformations of A ξ , similarly to (31), we have div (N 1 F )x T r (A ξ )Z ξ d ω ⊥ x = div F (N 1 F )x (T r (A ξ )Z ξ − T r (A ξ )Z ξ , H ⊥ ) d ω ⊥ x . Observe that div(σ r+1 (ξ) ξ) = −σ 1 (ξ) σ r+1 (ξ) + ξ(σ r+1 (ξ)) for all ξ ∈ N 1 F. Then at a point x ∈ M we obtain div (N 1 F )x T r (A ξ )Z ξ + σ r+1 (ξ) ξ d ω ⊥ x = (N 1 F )x div F T r (A ξ ), Z ξ − (r + 2) σ r+2 (ξ) − T r (A ξ )Z ξ , H ⊥ + tr F (T r (A ξ ) R P ξ, ξ ) + p a=1 T r (A ξ )(∇ ea ξ) ⊤ , ∇ ξ e a d ω ⊥ x , see (21) for the underlined term. Using the Divergence Theorem yields (30). Applications of main results Here, we apply results of Section 5 to foliations with small k or r, with restrictions on the curvature and extrinsic geometry and to codimension-one foliations. Let (M, D, F, g) be a foliated sub-Riemannian manifold with D = T F ⊕ N F, then (i) N F will be called P -auto-parallel, if (∇ X Y ) ⊤ = 0 for all X, Y ∈ N F, (ii) F will be called P -harmonic, if σ 1 (ξ) = 0 for all ξ ∈ N 1 F, (iii) F will be called P -totally umbilical, if A ξ = (σ 1 (ξ)/n) id T F , ξ ∈ N 1 F.(32) Obviously, P -auto-parallel, P -harmonic and P -totally umbilical distributions are auto-parallel, harmonic and totally umbilical, respectively, but the opposite is not true. The mixed scalar P -curvature S P mix -the simplest curvature invariant of a foliated sub-Riemannian manifold -is defined as an averaged sum of sectional P -curvatures of mixed planes (i.e., 2-planes that non-trivially intersect with each of the distributions T F and N F); for an adapted orthonormal frame, we have S P mix = p a=1 tr F R P ea,ea = p a=1 n i=1 R P (e i , e a )e a , e i . Let h ⊥ , T ⊥ : N F × N F → (N F) ⊥ and H ⊥ = tr g h ⊥ be the second fundamental form, the integrability tensor and the mean curvature vector field of N F, and similarly for T F; in our case of a foliation, we have T = 0. The following corollary generalizes (3). Corollary 1. From (29) for A ξ = id T F we get the following integral formula over M : M S P mix + P • h 2 + P • h ⊥ 2 − P H 2 − P H ⊥ 2 − P • T ⊥ 2 d vol g = 0.(33) Proof. From (29) with A ξ = id T F , we get N 1 F τ 2 1 (ξ) − τ 2 (ξ) − tr F R P ξ,ξ + Z ξ , H ⊥ − p a=1 (∇ ea ξ) ⊤ , ∇ ξ e a d ω ⊥ = 0.(34) Note that τ 2 1 (ξ) − τ 2 (ξ) = 2 σ 2 (ξ). The same (34) we get from (30) with r = 0. Let ξ = p a=1 y a e a , where y a ∈ R, be any unit vector field in N F. For a 2-homogeneous on ξ function f (ξ, ξ) = p a,b=1 f (e a , e b ) y a y b , as is integrand of (34), we have (N 1 F )x f (ξ, ξ) d ω ⊥ x =Ĩ 2 p a=1 f (e a , e a ), whereĨ 2 = (N 1 F )x y 2 a d ω ⊥ x = 2π p−1 2 Γ(3/2) Γ(p/2 + 1) , see Remark 3. Applying this to the terms of (34), we find (N 1 F )x τ 2 (ξ) − τ 2 1 (ξ) d ω ⊥ x =Ĩ 2 ( P • h 2 − P H 2 ), (N 1 F )x Z ξ , H ⊥ d ω ⊥ x =Ĩ 2 P H ⊥ 2 , (N 1 F )x tr F R P ξ,ξ d ω ⊥ x =Ĩ 2 p a=1 tr F R P ea,ea =Ĩ 2 S P mix , (N 1 F )x p a=1 (∇ ea ξ) ⊤ , ∇ ξ e a d ω ⊥ x =Ĩ 2 ( P • h ⊥ 2 − P • T ⊥ 2 ). Thus, (34) reduces to follows (33). Example 2. We shall look at first members of series (29). For A ξ = A 2 ξ , we get N 1 F div F A 2 ξ , Z ξ + τ 4 (ξ) − 1 3 τ 3 (ξ) τ 1 (ξ) + tr F (A 2 ξ R P ξ,ξ ) − A 2 ξ Z ξ , H ⊥ + p a=1 A 2 ξ (∇ ea ξ) ⊤ , ∇ ξ e a d ω ⊥ = 0.(35) Next, we look at first members of (30). For r = 2, (30) gives N 1 F 4 σ 4 (ξ) + T 2 (A ξ )Z ξ , H ⊥ − tr F T 2 (A ξ )R P ξ, ξ + T 1 (A ξ )R P Z ξ , ξ − R P A ξ Z ξ , ξ − p a=1 T 2 (A ξ )(∇ ea ξ) ⊤ , ∇ ξ e a d ω ⊥ = 0.(36) If the distribution N F is a P -auto-parallel, then (35) and (36) shorten to the formulas N 1 F τ 4 (ξ) − 1 3 τ 3 (ξ) τ 1 (ξ) + tr F (A 2 ξ R P ξ,ξ ) d ω ⊥ = 0,(37)N 1 F σ 4 (ξ) − 1 4 tr F (T 2 (A ξ ) R P ξ, ξ ) d ω ⊥ = 0.(38) Similarly to (34), one can transform (35)-(38) to the integrals over M . Corollary 2. Let N F be P -auto-parallel, H = 0 and S P mix ≥ 0. Then F has no compact P -harmonic leaves. Proof. We simplify (26) for A ξ = id T F (the same gives (27) for r = 0) as N 1 F | L τ 2 (ξ) − ξ(τ 1 (ξ)) + tr F R P ξ,ξ + p a=1 (∇ ea ξ) ⊤ , ∇ ξ e a d ω ⊥ L = 0. Thus, the claim follows. Now, we apply results of Section 5 to foliated sub-Riemannian manifolds with restrictions on the extrinsic geometry of F. By (33), we easily obtain the following. Corollary 3. A closed sub-Riemannian manifold (M, D, g) with a harmonic distribution D does not admit (i) P -harmonic foliations F with P -auto-parallel N F and condition S P mix > 0. (ii) P -totally umbilical foliations F with P -auto-parallel N F and condition S P mix < 0. From Theorem 2 we obtain the following. N 1 F τ k+2 (ξ) − 1 k + 1 τ k+1 (ξ) τ 1 (ξ) + tr F (A k ξ R P ξ, ξ ) d ω ⊥ = 0,(39)N 1 F (r + 2) σ r+2 (ξ) − tr F (T r (A ξ ) R P ξ, ξ ) d ω ⊥ = 0.(40) The total k-th mean curvature of F are defined by σ k (F) = N 1 F σ k (ξ) d ω ⊥ , τ k (F) = N 1 F τ k (ξ) d ω ⊥ . Note that σ 2s+1 (F) = τ 2s+1 (F) = 0. We have, see Remark 3, Suppose that N F is P -auto-parallel, D is harmonic, and condition (20) is satisfied. Then σ r (F) depends on r, n, p, c and the volume of (M, g) only, i.e., the following integral formula is valid: σ r (F) = 2 π p/2 Γ(p/2) n/2 r/2 c r/2 Vol(M, g), n and r even, 0, either n or r odd. σ 0 (F) = τ 0 (F) = n 2π p/2 Γ(p/2) Vol(M, g). Moreover, if F is P -harmonic, then τ k (F) = 2 π p/2 Γ(p/2) (−c) k/2 Vol(M, g), n and k even, 0, either n or k odd. Proof. By (20), R P ξ,ξ = c id F , hence by Lemma 1, tr F (A k ξ R P ξ, ξ ) = c τ k (ξ), tr F (T r (A ξ ) R P ξ, ξ ) = c (n − r) σ r (ξ). By (39) and conditions, we get τ k+2 (F) = −c τ k (F). By (40), we get σ r+2 (F) = c(n − r) r + 2 σ r (F). Then (by induction), from the above we obtain (41) and (42). We get similar formulas when (M, D, g) is a P -Einstein manifold, i.e., has the property tr F R P X,ξ = C X, ξ , X ∈ D, ξ ∈ N 1 F (43) for some C ∈ R, and a foliation F is P -totally umbilical, see (32). Note that for D = T M , condition (43) is satisfied for Einstein manifolds (M, g). The following corollary of (30) generalizes result in [14,Example 4.4] (see also [15,Corollary 2] and [2, Section 4.2] for codimension-one foliated manifolds). Corollary 6. Let (M, D, g) be a closed sub-Riemannian manifold with D = T F ⊕ N F. Suppose that N F is P -auto-parallel, D is harmonic, and conditions (32) and (43) are satisfied. Then σ r (F) depends on r, n, p, C and the volume of (M, g) only, i.e., the following integral formula is valid: σ r (F) = (C/n) r/2 n/2 r/2 Vol(M, g), n, r even, 0, r odd. Proof. In this case, T r (A ξ ) has the form T r (A ξ ) = n − r n σ r (ξ) id T F . For a P -Einstein manifold, by (43) we get tr F R P Z ξ ,ξ = 0, tr F R P ξ,ξ = C, where C ≥ 0 by Corollary 3. Thus, (30) becomes σ r+2 (F) = C n · n − r r + 2 σ r (F). Using induction similarly to Corollary 5, yields (44). Finally, we will briefly discuss the case of a codimension-one transversally orientable foliation F in D with a unit normal N , i.e., D = T F ⊕ span(N ), on a sub-Riemannian manifold with a harmonic distribution D, see also [15]. Put Z = ∇ P N N, R P X = R P X,N , A = A N , σ r+2 = σ r+2 (N ). The leafwise divergence of T r (A) (r > 0) satisfies, compare with (14), div F T r (A), X = 1≤j≤r tr F T r−j (A) R P (−A) j−1 X for any vector field X tangent to F. This yields the following corollaries. where Ric P N,N = tr F R P N = 1≤i≤n R P (e i , N )N, e i is the Ricci P -curvature is the Ndirection. Thus, a codimension-one F with τ 1 = const and Ric P N,N > 0 has no compact leaves. (b) For r = 0, by Corollary 8, we obtain the following generalization of (1): M 2 σ 2 − Ric P N,N d vol g = 0.(45) Let dim F = 1. The Gaussian P -curvature of D is a function on M defined by K P = R P (X, N, N, X), where X is a unit (local) vector tangent to F. Then σ 2 = 0 and (45) reduces to the zero integral of the Gaussian P -curvature: M K P d vol g = 0. Conclusion We suggest that integral formulas (26), (27), (29), (30), and, in particular, (33), are a very good tool for understanding the geometry of (sub-)Riemannian manifolds equipped with foliations or distributions. We delegate the following for further study. 1. Our integral formulas can be extended for the case of D = D 1 ⊕ D 2 -the sum of two smooth distributions, see [17] in relation to (33) and [14] for D = T M . In other words, we take a non-integrable distribution D 1 instead of a foliation F and D 2 instead of N F. This naturally appears when D ⊂ T M is a hyperdistribution, whose shape operator has an eigenvalue of constant multiplicity, and D 1 is the corresponding eigen-distribution. 2. Our integral formulas can be extended for foliations and distributions defined outside of a "singularity set" Σ (a finite union of pairwise disjoint closed submanifolds of codimension greater than 2 of a closed manifold M ) under additional assumption of convergence of certain integrals. Then, instead of the Divergence theorem, we apply the following, see [11,Lemma 2]: if X is a vector field on M \ Σ such that M X 2 d vol g < ∞, then M div X d vol g = 0. 3. Mathematicians have shown a specific interest in manifolds equipped with several distributions, e.g., webs composed of different foliations and multiply warped products. In [16], we introduced the mixed scalar curvature for k > 2 pairwise orthogonal complementary distributions on (M, g) and proved the integral formula similar to (3) and (33) for this case. This study was continued in [18] for the case of a linear connection instead of the Levi-Civita connection. We suggest that our integral formulas can be extended for sub-Riemannian manifolds equipped with several orthogonal foliations (in D) and certainly defined S P mix . 4. Finally, one can extend our integral formulas for holomorphic foliations of complex sub-Riemannian manifolds, see [23] for the case of D = T M and (3). and the following equalities at x ∈ M :P ∇ e i ξ = n j=1 ∇ e i ξ, e j e j , P ∇ ξ e i = p a=1 ∇ ξ e i , e a e a , A 2 ξ e i , e j = − n k=1 ∇ e i ξ, e k ∇ e k e j , ξ ,we simplify the last line in(25)as∇ e i Z ξ , e j − n k=1 Remark 4 . 4For any function f : N 1 F → R, we have the Fubini formula, see [5, Note 7.1.1.1], The next theorem with integral formulas along a compact leaf of a foliation follows from Proposition 2 and generalizes [14, Theorem 3.1].Theorem 1. Let (M, D, F, g) be a foliated sub-Riemannian manifold with D = T F ⊕ N F. Then for any compact leaf L of F we have Theorem 2 . 2Let (M, D, F, g) be a foliated closed sub-Riemannian manifold with D = T F ⊕ N F and a harmonic distribution D. Then the following integral formula is valid: Corollary 4 . 4Let (M, D, g) be a closed sub-Riemannian manifold with D = T F ⊕ N F, a harmonic distribution D and a P -auto-parallel distribution N F. Then the following integral formulas are valid: Corollary 5 . 5Let (M, D, g) be a closed sub-Riemannian manifold with D = T F ⊕ N F. Corollary 7 . 7For any compact leaf L of a codimension-one foliated sub-Riemannian manifold, (27) reads asL div F T r (A), Z − N (σ r+1 ) + σ 1 σ r+1 − (r + 2)σ r+2 + tr F (T r (A)R P N ) + T r (A)Z, Z d vol L = 0.Corollary 8. For a closed sub-Riemannian manifold (M, D, g) with D = T F ⊕ span(N ) and a harmonic distribution D, the following integral formula is valid: M div F T r (A), Z − (r + 2) σ r+2 + tr F (T r (A)R P N ) d vol g = 0. The following result generalizes [2, Lemma 2.2], see also[14, Lemmas 2.4 and 2.5]. Riemannian manifolds, see also[3,19] (and [15, Corollary 1], [2, Section 4.1] for codimension-one foliated manifolds).The following corollary of Theorem 1 generalizes [7, Theorem 1.1] and [14, Corollary 4.3] on foliated K Andrzejewski, V Rovenski, P Walczak, Integral formulas in foliations theory. Springer72Springer Proc. in Math. and StatisticsK. Andrzejewski, V. Rovenski and P. Walczak, Integral formulas in foliations theory, 73-82, in Geometry and Its Applications, Springer Proc. in Math. and Statistics, 72, Springer, 2014. The Newton transformation and new integral formulae for foliated manifolds. K Andrzejewski, P Walczak, Ann. Global Anal. Geom. 372K. Andrzejewski and P. Walczak, The Newton transformation and new integral formulae for foliated manifolds, Ann. Global Anal. Geom. 37, No. 2 (2010), 103-111. Extrinsic curvatures of distributions of arbitrary codimension. K Andrzejewski, P Walczak, J. Geom. Phys. 605K. Andrzejewski, and P. Walczak, Extrinsic curvatures of distributions of arbitrary codi- mension, J. Geom. Phys., 2010, 60, No. 5, 708-713. Foliations and geometric structures. A Bejancu, H Farran, Springer-VerlagA. Bejancu and H. Farran, Foliations and geometric structures. Springer-Verlag, 2006. A Panoramic View of Riemannian Geometry. M Berger, Springer-VerlagM. Berger, A Panoramic View of Riemannian Geometry. Springer-Verlag, 2002. F Baudoin, E Grong, G Molino, L Rizzi, arXiv:1909.03532Comparison theorems on H-type sub-Riemannian manifolds. math.DGF. Baudoin, E. Grong, G. Molino, and L. Rizzi, Comparison theorems on H-type sub- Riemannian manifolds, arXiv:1909.03532 [math.DG]. Intégrales de courbure sur des variétés feuilletées. F Brito, R Langevin, H Rosenberg, J. Diff. Geom. 16F. Brito, R. Langevin and H. Rosenberg, Intégrales de courbure sur des variétés feuilletées, J. Diff. Geom., 1981, 16, 19-50. Total extrinsic curvature of certain distributions on closed spaces of constant curvature. F Brito, A Naveira, Ann. Global Anal. Geom. 18F. Brito and A. Naveira, Total extrinsic curvature of certain distributions on closed spaces of constant curvature, Ann. Global Anal. Geom., 2000, 18, 371-383. Pseudo-Riemannian almost product manifolds and submersions. A Gray, J. Math. Mech. 167A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech., 16, No. 7 (1967) 715-737. Riemannian geometry and geometric analysis. J Jost, SpringerUniversitext7th ed.J. Jost, Riemannian geometry and geometric analysis, 7th ed., Universitext, Springer, 2017 New integral formulae for two complementary orthogonal distributions on Riemannian manifolds. M Lużyńczyk, P Walczak, Ann. Glob. Anal. Geom. 48M. Lużyńczyk and P. Walczak, New integral formulae for two complementary orthogonal distributions on Riemannian manifolds, Ann. Glob. Anal. Geom. 48 (2015), 195-209. Sur la courboure moyenne des variétés intégrales d'uneéquation de Pfaff ω = 0. G Reeb, C. R. Acad. Sci. Paris. 231G. Reeb, Sur la courboure moyenne des variétés intégrales d'uneéquation de Pfaff ω = 0, C. R. Acad. Sci. Paris 231 (1950), 101-102. V Rovenski, Foliations on Riemannian Manifolds and Submanifolds. BirkhäuserV. Rovenski, Foliations on Riemannian Manifolds and Submanifolds, Birkhäuser, 1998. Integral formulae for a Riemannian manifold with two orthogonal distributions. V Rovenski, Central European J. Math. 93V. Rovenski, Integral formulae for a Riemannian manifold with two orthogonal distri- butions, Central European J. Math. 9, No. 3 (2011), 558-577. Integral formulas for a foliation with a unit normal vector field. V Rovenski, 10.3390/math9151764Mathematics. 2021151764V. Rovenski, Integral formulas for a foliation with a unit normal vector field, Mathe- matics 2021, 9(15), 1764. https://doi.org/10.3390/math9151764 Integral formulas for a Riemannian manifold with several orthogonal complementary distributions. V Rovenski, Global J. of Advanced Research on Classical and Modern Geometries. 10V. Rovenski, Integral formulas for a Riemannian manifold with several orthogonal com- plementary distributions. Global J. of Advanced Research on Classical and Modern Ge- ometries, 10, Issue 1 (2021) 32-42. On a Riemannian manifold with two orthogonal distributions, 8 pages. V Rovenski, Proceedings of the Contemporary Mathematics in. the Contemporary Mathematics inKielce; Poland2021V. Rovenski, On a Riemannian manifold with two orthogonal distributions, 8 pages. In: Proceedings of the Contemporary Mathematics in Kielce 2020, Poland, 2021 On a metric affine manifold with several orthogonal complementary distributions. V Rovenski, S E Stepanov, Mathematics. 93229V. Rovenski and S. E. Stepanov, On a metric affine manifold with several orthogonal complementary distributions. Mathematics, 9(3), 229 (2021) Integral fomulae for foliations on Riemannian manifolds. V Rovenski, P Walczak, Proc. of 10th Int. 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[ "Gravitational Analogues, Geometric Effects and Gravitomagnetic Charge *", "Gravitational Analogues, Geometric Effects and Gravitomagnetic Charge *" ]
[ "Jian Qi Shen \nZhejiang Institute of Modern Physics\nDepartment of Physics\nZhejiang University\n310027Hangzhou, Peoples Republic of China\n" ]
[ "Zhejiang Institute of Modern Physics\nDepartment of Physics\nZhejiang University\n310027Hangzhou, Peoples Republic of China" ]
[]
This essay discusses some geometric effects associated with gravitomagnetic fields and gravitomagnetic charge as well as the gravity theory of the latter. Gravitomagnetic charge is the duality of gravitoelectric charge ( mass ) and is therefore also termed the dual mass which represents the topological property of gravitation. The field equation of gravitomagnetic matter is suggested and a static spherically symmetric solution of this equation is offered. A possible explanation of the anomalous acceleration acting on Pioneer spacecrafts are briefly proposed.
10.1023/a:1020082903104
[ "https://export.arxiv.org/pdf/gr-qc/0301067v2.pdf" ]
118,247,350
gr-qc/0301067
3caa17f158aef0432d5e48dc748041307673f13b
Gravitational Analogues, Geometric Effects and Gravitomagnetic Charge * 15 Feb 2003 Jian Qi Shen Zhejiang Institute of Modern Physics Department of Physics Zhejiang University 310027Hangzhou, Peoples Republic of China Gravitational Analogues, Geometric Effects and Gravitomagnetic Charge * 15 Feb 2003(March 3, 2022)Gravitational analoguegravitomagnetic chargefield equation This essay discusses some geometric effects associated with gravitomagnetic fields and gravitomagnetic charge as well as the gravity theory of the latter. Gravitomagnetic charge is the duality of gravitoelectric charge ( mass ) and is therefore also termed the dual mass which represents the topological property of gravitation. The field equation of gravitomagnetic matter is suggested and a static spherically symmetric solution of this equation is offered. A possible explanation of the anomalous acceleration acting on Pioneer spacecrafts are briefly proposed. I. INTRODUCTION Considering the following gravitational analogues of electromagnetic phenomena is of physical interest: (1) in electrodynamics a charged particle is acted upon by the Lorentz magnetic force and, in the similar fashion, a particle is also acted upon by the gravitational Lorentz force in weak-gravity theory [1,2]. According to the principle of equivalence, further analysis shows that in the non-inertial rotating reference frame, this gravitational Lorentz force is just the fictitious Coriolis force; (2) there exists Aharonov-Bohm effect in electrodynamics [3], accordingly, the so-called gravitational Aharonov-Bohm effect, i.e., the gravitational analogue of Aharonov-Bohm effect also exists in the theory of gravitation, which is now termed the Aharonov-Carmi effect [4][5][6]; (3) a particle with intrinsic spin possesses a gravitomagnetic moment [7] of such magnitude that it equals the spin of this particle. The interaction of spinning gravitomagnetic moment with the gravitomagnetic field is called spin-gravity coupling, which is similar to the interaction between spinning magnetic moment and magnetic fields in electrodynamics; (4) in the rotating reference frame, the rotating frequency relative to the fixing frame may be considered the effective gravitomagnetic field strength that is independent of the Newtonian gravitational constant, G, in accordance with the principle of equivalence. This, therefore, means that the nature of spin-rotation coupling [8,9] is the interaction of spinning gravitomagnetic moment with gravitomagnetic fields; (5) it is well known that geometric phase reflecting the global and topological properties of evolution of the quantum systems [10,11] appears in systems whose Hamiltonian is time-dependent or possessing evolution parameters. Apparently, the geometric phase in the Aharonov-Bohm effect and Aharonov-Carmi effect results from the presence of the evolution parameter in the Hamiltonian. We suggested another geometric phase [1] that exists in the spin-rotation coupling system where the rotating frequency of the rotating frame is time-dependent. Investigation of this geometric phase is believed to be a potential application to the Earth , s time-dependent rotating frequency ( frequency fluctuations ), namely, by measuring the geometric phase difference of spin polarized vertically down and up in the neutron-gravity interferometry experiment, one may obtain the information concerning the variation of the Earth , s rotation. For the present, it is possible to investigate quantum mechanics in weak-gravitational fields [12,13], with the development of detecting and measuring technology, particularly laser-interferometer technology, low-temperature technology, electronic technology and so on. These investigations enable physicists to test validity or universality of fundamental laws and principles of general relativity. It should be noted that the Aharonov-Carmi effect and the geometric phase factor in the time-dependent spin-rotation coupling reflect the aspects of geometric properties in gravity. Both of them are related to the gravitomagnetic fields. In this essay, the author discusses another geometric or topological aspect of the gravity, i.e., the gravitomagnetic charge that is the gravitational analogue of magnetic monopole in electrodynamics. In electrodynamics, electric charge is a Noether charge while its dual charge ( magnetic charge ) is a topological charge, since the latter relates to the singularity of non-analytical vector potentials. Magnetic monopole [14] attracts attentions of many physicists in various fields such as gauge field theory, grand unified theory, particle physics and cosmology [15][16][17][18][19][20]. In the similar fashion, it is also interesting to consider the gravitomagnetic charge, which is the source of gravitomagnetic field just as mass ( gravitoelectric charge ) is the source of gravitoelectric field ( Newtonian gravitational field ). In this sense, gravitomagnetic charge is also termed dual mass. It should be noted that the concept of mass is of no significance for the gravitomagnetic charge and it is therefore of interest to investigate the relativistic dynamics and gravitational effects of this topological dual mass. The paper is organized as follows: The gravitational field equation of gravitomagnetic matter is derived in Sec. II, the form of field equation in the weak-field approximation is given in Sec. III and the exact static spherically symmetric solution is obtained in Sec. IV. In Sec. V, two related problems, i.e., the geometric phase factor possessed by a photon propagating in the gravitomagnetic field, and a potential interpretation, by means of the mechanism of gravitational Meissner effect, of the anomalous acceleration acting on Pioneer spacecrafts [21] are briefly proposed. In Sec. VI, the author concludes with some remarks. II. GRAVITATIONAL FIELD EQUATION OF GRAVITOMAGNETIC MATTER In order to obtain the gravitational field equation of gravitomagnetic charge, we should construct the dual Einstein , s tensor. By using the variational principle, we can obtain the following equation δ Ω √ −gRdΩ = Ω √ −gG µν δg µν dΩ (1) withR = g στR στ ,R υγ = g µδ (ǫ αβ µυ R γδαβ + ǫ αβ γδ R µυαβ ) (2) andG µν = ǫ λστ µ R νλστ − ǫ λστ ν R µλστ ,(3) where 1 ǫ λστ µ = g µν ǫ νλστ with ǫ νλστ being the completely antisymmetric Levi-Civita tensor, and the dual scalar curvatureR is assumed to be the Lagrangian density of the interaction of metric fields with gravitomagnetic matter. Since the dual Einstein , s tensor,G µν , is an antisymmetric tensor, we construct the following antisymmetric tensor for the Fermi field K µν = iψ(γ µ ∂ ν − γ υ ∂ µ )ψ, H µν = ǫ αβ µν K αβ(4) and regard them as the source tensors in the field equation of gravitomagnetic charge, where γ , µ s are general Dirac matrices with respect to x µ and satisfy γ µ γ v + γ ν γ µ = 2g µν . Then the field equation of gravitational field produced by the gravitomagnetic charge may be given as follows 2 G µν = κ 1 K µν + κ 2 H µν(5) with κ 1 , κ 2 being the coupling coefficients between gravitomagnetic matter and gravity. It should be noted that G µν ≡ 0 in the absence of gravitomagnetic matter since no singularities associated with topological charge exist in the metric functions and therefore Ricci identity still holds. However, once the metric functions possess non-analytic properties in the presence of gravitomagnetic matter ( should such exist ),G µν , is no longer vanishing due to the violation of Ricci identity. Additionally, further investigation shows that the cosmological term of Fermi field in Eq. (5) can be written as the linear combination of the antisymmetric tensors iψ(γ µ γ ν − γ υ γ µ )ψ and iǫ αβ µνψ (γ α γ β − γ β γ α )ψ. It is believed that there would exist formation (and creation) mechanism of gravitomagnetic charge in the gravitational interaction, just as some prevalent theories [18] provide the theoretical mechanism of existence of magnetic monopole in various gauge interactions. Magnetic monopole in electrodynamics and gauge field theory has been discussed and sought after for decades, and the existence of the , t Hooft-Polyakov monopole solutions [18,22,23] has spurred new interest of both theorists and experimentalists. Similar to magnetic monopole, gravitomagnetic charge is believed to give rise to such situations. If it is indeed present in universe, it will also lead to significant consequences in astrophysics and cosmology. We emphasize that although it is the classical solution to the field equation as discussed above, this kind of topological gravitomagnetic monopoles may arise not as fundamental entities in gravity theory. III. LOW-MOTION WEAK-FIELD APPROXIMATION In what follows the low-motion weak-field approximation is applied to the following gravitational field equation of gravitomagnetic matterG µν = S µν(6) with the source tensor S µν = κ 1 K µν + κ 2 H µν . First we considerG 01 = ǫ 0αβγ R 1 αβγ − ǫ 1αβγ R 0 αβγ by the linear approximation. The following expressions may be given −ǫ 1αβγ R 0 αβγ ≃ 2(R 0302 + R 0230 ), 2R 0302 ≃ ∂ 2 g 02 ∂x 3 ∂x 0 + ∂ 2 g 30 ∂x 0 ∂x 2 − ∂ 2 g 00 ∂x 3 ∂x 2 − ∂ 2 g 32 ∂x 0 ∂x 0 , 2R 0230 ≃ ∂ 2 g 00 ∂x 2 ∂x 3 + ∂ 2 g 23 ∂x 0 ∂x 0 − ∂ 2 g 03 ∂x 2 ∂x 0 − ∂ 2 g 02 ∂x 0 ∂x 3 ,(7) where the nonlinear terms (the products of two Christoffel symbols) are ignored and use is made of ǫ 1023 = ǫ 1302 = ǫ 1230 ≃ −1 and R 0 023 ≃ R 0023 = 0, R 0 302 ≃ R 0302 , R 0 230 ≃ R 0230 . If metric functions g µν are analytic, then 2(R 0302 + R 0230 ) is therefore vanishing. But once gravitomagnetic charge is present in spacetime and thus the metric functions possess the singularities, this result does not hold. Taking the gravitomagnetic vector potential g = (g 01 , g 02 , g 03 ), −g = (g 01 , g 02 , g 03 ), one can arrive at −ǫ 1αβγ R 0 αβγ = − ∂ ∂x 0 (∇ × g) 1 − ∇ × (∇g 00 − ∂g ∂x 0 ) 1 .(8) When we utilize the linear approximation for the field equation of dual matter ( gravitomagnetic matter ), we are concerned only with the space-time derivatives of gravitational potentials g µ = ( g 00 −1 2 , g 01 , g 02 , g 03 ) rather than that of g ii and g ij with i, j = 1, 2, 3, since the latter is either the analytic functions or the small terms. This, therefore, implies that the contribution of ǫ 0αβγ R 1 αβγ toG 01 vanishes. Eq. (8) is readily generalized to the three-dimensional case, and the combination of Eq. (6) and Eq. (8) yields ∇ × (∇g 00 − ∂ ∂x 0 g) = − ∂ ∂x 0 (∇ × g) + S,(9) where S is defined to be S i0 (i = 1, 2, 3). It is apparently seen that Eq. (9) is exactly analogous to the Faraday , s law of electromagnetic induction in the presence of current density of magnetic monopole in electrodynamics. This, therefore, implies that Eq. (6) is indeed the field equation of gravitation of gravitomagnetic matter. It is also of interest to discuss the motion of gravitomagnetic monopole in curved spacetime. Although Ricci identity is violated due to the non-analytic properties caused by the gravitomagnetic charge, Bianchi identity still holds in the presence of gravitomagnetic charge. It follows that the covariant divergence ofG µν vanishes, namely, G µν ;ν = 0.(10) Then in terms of the following field equationG µν = S µν(11) with the antisymmetric source tensor of gravitomagnetic matter S µν being κ 1 K µν + κ 2 H µν , one can arrive at S µν ;ν = 0 (12) which may be regarded as the equation of motion of gravitomagnetic charge in the curved spacetime. It is useful to obtain the low-motion and weak-field-approximation form of Eq. (12), which enables us to guarantee that Eq. (12) is the equation of motion of gravitomagnetic monopole indeed. The general Dirac matrices in the weak-field approximation may be obtained via the relations γ µ γ v + γ ν γ µ = 2g µν and the results are given by γ 0 = (1 + g 0 )β, γ i = g i β + γ i M ,(13) where i = 1, 2, 3; β = γ 0 M . γ 0 M and γ i M are the constant Dirac matrices in the flat Minkowski spacetime. Note that the gravitoelectric potential is defined to be g 0 = g 00 −1 2 , and gravitomagnetic vector potentials are g i = g 0i (i = 1, 2, 3). In the framework of dynamics of point-like particle, the source tensor is therefore rewritten as S µν = ρ κ 1 (g µ U ν − g ν U µ ) + κ 2 ǫ µναβ (g α U β − g β U α ) ,(14) where ρ denotes the density of gravitomagnetic matter. It follows from Eq. (12) and Eq. (14) that there exists the gravitational Lorentz force density in the expression for the force acting on the gravitomagnetic charge, namely, the equation of motion of gravitomagnetic charge is of the form ∂ ∂x 0 v = [∇ × g − v × (∇g 0 − ∂ ∂x 0 g)],(15) where some small terms are ignored and the relation, κ 1 g 0 = 2κ 2 , between the coupling coefficients, κ 1 and κ 2 is assumed ( one may be referred to the Appendix for the consideration why this assumption is reasonable ). Note, however, that the relation of the two coupling parameters suggested above holds only when weak-field approximation is employed ( see Appendix for more detailed analysis ). This connection between κ 1 and κ 2 gives us a helpful insight into the generally covariant relation between them. In view of what has been discussed, it can be seen that, in the weak gravitational field, the gravitomagnetic charge behaves like the magnetic charge. This, therefore, implies that gravitomagnetic charge proposed above is the gravitational analogue of magnetic charge in electrodynamics. IV. EXACT SOLUTION OF STATIC SPHERICALLY SYMMETRIC GRAVITOMAGNETIC FIELD A static spherically symmetric solution is exactly obtained by supposing that when a point-like gravitomagnetic charge is fixed at the origin of the spherical coordinate system, the exterior spacetime interval is of the form ds 2 = (dx 0 ) 2 − dr 2 − r 2 (dθ 2 + sin 2 θdϕ 2 ) + 2g 0ϕ (θ)dx 0 dϕ,(16) where the gravitomagnetic potential g 0ϕ is assumed to be the function with respect only to θ. Thus we obtain all the Christoffel symbols with non-vanishing values as follows: Γ 0,ϕθ = Γ 0,θϕ = 1 2 ∂g 0ϕ ∂θ , Γ ϕ,0θ = Γ ϕ,θ0 = 1 2 ∂g 0ϕ ∂θ , Γ θ,0ϕ = Γ θ,ϕ0 = − 1 2 ∂g 0ϕ ∂θ .(17) Since the field equation of gravitomagnetic matter is the antisymmetric equation, we might as well take into account a simple case of the following equation ǫ 0αβγ R 0 αβγ = M δ(x i )(18) with M being the parameter associated with the coupling parameters and gravitomagnetic charge. It is therefore apparent that Eq. (18) agrees with Eq. (11). Hence, the solution of the former equation also satisfies the latter. For the reason of the completely antisymmetric property of the Levi-Civita tensor, the contravariant indices α, β, γ should be respectively taken to be x, y, z of three-dimensional space coordinate, namely, we have ǫ 0αβγ R 0 αβγ = 2ǫ 0xyz (R 0 xyz + R 0 zxy + R 0 yzx ).(19) There exist the products of two Christoffel symbols, i.e., g στ (Γ τ,αγ Γ λ,σβ −Γ τ,αβ Γ λ,σγ ) in the definition of the Riemann curvature, R λαβγ . Apparently, the products of two Christoffel symbols (the nonlinear terms of field equation) contain the total indices of three-dimensional space coordinate (namely, these indices are taken the permutations of r, θ, ϕ ) and therefore vanish, in the light of the fact that the Christoffel symbol with index r is vanishing in terms of Eq. (17). In view of the above discussion, one can conclude that Eq. (18) can be exactly reduced to a linear equation. It is easily verified that R λαβγ (λ = r, θ, ϕ) vanishes with the help of the linear expression for R λαβγ given by R λαβγ = 1 2 ( ∂ 2 g λγ ∂x α ∂x β + ∂ 2 g αβ ∂x λ ∂x γ − ∂ 2 g λβ ∂x α ∂x γ − ∂ 2 gαγ ∂x λ ∂x β ) and the linear element expressed by Eq. (16). We thus obtain that R 0 αβγ = g 00 R 0αβγ . By the aid of the following expression R 0αβγ = 1 2 ∂ ∂x α ( ∂g 0γ ∂x β − ∂g 0β ∂x γ ),(20) one can arrive at ǫ 0αβγ R 0 αβγ = − g 00 √ −g ∇ · (∇ × g),(21) where the gravitomagnetic vector potentials, g, are defined to be g = (−g 0x , −g 0y , −g 0z ). Substitution of Eq. (21) into Eq. (18) yields ∇ · (∇ × g) = − √ −g g 00 M δ(x i ).(22) Note that Eq. (22) is the exact static gravitational field equation of gravitomagnetic matter derived from Eq. (11), where use is made of the expression (16) for the spacetime interval. It follows from Eq. (22) that the static spherically symmetric solution is given as follows 2g 0ϕ dx 0 dϕ = ∓ 2c 4π · 1 ± cos θ r sin θ · r sin θdx 0 dϕ,(23) where c is defined to be determined by the metric functions of the origin of the spherical coordinate system, i.e., c = −M ( which is the inverse matrix of the metric (g µν ) and we thus obtain the contravariant metric g µν . The gravitomagnetic field strength takes the form B i g = c 4π x i r 3 ,(25) provided that use is made of B g = ∇ × A with A = (0, 0, ∓ 2c 4π 1±cos θ r sin θ ) expressed in spherical coordinate system. From what has been discussed above, similar to the magnetic charge in electrodynamics, gravitomagnetic charge is a kind of topological charge that is the duality of mass of matter. In this sense, gravitomagnetic charge is also called dual mass. From the point of view of the classical field equation, matter may be classified into two categories: gravitomagnetic matter and gravitoelectric matter, of which the field equation of the latter is Einstein , s equation of general relativity. Due to their different gravitational features, the concept of mass is of no significance for the gravitomagnetic matter. V. TWO RELATED PROBLEMS (1) It is worthwhile to take into account the motion of photon in gravitomagnetic fields. Consider a weak gravitomagnetic field where the adiabatic approximation can be applicable to the motion of a photon, a conclusion can be drawn that the eigenvalue of the helicity k(t) k · J of the photon is conserved in motion and the helicity operator k(t) k · J is an invariant in terms of the invariant equation ( i.e., the Liouville-Von Neumann equation ) in Lewis-Riesenfeld theory [24] ∂I(t) ∂t + 1 i [I(t), H(t)] = 0,(26) where the invariant I(t) = k(t) k · J. From Eq. (26), simple calculation yields H(t) = k × d dt k k 2 · J,(27) which is considered an effective Hamiltonian governing the motion of photon in gravitomagnetic fields. Hence, the equation of motion of a photon in gravitomagnetic fields is written k × d dt k k 2 = B g ,(28) where the gravitomagnetic field strength B g is determined by the field equations of gravitation such as Eq. (11) and Einstein , s equation of general relativity. It follows from the geodesic equation in the weak-field approximation that the acceleration due to gravitational Lorentz force is 1 k d dt k = − k k × (∇ × g)(29) with k k being the velocity vector of the photon, and the gravitomagnetic field strength is B g = ∇ × g, where g = (g 01 , g 02 , g 03 ). Substitution of Eq. (28) into Eq. (29) yields 1 k d dt k = − k k × k × d dt k k 2 .(30) Since k · k = k 2 , k · d dt k = 0,(31) Eq. (30) is proved to be an identity. This, therefore, implies that Eq. (28) is truly the equation of motion of a photon in gravitomagnetic fields. For the time-dependent gravitomagnetic fields, similar to the case of the photon propagating inside the noncoplanarly curved optical fiber that is wound smoothly on a large enough diameter [25][26][27], the photon propagating in the gravitomagnetic field would also give rise to a geometric phase, which can be calculated by making use of the Lewis-Riesenfeld invariant theory [24] and the invariant-related unitary transformation formulation [28], and the result is φ (g) ± = ± t 0γ (t ′ )[1 − cos λ(t ′ )]dt ′(32) with ± corresponding to the eigenvalue σ = ±1 of the helicity k(t) k · J of the photon. The time-dependent parameters, γ and λ, are so defined that k(t) k = (sin λ(t) cos γ(t), sin λ(t) sin γ(t), cos λ(t)) . Differing from the dynamical phase that is related to the energy, frequency or velocity of a particle or a quantum system, geometric phase is dependent only on the geometric nature of the pathway along which the system evolves. For the case of adiabatic process where λ does not explicitly involve time t, Eq. (32) is reduced to ∆θ = ±2π(1 − cos λ) (33) in one cycle over the parameter space of the helicity k(t) k · J. It follows from Eq. (14) that 2π(1 − cos λ) is the expression for a solid angle in k(t) space, which presents the geometric properties of time evolution of the interaction between the gravitomagnetic field and photon spin ( gravitomagnetic moment ). (2) Taking the effects of gravitomagnetic fields and gravitomagnetic charges into consideration is believed to be of essential significance in resolving some problems and paradoxes. An illustrative example that would be briefly discussed in what follows may be regarded as an application of gravitomagnetic fields ( matter ) to the cosmological constant problem. The gravitational analogue of Meissner effect in superconductivity is gravitational Meissner effect. Due to the conservation of momentum, mass-current density is conserved before and after the scatterings of particles in perfect fluid, which is analogous to the superconductivity of superconducting electrons in superconductors cooled below T c . Since gravitational field equation of linear approximation is similar to the Maxwell's equation in electrodynamics, one can predict that gravitational Meissner effect arises also in perfect fluid. The author think that the investigation of both the effect of gravitomagnetic matter and gravitational Meissner effect may provide us with a valuable insight into the problem of cosmological constant and vacuum gravity [29][30][31][32]. Given that the vacuum matter is perfect fluid, the gravitoelectric field ( Newtonian gravitational field ) produced by the gravitoelectric charge ( mass ) of the vacuum quantum fluctuations is exactly cancelled by the gravitoelectric field due to the induced current of the gravitomagnetic charge of the vacuum quantum fluctuations; the gravitomagnetic field produced by the gravitomagnetic charge ( dual mass ) of the vacuum quantum fluctuations is exactly cancelled by the gravitomagnetic field due to the induced current of the gravitoelectric charge ( mass current ) of the vacuum quantum fluctuations. Thus, at least in the framework of weak-field approximation, the extreme space-time curvature of vacuum caused by its large energy density does not arise, and the gravitational effects of cosmological constant 3 is eliminated by the contributions of the gravitomagnetic charge ( dual mass ). Additionally, in 1998, Anderson et al. reported that, by ruling out a number of potential causes, radio metric data from the Pioneer 10/11, Galileo and Ulysses spacecraft indicate an apparent anomalous, constant, acceleration acting on the spacecraft with a magnitude ∼ 8.5× 10 −8 cm/s 2 directed towards the Sun [21]. Is it the effects of dark matter or a modification of gravity? Unfortunately, neither easily works. By taking the cosmic mass, M = 10 53 kg, and cosmic scale, R = 10 26 m, calculation shows that this acceleration is just equal to the value of field strength on the cosmic boundary due to the total cosmic mass. This fact leads us to consider a theoretical mechanism to interpret this anomalous phenomenon. The author favors that the gravitational Meissner effect may serve as a possible interpretation. Here we give a rough analysis, which contains only the most important features rather than the precise details of this theoretical explanation. Parallel to London , s electrodynamics of superconductivity, it shows that gravitational field may give rise to an effective rest mass m g =h c 2 √ 8πGρ m due to the self-induced charge current [33], where ρ m is the mass density of the universe. Then one can obtain thath mgc ≃ 10 26 m that approximately equals R, where the mass density of the universe is taken to be ρ m = 0.3ρ c [34] with ρ c ≃ 2 × 10 −26 kg/m 3 being the critical mass density. An added constant acceleration, a, may result from the Yukawa potential and can be written as 4 a = GM 2 m g c h 2 ≃ GM R 2 .(34) Note, however, that it is an acceleration of repulsive force directed, roughly speaking, from the center of the universe. By analyzing the NASA , s Viking ranging data, Anderson, Laing, Lau et al. concluded that the anomalous acceleration does not act on the body of large mass such as the Earth and Mars. If gravitational Meissner effect only affected the gravitating body of large mass or large scale rather than spacecraft ( perhaps the reason lies in that small-mass flow cannot serve as self-induced charge current, which deserves to be further investigated ), then seen from the Sun or Earth, there exists an added attractive force acting on the spacecraft. This added force give rise to an anomalous, constant, acceleration directed towards the Sun or Earth. It should be emphasized that the above theoretical theme is only a potential interpretation of this anomalous gravitational phenomenon. However, the sole reason that the above resolution of the anomalous acceleration is somewhat satisfactory lies in that no adjustable parameters exist in this theoretical framework. It is one of the most important advantages in the above mechanism of gravitational Meissner effect, compared with some possible theories of modification of gravity [35], which are always involving several parameters that cannot be determined by theory itself. These theories of modification of gravity were applied to the problem of the anomalous acceleration but could not calculate the value of the anomalous acceleration. 3 For the more present considerations regarding the cosmological constant problem, see, for example, hep-ph/0002297 ( by E. Witten ) and astro-ph/0005265 ( by S. Weinberg ). 4 It may also be calculated as follows: a = GM 2 mg c h 2 = GM c 2 (4πρmG) = GM c 2 R G(4πR 3 ρm) R 2 ≃ GM R 2 , where use is made of GM c 2 R ≃ O(1), 4πR 3 ρm ≃ M , which holds when the approximate estimation is performed. VI. CONCLUDING REMARKS In summary, in the present paper the author investigates some geometric effects, gravitational analogues of electromagnetic phenomena, and the field equation of gravitomagnetic matter ( dual matter ) as well as its static spherically symmetric solution. Differing from the symmetric property ( with respect to the indices of tensors ) of gravitational field equation of gravitoelectric matter, the field equation of gravitomagnetic matter possesses the antisymmetric property. This, therefore, implies that the number of the non-analytic metric functions is no more than 6. Although we have no observational evidences for the existence of gravitomagnetic charge, it is still of essential significance to investigate the gravity theory of the topological dual mass 5 . Some physically interesting problems associated with gravitomagnetic fields are proposed, of which the most interesting investigation is the potential solution to the anomalous acceleration acting on Pioneer spacecrafts by means of the mechanism of gravitational Meissner effect. The theoretical resolution of this problem is not very definite at present, for it cannot account for why the gravitational Meissner effect does not explicitly affect the body of small mass. This curiosity deserves further considerations. Since with foreseeable improvements in detecting and measuring technology, it is possible for us to investigate quantum mechanics in weak-gravitational fields, the above effects and phenomena deserve further detailed investigations. Work in this field is under consideration and will be published elsewhere. APPENDIX It is necessary to analyze the relation between the coupling parameters, κ 1 and κ 2 , under the low-motion and weak-field approximation. Substitution of the expression (14) for S µν into Eq. (12), where S µν ;ν = ∂S µν ∂x ν + 1 2 S µλ g στ ∂g στ ∂x λ ,(A1) yields κ 1 g 0 ∂ ∂x 0 v = 2κ 2 [∇ × g − v × (∇g 0 − ∂ ∂x 0 g)] − 2κ 2 g × ( ∂ ∂x 0 v + ∇g 0 ) +κ 1 g 0 ∂ ∂x 0 g − 2κ 2 g 0 (∇g 0 × v)(A2) with v being the velocity of the tested gravitomagnetic monopole. It is apparent that ∇ × g − v × (∇g 0 − ∂ ∂x 0 g) is the expression associated with gravitational Lorentz force density. Note that in Eq. (A2), κ 1 , κ 2 are considered coupling constants. However, further analysis shows that at least one of them is not a constant, and if the relation κ 1 g 0 = 2κ 2 (A3) between them is assumed, then Eq. (A2) may be rewritten as ∂ ∂x 0 v = [∇ × g − v × (∇g 0 − ∂ ∂x 0 g)],(A4) where we temporarily ignore the second and third terms, which can be considered the small terms, on the right-hand side of Eq. (A2) and the derivative term of coupling coefficients with respect to space-time coordinates x µ . It is well known that the form of Eq. (A4) is the equation of motion of a particle acted upon by the Lorentz force. Hence, Eq. (12) is believed to be the generally relativistic equation of motion of gravitomagnetic monopole in the Riemann space-time. When it comes to the description problem of fields arising from the dual charges, the dual tensorGµν in gravity theory is analogous to the dual field tensorF µν = 1 2 ǫ µναβ F αβ in electrodynamics.2 One of the advantages of this field equation is that it does not introduce extra tensor potentials when allowing for gravitomagnetic monopole densities and currents. This fact is in analogy with that in electrodynamics, where the equation ∂νF µν = J µ M governs the motion of electromagnetic fields produced by magnetic monopoles. 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[]
[ "Analysis of User Dwell Time by Category in News Application", "Analysis of User Dwell Time by Category in News Application" ]
[ "Yoshifumi Seki [email protected] \nGunosy Inc. Minato-ku\nTokyoJapan\n", "Mitsuo Yoshida [email protected] \nToyohashi University of Technorogy Toyohashi\nAichiJapan\n" ]
[ "Gunosy Inc. Minato-ku\nTokyoJapan", "Toyohashi University of Technorogy Toyohashi\nAichiJapan" ]
[]
Dwell time indicates how long a user looked at a page, and this is used especially in fields where ratings from users such as search engines, recommender systems, and advertisements are important. Despite the importance of this index, however, its characteristics are not well known. In this paper, we analyze the dwell time of news pages according to category in smartphone application. Our aim is to clarify the characteristics of dwell time and the relation between length of news page and dwell time, for each category. The results indicated different dwell time trends for each category. For example, the social category had fewer news pages with shorter dwell time than peaks, compared to other categories, and there were a few news pages with remarkably short dwell time. We also found a large difference by category in the correlation value between dwell time and length of news page. Specifically, political news had the highest correlation value and technology news had the lowest. In addition, we found that a user tends to get sufficient information about the news content from the news title in short dwell times.
10.1109/wi.2018.000-3
[ "https://arxiv.org/pdf/1908.08690v1.pdf" ]
58,005,599
1908.08690
2437d43f1fec7a39ea207c5db25b9005fc84d469
Analysis of User Dwell Time by Category in News Application Yoshifumi Seki [email protected] Gunosy Inc. Minato-ku TokyoJapan Mitsuo Yoshida [email protected] Toyohashi University of Technorogy Toyohashi AichiJapan Analysis of User Dwell Time by Category in News Application Index Terms-news pagesmartphonedwell timeuser be- haviorclick bait Dwell time indicates how long a user looked at a page, and this is used especially in fields where ratings from users such as search engines, recommender systems, and advertisements are important. Despite the importance of this index, however, its characteristics are not well known. In this paper, we analyze the dwell time of news pages according to category in smartphone application. Our aim is to clarify the characteristics of dwell time and the relation between length of news page and dwell time, for each category. The results indicated different dwell time trends for each category. For example, the social category had fewer news pages with shorter dwell time than peaks, compared to other categories, and there were a few news pages with remarkably short dwell time. We also found a large difference by category in the correlation value between dwell time and length of news page. Specifically, political news had the highest correlation value and technology news had the lowest. In addition, we found that a user tends to get sufficient information about the news content from the news title in short dwell times. I. INTRODUCTION The most widely known index in webpage rating is page view, because increasing page view generically leads to an increase in revenue from advertisements. However, there are limits to this index. Dwell time, on the other hand, is a wellknown indicator that is also used as an index for webpage rating [1]. Dwell time indicates how long a user looked at a page, and it is used especially in fields where ratings from users such as search engines [2], [3], recommender systems [4], and advertisements [5], [6] are important. Despite the importance of this index, however, its characteristics are not well known. In addition, previous studies have reported that trends vary greatly between PCs and smartphones [4], [5], but there has not been much analysis focusing on smartphones. In this study, we analyze the characteristics of the dwell time of news pages in smartphone applications according to category. Due to growing interest in the social issues of fake news [7], [8] and clickbait news [9], [10], the question of quality of news on the web is becoming more important. An increasing number of users are concerned about the quality of news pages on the web, and news media publishers have begun to rely on dwell times to indicate the high value of their media 1 . However, in order to discuss the quality of news pages based on dwell time, we need to understand the characteristics of dwell time in news pages, but there is not enough research on this subject. For example, Yi et al. [4] visualize trends in dwell time in the news recommender system, but they show results for only some categories and do not conduct an overall analysis. We show the characteristics of the dwell time of news pages using actual data in order to create a basis for discussing the quality of news pages in relation to user behavior such as dwell times. We focus on the topic category of news when analyzing dwell time. News categories are important, as Bandari et al. [11] reported that category was the most important feature when predicting the number of news clicks. In addition, Liu et al. [12] report that the variation in parameter for each category is large when modeling dwell time. Lagun and Lalmas [13] predicted dwell time by extending Latent Dirichlet Allocation (LDA) [14]. Latent topics in LDA are also known to represent categories of documents. It is suggested that there is a strong relationship between dwell time and category in this way, but the nature of this relationship has not been sufficiently discussed. In this paper, we analyze the dwell time of news pages according to category in smartphone application. Our aim is to clarify the characteristics of dwell time and the relation between length of news page and dwell time, for each category. To this end, we address the following three research questions: RQ1 Dwell time by category: Is there a difference in the dwell time for each category? RQ2 Dwell time and length of news page: Is there a correlation between dwell time and length of news page? RQ3 News content with short dwell time: What kind of content on is a news page with a short dwell time? The results indicated different dwell time trends for each category. For example, the social category had fewer news pages with shorter dwell time than peaks, compared to other categories, and there were a few news pages with remarkably short dwell time. We also found a large difference by category in the correlation value between dwell time and length of news page. Specifically, political news had the highest correlation value and technology news had the lowest. In addition, we found that a user tends to get sufficient information about the news content from the news title in short dwell times. II. DATA In this study, we used the browsing data of news pages from July 1 to 7, 2017. This data was gathered using Gunosy 2 , an information curation service for smartphones. Gunosy is one of the well-known famous news applications for iOS and Android in Japan. Our data consists of two types of data: news page data and user behavior data for news pages. The news page data includes categories, titles, news articles, and thumbnail images. In Gunosy, news pages are displayed by category which is determined by several heuristic rules and supervised machine learning. In this study, we used news pages from eight categories: "politics", "economy", "society", "international", "technology", "sports", "entertainment", and "column". The user behavior data includes the dwell time on the target news page and the length of the news page. If a user browses the same news page more than once, the maximum dwell time is taken as the representative value of the user. The length of the news page is included in the user behavior data because the length varies depending on the device used by the user. Dwell time should be related to news page. We regard the dwell time of a news page as the median of the dwell time of the user who viewed the news page. If the user leaves the application launched, a very long dwell time will be recorded. Therefore, it is not appropriate to use the average value as a representative value of dwell time. We analyze the dwell time of news pages, but this may be an unreliable measure for pages that are rarely viewed. Also, the page view tendency differs for each category, and there are categories that are frequently viewed and those that are rarely viewed. Therefore, only news pages in the top 10% of page view for each category are used. III. ANALYSIS AND RESULTS A. Dwell time by category The length of dwell time has been used as an index for rating web pages. Previous studies have suggested a strong relationship between dwell time and category, but the nature of this relationship has not been sufficiently discussed. This section addresses the following research question: Is there a difference in the dwell time for each category? A histogram of the dwell time of all news pages is shown in Fig. 1. The values of the x-axis and the y-axis are equally spaced 3 . The visualization does not include e do not use news pages whose dwell time is over a certain threshold for visualization, but these excluded news pages are account for less than 0.5% of the total. There is a peak at a relatively short dwell time, and the frequency decreases from there. Fig. 2 shows the histogram of dwell time by category. The xaxis is common to all categories, as shown in Fig. 1. The y-axis differs according to the maximum frequency of each category. The peak of thedwell time in the economy category in Fig. 2 is short compared to the histogram in Fig. 1, but the decay of the value is not steep, and there are a lot of news pages with relatively long dwell times. The social category has fewer news pages with shorter dwell time than peaks, compared to other categories, and there are a few news pages with remarkably short dwell time. The entertainment category tends to have many more news pages with shorter dwell times than other categories do. The column category is similar in tendency to the social category, but the decay of the value is not steep. As described above, the characteristics of the histogram are different for each category. B. Dwell time and length of news page Assuming that the user reads all news pages to the end, the dwell time correlates with the length of the news page. This section addresses the following research question: Is there a correlation between dwell time and length of news page? To confirm this, we obtain the correlation between the dwell time and the length of news page for each category by using Pearson's correlation coefficient. The results are shown in TABLE I. The correlation coefficient of the whole is not as high as about 0.3. This is dragged by correlation coefficients in some categories because the number of news pages has a bias for each category. By category, the correlation coefficients of politics, international, and sports are as high as about 0.55, and there is a relatively strong correlation between dwell time and length of news page. On the other hand, the correlation coefficients of technology, entertainment and column are as low as about 0.3, and the correlation between dwell time and length of news page is low. In this way, correlation coefficients have a bias for each category. C. News content with short dwell time The previous section showed that there was not much correlation between dwell time and length of news page. We Fig. 3. Image with thumbnail photo trimmed: Users who only want to enlarge the photo will have a shorter stay. been "No, I am not satisfied." However, a user can understand the content from the title of the news page and may not have read the news text. In addition, we focus on the length of the news page, not the length of the news text. Photos increase the length of the news page, but it takes no time to read a photo. We hypothesized that the user is satisfied with the news page even when the dwell time is short. We assume the following about news pages with remarkably short dwell time: (a) The title has a sufficient amount of information, and the title matches the content of the text. (b) The information expected from the title is not in the content of the text. (c) The title recalls photos, and the desire for information is satisfied by viewing large photos. (d) The title recalls photos, but the expected photos do not appear in the body. The case of (a) is evident in news stories that simply convey facts, for example, a news article that states, "Mr. A won in City B mayoral election." Since there is not much information in the text, the dwell time is short, but the user will be satisfied with the news page. In the case of (b), there are many news pages in which only a part of the events are indicated in the title. For example, in a news article titled "Mr. A loved Ms. B," if the content of this news is a press release of a movie, Mr. A did not actually love Ms. B, but it is a story in a movie. Such titles are dishonest, and users may be dissatisfied with the news. The case of (c) is often seen in news pages on album releases with the musician's photo. In Gunosy, especially, when displaying the thumbnail photo on the news list, as shown in Fig. 3, if the photo is of a person, the face of the person is extracted as the center. This may motivate a user to view the entire photo. Since a user only enlarges the image, the dwell time is short, but the user will be satisfied with the news page. The case of (d) is often seen in news pages on album releases with the musician's photo, as in (c). A photos of the musician is used in the title. A user expects such a photo in the body, but a photo of a press conference has been used in the body instead, and users may be dissatisfied with the news. We analyzed the news articles in relation to these four cases. We divided the four cases into groups: (a) and (b) for text and (c) and (d) for photos. For (a) and (b), it is important whether the title matches the content of the text. However, there are cases where there is not enough information in the title. Therefore, we consider whether the title has enough information and whether the title matches the content of the text. For (c) and (d), we consider whether the title recalls the photo and whether the photo matches the recalled content. Based on the above, we prepared the following four questions for news pages. 1) Does the title have enough information? 2) Are the title and body sufficiently matched? 3) Does the title recall photos? 4) Does a proper photo appear in the body? The author answered the four questions for news pages whose dwell time is not over a certain limit. There were 321 target news pages. The results of this investigation are shown in TABLE II. The answer of "Yes" to each question holds for news pages that satisfy the user. About two-thirds of all news pages answer "Yes," and even if the dwell time is short, users may be satisfied with the news page. In addition, 85% of news pages recalled photos. Many news pages with short dwell times had titles to recall photos. IV. CONCLUSION We analyzed the dwell time of news pages according to category in smartphone application. Our aim was to clarify the characteristics of dwell time and the relation between length of news page and dwell time, for each category. To this end, we addressed three research questions. The results indicated different dwell time trends for each category. For example, the social category had fewer news pages with shorter dwell time than peaks, compared to other categories, and there were a few news pages with remarkably short dwell time. We also found a large difference by category in the correlation value between dwell time and length of news page. Specifically, political news had the highest correlation value and technology news had the lowest. In addition, we found that a user tends to get sufficient information about the news content from the news title in short dwell times. Fig. 1 . 1Histogram of dwell time: The x-axis indicates the dwell time by news page. The y-axis indicates the number of such news pages. There are many news pages with a relatively short dwell time. Fig. 2 . 2Histogram of dwell time by category: The x-axis indicates the dwell time by news page. The y-axis indicates the number of such news pages. Each category has a different tendency. TABLE I CORRELATION ICOEFFICIENT BETWEEN DWELL TIME AND THE LENGTH OF NEWS PAGE: THE VALUES FOR POLITICS AND SPORTS ARE AS HIGH AS 0.55 OR MORE, BUT THE VALUE FOR TECHNOLOGY IS AS LOW AS BELOW 0.25.Category Correlation coefficient # of news politics 0.602 4.05% economy 0.432 5.72% society 0.454 9.52% international 0.527 4.05% technology 0.245 6.22% sports 0.561 13.42% entertainment 0.309 29.26% column 0.367 27.78% whole 0.291 also found a different trend for each category. In this section, we focus on dwell time and address the following research question: what kind of content on is a news page with a short dwell time? On news pages with short dwell time, was the user not satisfied? In previous studies, the answer to this question has News List Body Cl ick TABLE II RESULTS IIOF ANSWERS TO NEWS PAGES. Does a proper photo appear in the body? 209 112Question Yes No 1) Does the title have enough information? 253 68 2) Are the title and body sufficiently matched? 246 75 3) Does the title recall photos? 273 48 4) Digiday (2017): How The New York Times gets people to spend 5 minutes per visit on its site. http://gunosy.co.jp/en/ 3 Due to limitations because of the Gunosy privacy policy, we discussed using relative values instead of absolute values. The development and evaluation of a survey to measure user engagement. H L O&apos;brien, E G Toms, Journal of the American Society for Information Science and Technology. 611H. L. O'Brien and E. G. Toms, "The development and evaluation of a survey to measure user engagement," Journal of the American Society for Information Science and Technology, vol. 61, no. 1, pp. 50-69, 2010. Improving web search ranking by incorporating user behavior information. 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Tolomei, "Promoting Positive Post-Click Experience for In-Stream Yahoo Gemini Users," in Proceedings of the 21th ACM SIGKDD International Confer- ence on Knowledge Discovery and Data Mining, 2015, pp. 1929-1938. Predicting Pre-click Quality for Native Advertisements. K Zhou, M Redi, A Haines, M Lalmas, Proceedings of the 25th International Conference on World Wide Web. the 25th International Conference on World Wide WebK. Zhou, M. Redi, A. Haines, and M. Lalmas, "Predicting Pre-click Quality for Native Advertisements," in Proceedings of the 25th Interna- tional Conference on World Wide Web, 2016, pp. 299-310. Social Media and Fake News in the 2016 Election. H Allcott, M Gentzkow, Journal of Economic Perspectives. 312H. Allcott and M. Gentzkow, "Social Media and Fake News in the 2016 Election," Journal of Economic Perspectives, vol. 31, no. 2, pp. 211-236, 2017. From Clickbait to Fake News Detection: An Approach based on Detecting the Stance of Headlines to Articles. 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Lalmas, "Understanding User Attention and Engage- ment in Online News Reading," in Proceedings of the Ninth ACM International Conference on Web Search and Data Mining, 2016, pp. 113-122. Latent Dirichlet Allocation. D M Blei, A Y Ng, M I Jordan, Journal of Machine Learning Research. 3D. M. Blei, A. Y. Ng, and M. I. Jordan, "Latent Dirichlet Allocation," Journal of Machine Learning Research, vol. 3, no. Jan, pp. 993-1022, 2003.
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[ "Array Variate Skew Normal Random Variables with Multiway Kronecker Delta Covariance Matrix Structure", "Array Variate Skew Normal Random Variables with Multiway Kronecker Delta Covariance Matrix Structure" ]
[ "Deniz Akdemir \nDepartment of Statistics\nUniversity of Central Florida Orlando\n32816FL\n" ]
[ "Department of Statistics\nUniversity of Central Florida Orlando\n32816FL" ]
[]
In this paper, we will discuss the concept of an array variate random variable and introduce a class of skew normal array densities that are obtained through a selection model that uses the array variate normal density as the kernel and the cumulative distribution of the univariate normal distribution as the selection function.
null
[ "https://arxiv.org/pdf/1103.3795v2.pdf" ]
88,518,500
1103.3795
931fa2b33ee9acfa96e2056823a7178acfc1e95b
Array Variate Skew Normal Random Variables with Multiway Kronecker Delta Covariance Matrix Structure 22 Mar 2011 January 15, 2013 Deniz Akdemir Department of Statistics University of Central Florida Orlando 32816FL Array Variate Skew Normal Random Variables with Multiway Kronecker Delta Covariance Matrix Structure 22 Mar 2011 January 15, 2013 In this paper, we will discuss the concept of an array variate random variable and introduce a class of skew normal array densities that are obtained through a selection model that uses the array variate normal density as the kernel and the cumulative distribution of the univariate normal distribution as the selection function. Introduction The array variate random variable up to 2 dimensions has been studied intensively in [Gupta and Nagar(2000)] and by many others. For arrays observations of 3, 4 or in general i dimensions a suitable normal probability model has been recently proposed in [Akdemir and Gupta(2011)]. An elliptical generalization of the normal array variable can be found in [Akdemir(2011)]. In this paper, we will generalize the notion of elliptical random variables to the array case. In Section 2, we first study the algebra of arrays, and introduce the concept of an array variable random variable. In Sections 3, the density of the normal array random variable is provided. We finally provide the definition for an array random variable with skew normal density in Section 4. Array Algebra In this paper we will only study arrays with real elements. We will write X to say that X is an array. When it is necessary we can write the dimensions of the array as subindices, e.g., if X is a m 1 × m 2 × m 3 × m 4 dimensional array in R m1×m2×...×mi , then we can write X m1×m2×m3×m4 . Arrays with the usual element wise summation and scalar multiplication operations can be shown to be a vector space. To refer to an element of an array X m1×m2×m3×m4 , we write the position of the element as a subindex to the array name in parenthesis, ( X) r1r2r3r4 . If we want to refer to a specific column vector obtained by keeping all but an indicated dimension constant, we indicate the constant dimensions as before but we will put ':' for the non constant dimension, e.g., for X m1×m2×m3×m4 , ( X) r1r2:r4 refers to the the column vector ((X) r1r21r4 , (X) r1r22r4 , . . . , (X) r1r2m3r4 ) ′ . We will now review some basic principles and techniques of array algebra. These results and their proofs can be found in Rauhala [Rauhala(1974)], [Rauhala(1980)] and Blaha [Blaha(1977)]. Definition 2.1. Inverse Kronecker product of two matrices A and B of dimensions p × q and r × s correspondingly is written as A ⊗ i B and is defined as A ⊗ i B = [A(B) jk ] pr×qs = B ⊗ A, where ′ ⊗ ′ represents the ordinary Kronecker product. The following properties of the inverse Kronecker product are useful: • 0 ⊗ i A = A ⊗ i 0 = 0. • (A 1 + A 2 ) ⊗ i B = A 1 ⊗ i B + A 2 ⊗ i B. • A ⊗ i (B 1 + B 2 ) = A ⊗ i B 1 + A ⊗ i B 2 . • αA ⊗ i βB = αβA ⊗ i B. • (A 1 ⊗ i B 1 )(A 2 ⊗ i B 2 ) = A 1 A 2 ⊗ i B 1 B 2 . • (A ⊗ i B) −1 = (A −1 ⊗ i B −1 ). • (A ⊗ i B) + = (A + ⊗ i B + ), where A + is the Moore-Penrose inverse of A. • (A ⊗ i B) − = (A − ⊗ i B − ), where A − is the l-inverse of A defined as A − = (A ′ A) −1 A ′ . • If {λ i } and {µ j } are the eigenvalues with the corresponding eigenvectors {x i } and {y j } for matrices A and B respectively, then A ⊗ i B has eigenvalues {λ i µ j } with corresponding eigenvectors {x i ⊗ i y j }. • Given two matrices A n×n and B m×m |A ⊗ i B| = |A| m |B| n , tr(A ⊗ i B) = tr(A)tr(B). • A ⊗ i B = B ⊗ A = U 1 A ⊗ BU 2 , for some permutation matrices U 1 and U 2 . It is well known that a matrix equation AXB ′ = C can be rewritten in its monolinear form as A ⊗ i Bvec(X) = vec(C).(1) Furthermore, the matrix equality A ⊗ i BXC ′ = E obtained by stacking equations of the form (1) can be written in its monolinear form as (A ⊗ i B ⊗ i C)vec(X) = vec(E). This process of stacking equations could be continued and R-matrix multiplication operation introduced by Rauhala [Rauhala(1974)] provides a compact way of representing these equations in array form: Definition 2.2. R-Matrix Multiplication is defined element wise: ((A 1 ) 1 (A 2 ) 2 . . . (A i ) i X m1×m2×...×mi ) q1q2...qi = m1 r1=1 (A 1 ) q1r1 m2 r2=1 (A 2 ) q2r2 m3 r3=1 (A 3 ) q3r3 . . . mi ri=1 (A i ) qiri ( X) r1r2...ri . R-Matrix multiplication generalizes the matrix multiplication (array multiplication in two dimensions)to the case of k-dimensional arrays. The following useful properties of the R-Matrix multiplication are reviewed by Blaha [Blaha(1977)]: 1. (A) 1 B = AB. 2. (A 1 ) 1 (A 2 ) 2 C = A 1 CA ′ 2 . 3. Y = (I) 1 (I) 2 . . . (I) i Y . 4. ((A1) 1 (A2) 2 . . . (Ai) i )((B1) 1 (B2) 2 . . . (Bi) i ) Y = (A1B1) 1 (A2B2) 2 . . . (AiBi) i Y . The operator rvec describes the relationship between X m1×m2×...mi and its monolinear form x m1m2...mi×1 . Definition 2.3. rvec( X m1×m2×...mi ) = x m1m2...mi×1 where x is the column vector obtained by stacking the elements of the array X in the order of its dimensions; i.e., ( X) j1j2...ji = (x) j where j = (j i − 1)n i−1 n i−2 . . . n 1 + (j i − 2)n i−2 n i−3 . . . n 1 + . . . + (j 2 − 1)n 1 + j 1 . Theorem 2.1. Let L m1×m2×...mi = (A 1 ) 1 (A 2 ) 2 . . . (A i ) i X where (A j ) j is an m j × n j matrix for j = 1, 2, . . . , i and X is an n 1 × n 2 × . . . × n i array. Write l = rvec( L) and x = rvec( X). Then, l = A 1 ⊗ i A 2 ⊗ i . . . ⊗ i A i x. Therefore, there is an equivalent expression of the array equation in monolinear form. Definition 2.4. The square norm of X m1×m2×...mi is defined as X 2 = m1 j1=1 m2 j2=1 . . . mi ji=1 (( X) j1j2...ji ) 2 . Definition 2.5. The distance of X 1m 1×m2×...mi from X 2m 1×m2×...mi is defined as X 1 − X 2 2 . Example 2.1. Let Y = (A 1 ) 1 (A 2 ) 2 . . . (A i ) i X + E. Then E 2 is minimized for X = (A − 1 ) 1 (A − 2 ) 2 . . . (A − i ) i Y . Array Variate Normal Distribution By using the results in the previous section on array algebra, mainly the relationship of the arrays to their monolinear forms described by Definition 2.3 , we can write the density of the standard normal array variable. Definition 3.1. If Z ∼ N m1×m2×...×mi ( M = 0, Λ = I m1m2...mi ), then Z has array variate standard normal distribution. The pdf of Z is given by f Z ( Z) = exp (− 1 2 Z 2 ) (2π) m1m2...mi/2 .(2) For the scalar case, the density for the standard normal variable z ∈ R 1 is given as φ 1 (z) = 1 (2π) 1 2 exp(− 1 2 z 2 ). For the m 1 dimensional standard normal vector z ∈ R m1 , the density is given by φ m1 (z) = 1 (2π) m 1 2 exp(− 1 2 z ′ z). Finally the m 1 × m 2 standard matrix variate variable Z ∈ R m1×m2 has the density φ m1×m2 (Z) = 1 (2π) m 1 m 2 2 exp(− 1 2 trace(Z ′ Z)). With the above definition, we have generalized the notion of normal random variable to the array variate case. Definition 3.2. We write X ∼ N m1×m2×...×mi ( M , Λ m1m2...mi ) if rvec( X) ∼ N m1m2...mi (rvec( M ), Λ m1m2...mi ). Here, M is the expected value of X, and Λ m1m2...mi is the covariance matrix of the m 1 m 2 . . . m i -variate random variable rvec( X). The family of normal densities with Kronecker delta covariance structure are obtained by considering the densities obtained by the location-scale transformations of the standard normal variables. This kind of model is defined in the next. Definition 3.3. ( [Akdemir and Gupta(2011) ]) Let Z ∼ N m1×m2×...×mi ( M = 0, Λ = I m1m2...mi ). Define X = (A 1 ) 1 (A 2 ) 2 . . . (A i ) i Z + M where A 1 , A 2 , . . . , A i are non singular matrices of orders m 1 , m 2 , . . . , m i . Then the pdf of X is given by φ( X; M , A1, A2, . . . Ai) = exp (− 1 2 (A −1 1 ) 1 (A −1 2 ) 2 . . . (A −1 i ) i ( X − M ) 2 ) (2π) m 1 m 2 ...m i /2 |A1| j =1 m j |A2| j =2 m j . . . |Ai| j =i m j .(3) Distributional properties of a array normal variable with density in the form of Definition 3.3 variable can obtained by using the equivalent monolinear representation of the random variable. The moments, the marginal and conditional distributions, independence of variates can be studied considering the equivalent monolinear form of the array variable and the well known properties of the multivariate normal random variable. Array Variate Skew Normal Variable A very general definition of skew symmetric variable for the matrix case can be obtained from matrix variate selection models. Suppose X is a k × n random matrix with density f (X), let g(X) be a weight function. A weighted form of density f (X) is given by h(X) = f (X)g(X) R k×n g(X)f (X)dX .(4) When the sample is only a subset of the population then the associated model would be called a selection model. Chen and Gupta provide a skew normal matrix variate probability density function in the following form ( [Chen and Gupta(2005)]): f 1 (X, Σ ⊗ Ψ, b) = c * 1 φ k×n (X; Σ ⊗ Ψ)Φ n (X ′ b, Ψ)(5) where c * 1 = (Φ n (0, (1 + b ′ Σb)Ψ)) −1 , φ(.) and Φ(.) denote the density function the cumulative distribution of the standard normal random variable correspondingly. A drawback of this definition is that it allows independence only over its rows or columns, but not both. Harrar ([Harrar and Gupta(2008)]) give two more matrix variate skew normal densities: f 2 (X, Σ, Ψ, b, Ω) = c * 2 φ k×n (X; Σ ⊗ Ψ)Φ n (X ′ b, Ω)(6) and f 3 (X, Σ, Ψ, b, B) = c * 3 φ k×n (X; Σ ⊗ Ψ)Φ n (tr(B ′ X))(7) where c * 2 = (Φ n (0, (Ω + b ′ Σb)Ψ)) −1 , c * 3 = 2; Σ, Ψ, and Ω are positive definite covariance matrices of dimensions k, n and n respectively, B is a matrix of dimension k × n. Note that if Ω = Ψ then f 2 is the same as f 1 . Although, more general than f 1 , the densities f 2 and f 3 still do not permit independence of rows and columns simultaneously. In [Akdemir(2009)], the following density is given to overcome this problem: f (X, M, A, B, ∆) = φ k×n (A −1 (X − M )B −1 ) k j=1 n i=1 Φ(αjie ′ j (A −1 (X − M )B −1 )ci) |A| n |B| k 2 −kn .(8) Here, the parameter M is a k × n matrix. The scale parameters A and B are k × k and n× n positive definite matrices correspondingly. The shape parameter ∆ is a matrix of order k × n. Finally, e j is the unit length k dimensional vector with 1 at its jth row and c i is the unit length n dimensional vector with 1 at its ith row. We denote the distribution of a variable with density of the form (8) by msn k×n (M, A, B, ∆). Motivated by the model in (9) we first define a general class of selection models for the array variate random variables. h( X) = f ( X)g( X) R m 1 ×m 2 ×...×m i g( X)f ( X)d X .(9) When the sample is only a subset of the population then the associated model would be called a selection model. Here f (.) is called the kernel density and g(.) is called the weight or selection function. We will use the density for the array variate random variable in as the kernel density for the selection model. We obtain a class of skew normal array variate densities by using a selection function like the one in (8): Definition 4.2. A density for a skew normal array variate random variable X m1×m2×...×mi is given by f ( X; M , A 1 , A 2 , . . . A i , ∆) = φ( X; M , A 1 , A 2 , . . . A i )g( X; M , A 1 , A 2 , . . . A i , ∆) 2 −m1m2...mi (10) where g( X ; M , A 1 , A 2 , . . . A i , ∆) = m 1 j 1 =1 m 2 j 2 =1 . . . m i j i =1 Φ(( ∆) j 1 j 2 ...j i ((A −1 1 ) 1 (A −1 2 ) 2 . . . (A −1 i ) i ( X − M )) j 1 j 2 ...j i ), A 1 , A 2 , . . . , A i are non singular matrices of orders m 1 , m 2 , . . . , m i , M and ∆ are constant arrays of the same dimension as X. Discussion The models with Kronecker delta covariance structures provide a great deal of decrease in the number of parameters that has to be estimated. We will now discuss this in the context if principle component analysis. Principal components analysis is a useful statistical technique that has found applications in fields such as face recognition and image compression, and is a common technique for finding patterns in data of high dimension. The end product of PCA is a set of new uncorrelated variables ordered in terms of their variances obtained from a linear combination of the original variables. Definition 5.1. For the m 1 × m 2 × . . . × m i dimensional array variate random variable X, the principal components are defined as the principal components of the d = m 1 m 2 . . . m i -dimensional random vector rvec( X). The advantage in choosing a Kronecker structure is the decrease in the number of parameters. If {λ(A r ) rj } are the m j eigenvalues of A r A ′ r with the corresponding eigenvectors {(x r ) rj } for r = 1, 2, . . . , i and r j = 1, 2, . . . , m r , then (A 1 ⊗ i A 2 ⊗ i A i )(A 1 ⊗ i A 2 ⊗ i A i ) ′ will have eigenvalues {λ(A 1 ) r1 λ(A 2 ) r2 . . . λ(A i ) ri } with corresponding eigenvectors {(x 1 ) r1 ⊗ i (x 2 ) r2 ⊗ i . . . ⊗ i (x i ) ri }. By replacing A r by their estimators, we can estimate the eigenvalues and eigenvectors of the covariance of rvec( X) using this relationship. When applying the array variate skew normal model to real data, more parsimonious forms of the model in Equation (10) should be considered. For example, a model where ∆ is diagonal should be appropriate for most cases. One could further assume that many of these diagonal elements are zero. Definition 4 . 1 . 41Let f (.) be a density for a array random variable of dimensions m 1 × m 2 × . . . × m i and let g(.) be a weight function that takes an array of the same dimensions as its argument. A weighted form of density f (.) is given by A Class of Multivariate Skew Distributions: Properties and Inferential Issues. ; D Akdemir, Akdemir, Bowling Green State UniversityPhD thesisAkdemir(2009)] D. Akdemir. A Class of Multivariate Skew Distributions: Properties and Inferential Issues. PhD thesis, Bowling Green State Uni- versity, 2009. Array Variate Random Variables with Multiway Kronecker Delta Covariance Matrix Structure. Gupta ; D Akdemir, A K Akdemir, Gupta, 11Technical Report[Akdemir and Gupta(2011)] D. Akdemir and A.K. Gupta. Array Variate Ran- dom Variables with Multiway Kronecker Delta Covariance Matrix Structure. Technical Report, 11(2), 2011. A Few Basic Principles and Techniques of Array Algebra. ; G Blaha, Blaha, Journal of Geodesy. 513Blaha(1977)] G. Blaha. A Few Basic Principles and Techniques of Array Al- gebra. Journal of Geodesy, 51(3):177-202, 1977. Matrix variate skew normal distributions. Gupta ; J T Chen, A K Chen, Gupta, 0233-1888.doi:{10.1080/02331880500108593}Statistics. 393[Chen and Gupta(2005)] J.T. Chen and A.K. Gupta. Matrix variate skew nor- mal distributions. Statistics, 39(3):247-253, 2005. ISSN 0233-1888. doi: {10.1080/02331880500108593}. Matrix Variate Distributions. Nagar ; A K Gupta, D K Gupta, Nagar, Chapman & Hall CRC Monographs and Surveys in Pure and Applied Mathematics. Chapman & Hall[Gupta and Nagar(2000)] A.K. Gupta and D.K. Nagar. Matrix Variate Dis- tributions. Chapman & Hall CRC Monographs and Surveys in Pure and Applied Mathematics. Chapman & Hall, 2000. U.A. Rauhala. Array Algebra with Applications in Photogrammetry and Geodesy. Division of Photogrammetry. Gupta ; S W Harrar, A K Harrar, Gupta, Statistics. 422Royal Institute of TechnologyOn matrix variate skew-normal distributions[Harrar and Gupta(2008)] S.W. Harrar and A.K. Gupta. On matrix variate skew-normal distributions. Statistics, 42(2):179-194, 2008. [Rauhala(1974)] U.A. Rauhala. Array Algebra with Applications in Photogram- metry and Geodesy. Division of Photogrammetry, Royal Institute of Tech- nology, 1974. Introduction to Array Algebra. Photogrammetric Engineering and Remote Sensing. ; U A Rauhala, Rauhala, 46Rauhala(1980)] U.A. Rauhala. Introduction to Array Algebra. Photogrammet- ric Engineering and Remote Sensing, 46(2):177-192, 1980.
[]
[ "Nuclear and Particle Physics Proceedings Probing cold nuclear matter effects with weak gauge boson production in ultra-relativistic heavy-ion collisions", "Nuclear and Particle Physics Proceedings Probing cold nuclear matter effects with weak gauge boson production in ultra-relativistic heavy-ion collisions" ]
[ "Peng Ru \nSchool of Physics & Optoelectronic Technology\nDalian University of Technology\n116024DalianChina\n\nInstitute of Particle Physics\nKey Laboratory of Quark & Lepton Physics (MOE)\nCentral China Normal University\n430079WuhanChina\n", "Ben-Wei Zhang \nInstitute of Particle Physics\nKey Laboratory of Quark & Lepton Physics (MOE)\nCentral China Normal University\n430079WuhanChina\n" ]
[ "School of Physics & Optoelectronic Technology\nDalian University of Technology\n116024DalianChina", "Institute of Particle Physics\nKey Laboratory of Quark & Lepton Physics (MOE)\nCentral China Normal University\n430079WuhanChina", "Institute of Particle Physics\nKey Laboratory of Quark & Lepton Physics (MOE)\nCentral China Normal University\n430079WuhanChina" ]
[ "Nuclear and Particle Physics Proceedings" ]
Within the framework of pQCD, we systematically study the W ± /Z 0 boson production as a probe of cold nuclear matter effects and nuclear parton distributions in p+Pb and Pb+Pb collisions at the LHC and future colliders. A detailed analysis at partonic level is performed. Moreover, with a semi-microscopic KP model of NPDFs, in which several nuclear effects (e.g. Fermi motion and nuclear binding, the off-shell correction, the nuclear coherent correction, and the nuclear meson correction) are included, we study the vector boson rapidity distribution in p+Pb collisions at the LHC, and a very good agreement with the latest data is found, including the W-boson charge asymmetry.
10.1016/j.nuclphysbps.2017.05.043
[ "https://arxiv.org/pdf/1612.02899v1.pdf" ]
119,336,704
1612.02899
0383553d83d9008f3ca134cab4d9f9d0887d865e
Nuclear and Particle Physics Proceedings Probing cold nuclear matter effects with weak gauge boson production in ultra-relativistic heavy-ion collisions 2018 Peng Ru School of Physics & Optoelectronic Technology Dalian University of Technology 116024DalianChina Institute of Particle Physics Key Laboratory of Quark & Lepton Physics (MOE) Central China Normal University 430079WuhanChina Ben-Wei Zhang Institute of Particle Physics Key Laboratory of Quark & Lepton Physics (MOE) Central China Normal University 430079WuhanChina Nuclear and Particle Physics Proceedings Probing cold nuclear matter effects with weak gauge boson production in ultra-relativistic heavy-ion collisions Nuclear and Particle Physics Proceedings 002018weak gauge bosoncold nuclear matter effectsnuclear PDFsW charge asymmetry Within the framework of pQCD, we systematically study the W ± /Z 0 boson production as a probe of cold nuclear matter effects and nuclear parton distributions in p+Pb and Pb+Pb collisions at the LHC and future colliders. A detailed analysis at partonic level is performed. Moreover, with a semi-microscopic KP model of NPDFs, in which several nuclear effects (e.g. Fermi motion and nuclear binding, the off-shell correction, the nuclear coherent correction, and the nuclear meson correction) are included, we study the vector boson rapidity distribution in p+Pb collisions at the LHC, and a very good agreement with the latest data is found, including the W-boson charge asymmetry. Introduction The cold nuclear matter (CNM) effects or the nuclear parton distribution functions (NPDFs) are quite important for the hard-scattering processes in high-energy nuclear collisions. They also provide the baseline for the study of the quark-gluon-plasma (QGP) property with hard probes in relativistic nucleus-nucleus collisions [1,2]. Largely impeded by the non-perturbative mechanism therein, it is hard to fully compute the nuclear parton distributions from first principle. Conventionally, the NPDFs are extracted from global fits to the experimental (e.g. DIS, DY) data. With great efforts, several groups working on that [3,4,5] have achieved significant progresses. However the error bars of the obtained NPDFs and the differences among the CNM effects extracted by different groups are still considerable. With the running of the LHC, the production of the weak gauge boson (W ± /Z 0 ) in heavy-ion collisions provides a new excellent probe of the CNM effects. With the high-invariant mass (∼ 80 − 90 GeV), weak bosons will be produced in the very early stage (∼ 1/m W/Z ∼ 10 −3 fm/c) of the collisions, and can decay later (∼ 0.08 − 0.09 fm) to a colorless lepton pair in the final state (Drell-Yan), which carries a clean signal of the initial state (e.g PDFs) [6]. Even in the nucleus-nucleus collision (Fig. 1), the created hot and dense QCD medium will hardly pollute this signal. The weak boson production in nuclear collisions will open a unique opportunity to study the NPDFs at high Q 2 (∼ 100 2 GeV 2 ). Features of W/Z production: Clean signal of initial state of the collision， e.g. parton distributions (PDFs), and cold nuclear medium effects or NPDFs. In nucleus-nucleus collisions at LHC: The created hot/dense nuclear environment (after ~1fm/c) can hardly pollute the signal of initial CNM effect. In QCD perturbation theory, the cross section of the DY process in hadronic collision can be written as the convolution of the parton distribution and the partonic hard-scattering cross section, according to the factorization theorem. As the factorization is expected to hold in arXiv:1612.02899v1 [nucl-th] 9 Dec 2016 modifications. The experimental data are taken from Ref. [9]. More details can be seen in Ref. [10]. nuclear collisions, the cross section can be calculated by utilizing the nuclear PDFs. In this work, we study the vector boson production in nuclear collisions at NLO and NNLO with the program DYNNLO [7,8] by incorporating nuclear parton distributions. Vector boson in nuclear collisions at the LHC In this section we focus on the vector boson production in nuclear collisions at the LHC. We show in Fig. 2 the Z 0 rapidity distribution in Pb+Pb collisions at √ s NN = 2.76 TeV. Both the results at NLO and NNLO with EPS09 NPDFs agree well with the CMS data, and the NNLO correction is rather small in the studied rapidity region. In Ref. [10], the NLO result is also found to well describe the Z 0 transverse momentum distribution in large-p T ( 10 GeV) region in Pb+Pb collisions. In the last panel of Fig. 3, we show the nuclear modification factor R AA (y) corresponding to Fig. 2. It is observed that the predictions with EPS09 and DSSZ NPDF sets show obvious difference, especially in the mid-rapidity region. This can be understood by performing a partonic level analysis. The momentum fraction carried by the initial parton can be estimated at LO as x 1,2 = m Z e ±y Z / √ s NN . At the LHC, the kinematic region related to the mid-rapidity corresponds to the transition between the valence-dominated and the sea-quark-dominated regions (x ∼ 0.033). Since gluons do not give the LO contribution, the boson rapidity distribution is much sensitive to the valence and sea quark distributions. In Fig. 3 we show the partonic nuclear modification factor R ff (y Z ) simply defined as 1 2 Figure 3: Partonic nuclear modification factor R ff (y Z ) for different flavors, and the R AA (y Z ) for Z 0 rapidity distribution at NLO (the last panel), in Pb+Pb collisions at √ s NN = 2.76 TeV. More details can be seen in Ref. [10]. R f (x 1 )Rf (x 2 ) + 1 2 R f (x 2 )Rf (x 1 ) by considering the 0.9 1 1.1 EPS09 DSSZ -1 0 1 2 -2 -1 0 1 2 0.9 1 1.1 y Z R f f (y Z , µ=m Z ) u+u d+d s+s R AA (y) NLO 1LO process qq → Z 0 , where the R f (x) ≡ f Pb (x)/ f p (x) is the nucelar modification ratio given by NPDFs and the LO relation between y Z and x 1,2 is used. The R AA show similarity with the R ff and is an excellent probe of the CNM effects on valence and sea quark distributions. However we note the precision of the measurement until now is not sufficient to exclude neither EPS09 nor DSSZ. In p+Pb collisions, the CNM effects usually result in an asymmetric nuclear modification on Z-boson rapidity distribution, as is shown in the last panel of Fig. 4. The partonic level analysis is easier than that in Pb+Pb collisions, since the nuclear partons come from only one direction. In Fig. 4 we show the partonic factor R f (y Z ) ≡ R f (x Pb ) given by NPDFs. We found R pA (y Z ) provides an image of the nuclear modifications on the valence and sea quark distributions. The suppressions in the forward region is mainly the result of the shadowing effect on the sea quark distributions, while the enhancements in the backward is largely due to the antishadowing effect on the valence quark distributions. See also the parton contributions in the first panel of Fig. 5. The nuclear modification on Z 0 -boson p T spectrum in heavy-ion collisions at the LHC is also studied [10]. Here we just emphasize that, compared to the rapidity distribution, the p T spectrum will provide more information on the nuclear gluon distribution since the gluon can give the lowest-order contributions (e.g. qg → Z 0 q). See also the right-top panel in Fig. 5. Besides the similar nuclear modifications as observed in Z 0 production, those on W ± -boson production show some unique characteristics, due to the flavor dependence (ud → W + , dū → W − ). For example the strong isospin effect [10] and the sensitivity to the flavor dependent nuclear modifications (e.g. r u r d ). Vector boson production in future HIC In future HIC with much higher energies [11], the vector boson production will probe CNM effects in new kinematic regions. The momentum fraction carried by the nuclear partons will decrease a lot. The shadowing effects on sea quarks and gluons would be more important, while the valence quark contribution and the The momentum fraction carried by the nuclear parton estimated at LO are shown on the top axis. Bottom panels: contributions to the nuclear correction from each individual nuclear effects. More details can be seen in Ref. [14]. isospin effect should be small. See also the parton contributions for Z 0 rapidity and p T distributions in p+Pb collisions at both LHC and future colliders in Fig. 5. By combining the LHC and the future measurements, we may give new powerful constraints on the NPDFs [11]. W/Z production in p+Pb with KP NPDFs Different from the conventional NPDFs obtained by fitting data, a semi-microscopic-model based NPDFs have been recently developed by S. A. Kulagin and R. Petti (KP) [12,13]. The KP NPDFs have been validated with the data from a wide range of processes (e.g. DIS, DY), and the offered insights on the underlying nuclear physics mechanisms make it more valuable. In KP model, different nuclear effects on the parton distribution are taken into account: the Fermi motion and nuclear binding (FMB), the off-shell effect (OS), 5.02 TeV. Bottom panel: differences between each result and that without nuclear modification (ABMP15). The CMS data are taken from Ref. [15]. More details can be seen in Ref. [14]. the coherent nuclear interaction related to the nuclear shadowing (NS) and the nuclear meson exchange current (MEC) correction. In Ref. [14] we study the rapidity dependence of vector boson production in p+Pb collisions at the LHC with KP NPDFs. The NLO results with KP NPDFs is found to show an excellent agreement with the latest data from both CMS and ATLAS. Much meaningfully, we study how each individual CNM effect play a role on W/Z production in p+Pb collisions at the LHC (see Fig. 6), through which the experimental data are better understood. It is found that the full nuclear correction in the rapidity region measured by CMS and ATLAS is the result of an interplay of different mechanisms. In Fig. 7, we show the interesting result for W-boson charge asymmetry. It is found the prediction with KP NPDFs show a good agreement with the CMS data. At the same time we note the difference between the predictions with KP and EPS09 is largely due to their corresponding proton PDFs (ABMP15 [16] and CT10 [17]). It is also noteworthy that for the observable sensitive to the CNM effects rather than the proton PDFs, e.g. the Z 0 forward-backward asymmetry, the results with KP NPDFs show a good performance with the CMS data [14]. Conclusion We systematically study the W ± /Z 0 boson production as a probe of the CNM effects in nuclear collisions at the LHC and future colliders, within the framework of pQCD. Differences among various state-of-art NPDFs sets are found in several observable, and a partonic-level analysis is performed. With the KP NPDFs we show how the underlying nuclear physics mechanisms play a role on the vector boson rapidity distributions in p+Pb collisions at the LHC. The predictions with KP NPDFs show an excellent performance with latest LHC p+Pb data. 4 Peng Ru DUT / CCNU 24 th Sept. . Figure 1 : 1A schematic diagram for Z 0 -boson production through the Drell-Yan (DY) mechanism in nucleus-nucleus collisions. Figure 2 : 2The Z 0 -boson rapidity distribution in Pb+Pb collisions at √ s NN = 2.76 TeV, calculated at NNLO and NLO with EPS09 nuclear Figure 4 : 4Partonic nuclear modification factor R f (y Z ) for different flavors, and the R pA (y Z ) for Z 0 rapidity distribution at NLO (the last panel), in p+Pb collisions at √ s NN = 5.02 TeV. More details can be seen in Ref.[10]. Figure 5 : 5Nuclear parton contributions to the Z 0 boson rapidity (left) and transverse momentum (rght) distributions, in p+Pb collisions at √ s NN = 5.02 TeV (LHC, top panels) and √ s NN = 63 TeV (future, bottom panels), calculated at LO. More details can be seen in Ref.[11]. Figure 6 : 6Top panel: nuclear corrections on charged-lepton pseudorapidity distribution with different combinations of nuclear effects in KP model with respect to the prediction without nuclear modification (ABMP15), calculated at NLO for W + production in p+Pb collisions at √ s NN = 5.02 TeV. The CMS data are taken from Ref.[15]. Figure 7 : 7Top panel: the W ± -boson charge asymmetry as a function of the charged-lepton pseudo-rapidity in p+Pb collisions at √ s NN = 4 . 4I. Framework of the studyVector boson as a probe of Cold Nuclear Medium effect0 Z q q l  l  Hot/Dense QCD matter Nucleus A Nucleus B Drell-Yan process High-invariant mass Produced in early stage Decay later Colorless lepton pair in final state 3 W/Z 1~10 fm/c m  W/Z~8 0 90 GeV m  0.08 0.09 fm/c  . J L Albacete, arXiv:1301.3395Int. J. Mod. Phys. E. 221330007hep-phJ. L. Albacete et al., Int. J. Mod. Phys. E 22 (2013) 1330007 [arXiv:1301.3395 [hep-ph]]. . J L Albacete, arXiv:1605.09479Int. J. Mod. Phys. E. 2591630005hep-phJ. L. Albacete et al., Int. J. Mod. Phys. E 25 (2016) no.9, 1630005 [arXiv:1605.09479 [hep-ph]]. . K J Eskola, H Paukkunen, C A Salgado, arXiv:0902.4154JHEP. 090465hep-phK. J. Eskola, H. Paukkunen and C. A. Salgado, JHEP 0904 (2009) 065 [arXiv:0902.4154 [hep-ph]]. . D De Florian, R Sassot, P Zurita, M Stratmann, arXiv:1112.6324Phys. Rev. D. 8574028hep-phD. de Florian, R. Sassot, P. Zurita and M. Stratmann, Phys. Rev. D 85 (2012) 074028 [arXiv:1112.6324 [hep-ph]]. . I Schienbein, J Y Yu, K Kovarik, C Keppel, J G Morfin, F Olness, J F Owens, arXiv:0907.2357Phys. Rev. D. 8094004hep-phI. Schienbein, J. Y. Yu, K. Kovarik, C. Keppel, J. G. Morfin, F. Olness and J. F. Owens, Phys. Rev. D 80 (2009) 094004 [arXiv:0907.2357 [hep-ph]]. . Z Conesa Del Valle, arXiv:0903.1432Eur. Phys. J. C. 61729hep-exZ. Conesa del Valle, Eur. Phys. J. C 61 (2009) 729 [arXiv:0903.1432 [hep-ex]]. . S Catani, M Grazzini, hep-ph/0703012Phys. Rev. Lett. 98222002S. Catani and M. Grazzini, Phys. Rev. Lett. 98 (2007) 222002 [hep-ph/0703012]. . S Catani, L Cieri, G Ferrera, D De Florian, M Grazzini, arXiv:0903.2120Phys. Rev. Lett. 10382001hep-phS. Catani, L. Cieri, G. Ferrera, D. de Florian and M. Grazzini, Phys. Rev. Lett. 103 (2009) 082001 [arXiv:0903.2120 [hep-ph]]. . S Chatrchyan, CMS CollaborationarXiv:1410.4825JHEP. 150322nucl-exS. Chatrchyan et al. [CMS Collaboration], JHEP 1503 (2015) 022 [arXiv:1410.4825 [nucl-ex]]. . P Ru, B W Zhang, L Cheng, E Wang, W N Zhang, arXiv:1412.2930J. Phys. G. 42885104nucl-thP. Ru, B. W. Zhang, L. Cheng, E. Wang and W. N. Zhang, J. Phys. G 42 (2015) no.8, 085104 [arXiv:1412.2930 [nucl-th]]. . P Ru, B W Zhang, E Wang, W N Zhang, arXiv:1505.08106Eur. Phys. J. C. 759426nucl-thP. Ru, B. W. Zhang, E. Wang and W. N. Zhang, Eur. Phys. J. C 75 (2015) no.9, 426 [arXiv:1505.08106 [nucl-th]]. . S A Kulagin, R Petti, hep- ph/0412425Nucl. Phys. A. 765126S. A. Kulagin and R. Petti, Nucl. Phys. A 765 (2006) 126 [hep- ph/0412425]. . S A Kulagin, R Petti, arXiv:1405.2529Phys. Rev. C. 90445204hep-phS. A. Kulagin and R. Petti, Phys. Rev. C 90 (2014) no.4, 045204 [arXiv:1405.2529 [hep-ph]]. . P Ru, S A Kulagin, R Petti, B W Zhang, arXiv:1608.06835nucl-thP. Ru, S. A. Kulagin, R. Petti and B. W. Zhang, arXiv:1608.06835 [nucl-th]. . V Khachatryan, CMS CollaborationarXiv:1503.05825Phys. Lett. B. 750565nucl-exV. Khachatryan et al. [CMS Collaboration], Phys. Lett. B 750 (2015) 565 [arXiv:1503.05825 [nucl-ex]]. . J S. Alekhin, S Bluemlein, R Moch, Placakyte, arXiv:1508.07923hep-phS. Alekhin, J. Bluemlein, S. Moch and R. Placakyte, arXiv:1508.07923 [hep-ph]. . J Gao, arXiv:1302.6246Phys. Rev. D. 89333009hep-phJ. Gao et al., Phys. Rev. D 89 (2014) no.3, 033009 [arXiv:1302.6246 [hep-ph]].
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[ "Aggregating Predictions on Multiple Non-disclosed Datasets using Conformal Prediction", "Aggregating Predictions on Multiple Non-disclosed Datasets using Conformal Prediction" ]
[ "Ola Spjuth es:[email protected] \nDepartment of Pharmaceutical Biosciences\nUppsala University Uppsala\nSweden\n", "Lars Carlsson <[email protected] \nDepartment of Computer Science\nRoyal Holloway\nUniversity of London\nEgham Hill, EghamSurreyUnited Kingdom\n\nStena Line\nGothenburgSweden\n", "Niharika Gauraha [email protected] \nDepartment of Pharmaceutical Biosciences\nUppsala University Uppsala\nSweden\n", "Ola Spjuth ", "Lars Carlsson " ]
[ "Department of Pharmaceutical Biosciences\nUppsala University Uppsala\nSweden", "Department of Computer Science\nRoyal Holloway\nUniversity of London\nEgham Hill, EghamSurreyUnited Kingdom", "Stena Line\nGothenburgSweden", "Department of Pharmaceutical Biosciences\nUppsala University Uppsala\nSweden" ]
[]
Conformal Prediction is a machine learning methodology that produces valid prediction regions under mild conditions. In this paper, we explore the application of making predictions over multiple data sources of different sizes without disclosing data between the sources. We propose that each data source applies a transductive conformal predictor independently using the local data, and that the individual predictions are then aggregated to form a combined prediction region. We demonstrate the method on several data sets, and show that the proposed method produces conservatively valid predictions and reduces the variance in the aggregated predictions. We also study the effect that the number of data sources and size of each source has on aggregated predictions, as compared with equally sized sources and pooled data.
null
[ "https://arxiv.org/pdf/1806.04000v2.pdf" ]
47,016,852
1806.04000
bafa660e6563af7d36e38aad58f112925618c58d
Aggregating Predictions on Multiple Non-disclosed Datasets using Conformal Prediction 14 Jun 2018 Ola Spjuth es:[email protected] Department of Pharmaceutical Biosciences Uppsala University Uppsala Sweden Lars Carlsson <[email protected] Department of Computer Science Royal Holloway University of London Egham Hill, EghamSurreyUnited Kingdom Stena Line GothenburgSweden Niharika Gauraha [email protected] Department of Pharmaceutical Biosciences Uppsala University Uppsala Sweden Ola Spjuth Lars Carlsson Aggregating Predictions on Multiple Non-disclosed Datasets using Conformal Prediction 14 Jun 2018Preprint submitted to Elsevier June 15, 2018(Niharika Gauraha)conformal predictionmachine learningaggregated predictionsprivacy preservationnon-disclosed data Conformal Prediction is a machine learning methodology that produces valid prediction regions under mild conditions. In this paper, we explore the application of making predictions over multiple data sources of different sizes without disclosing data between the sources. We propose that each data source applies a transductive conformal predictor independently using the local data, and that the individual predictions are then aggregated to form a combined prediction region. We demonstrate the method on several data sets, and show that the proposed method produces conservatively valid predictions and reduces the variance in the aggregated predictions. We also study the effect that the number of data sources and size of each source has on aggregated predictions, as compared with equally sized sources and pooled data. Introduction The increasing volumes of data generated in virtually all scientific and industrial domains presents formidable challenges to store and analyze. Of particular interest is to make use of the information in statistical learning systems with the objective to make predictions on future objects. If data reside in multiple data sources, possibly in different databases or geographical locations, the most common approach is to collect all data intended for model building in a single location, such as a data warehouse or a file system, after which it is subjected to learning algorithms and subsequent predictions (see Figure 1a). However, if data is large or if the data owners do not allow such pooling of data, this strategy may not be possible. One example is in the pharmaceutical industry where large databases are available at companies, each holding results on e.g. chemical compounds tested against different endpoints in drug discovery projects. This information is valuable and sensitive for these companies, but at the same time there are occasions where companies would want to contribute to predictive models without disclosing the data to others, such as in collaborative efforts or precompetitive alliances. There are approaches that have been developed towards integrated analysis that do not require sharing of original data, but these come with limitations. For example, methods for integrated analysis of non-sensitive availability data derived from original data has been developed (Spjuth et al., 2016) but these are not suitable in machine learning contexts. Another example is dataSHIELD (Gaye et al., 2014) which comprises a technically advanced computational infrastructure and uses distributed computing and parallelized analysis to enable joint analysis, but does not support machine learning models. Federated learning models such as the ones proposed by Shokri and Shmatikov (2015) for deep learning are not generally applicable to other machine learning methods and also complex to implement. In this manuscript, we propose a light-weight approach to improve predictions over different sources without explicitly sharing the data, by means of aggregating conformal predictions computed at individual locations (see Figure 1b). Conformal predictors provide a layer on top of underlying machine learning algorithms, and produce results with valid measures of confidence Vovk et al. (2005). Here we extend the basic conformal prediction framework to handle multiple data sources and without sharing of data between sources. We propose to aggregate conformal predictions from multiple sources, where transductive conformal predictors (TCP) are applied on the multiple data sources and their individual predictions are aggregated to form a single prediction on a new example. The advantages of this approach are two-fold: Firstly, it extends the existing framework of conformal prediction to multiple data sources that do not require sharing of data between the sources. Secondly, combined learning produces more efficient predictions than individual learning. The method is flexible in the sense that it supports flexible number and sizes of data sources. Our main research objectives in this paper are summarized as follows: 1. to investigate if and how the number of data sources and size of the sources affect the aggregated efficiency and validity 2. to evaluate how good both "aggregated equally partitioned" and "aggregated randomly partitioned" perform when compared to the whole (pooled) data set. 3. to evaluate if and under what conditions aggregated TCP delivers acceptable results when compared to pooled data Figure 1: a) The most common approach is to collect data from different data sources (D 1 − D 3 ) into a single dataset, which then is used to train a model M that can be used to make a prediction P on a query object Q. b) The aggregated TCP approach implies that a model M n is trained at each data source D n , and the query object Q is passed on to each model, and predictions P n are then aggregated to deliver the resulting prediction P . The gray wireframes are used to visualize the different actors taking part in the procedure independent of each other. The organization of the paper is as follows. In section 2, we introduce the background concepts and notations used throughout the paper. In Section 3, we introduce the concept of aggregating conformal predictions from multiple sources. In Section 4, we discuss the statistical properties of aggregated conformal predictions from multiple sources. In Section 5, we perform numerical analysis on simulated and real datasets. Finally, in Section 6, a summary of the paper is provided. Methods Conformal prediction The object space is denoted by X ∈ R p , where p is the number of features, and label space is denoted by Y ∈ {0, 1}. We assume that each example consists of an object and its label, and its space is given as Z := X × Y. The typical classification problem is, given a training dataset Z = {z 1 , ..., z n }where n is the number of examples in the training set, and each example z i = (x i , y i ) are labeled examples -we want to predict the label of a new object x new whose label is unknown. We also assume the exchangeability of examples throughout the paper. The nonconformity measure is the score from a function that measures the strangeness of an example. In this study we use the noncomformity measure from a random forest classifier (RFC) which outputsỹ = F RF (x) wherẽ y ∈ [0, 1]. Furthermore the noncomformity scores are for 0-label α i = 1 −ỹ i and the 1-label α i =ỹ i . However, the methodology is general and other underlying machine learning methods and nonconformity scores are equally applicable.To compute the corresponding p-values, we use the smoothed mondrian approach (Vovk et al., 2005), where the taxonomy is defined by the labels. Conformal prediction provides a layer on top of an existing machine learning method and uses available data to determine valid prediction regions for new objects (Vovk et al., 2005). The predicted region of an object is a subset of Y , denoted as Γ = {y | p y > }, at a significance level . In the transductive approach (Algorithm 1), the underlying model must be retrained each time an object is to be predicted. For further details on TCP, we refer to Vapnik and Vapnik (1998), Shafer and Vovk (2008), Vovk et al. (2005) and Balasubramanian et al. (2014). Algorithm 1: Mondrian TCP with random forest Input: (training dataset:Z, object to predict:x new , label set:Y , a nonconformity measure:F RF Output: p-values for each y ∈ Y do z n+1 = (x new , y); Z * = (Z, z n+1 ) ; Create F RF using Z * if y i = y then for each i ∈ n + 1 do if y i = 1 then α i = F RF (x i ) if y i = 0 : α i = 1 − F RF (x i ) end end Compute p y end return p 0 new , p 1 new ; To assess the quality of a conformal predictor we consider validity and efficiency. A predictor makes an error when the predicted region does not contain the true label y ∈ Γ . Given a training dataset Z and an external test set T , and |T | = m. Suppose that the conformal predictor gives prediction regions as Γ 1 , ...., Γ m , then the error rate is defined as Definition 1 (Error rate). ER = 1 m m i=1 I {y i ∈Γ i } ,(1) where y i is the true class label of the i th test case and I is an indicator function. In the following, we consider a way of assessing validity of a conformal predictor in terms of deviation of observed from expected error. The deviation from exact validity can be computed as the Euclidean norm of the difference of the observed error and the expected error for a given set of predefined significance levels (Carlsson et al., 2017). Let us assume a set of significance levels = { 1 , ..., k }, then the formula for the validity can be given as follows. Definition 2 (Validity). V AL = k i=1 (ER i − i ) 2(2) We use observed fuzziness (Vovk et al., 2016) as our measure of efficiency, defined as the sum of all p-values for the incorrect class labels. Definition 3 (Efficiency). EF F = 1 m m i=1 y i =y p y i ,(3) We note that for the above measure of validity and efficiency, smaller values are preferable. Non-Disclosed aggregated Conformal Prediction Suppose we have K data sources, each with a training dataset D k of arbitrary sizes where k ∈ 1, ..., K. For a new object x new , the objective is to aggregate p-values at the location A that were computed in each data source using Mondrian TCP with random forest (Algorithm 1). We name this aggregated predictor Non-Disclosed aggregated Conformal Predictor (NDCP), with details described in Algorithm 2. The result is a set of aggregated pvalues, where no data training data is disclosed between the data sources and between the data sources and location A, but the only information that is transmitted between data sources and A is the object to predict and the resulting p-values. Algorithm 2: Non-Disclosed Conformal Prediction (NDCP) Input: D 1 , ...D K ; x new Output: Aggregated p-values Steps; for each D k , k ∈ {1, ..., K} do Compute p 0 and p 1 using Algorithm 1 at each data source Transfer p 0 and p 1 to location Ā p 0 =p 0 + p 0 p 1 =p 1 + p 1 end p 0 new =p 0 /K p 1 new =p 1 /K returnp 0 new ,p 1 new Evaluation We evaluate NDCP on five binary classification datasets from the UCI repository (Table 1) that are randomly partitioned into a training set (80%) and a test set (20%). The steps are described below, and also illustrated in Figure 2. 1. The training data set is randomly split into K parts (disjointly) with varying sizes. For example, Let Z = {z 1 , ..., z n } be the data set, then we divide the dataset into S 1 , ..., S K such that Z = K i=1 S i , k i = |S i | and n = k 1 + ... + k K . More specifically, the following types of partitions were made to create different scenarios, with one to compare with unpartitioned data (a TCP trained on all data): (a) Pooled source (pooled): Entire training set is considered as one single data source. (b) Equally sized sources (EqSrc): Training set is randomly partitioned into equally sized sources and each partition is considered as a proper training set to model and compute p-values, and then p-values are aggregated for all sources. We consider 2, 4 and 6 equal sized sources. (c) Unequally sized sources (RandSrc): Training set is randomly partitioned into unequally sized sources and each partition is considered as a proper training set to model and compute p-values, and then p-values are aggregated for all sources. We consider 2, 4 and 6 unequally sized sources, and we repeat it five times to get five different set of sizes for each source. 2. Predict p-values for the different scenarios using Algorithm 2 3. Perform evaluation based on validity and efficiency 4. Repeat step 1 to 3 five times. Results were combined for each scenario over all datasets, and then a pairwise comparison using a Wilcoxon signed-rank test on validity and efficiency for all scenarios were performed. Dataset Results and discussion The results from the pairwise comparison of validity and efficiency (observed fuzziness) are shown in Figure 3. To illustrate the quantitative difference between the scenarios, box plots are presented in Figure 4. Results in Figure 3 show that pooled is significantly more efficient than all other models, as would be expected, but in absolute numbers the decrease in efficiency is not so large when using NDCP. When comparing NDCP with individual smallTCP, we do not see a significant improvement in efficiency using NDCP but we observe a reduced variance, consistent with previous work (Carlsson et al., 2014), when there are 4 or more partitions. We also observe in Figure 3 that there is no significant difference between 'aggregated equally partitioned' and 'aggregated randomly partitioned', which would make the method generally applicable regardless of the sizes of individual training sets. Regarding validity, we observe that the pooled model is always valid, as an example see Figure 5a for the Spambase dataset. Further, we see that individual small models are also valid, see Figure 5b for randomly partitioned small TCPs for Spambase dataset. Consistent with previous work by Linusson et al Linusson et al. (2017) and Carlsson et al. Carlsson et al. (2014), NDCP is less valid overall, see Figure 5c for randomly partitioned NDCPs for Spambase dataset, but calibration plot shows conservative validity for the significance levels 0 to 0.5 which is the interesting region for predictions. This is a known issue that requires further research; we settle here with the observation that validity does not seem to be a practical problem for NDCP in the interesting significance region. Conclusions We present a method to aggregate conformal predictions from multiple sources while preserving data privacy. The method is a generalization of the basic conformal prediction framework to handle multiple data sources without disclosing data between the data sources. Due to its low complexity for implementation, we believe the method will be useful for organizations that wish to make predictions over combined data without disclosing data to each other, such as for drug discovery problems when pharmaceutical companies wishes to establish predictive models of drug safety. Figure 2 : 2Evaluation of NDCP Algorithm for a given dataset Figure 3 : 3Results of Wilcoxon signed-rank tests for two alternative hypotheses relating validity (a) and observed fuzziness (b) with combining all the datasets. The p-values are shown for the scenarios in the right column having greater values than the scenarios in the rows. All significant p-values are marked in red. Pooled: Unpartitioned dataset. EqSrc: equally partitioned data sources, RandSrc: randomly partitioned data sources. smallTCP: a single TCP model. Figure 4 : 4Box plot of observed fuzziness for aggregating 0, 2, 4, and 6 non-disclosed data sources. Pooled: Unpartitioned dataset. EqSrc: equally partitioned data sources, RandSrc: randomly partitioned data sources. smallTCP: a single TCP model. Figure 5 : 5Calibration plot for various models. a) Calibration plot of TCP for one fold of Spambase dataset. b) Calibration plot of randomly partitioned small TCPs for one fold of Spambase dataset. Blue, orange and green line indicate each small TCP from two, four and six source random partitions respectively. c) Calibration plot of NDCPs for one fold of Spambase dataset. Blue, orange and green line indicate NDCP from two, four and six source random partitions respectively Table 1 : 1Description of the datasets from UCI repository that are used in the evaluation.Dataset Observations # Features Spambase (SB) 4601 57 Breast Cancer Wisconsin (BC) 699 10 Mushroom (Mush) 8124 22 First-order theorem proving (FOTP) 6118 51 Phishing Websites (Phish) 2456 30 V Balasubramanian, S.-S Ho, V Vovk, Conformal Prediction for Reliable Machine Learning: Theory, Adaptations and Applications. NewnesBalasubramanian, V., Ho, S.-S., Vovk, V., 2014. Conformal Prediction for Reliable Machine Learning: Theory, Adaptations and Applications. Newnes. Comparing performance of different inductive and transductive conformal predictors relevant to drug discovery. L Carlsson, C Bendtsen, E Ahlberg, Conformal and Probabilistic Prediction and Applications. Carlsson, L., Bendtsen, C., Ahlberg, E., 2017. Comparing performance of different inductive and transductive conformal predictors relevant to drug discovery. In: Conformal and Probabilistic Prediction and Applications. pp. 201-212. Aggregated conformal prediction. L Carlsson, M Eklund, U Norinder, L Iliadis, I Maglogiannis, H Papadopoulos, S Sioutas, Artificial Intelligence Applications and Innovations. Makris, C.Berlin Heidelberg; Berlin, HeidelbergSpringerCarlsson, L., Eklund, M., Norinder, U., 2014. Aggregated conformal pre- diction. In: Iliadis, L., Maglogiannis, I., Papadopoulos, H., Sioutas, S., Makris, C. (Eds.), Artificial Intelligence Applications and Innovations. Springer Berlin Heidelberg, Berlin, Heidelberg, pp. 231-240. . A Gaye, Y Marcon, J Isaeva, P Laflamme, A Turner, E M Jones, J Minion, A W Boyd, C J Newby, M.-L Nuotio, R Wilson, O Butters, B Murtagh, I Demir, D Doiron, L Giepmans, S E Wallace, I Budin-Ljøsne, C Oliver Schmidt, P Boffetta, M Boniol, M Bota, K W Carter, N Deklerk, C Dibben, R W Francis, T Hiekkalinna, K Hveem, K Kvaløy, S Millar, I J Perry, A Peters, C M Phillips, F Popham, G Raab, E Reischl, N Sheehan, M Waldenberger, M Perola, E Van Den Heuvel, J Macleod, B M Knoppers, R P Stolk, I Fortier, J R Harris, B H R Woffenbuttel, M J Murtagh, V Ferretti, P R Burton, Int J Epidemiol. 436Datashield: taking the analysis to the data, not the data to the analysisGaye, A., Marcon, Y., Isaeva, J., LaFlamme, P., Turner, A., Jones, E. M., Minion, J., Boyd, A. W., Newby, C. J., Nuotio, M.-L., Wilson, R., Butters, O., Murtagh, B., Demir, I., Doiron, D., Giepmans, L., Wallace, S. E., Budin-Ljøsne, I., Oliver Schmidt, C., Boffetta, P., Boniol, M., Bota, M., Carter, K. W., deKlerk, N., Dibben, C., Francis, R. W., Hiekkalinna, T., Hveem, K., Kvaløy, K., Millar, S., Perry, I. J., Peters, A., Phillips, C. M., Popham, F., Raab, G., Reischl, E., Sheehan, N., Waldenberger, M., Perola, M., van den Heuvel, E., Macleod, J., Knoppers, B. M., Stolk, R. P., Fortier, I., Harris, J. R., Woffenbuttel, B. H. R., Murtagh, M. J., Ferretti, V., Burton, P. R., Dec 2014. Datashield: taking the analysis to the data, not the data to the analysis. Int J Epidemiol 43 (6), 1929-44. On the calibration of aggregated conformal predictors. H Linusson, U Norinder, H Boström, U Johansson, T Löfström, A Gammerman, V Vovk, Z Luo, Proceedings of the Sixth Workshop on Conformal and Probabilistic Prediction and Applications. Papadopoulos, H.the Sixth Workshop on Conformal and Probabilistic Prediction and ApplicationsStockholm, Sweden60of Proceedings of Machine Learning Research. PMLRLinusson, H., Norinder, U., Boström, H., Johansson, U., Löfström, T., 13-16 Jun 2017. On the calibration of aggregated conformal predictors. In: Gam- merman, A., Vovk, V., Luo, Z., Papadopoulos, H. (Eds.), Proceedings of the Sixth Workshop on Conformal and Probabilistic Prediction and Ap- plications. Vol. 60 of Proceedings of Machine Learning Research. PMLR, Stockholm, Sweden, pp. 154-173. URL http://proceedings.mlr.press/v60/linusson17a.html A tutorial on conformal prediction. G Shafer, V Vovk, Journal of Machine Learning Research. 9Shafer, G., Vovk, V., 2008. A tutorial on conformal prediction. Journal of Machine Learning Research 9 (Mar), 371-421. Privacy-preserving deep learning. R Shokri, V Shmatikov, 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton). Shokri, R., Shmatikov, V., Sept 2015. Privacy-preserving deep learning. In: 2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton). pp. 909-910. Harmonising and linking biomedical and clinical data across disparate data archives to enable integrative cross-biobank research. O Spjuth, M Krestyaninova, J Hastings, H.-Y Shen, J Heikkinen, M Waldenberger, A Langhammer, C Ladenvall, T Esko, M.-Å Persson, J Heggland, J Dietrich, S Ose, C Gieger, J S Ried, A Peters, I Fortier, E J C De Geus, J Klovins, L Zaharenko, G Willemsen, J.-J Hottenga, J.-E Litton, J Karvanen, D I Boomsma, L Groop, J Rung, J Palmgren, N L Pedersen, M I Mccarthy, C M Van Duijn, K Hveem, A Metspalu, S Ripatti, I Prokopenko, J R Harris, Eur J Hum Genet. 244Spjuth, O., Krestyaninova, M., Hastings, J., Shen, H.-Y., Heikkinen, J., Waldenberger, M., Langhammer, A., Ladenvall, C., Esko, T., Persson, M.-Å., Heggland, J., Dietrich, J., Ose, S., Gieger, C., Ried, J. S., Peters, A., Fortier, I., de Geus, E. J. C., Klovins, J., Zaharenko, L., Willemsen, G., Hottenga, J.-J., Litton, J.-E., Karvanen, J., Boomsma, D. I., Groop, L., Rung, J., Palmgren, J., Pedersen, N. L., McCarthy, M. I., van Duijn, C. M., Hveem, K., Metspalu, A., Ripatti, S., Prokopenko, I., Harris, J. R., Apr 2016. Harmonising and linking biomedical and clinical data across disparate data archives to enable integrative cross-biobank research. Eur J Hum Genet 24 (4), 521-8. Statistical learning theory. V N Vapnik, V Vapnik, Wiley1New YorkVapnik, V. N., Vapnik, V., 1998. Statistical learning theory. Vol. 1. Wiley New York. Criteria of efficiency for conformal prediction. V Vovk, V Fedorova, I Nouretdinov, A Gammerman, Symposium on Conformal and Probabilistic Prediction with Applications. SpringerVovk, V., Fedorova, V., Nouretdinov, I., Gammerman, A., 2016. Criteria of efficiency for conformal prediction. In: Symposium on Conformal and Probabilistic Prediction with Applications. Springer, pp. 23-39. Algorithmic learning in a random world. V Vovk, A Gammerman, G Shafer, Springer Science & Business MediaVovk, V., Gammerman, A., Shafer, G., 2005. Algorithmic learning in a ran- dom world. Springer Science & Business Media.
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[ "Weak localization of disordered quasiparticles in the mixed superconducting state", "Weak localization of disordered quasiparticles in the mixed superconducting state" ]
[ "R Bundschuh \nDepartment of Physics\nUCSD\nLa Jolla92093-0319CAU.S.A\n", "C Cassanello \nInstitut für Theoretische Physik\nUniversität zu Köln\nZülpicher Str. 77D-50937KölnGermany\n", "D Serban \nService de Physique Théorique\nCE-Saclay\nF-91191Gif-Sur-YvetteFrance\n", "M R Zirnbauer \nInstitut für Theoretische Physik\nUniversität zu Köln\nZülpicher Str. 77D-50937KölnGermany\n" ]
[ "Department of Physics\nUCSD\nLa Jolla92093-0319CAU.S.A", "Institut für Theoretische Physik\nUniversität zu Köln\nZülpicher Str. 77D-50937KölnGermany", "Service de Physique Théorique\nCE-Saclay\nF-91191Gif-Sur-YvetteFrance", "Institut für Theoretische Physik\nUniversität zu Köln\nZülpicher Str. 77D-50937KölnGermany" ]
[]
Starting from a random matrix model, we construct the low-energy effective field theory for the noninteracting gas of quasiparticles of a disordered superconductor in the mixed state. The theory is a nonlinear σ model, with the order parameter field being a supermatrix whose form is determined solely on symmetry grounds. The weak localization correction to the field-axis thermal conductivity is computed for a dilute array of s-wave vortices near the lower critical field Hc1. We propose that weak localization effects, cut off at low temperatures by the Zeeman splitting, are responsible for the field dependence of the thermal conductivity seen in recent high-Tc experiments by Aubin et al.
10.1103/physrevb.59.4382
[ "https://export.arxiv.org/pdf/cond-mat/9808297v3.pdf" ]
119,394,414
cond-mat/9808297
62435db334bca236ce77125f9382a838a2843b33
Weak localization of disordered quasiparticles in the mixed superconducting state 23 Feb 1999 (September 24, 1998) R Bundschuh Department of Physics UCSD La Jolla92093-0319CAU.S.A C Cassanello Institut für Theoretische Physik Universität zu Köln Zülpicher Str. 77D-50937KölnGermany D Serban Service de Physique Théorique CE-Saclay F-91191Gif-Sur-YvetteFrance M R Zirnbauer Institut für Theoretische Physik Universität zu Köln Zülpicher Str. 77D-50937KölnGermany Weak localization of disordered quasiparticles in the mixed superconducting state 23 Feb 1999 (September 24, 1998) Starting from a random matrix model, we construct the low-energy effective field theory for the noninteracting gas of quasiparticles of a disordered superconductor in the mixed state. The theory is a nonlinear σ model, with the order parameter field being a supermatrix whose form is determined solely on symmetry grounds. The weak localization correction to the field-axis thermal conductivity is computed for a dilute array of s-wave vortices near the lower critical field Hc1. We propose that weak localization effects, cut off at low temperatures by the Zeeman splitting, are responsible for the field dependence of the thermal conductivity seen in recent high-Tc experiments by Aubin et al. I. INTRODUCTION The long wave length physics of phases of matter with spontaneously broken symmetries is commonly described by an effective field theory for the relevant order parameter. For the problem of localization and transport in disordered metallic systems at low temperature, the appropriate "order parameter" is known [1] to be a supermatrix (or a matrix of dimension zero if the replica trick is used), conventionally denoted by Q. Three universality classes, differing by their behavior under time reversal and spin rotations, are widely known [2]. They are labeled by an index β = 1, 2, 4, and are traditionally referred to as the classes with orthogonal, unitary, and symplectic symmetry. We denote them by AI, A, and AII for short [3]. In each case, the field theory for Q belongs to the general family of nonlinear σ models. The field Q contains the Goldstone modes of a hidden symmetry [4] connecting retarded and advanced single-electron Green functions, which is broken by a nonzero density of states. At tree level one recovers the classical diffusion approximation, which neglects quantum interference corrections due to electron paths with loops. Anharmonic terms in the field theory represent "interactions" between the diffusion modes, giving rise to so-called weak localization corrections to diffusion. For the classes AI and A in dimension d ≤ 2, these interactions become strong at large distance scales and thus cause localization of all states, regardless of the strength of the disorder. In recent years it was found that the β = 1, 2, 4 classification is not exhaustive: for systems with symmetries of the particle-hole (ph) type, the invariance group of the order parameter field Q becomes enlarged in the vicinity of the ph-symmetric point. One instance of such symmetry enhancement are Dirac fermions in a random gauge field [5,6], another are disordered quasiparticles that exchange charge (but no energy) with a superconducting condensate [3]. Such systems exhibit novel spectral statistics and transport properties. Seven symmetryenhanced universality classes have been identified, and the corresponding order parameters Q were constructed from their random matrix limit in [7]. The main message is that Q lives on a symmetric space -in precise technical language: on a Riemannian symmetric superspacein all cases. The nonlinear σ models defined over such spaces are known [8] to be attractive under the flow of the renormalization group. Therefore the order parameter of a given disordered single-particle system, and in most cases its low-energy effective field theory as well, can be inferred quite simply by investigating the ergodic (or random matrix) limit. The present report focusses on class C, which emerges for noninteracting low-energy quasiparticles in a magnetic field and in contact with a spin-singlet superconductor. The defining condition [3] is that the quasiparticle Hamiltonian be invariant under SU(2) rotations of the electron spin, whereas time reversal invariance has to be broken. Since a superconductor screens magnetic fields, this universality class can only be realized in an inhomogeneous superconducting state -unless time reversal invariance is broken spontaneously. In the very recent literature the following realizations have appeared [9]: i) a metallic quantum dot in the form of a chaotic billiard, subject to a magnetic flux and bordering on a superconductor [10,11]; ii) quasiparticles in the core of an isolated vortex in a disordered s-wave superconductor [12]; and iii) a (quantum) disordered version of a d x 2 −y 2 superconductor with orbital coupling to a magnetic field [13]. The hallmark of class C is that, in contrast with the metallic class A, the weak localization correction does not vanish [14], in spite of the presence of a magnetic field. The persistence of weak localization in a field is caused by nonstandard modes of quantum interference that appear when impurity and Andreev scattering are simultaneously present. In a semiclassical picture, the effect can be understood as being due [3] to quasiparticle paths in which a loop is circled twice, with the charge states during the first and second looping being exactly opposite to each other. To identify the order parameter field Q and its lowenergy effective theory for class C, one may proceed in several ways. The direct method, due to [15] and worked out in detail for isolated vortices of an s-wave superconductor in Ref. [12], is to start from the BCS mean field Hamiltonian for the quasiparticles, set up a supersymmetric generating functional for the Gorkov Green function, introduce a composite field Q to decouple the 4-vertices produced by averaging over the disorder, integrate out the quasiparticle fields, solve two saddle point equations for Q in sequence (the second of which turns out to coincide with the Usadel equation [16]), and finally expand in gradients of Q to obtain the low-energy effective theory. The field theory so obtained is a nonlinear σ model, with Q taking values in a Riemannian symmetric superspace of type DIII|CI, in agreement with the random matrix analysis of [7]. Its coupling constant has the universal meaning of a conductivity for the conserved probability (or energy) current transported by the quasiparticles. Because quasiparticles also carry spin, the coupling constant may be reinterpreted [13] as a spin conductivity in the present context. (The latter interpretation fails for systems with spin-orbit scattering or magnetic impurities, where spin is not conserved.) Given the proper identification of the order parameter field Q, a few qualitative conclusions are immediate. According to the renormalization theory of nonlinear σ models [8], the sign of the one-loop renormalization group beta function in two space dimensions is completely determined by the sign of the (Ricci) curvature tensor relative to the metric tensor. Since the curvature of the Riemannian symmetric superspace of type DIII|CI is positive [17], the weakly coupled two-dimensional theory renormalizes by logarithmic corrections towards strong coupling (i.e. strong disorder), which ultimately leads to localization of all quasiparticle states at T = 0. This localized phase was called a "spin insulator" in [13]. In dimension d = 1 the same corrections are present, but with a linear dependence on the cutoff length. In d = 3 the theory supports a delocalization transition to a phase of extended states, the "spin metal" [13]. The addition of random classical Heisenberg impurity spins (at subcritical concentration, so as to maintain superconductivity) causes crossover from class C to class D [3], with the nonlinear σ model changing to type CI|DIII [7]. In the process, the sign of the symmetric space curvature gets reversed, whence weak localization turns into weak antilocalization, making it possible for extended states to exist already in two dimensions. Our goal here is to extend the treatment of [12] and illustrate some of the above general facts at the thermal transport of the class C quasiparticles of a disordered s-or d-wave superconductor in the mixed state [18]. We assume the presence of (nondescript) nonmagnetic impurities, which disorder the vortex array and cause elastic scattering of the quasiparticles. To tackle this problem we will use a coarse grained or random matrix type of approach, placing the emphasis on symmetry considerations. II. EFFECTIVE FIELD THEORY FROM AN N -ORBITAL MODEL We begin our treatment by partitioning the superconductor into cells of equal size, with each cell containing one vortex segment with a length of the order of the elastic mean free path ℓ. Within each cell we introduce (in the spirit of the real-space renormalization group) a basis of N quasiparticle wavefunctions that comprise the relevant low-energy configurations. The matrix of the Hamiltonian in such a basis assumes a sparse block structure, with one block on the diagonal for each cell, and with off-diagonal blocks that couple neighboring cells. If i labels the cells and a = 1, ..., N the orbitals inside a cell, the "coarse grained" Hamiltonian is of the form H = i,j ab h ia,jb (c † ia↑ c jb↑ + c † ia↓ c jb↓ ) +∆ ia,jb (c † ia↑ c † jb↓ − c † ia↓ c † jb↑ )/2 + h.c. , where the sum over i, j is restricted to i = j and pairs of neighboring cells. The spin-singlet nature (↑↓ − ↓↑) of the coupling to the pairing field is dictated by conservation of spin. Fermi statistics then requires the complex matrix ∆ to be symmetric: ∆ ia,jb = ∆ jb,ia [19]. If we temporarily suppress the cell and orbital indices, H can be written in the schematic form H = Tr(Hcc) + const, wherẽ c = c † ↑ c ↓ , c = (c ↑ c † ↓ ) , H = h T ∆ † ∆ −h . The symmetries of the Hamiltonian matrix H are summarized by the equation H = −CH T C −1 , with C being the symplectic unit C = iσ 2 ⊗ 1. Note that when the Zeeman energy H Z = µB ia (c † ia↑ c ia↑ − c † ia↓ c ia↓ )/2 is taken into account, the SU(2) spin rotation invariance of H is broken down to a U(1) symmetry. Disorder in the microscopic Hamiltonian gives rise to randomness in H. Because the universal properties at long wave lengths are insensitive to the microscopic details, we have considerable freedom in choosing the random Hamiltonian H. The simplest choice is an N -orbital model with locally gauge-invariant disorder of the type invented by Wegner [20] for the purpose of describing the universal physics of the Anderson localization transition for β = 1, 2, 4. The crucial new feature in the present case is the relation H = −CH T C −1 , which is invariant under symplectic transformations H → SHS −1 , S T CS = C. We therefore adopt a model with local Sp(2N ) gauge invariance: the elements of the matrix H are taken to be Gaussian distributed uncorrelated random variables with zero mean, H = 0, and second moments specified by TrAH ij TrBH kl = w ij 2N δ lk ij TrAB − δ kl ij TrACB T C −1 where δ kl ij = δ ik δ jl , and w ij is a rapidly decreasing function of the distance between the cells i and j. Aside from respecting the symmetries and locality of the Hamiltonian, this choice has the virtue of maximizing the information entropy. The main benefit from using such a maximum entropy model is that the introduction of the supermatrix Q, usually a tricky step that requires some expertise, becomes straightforward as we now proceed to show. We replace the operatorsc, c by classical fieldsψ, ψ: c iaα c jbβ → σψ iaα,σ ψ σ,jbβ and integrate bilinears iñ ψ, ψ against exp iTr(H − E)ψψ in the usual way to generate the Gorkov Green function at energy E. The introduction of a bosonic partner (σ = B) for each fermionic field (σ = F) serves to cancel vacuum graphs by the mechanism of supersymmetry. There is one complication, however: the matrixψψ does not share the symplectic symmetry of the Hamiltonian. To remedy this mismatch, we introduce an extra quantum number ("pseudo charge") c = ±1, so that the quasiparticle fields expand to tensorsψ iaα,σc and ψ σc,iaα [7]. On imposing the conditionsψ = Cψ T γ −1 and ψ = −γψ T C −1 , where γ is a real orthogonal matrix that will be specified shortly, we have the symmetryψψ = −C(ψψ) T C −1 as desired. Since the order parameter Q is a local field, its nature can be uncovered by looking at the Hamiltonian truncated to a single cell. With this truncation temporarily in force, we introduce Q as follows: dH exp(−N TrH 2 /2w 0 + iTrHψψ) = dQ exp(−N STrQ 2 /2w 0 + iSTrQψψ) . The equality is verified by using the cyclic invariance of the (super)trace: Tr(ψψ) 2 = STr(ψψ) 2 . The Hubbard-Stratonovitch field Q is a 4 × 4 supermatrix which, by its coupling to ψψ, inherits the symmetry Q = −γQ T γ −1 .(1) The constraints relatingψ and ψ to one another are compatible only if γ 2 equals the superparity matrix (+1 on bosons, −1 on fermions). To meet this condition we put γ = E BB ⊗ σ 1 + E FF ⊗ iσ 2 = σ 1 0 0 iσ 2 . Relation (1) is the defining equation of an orthosymplectic Lie algebra and is invariant under Q → T QT −1 with γ(T −1 ) T γ −1 = T ∈ OSp(2|2). In the general case, where Green functions at n different energies are to be averaged, Q acquires matrix dimension 4n × 4n, and the symmetry group gets enlarged to OSp(2n|2n) ≡ G. Returning now to the full lattice problem, introducing Q i for every cell i and integrating over the quasiparticle fields ψ,ψ we arrive at the following action functional: S/N = ij w −1 ij STrQ i Q j /2 + i STr ln(Q i − ω ⊗ Σ 3 ) where (Σ 3 ) σc,σ ′ c ′ = δ σσ ′ (σ 3 ) cc ′ and ω is a diagonal matrix containing the energies at which the quasiparticle Green functions are to be evaluated. Variation of S yields the saddle point equation j w −1 ij Q j = (ωΣ 3 − Q i ) −1 , whose physical solution (dictated by causality of the Green function) at ω = 0 and homogeneous in space is Q 0 = ivΣ 3 with v −2 = j w −1 ij . Low-energy fluc- tuations result from setting Q i = T i Q 0 T −1 i and taking T i ∈ G to vary slowly with the position of the cell i. By expanding in gradients, the low-energy effective action for such configurations at ω = 0 is easily seen to be a nonlinear σ model, S 0 = − πν 8 d 3 x STr D ⊥ (∇ ⊥ Q) 2 + D (∇ Q) 2 ,(2) where we have switched to continuous coordinates x ⊥ = (x, y) and x = z. The parameter ν is the density of states of the superconductor, and D , D ⊥ are the fieldaxis and transverse effective diffusion constants of the quasiparticle gas. We are using unitsh = 1. At finite ω, the field theory action is perturbed by a term S ω = iπν 2 d 3 x STrωΣ 3 Q , S = S 0 + S ω .(3) We have rescaled the field to Q = T Σ 3 T −1 . Since this expression for Q is invariant under T → T k for k = Σ 3 kΣ 3 ∈ GL(n|n) ≡ K, the supermatrix Q lives on a coset space G/K. If we parametrize Q by Q = exp 0 X X 0 Σ 3 , positivity of S 0 or equivalently, stability of the functional integral, requiresX BB = +X † BB andX FF = −X † FF . In invariant mathematical language, this means Q BB ∈ SO * (2n)/U(n) [21] and Q FF ∈ Sp(2n)/U(n), which are symmetric spaces of type DIII and CI -hence the name DIII|CI for the present nonlinear σ model. The same effective theory (restricted to the FF sector due to the use of fermionic replicas) was obtained in Ref. [13], based on a quasiparticle Hamiltonian for a dirty d x 2 −y 2 superconductor with orbital coupling to a magnetic field. This is no surprise, as that system belongs to symmetry class C and the order parameter field Q and its low-energy effective theory are determined solely by symmetry. (Incidentally, the classification scheme of [3] assigns the quasiparticles of the d x 2 −y 2 superconductor in zero field to class CI. According to [7], the corresponding symmetric superspace is D|C, also in agreement with the findings of Ref. [13].) In order to break parity and account for the Hall angle, one would need to add to the Lagrangian a topological density proportional to ǫ kl STrQ∂ k Q∂ l Q, which is closely related to Pruisken's θ term [22] well known from the theory of the integer quantum Hall effect. In two dimensions this term integrates to a winding number and is nontrivial, since the fundamental group of U(n) is Π 1 (U(n)) = Z and there exists the topological identity Π 1 (U(n) = Π 2 (Sp(2n)/U(n)). However, such a topological term does not affect the results for the longitudinal spin and thermal conductivities presented below and will therefore be omitted. The maximum entropy derivation presented here does not supply microscopic expressions for the couplings νD and νD ⊥ . (We can express them in terms of the random matrix parameters w ij , but this is neither illuminating nor useful.) These parameters can either be calculated from (quasi)classical transport theory [23] or, better yet, taken from experiment. In the latter case we extract the (bare) coupling constants from experiments conducted at temperatures high enough so that the transport is classical, and then use the field theory (2,3) to predict the quantum corrections that emerge at lower temperatures. III. WEAK LOCALIZATION IN CLASS C The field theory (2) does not apply to charge transport, as the condensate carries charge and quasiparticle charge is not a constant of motion. The energy, however, and for class C also the spin of a quasiparticle are conserved, which allows to probe for quasiparticle transport and localization by measuring the thermal and spin transport. To obtain the relevant transport coefficients we start from the bilocal conductivity tensor τ ll (x, x ′ ; E) = αα ′ v (x) lα G R αα ′ (x, x ′ ; E)v (x ′ ) lα ′ G A α ′ α (x ′ , x; E), which describes the nonlocal linear response of the spin current to a perturbation due to the Zeeman coupling with an applied field. The quantities G R and G A are the retarded and advanced Gorkov Green functions and v lα = i(σ 3 ) αα ( ← ∂ l − → ∂ l ) − 2eA l /2m is the l-component of the velocity operator. We use the relation H = −CH T C −1 to express G A at energy E by G R at energy −E [12]. Disorder averaging and the mapping on the nonlinear σ model with n = 2 then turn the tensor τ ll into a correlation function of the conserved OSp(4|4) Noether current J = (Q∇Q) B11,B22 of the field theory: τ ll (x, x ′ ) = (πνD l ) 2 J l (x)J l (x ′ ) .(4) The second superscript in the expression for the Noether current refers to the pseudo charge, while the third distinguishes between the two Green functions. The symmetry breaking perturbation due to the quasiparticle en-ergy E is incorporated into the formalism by setting ω = diag(E + , −E − ), where E ± = E ± i0. Next, let σ ll = (2π) −1 τ ll (x, x ′ ) d 3 x ′ be the local "spin" conductivity for quasiparticles with fixed spin up or down. To compute this quantity from the correlator (4), we adopt a rational parametrization for Q, Q = 1 Z Z 1 1 0 0 −1 1 Z Z 1 −1 . Inserting this parametrization into the field theory action (2), and doing the functional integral in Gaussian approximation (tree level), we obtain σ ll = σ 0 ll with σ 0 ll = νD l , which is the result expected from quasiclassical transport theory. The weak localization correction to τ ll (x, x ′ ) arises from one-loop graphs of the kind shown in Fig. 1; see [12]. The basic element of this graph is a 4-vertex representing the fourth-order term in the Taylor expansion of S with respect to Z,Z. A double line oriented by an arrow stands for the bare propagator ZZ 0 . All one-loop graphs are composed of three propagators and one 4-vertex. Although these graphs appear as a calculational device for organizing the field-theoretic perturbation expansion, they do have a direct physical meaning, as follows. Each of the two single lines in Fig. 1 stands for a Feynman path contributing to the Gorkov Green function G R (x, x ′ ; ±E). Double lines represent sums of impurity ladders with an arbitrary number of Andreev scattering events inserted. It is seen from Fig. 1 that one of the two Green function lines proceeds directly from the point x ′ to the point x, whereas the other one makes an excursion in the form of a double loop. The propagator associated with the double loop is called the D-type cooperon [3]. What is essential here is that the charge of the quasiparticle during the second looping is exactly opposite to the charge during the first looping. This feature makes the D-type cooperon stable with respect to disorder averaging irrespective of the orbital coupling to a magnetic field, by canceling the Aharonov-Bohm phase A · dl accumulated in the loop. Fig. 1 also indicates the fact [10] that the present variant of the weak localization phenomenon already affects a single Green function and thus the density of states. By evaluating the one-loop graphs in a similar manner as in Ref. [12], we obtain δσ ll = − D l π Re d 3 k (2π) 3 D k 2 + D ⊥ k 2 ⊥ + 2iE −1 .(5) The full spin conductivity is σ ll = σ 0 ll + δσ ll + . . .. Note that the correction is formally similar [24] to that for class AI, except that it explicitly depends on energy. In fact, it disappears with increasing excitation energy, or temperature, in agreement with the fact [12] that moving up in energy causes crossover from class C to class A, where weak localization is absent. In dimension d ≤ 2 the integral over wave numbers is cut off in the infrared by the inverse of the dephasing length L ϕ due to inelastic (or quasielastic [25]) scattering, while for dimension d ≥ 2 it is UV-regularized by the inverse elastic mean free path. Next recall that the quasiparticle spin is assumed to be conserved, which allows to consider the sectors with spin up (s = +1/2) and spin down (s = −1/2) separately. Turning on the Zeeman coupling is equivalent to shifting the excitation energy E → E − sµB. As a result, the energy dependence of the weak localization correction translates into a field dependence. Note that this effect differs from weak localization in disordered metals [24], where the orbital coupling to a magnetic field causes class AI to cross over to class A. In that case, the field scale is set by B O = (eDτ ϕ ) −1 with τ ϕ being the dephasing time. In the present case the relevant field scale is B Z = (µτ ϕ ) −1 . To compute the thermal conductivity κ at temperature T , we use the relation κ ll = s=±1/2 ∞ 0 σ ll (E − sµB) ∂f T ∂T (E) E dE , where f T (E) = (1 + e E/T ) −1 is the Fermi-Dirac distribution, and our unit of temperature is such that k B = 1. If the energy dependence of σ 0 can be neglected in the range 0 < E < ∼ T , and if T < ∼ Max(µB, Γ ϕ ), i.e. δσ is cut off by the Zeeman energy or the dephasing rate Γ ϕ = τ −1 ϕ , rather than by the temperature, we may pull out σ ll (E) from under the integral sign, thus obtaining an analog of the Wiedemann-Franz law: κ ll (B) T = π 2 3 σ ll (µB) .(6) Here we have combined the spin up and spin down contributions, by assuming the quasiclassical term σ 0 ll to be unaffected by the Zeeman splitting. IV. ISOLATED VORTICES We now specialize to an extreme type-II s-wave superconductor in a weak magnetic field (but well into the mixed state so that the field is approximately homogeneous), where quasiparticles are bound to a dilute array of vortex cores and the amplitude to hop from one vortex to another is negligibly small. In this case the problem reduces to a set of decoupled one-dimensional theories, one for each vortex, and we formally set D ⊥ = 0. The parameters of the one-dimensional nonlinear σ model were calculated by solving the Usadel equation for a single vortex in [12], where we found D = C 2 v F ℓ/3C 1 , and ν d 2 x ⊥ = 2ν N πξ 2 C 1 if the integral extends over the area occupied by one vortex. The parameter ξ is the dirty coherence length, ν N is the density of states of the normal metal, and C 1 = 3.16 and C 2 = 1.20 are numerical constants dependent on the vortex profile. Using the fact that the total number of vortices equals the transverse area of the sample divided by half the square of the magnetic length l B = 2π/eB, we obtain σ 0 = 4πC 2 ν N (ξ/l B ) 2 v F ℓ/3 for the quasiclassical limit of the spin conductivity. The weak localization correction is given by δσ = − 2D πl 2 B Re dk 2π D k 2 + Γ ϕ + 2i(E − sµB) −1 , where inelastic events were incorporated by shifting the denominator by Γ ϕ . This result applies when the dephasing length L ϕ = D /Γ ϕ is shorter than the vortex length L . In the opposite, mesoscopic regime (L ≪ L ϕ ) the weak localization effect was worked out in [12]. The low-temperature behavior of the thermal conductivity depends on how Γ ϕ varies with T . If we assume a power law Γ ϕ ∼ T p with exponent p < 1 [25], then σ becomes constant in the energy range where df T (E)/dT is essentially different from zero, and we get the Wiedemann-Franz law (6) with σ (µB) = σ 0 − (πl 2 B ) −1 Re D /(Γ ϕ + iµB) . In the high-field regime µB > ∼ Γ ϕ the weak localization correction to the thermal conductivity is cut off by the Zeeman splitting, giving a characteristic dependence δκ /T ∼ −1/ √ B. On the other hand, if p > 1 then the relevant low-T regime is T ≫ Γ ϕ , and the weak localization effect is cut off by T for low fields. In that case, one finds κ /T = π 2 3 σ 0 − 3 4 π 2 ( √ 2−1)ζ(3/2)L T /πl 2 B , i.e. the quantum correction is determined by the thermal length L T = D /T . The above considerations apply to a vortex array in the dilute limit near H c1 . As the field is increased, the quasiparticle hopping rate between vortices in an swave superconductor grows strongly. When the field is tuned close to H c2 , where the system of vortex cores becomes dense, the diffusion constant D ⊥ gets large and the anisotropic field theory (2) three-dimensional. Since the quasiparticle states of the weakly disordered threedimensional system are extended, a delocalization transition must take place with increasing field. Note that this transition is not in a new universality class, as the breaking of spin rotation invariance by the Zeeman coupling reduces class C to class A [13]. Nevertheless, the occurrence of such a delocalization transition may be of experimental interest, for it can be observed by varying the magnetic field (instead of the disorder strength or the chemical potential). V. WEAK LOCALIZATION IN THE CUPRATES We now adapt our results to the very interesting case of quasi two-dimensional d-wave superconductors such as the cuprates. As was stated before, the low-energy quasiparticles of a dirty d-wave superconductor in zero magnetic field belong to symmetry class CI. Weak localization effects in that class arise from two distinct modes of quantum interference [3]: the cooperon of type A, and the cooperon of type D. The former is the natural analog of the cooperon mode well known from the theory of disordered metals [24]. When time reversal symmetry is broken by a magnetic field penetrating the superconductor, the A-type cooperon becomes massive and disappears over a scale given by B O = (eDτ ϕ ) −1 . This crossover takes class CI into class C, while leaving weak localization due to the D-type cooperon intact. As we have seen, the latter mode is cut off only by the Zeeman energy, which becomes effective over the characteristic field scale B Z = (µτ ϕ ) −1 . Using µ = 2µ B = e/m and D ∼ k F ℓ/m we see that the two scales are separated by a large factor: B Z /B O ∼ k F ℓ, i.e. the elimination of the A-type cooperon by the orbital coupling to the magnetic field takes place at much smaller fields than does the removal of the D-type cooperon by the Zeeman energy. This justifies our explicitly retaining the Zeeman coupling, while burying the orbital coupling via the introduction of a maximum entropy model. In the following, we take the magnetic field to be applied along the c-axis, and assume the system to be well into the mixed state so that the field is approximately homogeneous. The cuprates are highly anisotropic materials, consisting of weakly coupled CuO 2 planes, for which D ≡ D c ≪ D ab ≡ D ⊥ . At weak interlayer coupling, the continuum approximation leading to (2) is not justified in the c-direction, and we need to restore the discrete layer structure. This is done by making the replacement D k 2 → 2t c (1 − cos k a), where a is the distance between layers and t c is the interlayer hopping rate. Then, by performing the integral in (5) over the domain L −1 ϕ < |k ⊥ | < ℓ −1 and −π/a < k < π/a, we obtain δσ ⊥ = −(2π 2 a) −1 Re ln F s (Γ)/F s (Γ ϕ ) ,(7)F s (ε) = ε + 4t c + 2i(E − sµB) + ε + 2i(E − sµB) , where Γ = D ⊥ /ℓ 2 and Γ ϕ = D ⊥ /L 2 ϕ are the elastic and inelastic scattering rates. To evaluate the consequences of this general formula, one needs to distinguish cases. For brevity, we concentrate on the limit defined by the condition that elastic scattering sets the largest energy scale: Γ ≫ Max(4t c , 2E, µB). Consider first the case 4t c < ∼ Max (2E, µB), which physically means that the coherence of the quantum interference modes is destroyed before quasiparticles have a chance to hop between layers. The layers then effectively decouple, yielding a twodimensional system, and the formula for δσ ⊥ becomes δσ ⊥ = −(4π 2 a) −1 Re ln Γ/(Γ ϕ + 2iE − 2isµB) . The appearance of a logarithm is characteristic of weak localization in two dimensions. For the in-plane thermal conductivity we get δκ ⊥ (B)/T = −(12a) −1 Re ln Γ/(Γ ϕ + iµB) , provided that the conditions of validity of the Wiedemann-Franz law (6) are satisfied. Note that in contrast with three-dimensional metals, where weak localization is a rather minute effect, the correction here can easily exceed 10% under experimental conditions. This is because the relative size δκ/κ is roughly given by the inverse of the dimensionless intralayer coupling constant 2πν 2d D ⊥ , whose value in zero field has been estimated [26,13] to be not much in excess of unity. In the opposite limit, where 4t c is much larger than µB and Γ ϕ , but still smaller than Γ, the field dependence of the weak localization correction to the thermal conductivity becomes three-dimensional: δκ ⊥ (B) T = − 1 6a ln Γ/t c − Re (Γ ϕ + iµB)/4t c , where again the law (6) was assumed. Note that the above expressions for δκ ⊥ (B) increase with B. To summarize, weak localization in class C, cut off by the Zeeman splitting, causes the thermal conductivity to increase with the magnetic field at sufficiently low temperatures. To make this more quantitative, we need to specify the field/temperature range where the effect becomes observable. The answer is provided by the value of the spin magnetic moment of the electron (with a g-factor of 2), which is 1.35 K/T in suitable units. As a result, if the field strength is of the order of 1 Tesla, the weak localization induced field dependence sets in at temperatures below one 1 Kelvin (unless for some unexpected reason the dephasing rate Γ ϕ is anomalously large). We now wish to elucidate whether such an effect might already be visible in recent experiments. The discussion is somewhat complicated by an ongoing debate concerning the leading, quasiclassical term κ 0 . Let us summarize the current situation as we see it. Krishana et al. [27] measured the magnetic field dependence of the thermal conductivity in a BSCCO system for temperatures T ≥ 6 K. After an initial decrease at weak fields, they observed a sharp kink at B * ∼ √ T , followed by a wide plateau for B > B * . The nonanalyticity at B * has been interpreted [28] as a phase transition to a new ground state with a secondary id xy order parameter. We will not be particularly concerned with that issue here. (The addition of an id xy component to the order parameter is fully compatible with the symmetries of class C and, if disorder is present, the field theory (2) for the quasiparticle excitations remains qualitatively unchanged.) From the observation of field independence over a sizable range of temperatures, one deduces [27] that both the electronic and the phonon contribution to the thermal conductivity must be individually constant. The constancy of the electronic part was initially attributed to the d x 2 −y 2 + id xy state being fully gapped, i.e. to the complete absence of low-energy quasiparticle excitations. This explanation has been challenged by experimental data of Aubin et al. [29]. While confirming the results of Ref. [27] for T > 5 K, these data reveal the emergence of a positive thermal magnetoconductance at lower temperatures T < ∼ 1 K. (The data also show pronounced hysteresis effects whose interpretation remains controversial.) Taking the constancy of the phonon contribution for granted, the observation of such dependence strongly indicates a residual density of quasiparticle states at zero energy. The existence of such states is no surprise. Indeed, in the mixed state of a superconductor with d x 2 −y 2 wave symmetry a residual density of states is expected even in the absence of disorder, because some fraction of the lowenergy quasiparticles (close in momentum to the d-wave nodes) are Doppler shifted to zero energy by the supercurrent circulating around the vortices (the Volovik effect [30]), which leads to ν(E = 0, B) = ν N B/B c2 . (For a recent discussion of the same effect for a ground state with d x 2 −y 2 +id xy symmetry, see [31].) The residual density of states created by this mechanism is approximately constant in energy below the average Doppler shift scale E B , roughly estimated by E B /T c ≃ B/B c2 . Disorder can only broaden the range of energy independence of ν. Hence, assuming B c2 ∼ 100 T and a superconducting transition temperature T c ∼ 100 K, the energy scale E B is of order 10 K for fields of a magnitude of about 1 T. Now recall the experimental observations reported in [29]: an electronic thermal conductivity which is independent of the magnetic field for T > ∼ 5 K (and H > H * ), and begins to increase with B below T ≃ 1 K. (According to a footnote in [29], the same effect has been seen in YBCO.) The first point to address is the field independence at the higher temperatures. Franz [32] has recently proposed a model for the quasiclassical thermal conductivity κ 0 , in which the increase of ν with the field is exactly canceled by a concomitant decrease of the quasiparticle mean free path ℓ. The model assumes scattering from the superflow due to randomly positioned vortices. In contrast, another recent theory [33] argues in favor of the dominant scattering mechanism being impurities close to the unitarity limit. We will not pursue here the discussion as to which is the correct model to use. With the microscopic theory of the plateau effect being a subject of debate, our philosophy is to accept it as an experimental fact that the field variation of ν and ℓ is such as to cancel in κ 0 ∼ ν(B)ℓ(B). The question to address, then, is why a field dependence sets in when the temperature is lowered. We argue that this is at least in part due to weak localization. As we have seen, weak localization in the mixed state of dirty d-wave superconductors is a phenomenon on safe theoretical ground, is sizable in magnitude, and is expected to occur at the right temperature and field scales to match the experiment [29]. To preclude any confusion, we stress that the effect under consideration is distinct from weak localization in combination with Aslamasov-Larkin fluctuations, which have been invoked in [34] to explain the negative thermal magnetoconductance observed in a dirty LSCO system at much higher temperatures. Theories proposed by previous authors attribute the temperature variation of dκ(B, T )/dB to the leading (quasiclassical) term, κ 0 ∼ νℓ. Given the low-energy constancy of the density of states, such a variation would have to arise from an energy (or temperature) dependence of the elastic mean free path. Possible explanations are: i) low-energy transparency of d-wave vortices to quasiparticles [32], and ii) energy-dependence of the elastic scattering rate due to impurities near the unitarity limit [33]. The challenge to these scenarios is to explain why for fields B ∼ 1 T the effect sets in at temperatures around 1 K. In the weak localization scenario we have described, this comes about very naturally if Γ ϕ is determined by thermal broadening, since µ = 1.35 K/T. A clear difference is that weak localization effects continue to be enhanced with decreasing temperature -they ultimately drive the system to an insulator by localization of all quasiparticle states -whereas the energy dependence of the elastic mean free path saturates. To discriminate, it is therefore desirable to push the experimental measurements to the lowest temperatures possible. In order to achieve a quantitative description based on formula (7), it will be necessary to take the field dependence of ℓ into account. Our suggestion is to extract the density of states ν(B) from measurements of the specific heat, and then deduce ℓ(B) from the quasiclassical formula κ(B) ∼ ν(B)ℓ(B), valid at high temperatures (T > ∼ 5 K). As far as the temperature dependence of the dephasing rate Γ ϕ is concerned, a phenomenological model needs to be used. To our knowledge, a theory for this quantity in the mixed state of dirty d-wave supercon-ductors does not exist. In the long run, weak localization may turn out to be the appropriate tool to measure Γ ϕ , as is established practice in disordered metals [24,25]. VI. CONCLUSION Noninteracting electrons subject to disorder and a magnetic field are well known to belong to the standard universality class A (unitary symmetry, β = 2). When spin-singlet pairing correlations are added, the universality class of the low-energy quasiparticles changes to type C. It has been shown that the transport properties of these quasiparticles are unconventional. In particular, there exist modes of destructive quantum interference which survive the orbital coupling to a magnetic field. They are cut off at higher fields by the Zeeman coupling, thereby giving rise to a field dependent quantum (or weak localization) correction to the low temperature thermal conductivity, with the characteristic scale given by µ = 1.35 K/T. A good place to look for such corrections experimentally are disordered low-dimensional superconductors, such as the cuprates, in the mixed state. On general symmetry grounds, the low-energy effective field theory for quasiparticles in class C is predicted to be a nonlinear σ model of type DIII|CI. The Lagrangian of this field theory has a universal form, independent of the symmetry of the order parameter (s, d, etc.), as long as the superconductor conserves the quasiparticle spin and is penetrated by magnetic flux. The role of the superconducting ground state is merely to determine the values of the field theory coupling constants, their anisotropy, and their dependence on energy and magnetic field. Quantitative predictions for the weak localization corrections to transport can be made once the values of the couplings and their dependences have been obtained, either from quasiclassical transport theory or from experiment. We advocate the use of such predictions in understanding the low-temperature experiments of Aubin et al. FIG. 1. One-loop diagram contributing to the correlator τ ll (x, x ′ ) . The electric charge of the quasiparticle during the second looping (−α) is opposite to the charge during the first looping (α). Acknowledgment. This research was supported in part by the Deutsche Forschungsgemeinschaft, SFB 341 (Köln-Aachen-Jülich). One of the authors (M.R.Z.) thanks A. Freimuth for a discussion. . K B Efetov, Adv. Phys. 3253K.B. Efetov, Adv. Phys. 32, 53 (1983). . F J Dyson, J. Math. Phys. 3140F.J. Dyson, J. Math. Phys. 3, 140 (1962). . A Altland, M R Zirnbauer, Phys. Rev. B. 551142A. Altland and M.R. Zirnbauer, Phys. Rev. B 55, 1142 (1997). . F J Wegner, Z. Phys. 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[]
[ "Dynamic shapes of floppy vesicles enclosing active Brownian particles with membrane adhesion", "Dynamic shapes of floppy vesicles enclosing active Brownian particles with membrane adhesion" ]
[ "Priyanka Iyer ", "Gerhard Gompper ", "Dmitry A Fedosov " ]
[]
[]
Recent advances in micro-and nano-technologies allow the construction of complex active systems from biological and synthetic materials. An interesting example is active vesicles, which consist of a membrane enclosing self-propelled particles, and exhibit several features resembling biological cells. We investigate numerically the behavior of active vesicles, where the enclosed self-propelled particles can adhere to the membrane. A vesicle is represented by a dynamically triangulated membrane, while the adhesive active particles are modelled as active Brownian particles (ABPs) that interact with the membrane via the Lennard-Jones potential. Phase diagrams of dynamic vesicle shapes as a function of ABP activity and particle volume fraction inside the vesicle are constructed for different strengths of adhesive interactions. At low ABP activity, adhesive interactions dominate over the propulsion forces, such that the vesicle attains near static configurations, with protrusions of membrane-wrapped ABPs having ring-like and sheet-like structures. At moderate particle densities and strong enough activities, active vesicles show dynamic highly-branched tethers filled with string-like arrangements of ABPs, which do not occur in the absence of particle adhesion to the membrane. At large volume fractions of ABPs, vesicles fluctuate for moderate particle activities, and elongate and finally split into two vesicles for large ABP propulsion strengths. We also analyze membrane tension, active fluctuations, and ABP characteristics (e.g., mobility, clustering), and compare them to the case of active vesicles with non-adhesive ABPs. The adhesion of ABPs to the membrane significantly alters the behavior of active vesicles, and provides an additional parameter for controlling their behavior. arXiv:2301.07952v1 [cond-mat.soft]
10.1039/d3sm00004d
[ "https://export.arxiv.org/pdf/2301.07952v1.pdf" ]
255,999,857
2301.07952
5fd3affb00f5b3e26c0e812897c0f9f33aea55ad
Dynamic shapes of floppy vesicles enclosing active Brownian particles with membrane adhesion Priyanka Iyer Gerhard Gompper Dmitry A Fedosov Dynamic shapes of floppy vesicles enclosing active Brownian particles with membrane adhesion (Dated: January 20, 2023) Recent advances in micro-and nano-technologies allow the construction of complex active systems from biological and synthetic materials. An interesting example is active vesicles, which consist of a membrane enclosing self-propelled particles, and exhibit several features resembling biological cells. We investigate numerically the behavior of active vesicles, where the enclosed self-propelled particles can adhere to the membrane. A vesicle is represented by a dynamically triangulated membrane, while the adhesive active particles are modelled as active Brownian particles (ABPs) that interact with the membrane via the Lennard-Jones potential. Phase diagrams of dynamic vesicle shapes as a function of ABP activity and particle volume fraction inside the vesicle are constructed for different strengths of adhesive interactions. At low ABP activity, adhesive interactions dominate over the propulsion forces, such that the vesicle attains near static configurations, with protrusions of membrane-wrapped ABPs having ring-like and sheet-like structures. At moderate particle densities and strong enough activities, active vesicles show dynamic highly-branched tethers filled with string-like arrangements of ABPs, which do not occur in the absence of particle adhesion to the membrane. At large volume fractions of ABPs, vesicles fluctuate for moderate particle activities, and elongate and finally split into two vesicles for large ABP propulsion strengths. We also analyze membrane tension, active fluctuations, and ABP characteristics (e.g., mobility, clustering), and compare them to the case of active vesicles with non-adhesive ABPs. The adhesion of ABPs to the membrane significantly alters the behavior of active vesicles, and provides an additional parameter for controlling their behavior. arXiv:2301.07952v1 [cond-mat.soft] I. INTRODUCTION In recent years, there has been a growing interest in a variety of active matter systems which operate far from equilibrium and show rich dynamical behaviors and functions [1][2][3][4]. Examples include biological systems ranging from cells to tissues [5,6], collections of micro-swimmers [7,8], and active engineered systems [9,10]. The growing research interest has been nurtured by rapid developments in microscale and nanoscale technologies which already allow for a well-controlled construction of complex multicomponent active systems and materials [9,11,12]. A prominent example is cell-mimicking systems, which are generally constructed from cell-based biological constituents, and include active nematics made of driven biofilaments [13,14], and growing and dividing dropletbased or vesicle-based compartments [15,16]. In many other examples, biological materials are combined with active synthetic constituents with a hope to mimic various biological systems or even go beyond their functionality [17,18]. Here, an interesting example is a closed membrane enclosing biological micro-swimmers such as bacteria [19][20][21] or synthetic self-propelled particles [18,[22][23][24][25]. Active components inside the soft confinement exert forces on the surface, leading to highly dynamic nonequilibrium shape changes which resemble certain processes in living cells such as the formation of filopodia * Theoretical Physics of Living Matter, Institute of Biological Information Processing and Institute for Advanced Simulation, Forschungszentrum Jülich, 52425 Jülich, Germany Email: [email protected], [email protected], [email protected] and lamellipodia [5,26,27], and active shape fluctuations of the membrane [28][29][30]. The main features that differentiate active vesicles from various membrane structures in equilibrium [31,32] are active force generation due to the enclosed active components and dynamic shape changes of the membrane. For instance, swimming bacteria or motile synthetic particles within a vesicle induce the formation of tethers and protrusions which dynamically elongate and retract [18][19][20]. In equilibrium, string-of-pearls-like and tubular protrusions can be formed by amphipathic peptides or BAR domain proteins [32,33], but these structures are static and correspond to a minimum of total energy. Therefore, different physical mechanisms govern the formation of various membrane structures in equilibrium and in non-equilibrium active vesicles. In particular, the curvature-induced clustering of active particles [34][35][36] at the membrane leads to the concentration of active forces at spots with a high curvature. Moreover, there exists a positive feedback mechanism between the induction of strong curvature by active particles and their clustering in places with large curvature, so that the shape of active vesicles is altered dynamically and collectively [18,23]. Furthermore, active components within a vesicle give rise to a significant active tension due to the swim pressure exerted by the particles [37]. Apart from active forces, the deformation of a membrane can also occur as a consequence of adhesive interactions between the membrane and enclosed particles [38][39][40][41][42]. In particular, adhesive interactions result in partial or full wrapping of the particles by the membrane, which can significantly reduce the force required for tether formation. Furthermore, the adhesion of multiple particles to the membrane often induces membrane-mediated interactions between the particles, leading to a cooperative wrapping of particles by the membrane [41] and the formation of various particle structures at the membrane surface [43][44][45][46][47]. These interactions can enhance or reduce the clustering of active particles, potentially altering the behavior of active vesicles. In addition, it is plausible to expect that the adhesive interactions between the particles and the membrane can facilitate the existence of active forces away from the membrane (i.e., pulling forces), which is not possible for non-adhesive active particles which exert pushing forces toward the membrane. Finally, adhesive interactions of particles and pathogens with a cell membrane are essential for a variety of biological processes such as membrane translocation, viral budding, and phagocytosis [48][49][50][51]. In our study, we investigate numerically the combined effect of particle activity and adhesive interactions on the behavior of active vesicles. Fluid membrane vesicles are modeled as dynamically triangulated surfaces [52,53] enclosing a number of active Brownian particles (ABPs). Adhesive interactions between the ABPs and the membrane are incorporated through the Lenard-Jones potential, whose strength is varied to induce various degrees of ABP wrapping by the membrane. A phase diagram of dynamic vesicle shapes is constructed as a function of the ABP propulsion strength and the volume fraction of particles within the vesicle. The presence of ABP adhesion to the membrane leads to qualitative changes in the phase diagram in comparison to that for non-adhesive ABPs [18]. For a weak particle activity, the adhesion interactions dominate, yielding nearly static vesicle shapes, which are similar to those in equilibrium with only adhesive interactions present. For moderate particle activities and volume fractions, complex tether structures filled with string-like arrangements of ABPs are formed, and characterized by a number of branching points. In contrast, for non-adhesive ABPs, the formed tethers show no significant branching, and the ABPs generally cluster at the end of membrane tethers [18,23]. Finally, for a strong particle propulsion, active forces from the ABPs dominate over the adhesion interaction, and the resulting behavior of active vesicles is similar to those with non-adhesive ABPs. Also, membrane properties of the active vesicles and the characteristics of ABP clustering and mobility are analysed and compared to those of nonadhesive ABPs. The article is organized as follows. Section II provides all necessary details about the employed methods and models, including the parameters used in simulations. Section III A presents dynamic shape diagrams for two strengths of the ABP adhesion to the membrane. Membrane tension and the importance of ABP adhesion are discussed in Section III B. Vesicle shape fluctuations are analysed in Section III C, and ABP characteristics are presented in Section III D. Finally, we conclude in Section IV. II. METHODS AND MODELS An active vesicle is represented by a closed fluid membrane of spherical topology with radius R, enclosing N p active Brownian particles (ABPs). The activity of the particles is described by the dimensionless Peclet number Pe = σv p /D t , where σ is the particle diameter, v p is the propulsion velocity, and D t is the translational diffusion coefficient. Note that Pe is a measure of the propulsion force f p of ABPs, with v p = f p /γ p and D t = k B T /γ p , where γ p is the translational friction coefficient, so that Pe = f p σ/k B T . Particle volume fraction within the vesicle is given by φ = N p (σ/2R) 3 . Table I ABPs are modeled as active spherical particles without hydrodynamic interactions. Each ABP experiences a propulsion force f p that acts along an orientation vector e i . The force results in a propulsion velocity v p = f p /γ p . The orientation vector e i is subject to orientational diffusionė i = ζ i × e i , where ζ i is a Gaussian random process with ζ i (t) = 0 and ζ i (t)ζ j (t ) = 2D r δ ij δ(t − t ) with a rotational diffusional coefficient D r . D r is related to the ABP size σ and translational diffusion coefficient D t as D t = D r σ 2 /3. The ABPs repel each other, which is implemented through the repulsive part of the 12-6 Lennard-Jones (LJ) potential, with the potential minimum and cut-off at r p−p m = r p−p c = 2 1/6 σ for ABP-ABP interactions. Furthermore, the ABPs are attracted to the membrane, which is implemented by the full 12-6 LJ potential with a minimum at r p−m m = 2 1/6 σ/2. The potential cut-off for ABP-membrane interactions is set to r p−m c = 2 1/6 σ. B. Membrane model The vesicle is modeled by a dynamically triangulated membrane of spherical topology consisting of N v linked vertices [52,53]. The interaction between linked vertices is controlled via a tethering potential [52,54] that is a combination of attractive and repulsive parts U att (r) =    k b exp[1/(l c0 − r)] l max − r , if r > l c0 , 0, if r ≤ l c0 ,(1)U rep (r) =    k b exp[1/(r − l c1 )] r − l min , if r < l c1 , 0, if r ≥ l c1 .(2) Here, k b is the bond stiffness, l min and l max are the minimum and maximum bond lengths, and l c0 and l c1 are the potential cutoff lengths. The membrane bending elasticity is modeled by the Helfrich curvature energy [55], U bend = 2κ c c 2 dA,(3) where κ c is the bending rigidity andc = (c 1 +c 2 )/2 is the mean local curvature at the membrane surface element dA. In the discretized form, it becomes [56,57] U bend = 2κ c Nv i=1 σ ic 2 i ,(4) wherec i = n i · j(i) σ ij r ij /(2σ i r ij ) is the discretized mean curvature at vertex i, n i is the unit normal at the membrane vertex i, σ i = j(i) σ ij r ij is the area corresponding to vertex i (the area of the dual cell), j(i) corresponds to all vertices linked to vertex i, and σ ij = r ij (cot θ 1 + cot θ 2 )/2 is the length of the bond in the dual lattice, where θ 1 and θ 2 are the angles at the two vertices opposite to the edge ij in the dihedral. In practice, since the dihedral terms corresponding to σ ij are additive, the local curvature at each vertex can be calculated by summing over contributions from all triangles containing that vertex. The area conservation is imposed locally to each triangle by the potential U A = k a 2 Nt i=1 (A i − A l ) 2 A l ,(5) where N t = 2(N v − 2) is the number of triangles, A l = A 0 /N t is the targeted local area (A 0 is the total membrane area), A i is the instantaneous local area, and k a is the local-area conservation coefficient. We do not impose any volume constraints, and therefore, the vesicle volume is free to change. Membrane fluidity is modelled by a stochastic flipping of bonds following a Monte-Carlo scheme. The bond shared by each pair of adjacent triangles can be flipped to connect the two previously unconnected vertices [52,57]. The flipping is performed with a frequency ω and probability ψ. [38]. Due to the long range of interactions between the particle and the membrane, the transition is gradual and the particle is only partially wrapped at c [41]. Partially wrapped states of a particle (black) by the membrane (red) for (b) = 2.5k B T and (c) = 3.5k B T . 1.0 A wrap / 2 (a) (b) (c) C. Equation of motion The system evolves in time according to the Langevin equation mr i = −∇ i U tot − γṙ i + 2γk B T ξ i (t),(6) where m is the mass of membrane particle or ABP,r i andṙ i represent the second and first time derivatives of particle positions, ∇ i is the spatial derivative at particle i, and U tot is the sum of all interaction potentials described above. The effect of a viscous fluid is mimicked by the friction co-efficient γ, whose value can be different for membrane particles and ABPs, see Table I. Thermal fluctuations are modelled as a Gaussian random process ξ i with ξ i (t) = 0 and ξ i (t)ξ j (t ) = δ ij δ(t − t ). Inertial effects are minimized by performing the simulations in the over-damped limit with m and γ such that τ t = m/γ < 1 τ r = D −1 r . The positions and velocities of all particles are integrated using the velocity-Verlet algorithm [59]. D. ABP adhesion and membrane wrapping Adhesion interactions between the ABPs and the membrane are mediated by the LJ potential whose strength is characterized by the potential depth . Adhesion strength determines the degree of particle wrapping by the membrane, with energetic costs due to membrane bending and tension. The ratio of the membrane bending modulus κ and the lateral tension λ defines a length l = κ/λ, below which membrane deformations are mainly controlled by the bending energy, while deforma-tions on length scales larger thanl are dominated by tension [39]. If tension is neglected and the membrane covers area A wrap ≤ πσ 2 of the particle (e.g., partial wrapping), the adhesion (E ad ) and bending (E bend ) energies are given by E ad = −ωA wrap , E bend = 8κA wrap /σ 2 ,(7) where ω is the adhesion strength per unit area. In this case, the minimum of the total energy corresponds to complete wrapping of the particle by the membrane (i.e., A wrap = πσ 2 ), which occurs for ω > ω min = 8κ/σ 2 [38]. Therefore, in the absence of membrane tension, the particle is in an unwrapped state for ω < ω min , while the particle is fully wrapped for ω > ω min with no energy barrier to overcome. However, in the presence of tension, particle adhesion shows a continuous transition from the unwrapped to partially wrapped state at ω min , while the transition to the fully wrapped state is discontinuous and has an energy barrier [39]. To relate the adhesion strength ω per unit area and the strength of the LJ potential in simulations, we consider the attraction of a single membrane vertex to an ABP, such that = 2ωA l with 2A l being the area of the vertex. For the parameters in Table I, the transition from the unwrapped to a wrapped state is expected at ω min = 8κ/σ 2 which implies c 4k B T . In our simulations, adhesive interactions between ABPs and the membrane are exerted up to a distance of σ/2 from the ABP surface, and are therefore long ranged. Theoretical predictions of particle wrapping for long-ranged adhesive interactions indicate that the transition to the fully wrapped state is gradual [41], which is consistent with the area A wrap of particle wrapping as a function of shown in Fig. 1(a). Thus, the fully wrapped state requires ad- Fluctuating Near-Equilibrium Near-Equilibrium hesion interactions with > c . For further simulations, we have selected two adhesion strengths of = 2.5k B T and = 3.5k B T , which correspond to a moderate degree of wrapping illustrated in Fig. 1(b,c). III. RESULTS A. Dynamic phase diagram Figure 2 presents phase diagrams of dynamic shape changes of active vesicles as a function of Pe and φ for two different adhesion strengths (see also Movies S1-S4). At small Pe 50, the formation of buds for low particle densities, ring-like aggregates of ABPs for intermediate φ values, and ABP aggregates with a hexagonal closed-packed (HCP) structure for large φ are observed and illustrated in Fig. 3 for Pe = 15. Some of these struc-tures have previously been observed in studies of passive particles adhering to a membrane [43][44][45][46]. Furthermore, for Pe 50, ABPs adhered to the membrane show little dynamics, suggesting that adhesive forces dominate over particle activity. As a result, active vesicles for Pe 50 are close to an equilibrium state, with ABP aggregate structures similar to those of equilibrium systems at Pe = 0. In the near-equilibrium regime at low ABP densities, both individually wrapped particles and short strings of several ABPs within membrane tubes [see Fig. 3(a)] are observed due to the competition between repulsive curvature-mediated interactions [47] and the in-plane motion from ABP propulsion. As the ABP volume fraction is increased, strong cooperative wrapping of ABPs is observed, as shown in Fig. 3(b-c). Here, it is likely that the gain in energy due to the cooperative wrapping overcomes the curvature-mediated repulsion. Cooperative wrapping of several particles is also enhanced by the interaction range of an adhesion potential [41]. Furthermore, the vesicle is free to change its volume in our simulations, and therefore, the area fraction of adhered membrane can be large. As a result, extreme deformations of the vesicle with protruding ring-like and sheet-like structures are observed and illustrated in Fig. 3(b-c). For the largest volume fraction of ABPs (φ = 0.18), membrane deformations are reduced [see Fig. 3(d)] in comparison to the cases of φ = 0.04 and φ = 0.12, because the gain in adhered membrane area is restricted at some point by the volume of the ABP content. Therefore, if the vesicle volume were constrained to near-spherical values, membrane deformations are expected to be reduced, since the gain in adhered membrane would be restricted by an increase in membrane tension due to the constrained vesicle volume. Note that for the lower adhesion strength of = 2.5k B T , membrane deformations are less pronounced than in the case of = 3.5k B T (see Fig. 2 for Pe 50) due to the competition between adhesion and bending energies. As Pe is increased, ABP propulsion starts to dominate over the adhesive forces, and the non-equilibrium nature of active vesicles becomes apparent. At low particle densities (φ 0.07), ABP activity leads to the formation of dynamic tether-like structures, which are filled by stringlike arrangements of ABPs. This behavior is qualitatively different from the tether formation by ABPs in the absence of adhesive interactions, where particle clustering takes place at the end of a tether [18,23]. Note that the string-like arrangement of particles in membrane tubes is favored by long-ranged adhesive interactions [41,60]. Another qualitative difference of the formed tethered structures by adhesive ABPs in comparison to those by nonadhesive active particles [18,23] is that the tethered structures in Fig. 2 are often highly branched. Since ABPs spend a considerable time in string-like configurations within membrane tethers, ABPs can change their orientation due to rotational diffusion and initiate branch formation from the existing tether. In the absence of adhesive interactions, ABPs quickly travel between the base of a tether and its end (or vise versa), and thus cannot easily initiate branched tethers [18,23]. Therefore, adhesive interactions promote the formation of branched tether structures and stabilize them. At the lower adhesion strength of = 2.5k B T , ABPs cluster more at tether ends than for the case of = 3.5k B T , and result in less branched structures, as shown in Fig. 2. A similar effect is observed with increasing particle activity (or Pe), suggesting that branched tether structures and string-like arrangements of ABPs are indeed a consequence of particle adhesion to the membrane, which is lost when ABPs have a sufficient force to detach from the membrane. Note that the tethering regime for adhesive ABPs occurs at significantly lower Pe numbers when compared to the non-adhesive ABP case [18,23] because particle adhesion facilitates wrapping, reducing the energy barrier required for the formation of tethers. At large particle densities (φ 0.07) and for Pe val-ues beyond the near-equilibrium regime, a fluctuating phase first develops, where shape changes of the vesicle are moderate and resemble membrane fluctuations. In Section III C, we will show that vesicle shape fluctuations for adhesive ABPs are different from those for the non-adhesive ABP case [18]. As Pe is further increased for φ 0.07, the ABPs form large clusters which can push in opposing directions and result in vesicle elongation or even splitting into two vesicles, similar to the non-adhesive ABP case [18]. Thus, the effect of adhesive interactions is prevalent only for low to intermediate Pe values, where the adhesive forces are larger than or comparable to ABP propulsion forces. B. Membrane tension The mean vesicle tensionλ computed from local membrane stresses (see Appendix A) for different adhesion energies is shown in Fig. 4(a). For small Pe 50 in the case of adhesive ABPs, the mean local tension of the vesicle is slightly negative, which indicates local compression of the membrane vertices. The local contraction of the membrane is facilitated by adhesive interactions, which are relatively long ranged, and favor the adhesion of more membrane vertices to the ABPs, leading to local compression of the membrane within the adhesion area. Furthermore, at small Pe, active particles generate a relatively low swim pressure at the membrane, so that the membrane tension remains slightly negative. Membrane tensionλ for > 0 in Fig. 4(a) exhibits two different regimes. For Pe < 100, the dependence ofλ is non-linear, while for Pe > 100,λ increases linearly with increasing Pe, similar to the case of = 0. For active vesicles with non-adhesive ABPs, the linear growth inλ is determined by the swim pressure of ABPs on the membrane, such thatλ/λ 0 = χPeφ, where λ 0 = R 2 k B T /(πσ 4 ) is a normalization factor and χ is the active tension weight related to the alignment of propulsion direction with the membrane normal [23]. Therefore, the linear regime ofλ for active vesicles with adhesive ABPs is also due to the swim pressure of ABPs on the membrane, because for large Pe, the ABP propulsion force dominates over adhesion interactions. However, the non-linear dependence ofλ for Pe < 100 and > 0 is due to the interplay of swim pressure and particle adhesion to the membrane. The location of the transition from the non-linear to the linear increase inλ with increasing Pe can be estimated using a simple model, where an adhesive particle placed at a distance z 0 from a flat membrane attempts to escape the surface, see Fig. 4(b). The attractive force exerted on the particle due to the membrane patch at a distance r = r 2 + z 2 0 with an area 2πrdr is given by dF = 24 2 σ 2r 12 − σ 2r 6 n z 0 r 2πr r dr,(8) where n = N v /4πR 2 is the number density of vertices at the membrane, and the factor z 0 /r is due to the projection of the force onto the normal-to-the-surface direction. When the propulsion direction e of the particle points away from the membrane along the normal, force balance implies Pe = σ k B T √ (r p−m c ) 2 −z 2 0 0 dF = 8π nσz 0 k B T 2r σ −12 − 2r σ −6 r p−m c z0 .(9) This expression allows the calculation of a maximum Pe required for ABP detachment from the membrane, yielding Pe max ≈ 385 for = 3.0k B T and z 0 1.14σ/2 (i.e., z 0 is the distance from the flat membrane at which the maximum in Pe is obtained). Here, the local curvature of the membrane is neglected, which would result in an increase of the detachment force. From simulations with a frozen membrane, the detachment of an ABP with = 3.0k B T takes place at Pe max ≈ 390, in good agreement with the analytical estimate. However, for a deformable membrane, thermal undulations lead to a steric repulsion of the ABP from the membrane [61], which causes a decrease in the detachment force. The repulsive force exerted on the ABP by the fluctuating membrane can be estimated in terms of Pe as [62,63] Pe noise = σ k B T (2πR eff ) c(k B T ) 2 κh 2 ,(10) where R eff = 2 1/6 σ/2 is the effective radius of the particle (here, the equilibrium distance between the particle and membrane vertices), h = z 0 − R eff 0.02σ/2 is the distance between the ABP surface and the membrane, and c is a constant in the range (0.01, 0.23) [61,[63][64][65]. Note that the magnitude of h in our case is similar to the average fluctuation amplitudeh σ of a flat tensionless membrane on a length scale of the particle size σ, whereh σ = 2k B T /κ/(2π) 3 σ/2 0.02σ/2. The range of c ∈ (0.01, 0.23) corresponds to a broad range of Pe noise ∈ (20, 500). Recent experiments of particle wrapping by a lipid membrane [63] suggest a much narrower range of c ∈ (0.03, 0.06), corresponding to Pe noise ∈ (65, 130). Taking the median value of Pe noise 100 for c = 0.045, Pe required for the ABP detachment becomes Pe detach = Pe max − Pe noise 290. From simulations of a single ABP adhered to a fluctuating membrane with = 3.0k B T , the detachment force corresponds to Pe 200. This Pe value is in a reasonable agreement with the theoretical estimate of Pe detach , taking into account that Pe noise is very sensitive to the choice of h and c. The transition from the non-linear to linear increase inλ in Fig. 4(a) corresponds to Pe ≈ 100. This value is lower than the theoretical estimate, which is likely due to the presence of frequent inter-ABP collisions at φ = 0.18, and enhanced membrane fluctuations facilitated by active particles (see Section III C). Interestingly, a shift between theλ curves for = 0 and > 0 in Fig. 4(a) also corresponds to about Pe 100. Since membrane tension is affected by the ABP adhesion, we also compute the fraction ρ of particles which are in a direct contact with the membrane. Figure 4(c) shows that ρ for adhesive ABPs is nearly twice larger than for non-adhesive particles at low Pe. As Pe is increased, ρ rapidly approaches unity for the cases with > 0 and levels off for Pe > 100, while in the absence of adhesion, ρ reaches a value of 0.88 only at Pe = 400. Therefore, adhesive interactions make a difference even at large Pe. Although the fraction ρ of near-membrane ABPs seem to follow the same trend for = 0 and > 0, the physical mechanisms are different. For = 0, an increase in Pe leads to an increase in the number of ABPs at the membrane due to activity-induced accumulation of ABPs at surfaces [34,66]. ABPs spend on average more time at the surface with increasing Pe, since the escape times decrease with decreasing rotational diffusion, leading to an increase in ρ. Furthermore, there exists a feedback mechanism between particle accumulation and membrane curvature [18,23], as the propulsion force exerted on the membrane induces a larger local curvature and ABPs accumulate in regions of the large curvature [35,36]. For the cases with > 0, this mechanism is also partially relevant, however, already at low Pe, most of the particles are located at the membrane due to adhesive interactions. The fraction ρ at low Pe for adhesive ABPs in Fig. 4(c) does not reach unity because of the strong wrapping of particles by the membrane, whose area is insufficient to all ABPs at φ = 0.18. As Pe is increased and ABPs have a sufficient force to detach from the membrane, near-equilibrium 'frozen' structures with strong particle wrapping dissolve and the activityinduced accumulation of ABPs results in ρ to approach unity. Note that even though the fraction of ABPs at the membrane for > 0 is larger than that for = 0, it does not contribute in the same way to membrane tension. For = 0, the larger is the fraction ρ, the larger is the mean membrane tensionλ due to an increasing swim pressure. For > 0, even though an increase in Pe leads to an increase inλ for the same reason, ABP adhesion reduces mean membrane tension because of long-ranged adhesive interactions discussed above. Moreover, at low Pe, a number of adhered ABPs may temporarily be oriented away from the membrane without detaching from it, which results in a reduction of the total swim pressure. C. Vesicle shape fluctuations In the fluctuating regime, we analyse vesicle shape changes by computing the fluctuation spectrum of a membrane cross-section, as outlined in Appendix B. Fluctuation spectra of active vesicles at φ = 0.18 are presented in Fig. 5 for various and Pe values. The fluctuation spectra can be divided into the three regimes with respect to the mode number l: (i) low l 10 where the ABP activity or adhesion dominate, (ii) intermediate 10 l l σ where the competition between the ABP propulsion and adhesion is important, with l σ = 2πR/σ 50 being a wavelength of the ABP size, and (iii) large l l σ where passive bending rigidity of the membrane dominates. At low mode numbers l 10 and small Pe 50 values, a plateau-like region is observed, which is more pronounced for large adhesion strengths. This indicates that large-wavelength fluctuations are suppressed in the presence of adhesion at low Pe due to nearly non-dynamic ABP clusters adhered to the membrane. The suppression of large wavelength fluctuations has also been observed in cells due to the presence of an underlying cytoskeleton [67,68]. Thus, the adhesion of particles to the membrane leads to a membrane confinement effect, significantly reducing fluctuations at low l modes. As Pe is increased, the ABPs attain sufficient propulsion force to detach from the membrane, accompanied by the disappearance of the plateau region at low l. Furthermore, the exponent β of fluctuation modes at low l ∈ [2, 8] becomes β < −1 [see the inset in Fig. 5(a)], which is a clear signature of active membrane fluctuations [18,19]. β decreases as a function of Pe, demonstrating the enhancement of low-mode fluctuations due to ABP activity. A shift in the fluctuation-spectrum curves for different adhesion strengths and intermediate l values at Pe = 50 in Fig. 5(b) is likely due to the fact that a number of adhered ABPs can enhance membrane fluctuations by exerting temporary forces in the direction away from the membrane without detaching from it. Note that a reduction in tension for > 0 cannot significantly contribute to this shift in fluctuation spectrum, because the effect of membrane tension is expected to be present for l 10 − 15 [18], while the observed shift extends significantly beyond those l values. Furthermore, for Pe = 200 and = 2.5k B T in Fig. 5(b), the shift in fluctuation spectrum nearly disappears despite the fact that the mean membrane tension is significantly larger than in the case of Pe = 50 and = 0 [see Fig. 4(a)]. This suggests that the combination of ABP activity (i.e., applied forces in the direction away from the membrane) and adhesion is responsible for the shift in fluctuation spectrum for Pe 100. Another interesting feature in the fluctuation spectra in Fig. 5 for adhesive ABPs with > 0 is the enhancement of amplitudes a 2 l at l 40−60 corresponding to the ABP size, since l σ = 2πR/σ 50. This local enhancement in a 2 l represents the wrapping of adhesive ABPs by the membrane, as it is consistently reduced at Pe = 200 in comparison to Pe = 50. Finally, at large l, the squared fluctuation amplitudes decay as l −3 irrespective of ABP adhesion or Pe, corresponding to the bending-dominated regime of membrane fluctuations. D. ABP characteristics Adhesion of ABPs to the membrane must decrease their overall mobility. Figure 6(a) presents distributions of fixed-time displacements δ of single ABPs for various Pe and values in the tethering regime. As expected, ABP mobility is significantly reduced for the cases of > 0 in comparison to non-adhesive ABPs, and the reduction in particle mobility is more pronounced at low Pe, since adhesion interactions dominate over the ABP activity. The mobility of active particles can also be reduced due to the formation of ABP clusters inside the vesicle. Figure 6(b) shows distributions of cluster sizes N c (i.e, the number of ABPs per single cluster) at large φ for various Pe and in the fluctuating regime. In the absence of adhesion ( = 0), an increase in Pe leads to an increased accumulation of ABPs at the membrane, such that large clusters are formed through a reduction in the number of small clusters, as can be seen through the emergence of a peak at large N c for Pe = 100 in Fig. 6(b). For > 0, ABP adhesion to the membrane further facilitates the membrane-mediated formation of large particle clusters, as in this case, a peak at large N c develops already at Pe = 50 in Fig. 6(b). As a result, adhesive interactions generally enhance cluster formation in comparison to the case of non-adhesive ABPs. We also compute cluster asphericity Ψ (see Appendix C for details) to quantify the effect of ABP adhesion on cluster shapes. Figure 6(c) presents Ψ as a function of Pe, and demonstrates that adhesive interactions cause an increase in the asphericity of ABP clusters. Thus, ABP clusters for > 0 attain shapes, which are further away from a spherical geometry, in agreement with the branched string-like arrangements of ABPs in the tethering regime discussed in Section III A. For = 0, ABPs primarily cluster at the end of tethers as nearly spherical aggregates. Interestingly, Ψ for the case of adhesive ABPs first increases and then decreases with increasing Pe. Characteristic vesicle shapes are illustrated in Fig. 7 Fig. 7(d)] with a reduced cluster asphericity. In conclusion, the results in Fig. 6 clearly show that adhesive interactions of ABPs with the membrane strongly alter the behavior of individual ABPs and their clusters. IV. SUMMARY AND CONCLUSIONS Vesicles enclosing active particles exhibit a variety of dynamic shape deformations, ranging from tethers to prolate and bola-like shapes. Adhesive interactions between particles and a vesicle in equilibrium can lead to strong, although static, deformations of the vesicle, such as the formation of buds and long tubular structures. In this work, we have combined the effects of particle activity and adhesion to study the deformation and properties of vesicles enclosing adhesive ABPs. At low propulsion forces of ABPs, adhesion interactions with the membrane dominate, leading to the formation of membrane structures (e.g., buds, tubes) which are similar to those in equilibrium. Furthermore, due to the absence of a volume constraint in our simulations, strong membrane deformations with ring-like and sheet-like ABP structures occur for moderate volume fractions of ABPs, which are governed by the balance of adhesive interactions and en-ergetic costs for membrane bending. As the propulsion of ABPs (or the Peclet number Pe) is increased, the particles are able to detach from the membrane, and the effects of adhesion become less dominant. A simple estimation for the detachment force of a single ABP adhered to the membrane based on theoretical arguments and simulations yields the adhesion-dominated regime for Pe 200. However, ABP-ABP collisions at large enough φ and enhanced membrane fluctuations due to the particle activity further lower the characteristic Pe 100 determining the adhesion-dominated regime. In the tethering regime, adhesion interactions between the membrane and ABPs significantly reduce the characteristic Pe for tether formation in comparison to non-adhesive ABPs. Furthermore, ABP adhesion favours the formation of long branched tether structures partially or fully filled with active particles for low to moderate volume fractions. At large φ, an increase in Pe first causes 'melting' of nearly frozen particle structures within the vesicle at low Pe, such that the vesicle attains a spherical shape with pronounced membrane fluctuations. A further increase in Pe results in elongated vesicle shapes or bola-like shapes which eventually split into two daughter vesicles. Different from active vesicles with non-adhesive ABPs, for which the fluctuating regime is observed at low Pe across all φ values, membrane fluctuations in the presence of ABP adhesion take place only at φ 0.07 and require some activation energy through a non-zero Pe. The fluctuation spectrum at low Pe has a plateau at low mode numbers because of a 'caging' effect due to the adhered particles. ABP adhesion to the membrane leads to local membrane compression with a slightly negative tension due to long-ranged adhesive interactions, so that the mean vesicle tension is lower in the case of adhesive ABPs than for non-adhesive particles. With increasing Pe, the mean membrane tension of the vesicle first has a non-linear dependence on Pe in the adhesion dominated regime, followed by a linear increase of the mean tension at large enough Pe 100, in agreement with theoretical predictions from the Young-Laplace equation in the case of non-adhesive ABPs [18,23]. Furthermore, the adhesion of ABPs to the membrane leads to a reduced particle mobility, but enhances ABP clustering through membrane-mediated interactions. Also, ABP clusters in the presence of adhesive interactions have larger cluster asphericities than those for non-adhesive ABPs, mainly due to the formation of branched string-like structures of ABPs within membrane tubes in the tethering regime. In conclusion, the presence of adhesive interactions between ABPs and the membrane affects not only the phase diagram of active vesicles, but also membrane characteristics (e.g., shape, tension) and ABP properties (e.g., mobilite, clustering). Therefore, particle adhesion serves as an additional parameter for the control and tuning of the behavior of active vesicles. APPENDIX A: CALCULATION OF MEMBRANE TENSION Membrane tension is calculated using the virial theorem [69]. The sum over virial contributions from the local area constraint is given by V av = α f a,α i r α i + f a,α j r α j + f a,α k r α k ,(11) where f a i,j,k are forces at the vertices i, j and k of a triangle within the membrane triangulation, and α = x, y, or z represents the three coordinates. For elastic bond forces f b , the virial contribution V b is V b = α f b,α i r α i + f b,α j r α j .(12) The total virial contribution from the forces at each vertex is then V = V av /3 + V b /2. The tension of the membrane is calculated as a spatial and temporal average of the local stresses as λ = 1 2a i (V i (t) + 2k B T ) i,t ,(13) where the factor two is due to the dimensionality of the membrane, V i (t) is the virial contribution at vertex i at time t, and a i is the area of the dual cell, which is approximated by considering that each neighbouring triangle to the vertex contributes roughly one third to the area. The contribution from momentum transfer (2k B T ) is approximated by using the equipartition theorem. The contributions from the bending energy, volume conservation, and global area conservation are not considered for membrane tension, because the bending forces mainly act perpendicular to the tension plane, while the global area and volume constraints have not been used in the simulations. APPENDIX B: MEMBRANE SHAPE FLUCTUATIONS The membrane shape fluctuations are measured by considering 2D sections of the vesicle contour in the x, y, and z directions. The local membrane position in these contours is given by r(θ m ), where θ m = 2πm/n and 2π/n is the angle for contour discretisation. The fluctuation mode amplitudes a l are given by the decomposition [70][71][72] a l = 1 n n−1 m=0 r(θ m ) exp −2πilm n . The complex modes a l are calculated using the opensource FFTW [73] library, and are averaged over different time frames. APPENDIX C: ASPHERICITY OF SPP CLUSTERS Shapes of ABP clusters are quantified by their asphericity. The asphericity is calculated from the gyration tensor G, which is based on the second moments of N particle positions as G xy ≡ 1 N N i=1 r i x r i y ,(15) where r is measured from the center of mass of the Nparticle system, i.e. N i=1 r i = 0. Let λ 1 , λ 2 , and λ 3 be the eigenvalues of G. Then, the asphericty Ψ is defined as [74] Ψ = (λ 1 − λ 2 ) 2 + (λ 2 − λ 3 ) 2 + (λ 1 − λ 3 ) 2 2(λ 1 + λ 2 + λ 3 ) 2 .(16) Values of Ψ range between 0 and 1, with Ψ = 0 for a perfectly spherical shape, Ψ = 1 for a long thin rod, and Ψ = 0.25 for a thin plate. N t = 2N v − 4 is the number of triangular faces in the vesicle discretization. An energetically favorable bond flip is accepted with a probability of p = 1. For an energetically unfavorable flip, the resulting change in energy due to an attempted bond flip ∆U = ∆U att + ∆U rep + ∆U A determines the probability of the flipping as p = exp[−∆U/k B T ]. The resulting membrane fluidity can be characterized by a 2D membrane viscosity for the selected frequency ω and flipping probability ψ[54,58]. FIG. 1 : 1(a) Fraction of wrapped area A wrap of the ABP as a function of . The dashed red line marks theoretical predictions of the critical c for the transition from unwrapped to fully wrapped state FIG. 2 : 2Phase diagrams of vesicle-shape changes as a function of Pe and φ for two different adhesion strengths (a) = 2.5k B T and (b) = 3.5k B T . Four regions are observed, including the tethering (blue symbols), bola/prolate (green symbols), fluctuating (yellow symbols), and near-equilibrium (red symbols) regimes. The points corresponding to the displayed snapshots have black outlines. The black lines provide an approximate demarcation of the different regimes, serving as a guide to the eye. For a visual illustration of dynamic shape changes of active vesicles, see also Movies S1-S4. FIG. 3 : 3Vesicle shapes in the near-equilibrium regime at Pe = 15 and = 3.5k B T for (a) φ = 0.009, (b) φ = 0.04 (see Movie S4), (c) φ = 0.12, and (d) φ = 0.18. Different nearly-frozen structures of the ABPs are observed, including ring-like and sheet-like arrangements. z₀ FIG. 4: (a) Mean local vesicle tensionλ as a function of Pe for different adhesion strengths at φ = 0.18. (b) Sketch of an ABP (blue) interacting with a flat membrane at a distance z 0 . Membrane vertices are depicted in red and the orientation vector e of the ABP is pointing away from the membrane. (c) Mean fraction ρ of the ABPs in contact with the membrane as a function of Pe for different values at φ = 0.18. FIG. 5 : 5Mode spectra of vesicle-shape fluctuations at (a) = 3.5k B T for different Pe values and at (b) Pe = 50 for different values. Large wavelength (low mode) fluctuations are suppressed at low Pe for a strong ABP adhesion, resulting in a plateau-like region at l 10. The inset in (a) shows the slope β of low-mode fluctuations with increasing Pe. The dashed lines indicate the mode number l σ = 2πR/σ 50, representing a wavelength of the ABP size. FIG. 6 :FIG. 7 : 67(a) Distributions of fixed-time displacements δ of single ABPs for different and Pe values in the tethering regime. (b) Distributions of cluster sizes N c in the fluctuating regime at φ = 0.18 for different Pe and . (c) Mean cluster asphericity Ψ as a function of Pe for = 0 and = 3.5k B T . Vesicle shapes for φ = 0.04 and = 3.5k B T at (a) Pe = 15 (see Movie S4), (b) Pe = 50, (c) Pe = 150 (see Movie S1), and (d) Pe = 300. Particle structures change from membrane-wrapped ring-like arrangements to membrane-wrapped (branched) tubular aggregates, as Pe is increased. A further increase in Pe leads to the detachment of ABPs from the membrane and their accumulation at the tether end. presents all simulation parameters.A. Model of adhesive active Brownian particles TABLE I : IParameters used for simulations of vesicles enclosing adhesive ABPs both in model and physical units. for different Pe. At low Pe, ring-like ABP clusters [Fig. 7(a)] in the near-equilibrium regime are observed and have the asphericity of about Ψ = 0.4. With increasing Pe, branched string-like clusters of ABPs within membrane tubes develop with Ψ > 0.4, see Fig. 7(c). At large Pe 200, ABP propulsion forces dominate over adhesive interactions, so that the string-like structures are destabilized and the ABPs cluster at the tether ends [ ACKNOWLEDGEMENTSWe thank Thorsten Auth and Roland G. Winkler for many helpful discussions. The authors gratefully acknowledge the computing time granted through JARA on the supercomputer JURECA[75]at Forschungszentrum Jülich.APPENDIX D: DESCRIPTION OF MOVIESAll movies are for an adhesion strength of = 3.5k B T . Movie S1: Formation of dynamic and highly branched tether structures at Pe = 150 and φ = 0.04. As ABP motion along the tether is limited due to their stringlike arrangement, rotational diffusion of the ABPs facil-itates tether branching in contrast to ABP escape from the tether for = 0. Movie S2: Tether formation at Pe = 300 and φ = 0.009. ABPs can escape from a tether and join new tethers due to their rotational diffusion. Movie S3: Vesicle elongation followed by splitting in the bola regime at Pe = 200 and φ = 0.12. Movie S4: Formation of nearly static ring-like structures of ABPs at Pe = 15 and φ = 0.04.AUTHOR CONTRIBUTIONSG.G. and D.A.F. conceived the research project. P.I. performed the simulations and analysed the obtained data. All authors participated in the discussions and writing of the manuscript.CONFLICTS OF INTERESTThere are no conflicts to declare. The mechanics and statistics of active matter. S Ramaswamy, Annu. Rev. Condens. Matter Phys. 1323S. Ramaswamy, The mechanics and statistics of active matter, Annu. Rev. Condens. Matter Phys. 1, 323 (2010). 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[ "DEEP LEARNING IN HUMAN ACTIVITY RECOGNITION WITH WEARABLE SENSORS: A REVIEW ON ADVANCES *", "DEEP LEARNING IN HUMAN ACTIVITY RECOGNITION WITH WEARABLE SENSORS: A REVIEW ON ADVANCES *" ]
[ "Shibo Zhang [email protected] \nNorthwestern University\nMcGill University\nGeorgia Institute of Technology\nNorthwestern University\nColumbia University\nNorthwestern University\nNorthwestern University\n\n", "Yaxuan Li [email protected] \nNorthwestern University\nMcGill University\nGeorgia Institute of Technology\nNorthwestern University\nColumbia University\nNorthwestern University\nNorthwestern University\n\n", "Shen Zhang [email protected] \nNorthwestern University\nMcGill University\nGeorgia Institute of Technology\nNorthwestern University\nColumbia University\nNorthwestern University\nNorthwestern University\n\n", "Farzad Shahabi [email protected] \nNorthwestern University\nMcGill University\nGeorgia Institute of Technology\nNorthwestern University\nColumbia University\nNorthwestern University\nNorthwestern University\n\n", "Stephen Xia \nNorthwestern University\nMcGill University\nGeorgia Institute of Technology\nNorthwestern University\nColumbia University\nNorthwestern University\nNorthwestern University\n\n", "Yu Deng \nNorthwestern University\nMcGill University\nGeorgia Institute of Technology\nNorthwestern University\nColumbia University\nNorthwestern University\nNorthwestern University\n\n", "Nabil Alshurafa \nNorthwestern University\nMcGill University\nGeorgia Institute of Technology\nNorthwestern University\nColumbia University\nNorthwestern University\nNorthwestern University\n\n" ]
[ "Northwestern University\nMcGill University\nGeorgia Institute of Technology\nNorthwestern University\nColumbia University\nNorthwestern University\nNorthwestern University\n", "Northwestern University\nMcGill University\nGeorgia Institute of Technology\nNorthwestern University\nColumbia University\nNorthwestern University\nNorthwestern University\n", "Northwestern University\nMcGill University\nGeorgia Institute of Technology\nNorthwestern University\nColumbia University\nNorthwestern University\nNorthwestern University\n", "Northwestern University\nMcGill University\nGeorgia Institute of Technology\nNorthwestern University\nColumbia University\nNorthwestern University\nNorthwestern University\n", "Northwestern University\nMcGill University\nGeorgia Institute of Technology\nNorthwestern University\nColumbia University\nNorthwestern University\nNorthwestern University\n", "Northwestern University\nMcGill University\nGeorgia Institute of Technology\nNorthwestern University\nColumbia University\nNorthwestern University\nNorthwestern University\n", "Northwestern University\nMcGill University\nGeorgia Institute of Technology\nNorthwestern University\nColumbia University\nNorthwestern University\nNorthwestern University\n" ]
[]
Mobile and wearable devices have enabled numerous applications, including activity tracking, wellness monitoring, and human-computer interaction, that measure and improve our daily lives. Many of these applications are made possible by leveraging the rich collection of low-power sensors found in many mobile and wearable devices to perform human activity recognition (HAR). Recently, deep learning has greatly pushed the boundaries of HAR on mobile and wearable devices. This paper systematically categorizes and summarizes existing work that introduces deep learning methods for wearables-based HAR and provides a comprehensive analysis of the current advancements, developing trends, and major challenges. We also present cutting-edge frontiers and future directions for deep learning-based HAR.
10.3390/s22041476
[ "https://arxiv.org/pdf/2111.00418v5.pdf" ]
240,354,685
2111.00418
4be02cb6668ca82618fff45370651db48ebe11d1
DEEP LEARNING IN HUMAN ACTIVITY RECOGNITION WITH WEARABLE SENSORS: A REVIEW ON ADVANCES * Shibo Zhang [email protected] Northwestern University McGill University Georgia Institute of Technology Northwestern University Columbia University Northwestern University Northwestern University Yaxuan Li [email protected] Northwestern University McGill University Georgia Institute of Technology Northwestern University Columbia University Northwestern University Northwestern University Shen Zhang [email protected] Northwestern University McGill University Georgia Institute of Technology Northwestern University Columbia University Northwestern University Northwestern University Farzad Shahabi [email protected] Northwestern University McGill University Georgia Institute of Technology Northwestern University Columbia University Northwestern University Northwestern University Stephen Xia Northwestern University McGill University Georgia Institute of Technology Northwestern University Columbia University Northwestern University Northwestern University Yu Deng Northwestern University McGill University Georgia Institute of Technology Northwestern University Columbia University Northwestern University Northwestern University Nabil Alshurafa Northwestern University McGill University Georgia Institute of Technology Northwestern University Columbia University Northwestern University Northwestern University DEEP LEARNING IN HUMAN ACTIVITY RECOGNITION WITH WEARABLE SENSORS: A REVIEW ON ADVANCES * Review · Human Activity Recognition · Deep Learning · Wearable Sensors · Ubiquitous Computing · Pervasive Computing Mobile and wearable devices have enabled numerous applications, including activity tracking, wellness monitoring, and human-computer interaction, that measure and improve our daily lives. Many of these applications are made possible by leveraging the rich collection of low-power sensors found in many mobile and wearable devices to perform human activity recognition (HAR). Recently, deep learning has greatly pushed the boundaries of HAR on mobile and wearable devices. This paper systematically categorizes and summarizes existing work that introduces deep learning methods for wearables-based HAR and provides a comprehensive analysis of the current advancements, developing trends, and major challenges. We also present cutting-edge frontiers and future directions for deep learning-based HAR. Introduction Since the first Linux-based smartwatch was presented in 2000 at the IEEE International Solid-State Circuits Conference (ISSCC) by Steve Mann, who was later hailed as the "father of wearable computing", the 21st century has witnessed a rapid growth of wearables. For example, as of January 2020, 21% of adults in the United States, most of whom are not opposed to sharing data with medical researchers, own a smartwatch [1]. In addition to being fashion accessories, wearables provide unprecedented opportunities for monitoring human physiological signals and facilitating natural and seamless interaction between humans and machines. Wearables integrate low-power sensors that allow them to sense movement and other physiological signals such as heart rate, temperature, blood pressure, and electrodermal activity. The rapid proliferation of wearable technologies and advancements in sensing analytics have spurred the growth of human activity recognition (HAR). As a general understanding of the HAR shown in Figure 1, HAR has drastically improved the quality of service in a broad range of applications spanning healthcare, entertainment, gaming, industry, and lifestyle, among others. Market analysts from Meticulous Research® [2] forecast that the global wearable devices market will grow at a compound annual growth rate of 11.3% from 2019, reaching $62.82 billion by 2025, with companies like Fitbit®, Garmin®, and Huawei Technologies® investing more capital into the area. In the past decade, deep learning (DL) has revolutionized traditional machine learning (ML) and brought about improved performance in many fields, including image recognition, object detection, speech recognition, and natural language processing. DL has improved the performance and robustness of HAR, speeding its adoption and application to a wide range of wearable sensor-based applications. There are two key reasons why DL is effective for many applications. First, DL methods are able to directly learn robust features from raw data for specific applications, whereas features generally need to be manually extracted or engineered in traditional ML approaches, which usually requires expert domain knowledge and a large amount of human effort. Deep neural networks can efficiently learn representative features from raw signals with little domain knowledge. Second, deep neural networks have been shown to be universal function approximators, capable of approximating almost any function given a large enough network and sufficient observations [3,4,5]. Due to this expressive power, DL has seen a substantial growth in HAR-based applications. Despite promising results in DL, there are still many challenges and problems to overcome, leaving room for more research opportunities. We present a review on deep learning in HAR with wearable sensors and elaborate on ongoing challenges, obstacles, and future directions in this field. (a) (b) (c) Figure 1: Wearable devices and their application. (a) Distribution of wearable applications [6]. (b) Typical wearable devices. (c) Distribution of wearable devices placed on common body areas [6]. Specifically, we focus on the recognition of physical activities, including locomotion, activities of daily living (ADL), exercise, and factory work. While DL has shown a lot of promise in other applications, such as ambient scene analysis, emotion recognition, or subject identification, we focus on HAR. Throughout this work, we present brief and high-level summaries of major DL methods that have significantly impacted wearable HAR. For more details about specific algorithms or basic DL, we refer the reader to original papers, textbooks, and tutorials [7,8]. Our contributions are summarized as followings. (i) Firstly, we give an overview of the background of the human activity recognition research field, including the traditional and novel applications where the research community is focusing, the sensors that are utilized in these applications, as well as widely-used publicly available datasets. (ii) Then, after briefly introducing the popular mainstream deep learning algorithms, we give a review of the relevant papers over the years using deep learning in human activity recognition using wearables. We categorize the papers in our scope according to the algorithm (autoencoder, CNN, RNN, etc.). In addition, we compare different DL algorithms in terms of the accuracy of the public dataset, pros and cons, deployment, and high-level model selection criteria. (iii) We provide a comprehensive systematic review on the current issues, challenges, and opportunities in the HAR domain and the latest advancements towards solutions. At last, honorably and humbly, we make our best to shed light on the possible future directions with the hope to benefit students and young researchers in this field. Methodology Research Question In this work, we propose several major research questions, including Q1 : What the real-world applications of HAR, mainstream sensors, and major public datasets are in this field, Q2 : What deep learning approaches are employed in the field of HAR and what pros and cons each of them have, and Q3 : What challenges we are facing in this field and what opportunities and potential solutions we may have. In this work, we review the state-of-the-art work in this field and present our answers to these questions. Deep Learning in Human Activity Recognition with Wearable Sensors: A Review on Advances This article is organized as follows: We compare this work with related existing review work in this field in Section 3. Section 4.1 introduces common applications for HAR. Section 4.2 summarizes the types of sensors commonly used in HAR. Section 4.3 summarizes major datasets that are commonly used to build HAR applications. Section 5 introduces the major works in DL that contribute to HAR. Section 6 discusses major challenges, trends, and opportunities for future work. We provide concluding remarks in Section 7. Research Scope In order to provide a comprehensive overview of the whole HAR field, we conducted a systematic review for human activity recognition. To ensure that our work satisfies the requirements of a high-quality systemic review, we conducted the 27-item PRISMA review process [9] and ensured that our work satisfied each requirement. We searched in Google Scholar with meta-keywords (We began compiling papers for this review in November 2020. As we were preparing this review, we compiled a second round of papers in November 2021 to incorporate the latest works published in 2021). (A) "Human activity recognition", "motion recognition", "locomotion recognition", "hand gesture recognition", "wearable", (B) "deep learning", "autoencoder" (alternatively "auto-encoder"), "deep belief network", "convolutional neural network" (alternatively "convolution neural network"), "recurrent neural network", "LSTM", "recurrent neural network", "generative adversarial network" (alternatively "GAN"), "reinforcement learning", "attention", "deep semi-supervised learning", and "graph neural network". We used an AND rule to get combinations of the above meta-keywords (A) and (B). For each combination, we obtained top 200 search results ranked by relevance. We didn't consider any patent or citation-only search result (no content available online). There are several exclusion criteria to build the database of the paper we reviewed. First of all, we omitted image or video-based HAR works, such as [10], since there is a huge body of work in the computer vision community and the method is significantly different from sensor-based HAR. Secondly, we removed the papers using environmental sensors or systems assisted by environmental sensors such as WiFi-and RFID-based HAR. Thirdly, we removed the papers with minor algorithmic advancements based on prior works. We aim to present the technical progress and algorithmic achievements in HAR, so we avoid presenting works that do not stress the novelty of methods. In the end, as the field of wearable-based HAR is becoming excessively popular and numerous papers are coming out, it is not a surprise to find that many papers share rather similar approaches, and it is almost impossible and less meaningful to cover all of them. Figure 2 shows the consort diagram that outlines step-by-step how we filtered out papers to arrive at the final 176 papers we included in this review. We obtained 8400 papers in the first step by searching keywords mentioned above on Google Scholar. Next, we removed papers that did not align with the topics in this review (i.e., works that do not utilize deep learning in wearable systems), leaving us with 870 papers. In this step, we removed 2194 papers that utilized vision, 2031 papers that did not use deep learning, and 2173 papers that did not perform human activity recognition. Then, we removed 52 review papers, 109 papers that did not propose novel systems or algorithms, and five papers that were not in English, leaving us with 704 papers. Finally, we selected the top 25% most relevant papers to review, leaving us with 176 papers that we reviewed for this work. We used the relevancy score provided through Google Scholar to select the papers to include in this systemic review. Therefore, we select, categorize, and summarize representative works to present in this review paper. We adhere to the goal of our work throughout the whole paper, that is, to give an overall introduction to new researchers entering this field and present cutting-edge research challenges and opportunities. However, we admit that the review process conducted in this work has some limitations. Due to the overwhelming amount of papers in this field in recent years, it is almost impossible to include all the published papers in the field of deep learning-based wearable human activity recognition in a single review paper. The selection of the representative works to present in this paper is unavoidably subject to the risk of bias. Besides, we may miss the very first paper initiating or adopting a certain method. At last, due to the nature of human-related research and machine learning research, many possibilities could cause heterogeneity among study results, including the heterogeneity in devices, heterogeneity from the demography of participants, and even heterogeneity from the algorithm implementation details. Taxonomy of Human Activity Recognition In order to obtain a straightforward understanding of the hierarchies under the tree of HAR, we illustrate the taxonomy of HAR as shown in Figure 3. We categorized existing HAR works into four dimensions: Sensor, application, DL approach, and challenge. Learning (DRL). In the end, we discuss the challenges our research community is facing and the state-of-the-art works are coping with, also shown in Figure 3. Related Work There are some existing review papers in the literature for deep learning approaches for sensor-based human activity recognition [11,12,13,14]. Nweke limitations up to year 2018 [12]. Similarly, Wang et al. conducted a thorough analysis on different sensor-based modalities, deep learning models, and their respective applications up to the year 2017 [11]. However, in recent years, due to huge advancements in the availability and computational power of computing resources and cutting-edge deep learning techniques, the applied deep learning area has been revolutionized and reached all-time-high performance in the field of sensor-based human activity recognition. Therefore, we aim to present the most recent advances and most exciting achievements in the community in a timely manner to our readers. In another work, Chen et al. provided the community a comprehensive review which has done an in-depth analysis of the challenges/opportunities for deep learning in sensor-based HAR and proposed a new taxonomy for the challenges ahead of the activity recognition systems [13]. In contrast, we view our work as more of a gentle introduction of this field to students and novices in the way that our literature review provides the community with a detailed analysis on most recent state-of-the-art deep learning architectures (i.e., CNN, RNN, GAN, Deep Reinforcement Learning, and hybrid models) and their respective pros and cons on HAR benchmark datasets. At the same time, we distill our knowledge and experience from our past works in this field and present the challenges and opportunities from a different viewpoint. Another recent work was presented by Ramanujam et al. , in which they categorized the deep learning architectures in CNN, LSTM, and hybrid methods and conducted an in-depth analysis on the benchmark datasets [14]. Compared with their work, our paper pays more attention to the most recent cutting-edge deep learning methods applied on HAR on-body sensory data, such as GAN and DRL. We also provide both new learners and experienced researchers with a profound resource in terms of model comparison, model selection and model deployment. In a nutshell, our review has thoroughly analysed most up-to-date deep learning architectures applied on various wearable sensors, elaborated on their respective applications, and compared performances on public datasets. What's more, we attempt to cover the most recent advances in resolving the challenges and difficulties and shed light on possible research opportunities. Human Activity Recognition Overview Applications In this section, we illustrate the major areas and applications of wearable devices in HAR. Figure 1a, taken from the wearable technology database [6], breaks down the distribution of application types of 582 commercial wearables registered since 2015 [6]. The database suggests that wearables are increasing in popularity and will impact people's lives in several ways, particularly in applications ranging from fitness and lifestyle to medical and human-computer interaction. Wearables in Fitness and Lifestyle Physical activity involves activities such as sitting, walking, laying down, going up or downstairs, jogging, and running [15]. Regular physical activity is increasingly being linked to a reduction in risk for many chronic diseases, such as obesity, diabetes, and cardiovascular disease, and has been shown to improve mental health [16]. The data recorded by wearable devices during these activities include plenty of information, such as duration and intensity of activity, which further reveals an individual's daily habits and health conditions [17]. For example, dedicated products such as Fitbit [18] can estimate and record energy expenditure on smart devices, which can further serve as an important step in tracking personal activity and preventing chronic diseases [19]. Moreover, there has been evidence of the association between modes of transport (motor vehicle, walking, cycling, and public transport) and obesity-related outcomes [20]. Being aware of daily locomotion and transportation patterns can provide physicians with the necessary information to better understand patients' conditions and also encourage users to engage in more exercise to promote behavior change [21]. Therefore, the use of wearables in fitness and lifestyle has the potential to significantly advance one of the most prolific aspects of HAR applications [22,23,24,25,26,27,28,29,30]. Energy (or calorie) expenditure (EE) estimation has grown to be an important reason why people care to track their personal activity. Self-reflection and self-regulation of one's own behavior and the habit has been important factor in designing interventions that prevent chronic diseases such as obesity, diabetes, and cardiovascular diseases. Wearables in Healthcare and Rehabilitation HAR has greatly impacted the ability to diagnose and capture pertinent information in healthcare and rehabilitation domains. By tracking, storing, and sharing patient data with medical institutions, wearables have become instrumental for physicians in patient health assessment and monitoring. Specifically, several works have introduced systems and methods for monitoring and assessing Parkinson disease (PD) symptoms [31,32,33,34,35,36]. Pulmonary disease, such as Chronic Obstructive Pulmonary Disease (COPD), asthma, and COVID-19, is one of leading causes of morbidity and mortality. Some recent works use wearables to detect cough activity, a major symptom of pulmonary diseases [37,38,39,40]. Other works have introduced methods for monitoring stroke in infants using wearable accelerometers [41] and methods for assessing depressive symptoms utilizing wrist-worn sensors [42]. In addition, detecting muscular activities and hand motions using electromyography (EMG) sensors has been widely applied to enable improved prostheses control for people with missing or damaged limbs [43,44,45,46]. Wearables in Human Computer Interaction (HCI) Modern wearable technology in HCI has provided us with flexible and convenient methods to control and communicate with electronics, computers, and robots. For example, a wrist-worn wearable outfitted with an inertial measurement unit (IMU) can easily detect the wrist shaking [47,48,49] to control smart devices to skip a song by shaking the hand, instead of bringing up the screen, locating, and pushing a button. Furthermore, wearable devices have played an essential role in many HCI applications in entertainment systems and immersive technology. One example field is augmented reality (AR) and virtual reality (VR), which has changed the way we interact and view the world. Thanks to accurate activity, gesture, and motion detection from wearables, these applications could induce feelings of cold or hot weather by providing an immersive experience by varying the virtual environment and could enable more realistic interaction between the human and virtual objects [43,44]. Wearable Sensors Wearable sensors are the foundation of HAR systems. As shown in Figure 1b, there are a large number of off-the-shelf smart devices or prototypes under development today, including smartphones, smartwatches, smart glasses, smart rings [50], smart gloves [51], smart armbands [52], smart necklaces [53,54,55], smart shoes [56], and E-tattoos [57]. These wearable devices cover the human body from head to toe with a general distribution of devices shown in Figure 1c, as reported by [6]. The advance of micro-electro-mechanical system (MEMS) technology (microscopic devices, comprising a central unit such as a microprocessor and multiple components that interact with the surroundings such as microsensors) has allowed wearables to be miniaturized and lightweight to reduce the burden on adherence to the use of wearables and Internet of Things (IoT) technologies. In this section, we introduce and discuss some of the most prevalent MEMS sensors commonly used in wearables for HAR. The summary of wearable sensors is represented as a part of Figure 3. Inertial Measurement Unit (IMU) Inertial measurement unit (IMU) is an integrated sensor package comprising of accelerometer, gyroscope, and sometimes magnetometer. Specifically, an accelerometer detects linear motion and gravitational forces by measuring the acceleration in 3 axes (x, y, and z), while a gyroscope measures rotation rate (roll, yaw, and pitch). The magnetometer is used to detect and measure the earth's magnetic fields. Since a magnetometer is often used to obtain the posture and orientation in accordance with the geomagnetic field, which is typically outside the scope of HAR, the magnetometer is not always included in data analysis for HAR. By contrast, accelerometers and gyroscopes are commonly used in many HAR applications. We refer to an IMU package comprising a 3-axis accelerometer and a 3-axis gyroscope as a 6-axis IMU. This component is often referred to as a 9-axis IMU if a 3-axis magnetometer is also integrated. Owing to mass manufacturing and the widespread use of smartphones and wearable devices in our daily lives, IMU data are becoming more ubiquitous and more readily available to collect. In many HAR applications, researchers carefully choose the sampling rate of the IMU sensors depending on the activity of interest, often choosing to sample between 10 and several hundred Hz. In [58], Chung et al. tested a range of sampling rates and gave the best one in his application. Besides, it's been shown that higher sampling rates allow the system to capture signals with higher precision and frequencies, leading to more accurate models at the cost of higher energy and resource consumption. For example, the projects presented in [59,60] utilize sampling rates above the typical rate. These works sample at 4 kHz to sense the vibrations generated from the interaction between a hand and a physical object. Electrocardiography (ECG) and Photoplethysmography (PPG) Electrocardiography (ECG) and photoplethysmography (PPG) are the most commonly used sensing modalities for heart rate monitoring. ECG, also called EKG, detects the heart's electrical activity through electrodes attached to the body. The standard 12-lead ECG attaches ten non-intrusive electrodes to form 12 leads on the limbs and chest. ECG is primarily employed to detect and diagnose cardiovascular disease and abnormal cardiac rhythms. PPG relies on using a low-intensity infrared (IR) light sensor to measure blood flow caused by the expansion and contraction of heart chambers and blood vessels. Changes in blood flow are detected by the PPG sensor as changes in the intensity of light; filters are then applied to the signal to obtain an estimate of heart rate. Since ECG directly measures the electrical signals that control heart activity, it typically provides more accurate measurements for heart rate and often serves as a baseline for evaluating PPG sensors. Electromyography (EMG) Electromyography (EMG) measures the electrical activity produced by muscle movement and contractions. EMG was first introduced in clinical tests to assess and diagnose the functionality of muscles and motor neurons. There are two types of EMG sensors: Surface EMG (sEMG) and intramuscular EMG (iEMG). sEMG uses an array of electrodes placed on the skin to measure the electrical signals generated by muscles through the surface of the skin [61]. There are a number of wearable applications that detect and assess daily activities using sEMG [44,62]. In [63], researchers developed a neural network that distinguishes ten different hand motions using sEMG to advance the effectiveness of prosthetic hands. iEMG places electrodes directly into the muscle beneath the skin. Because of its invasive nature, non-invasive wearable HAR systems do not typically include iEMG. Mechanomyography (MMG) Mechanomyography (MMG) uses a microphone or accelerometer to measure low-frequency muscle contractions and vibrations, as opposed to EMG, which uses electrodes. For example, 4-channel MMG signals from the thigh can be used to detect knee motion patterns [64]. Detecting these knee motions is helpful for the development of power-assisted wearables for powered lower limb prostheses. The authors create a convolutional neural network and support vector machine (CNN-SVM) architecture comprising a seven-layer CNN to learn dominant features for specific knee movements. The authors then replace the fully connected layers with an SVM classifier trained with the extracted feature vectors to improve knee motion pattern recognition. Moreover, Meagher et al. [65] proposed developing an MMG device as a wearable sensor to detect mechanical muscle activity for rehabilitation after stroke. Deep Learning in Human Activity Recognition with Wearable Sensors: A Review on Advances Other wearable sensors used in HAR include (but are not limited to) piezoelectric sensor [66,67] for converting changes in pressure, acceleration, temperature, strain, or force to electrical charge, barometric pressure sensor [68] for atmospheric pressure, temperature measurement [69], electroencephalography (EEG) for measuring brain activity [70], respiration sensors for breathing monitoring [71], ultraviolet (UV) sensors [72] for sun exposure assessment, GPS for location sensing, microphones for audio recording [39,73,74], and wearable cameras for image or video recording [55]. It is also important to note that the wearable camera market has drastically grown with cameras such as GoPro becoming mainstream [75,76,77,78] over the last few years. However, due to privacy concerns posed by participants related to video recording, utilizing wearable cameras for longitudinal activity recognition is not as prevalent as other sensors. Additionally, HAR with image/video processing has been extensively studied in the computer vision community [79,80], and the methodologies commonly used differ significantly from techniques used for IMUs, EEG, PPG, etc. For these reasons, despite their significance in applications of deep learning methods, this work does not cover image and video sensing for HAR. Major Datasets We list the major datasets employed to train and evaluate various ML and DL techniques in Table 1, ranked based on the number of citations they received per year according to Google Scholar. As described in the earlier sections, most datasets are collected via IMU, GPS, or ECG. While most datasets are used to recognize physical activity or daily activities [81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99], there are also a few datasets dedicated to hand gestures [100,101], breathing patterns [102], and car assembly line activities [103], as well as those that monitor gait for patients with PD [104]. [106]. One of the most commonly used datasets is the OPPORTUNITY dataset [90]. This dataset contains data collected from 12 subjects using 15 wireless and wired networked sensor systems, with 72 sensors and ten modalities attached to the body or the environment. Existing HAR papers mainly focus on data from on-body sensors, including 7 IMUs and 12 additional 3D accelerometers for classifying 18 kinds of activities. Researchers have proposed various algorithms to extract features from sensor signals and to perform activity classification using machine-learned models like K Nearest Neighbor (KNN) and SVM [22,110,111,112,113,114,115,116,117,118]. Another widely-used dataset is PAMAP2 [91], which is collected from 9 subjects performing 18 Deep Learning in Human Activity Recognition with Wearable Sensors: A Review on Advances different activities, ranging from jumping to house cleaning, with 3 IMUs (100-Hz sampling rate) and a heart rate monitor (9 Hz) attached to each subject. Other datasets such as Skoda [103] and WISDM [81] are also commonly used to train and evaluate HAR algorithms. In Figure 4, we present the placement of inertial sensors in 9 common datasets. Deep Learning Approaches In recent years, DL approaches have outperformed traditional ML approaches in a wide range of HAR tasks. There are three key factors behind deep learning's success: Increasingly available data, hardware acceleration, and algorithmic advancements. The growth of datasets publicly shared through the web has allowed developers and researchers to quickly develop robust and complex models. The development of GPUs and FPGAs have drastically shortened the training time of complex and large models. Finally, improvements in optimization and training techniques have also improved training speed. In this section, we will describe and summarize HAR works from six types of deep learning approaches. We also present an overview of deep learning approaches in Figure 3. Autoencoder The autoencoder, originally called "autoassociative learning module", was first proposed in the 1980s as an unsupervised pre-training method for artificial neural networks (ANN) [119]. Autoencoders have been widely adopted as an unsupervised method for learning features. As such, the outputs of autoencoders are often used as inputs to other networks and algorithms to improve performance [120,121,122,123,124]. An autoencoder is generally composed of an encoder module and a decoder module. The encoding module encodes the input signals into a latent space, while the decoder module transforms signals from the latent space back into the original domain. As shown in Figure 5, the encoder and decoder module is usually several dense layers (i.e., fully connected layers) of the form f θ (x) : z = σ (W e x + b e ) g θ (z) : x = σ (W d z + b d ) where θ = {W e , b e }, θ = {W d , b d } are the learnable parameters of the encoder and decoder. σ is the non-linear activation function, such as Sigmoid, tanh, or rectified linear unit (ReLU). W e and W d refer to the weights of the layer, while b e and b d are the bias vectors. By minimizing a loss function applied on x and x , autoencoders aim at generating the final output by imitating the input. Autoencoders are efficient tools for finding optimal codes, z, and performing dimensionality reduction. An autoencoder's strength in dimensionality reduction has been applied to HAR in wearables [34,121,125,126,127,128,129,130,131] and functions as a powerful tool for denoising and information retrieval. As such, autoencoders are most commonly used for feature extraction and dimensionality reduction [120,122,123,124,125,126,133,134,135,136,137,138,139,140,141]. Autoencoders are generally used individually or in a stacked architecture with multiple autoencoders. Mean squared error or mean squared error plus KL divergence loss functions are typically used to train autoencoders. Li et al. presents an autoencoder architecture where a sparse autoencoder and a denoising autoencoder are used to explore useful feature representations from accelerometer and gyroscope sensor data, and then they perform classification using support vector machines [125]. Experiments are performed on a public HAR dataset [82] from the UCI repository, and the classification accuracy is compared with that of Fast Fourier Transform (FFT) in the frequency domain and Principal Component Analysis (PCA). The result reveals that the stacked autoencoder has the highest accuracy of 92.16% and provides a 7% advantage over traditional methods with hand-crafted features. Jun and Choi [142] studied the classification of newborn and infant activities into four classes: Sleeping, moving in agony, moving in normal condition, and movement by an external force. Using the data from an parameterized by θ = {W e , b e } and θ = {W d , b d }, where σ is a non-linear activation function (e.g., rectified linear unit), W represents a weight coefficient matrix and b is a bias vector. The model weights are sometimes tied for regularization such that W d = W T e . Figure 2 provides graphical illustration of an autoencoder. Learning an autoencoder is an effective approach to perform dimensionality reduction and can be thought of as a strict generalization of PCA. Specifically, a 1-layer encoder with linear activation and mean squared error (MSE) loss (see Equation (3)) should be able to learn PCA transformation [38]. Nonetheless, deep models with several hidden layers and non-linear activation functions can learn better high-level and disentangled features from the original input data. L MSE (X, X ) = X − X 2 .(3) The classical autoencoder can be extended in several ways (see for a review [11]). For handling missing input data, a compelling strategy is to train an autoencoder with artificially corrupted inputx, which acts as an implicit regularization. Usually, the considered corruption includes isotropic Gaussian Figure 5: Illustration of an autoencoder network [132]. accelerometer attached to the body and a three-layer autoencoder combined with k-means clustering, they achieve 96% weighted accuracy in an unsupervised way. Additionally, autoencoders have been explored for feature extraction in domain transfer learning [143], detecting unseen data [144], and recognizing null classes [145]. For example, Prabono et al. [146] propose a two-phase autoencoder-based approach of domain adaptation for human activity recognition. In addition, Garcia et al. [147] proposed an effective multi-class algorithm that consists of an ensemble of autoencoders where each autoencoder is associated with a separate class. This modular structure of classifiers makes models more flexible when adding new classes, which only calls for adding new autoencoders instead of re-training the model. Furthermore, autoencoders are commonly used to sanitize and denoise raw sensor data [127,130,148], a known problem with wearable signals that impacts our ability to learn patterns in the data. Mohammed and Tashev in [127] investigated the use of sensors integrated into common pieces of clothing for HAR. However, they found that sensors attached to loose clothing are prone to contain large amounts of motion artifacts, leading to low mean signal-to-noise ratios (SNR). To remove motion artifacts, the authors propose a deconvolutional sequence-to-sequence autoencoder (DSTSAE). The weights for this network are trained with a weighted form of a standard VAE loss function. Experiments show that the DSTSAE outperforms traditional Kalman Filters and improves the SNR from −12 dB to +18.2 dB, with the F1-score of recognizing gestures improved by 14.4% and locomotion activities by 55.3%. Gao et al. explores the use of stacking autoencoders to denoise raw sensor data to improve HAR using the UCI dataset [82] [130]. Then, LightGBM (LBG) is used to classify activities using the denoised signals. Autoencoders are also commonly used to detect abnormal muscle movements, such as Parkinson's Disease and Autism Spectrum Disorder (ASD). Rad et al. in [34] utilizes an autoencoder to denoise and extract optimized features of different movements and use a one-class SVM to detect movement anomalies. To reduce the overfitting of the autoencoder, the authors inject artificial noise to simulate different types of perturbations into the training data. Sigcha et al. in [149] uses a denoising autoencoder to detect freezing of gait (FOG) in Parkinson's disease patients. The autoencoder is only trained using data labelled as a normal movement. During the testing phase, samples with significant statistical differences from training data are classified as abnormal FOG events. As autoencoders map data into a nonlinear and low-dimensional latent space, they are well-suited for applications requiring privacy preservation. Malekzadeh et al. developed a novel replacement autoencoder that removes prominent features of sensitive activities, such as drinking, smoking, or using the restroom [121]. Specifically, the replacement autoencoder is trained to produce a non-sensitive output from a sensitive input via stochastic replacement while keeping characteristics of other less sensitive activities unchanged. Extensive experiments are performed on Opportunity [90], Skoda [103], and Hand-Gesture [100] datasets. The result shows that the proposed replacement autoencoder can retain the recognition accuracy of non-sensitive tasks using state-of-the-art techniques while simultaneously reducing detection capability for sensitive tasks. Mohammad et al. introduces a framework called Guardian-Estimator-Neutralizer (GEN) that attempts to recognize activities while preserving gender privacy [128]. The rationale behind GEN is to transform the data into a set of features Deep Learning in Human Activity Recognition with Wearable Sensors: A Review on Advances containing only non-sensitive features. The Guardian, which is constructed by a deep denoising autoencoder, transforms the data into representation in an inference-specific space. The Estimator comprises a multitask convolutional neural network that guides the Guardian by estimating sensitive and non-sensitive information in the transformed data. Due to privacy concerns, it attempts to recognize an activity without disclosing a participant's gender. The Neutralizer is an optimizer that helps the Guardian converge to a near-optimal transformation function. Both the publicly available MobiAct [150] and a new dataset, MotionSense, are used to evaluate the proposed framework's efficacy. Experimental results demonstrate that the proposed framework can maintain the usefulness of the transformed data for activity recognition while reducing the gender classification accuracy to 50% (random guessing) from more than 90% when using raw sensor data. Similarly, the same authors have proposed another anonymizing autoencoder in [129] for classifying different activities while reducing user identification accuracy. Unlike most works, where the output to the encoder is used as features for classification, this work utilizes both the encoder and decoder outputs. Experiments performed on a self-collected dataset from the accelerometer and gyroscope showcased excellent activity recognition performance (above 92%) while keeping user identification accuracy below 7%. Deep Belief Network (DBN) A DBN, as illustrated in Figure 6, is formed by stacking multiple simple unsupervised networks, where the hidden layer of the preceding network serves as the visible layer for the next. The representation of each sub-network is generally the restricted Boltzmann machine (RBM), an undirected generative energy-based model with a "visible" input layer, a hidden layer, and intra-layer connections in between. The DBN typically has connections between the layers but not between units within each layer. This structure leads to a fast and layer-wise unsupervised training procedure, where contrastive divergence (a training technique to approximate the relationship between a network's weights and its error) is applied to every pair of layers in the DBN architecture sequentially, starting from the "lowest" pair. Figure 6: The greedy layer-wise training of DBNs. The first level is trained on triaxial acceleration data. Then, more RBMs are repeatedly stacked to form a deep activity recognition model [151]. The observation that DBNs can be trained greedily led to one of the first effective deep learning algorithms [152]. There are many attractive implementations and uses of DBNs in real-life applications such as drug discovery [153], natural language understanding [154], fault diagnosis [155], etc. There are also many attempts to perform HAR with DBNs. In early exploratory work back in 2011 [156], a five-layer DBN is trained with the input acceleration data collected from mobile phones. The accuracy improvement ranges from 1% to 32% when compared to traditional ML methods with manually extracted features. In later works, DBN is applied to publicly available datasets [151,157,158,159]. In [157], two five-layer DBNs with different structures are applied to the Opportunity dataset [90], USC-HAD dataset [94], and DSA dataset [87], and the results demonstrate improved accuracy for HAR over traditional ML methods for all the three datasets. Specifically, the accuracy for the Opportunity, USC-HAD, and DSA datasets are 82.3% (1.6% improvement over traditional methods), 99.2% (13.9% improvement), and 99.1% (15.9% improvement), respectively. In addition, Alsheikh et al. [151] tested the activity recognition performance of DBNs using different parameter settings. Instead of using the raw Deep Learning in Human Activity Recognition with Wearable Sensors: A Review on Advances acceleration data similar to [156], they used spectrogram signals of the triaxial accelerometer data to train the deep activity recognition models. They found that deep models with more layers outperform the shallow models, and the topology of layers having more neurons than the input layer is shown to be more advantageous, which indicates overcompete representation is essential for learning deep models. The accuracy of the tuned DBN was 98.23%, 91.5%, and 89.38% on the WISDM [81], Daphnet [104], and Skoda [103] benchmark datasets, respectively. In [158], a RBM is used to improve upon other methods of sensor fusion, as neural networks can identify non-intuitive predictive features largely from cross-sensor correlations and thus offer a more accurate estimation. The recognition accuracy with this architecture on the Skoda dataset reached 81%, which is around 6% higher than the traditional classification method with the best performance (Random Forest). In addition to taking advantage of public datasets, there are also researchers employing DBNs on human activity or health-related recognition with self-collected datasets [31,160]. In [31], DBNs are employed in Parkinson's disease diagnosis to explore if they can cope with the unreliable labelling that results from naturalistic recording environments. The data was collected with two tri-axial accelerometers, with one worn on each wrist of the participant. The DBNs built are two-layer RBMs, with the first layer as a Guassian-binary RBM (containing gaussian visible units) and the second layer as binary-binary (containing only binary units) (please refer to [161] for details). In [160], an unsupervised five-layer DBM-DNN is applied for the automatic detection of eating episodes via commercial bluetooth headsets collecting raw audio signals, and demonstrate classification improvement even in the presence of ambient noise. The accuracy of the proposed DBM-DNN approach is 94%, which is significantly better than SVM with a 75.6% accuracy. Convolutional Neural Network (CNN) A CNN comprises convolutional layers that make use of the convolution operation, pooling layers, fully connected layers, and an output layer (usually Softmax layer). The convolution operation with a shared kernel enables the learning process of space invariant features. Because each filter in a convolutional layer has a defined receptive field, CNN is good at capturing local dependency, compared with a fully-connected neural network. Though each kernel in a layer covers a limited size of input neurons, by stacking multiple layers, the neurons of higher layers will cover a larger more global receptive field. The pyramid structure of CNN contributes to its capability of gathering low-level local features into high-level semantic meanings. This allows CNN to learn excellent features as shown in [162], which compares the features extracted from CNN to hand-crafted time and frequency domain features (Fast Fourier Transform and Discrete Cosine Transform). CNN incorporates a pooling layer that follows each convolutional layer in most cases. A pooling layer compresses the representation it is learning and strengthens the model against noise by dropping a portion of the output to a convolutional layer. Generally, a few fully connected layers follow after a stack of convolutional and pooling layers that reduce feature dimensionality before being fed into the output layer. A softmax classifier is usually selected as the final output layer. However, as an exception, some studies explored the use of traditional classifiers as the output layer in a CNN [64,118]. Most CNNs use univariate or multivariate sensor data as input. Besides raw or filtered sensor data, the magnitude of 3-axis acceleration is often used as input, as shown in [26]. Researchers have tried encoding time-series data into 2D images as input into the CNN. In [163], the Short-time Fourier transform (STFT) for time-series sensor data is calculated, and its power spectrum is used as the input to a CNN. Since time series data is generally one-dimensional, most CNNs adopt 1D-CNN kernels. Works that use frequency-domain inputs (e.g., spectrogram), which have an additional frequency dimension, will generally use 2D-CNN kernels [164]. The choice of 1D-CNN kernel size normally falls in the range of 1 × 3 to 1 × 5 (with exceptions in [22,63,64] where kernels of size 1 × 8, 2 × 101, and 1 × 20 are adopted). To discover the relationship between the number of layers, the kernel size, and the complexity level of the tasks, we picked and summarized several typical studies in Table 2. A majority of the CNNs consist of five to nine layers [23,63,64,113,114,165,166,167,168], usually including two to three convolutional layers, two to three max-pooling layers, followed by one to two fully connected layers before feeding the feature representation into the output layer (softmax layer in most cases). Dong et al. [169] demonstrated performance improvements by leveraging both handcrafted time and frequency domain features along with features generated from a CNN, called HAR-Net, to classify six locomotion activities using accelerometer and gyroscope signals from a smartphone. Ravi et al. [170] used a shallow three-layer CNN network including a convolutional layer, a fully connected layer, and a softmax layer to perform on-device activity recognition on a resource-limited platform and shown its effectiveness and efficiency on public datasets. Zeng et al. [22] and Lee et al. [26] also used a small number of layers (four layers). The choice of the loss function is an important decision in training CNNs. In classification tasks, cross-entropy is most commonly used, while in regression tasks, mean squared error is most commonly used. Most CNN models process input data by extracting and learning channel-wise Deep Learning in Human Activity Recognition with Wearable Sensors: A Review on Advances features separately while Huang et al. [167] first propose a shallow CNN that considers cross-channel communication. The channels in the same layer interact with each other to obtain discriminative features of sensor data. Table 2: Summary of typical studies that use layer-by-layer CNN structure in HAR and their configurations. We aim to present the relationship of CNN kernels, layers, and targeted problems (application and sensors). Key: C-convolutional layer; P-max-pooling layer; FC-fully connected layer; S-softmax; S1-accelerometer; S2-gyroscope; S3magnetometer; S4-EMG; S5-ECG Study Architecture Kernel Conv. Application # Classes Sensors Dataset [26] C-P-FC-S 1 × 3, 1 × 4, 1 × 5 locomotion activities 3 S1 Self [171] C-P-C-P-S 4 × 4 locomotion activities 6, 12 S1 UCI, mHealth [22] C-P-FC-FC-S 1 × 20 daily activities, locomotion activities --Skoda, Opportunity, Actitracker [172] C-P-C-P-FC-S 5 × 5 locomotion activities 6 S1 WISDM [173] C-P-C-P-C-FC 1 × 5, 1 × 9 locomotion activities 12 S5 mHealth [174] C-P-C-P-FC-FC-S -daily activities, locomotion activities 12 S1, S2, S3 ECG mHealth [175] C-P-C-P-C-P-S 12 × 2 daily activities including brush teeth, comb hair, get up from bed, etc 12 S1, S2, S3 WHARF [23] C-P-C-P-C-P-S 12 × 2 locomotion activities 8 S1 Self [113] C-P-C-P-U-FC-S, U: unification layer 1 × 3, 1 × 5 daily activities, hand gesture 18 (Opp) 12 (hand) S1, S2 (1 for each) Opportunity Hand Gesture [63] C-C-P-C-C-P-FC 1 × 8 hand motion classification 10 S4 Rami EMG Dataset [114] C-C-P-C-C-P-FC-FC-S (one branch for each sensor) 1 × 5 daily activities, locomotion activities, industrial ordering picking recognition task 18 (Opp) 12 (PAMAP2) S1, S2, S3 Opportunity, PAMAP2, Order Picking [163] C-P-C-P-C-P-FC-FC-FC-S 1 × 4, 1 × 10, 1 × 15 locomotion activities 6 S1, S2, S3 Self The number of sensors used in a HAR study can vary from a single one to as many as 23 [90]. In [23], a single accelerometer is used to collect data from three locations on the body: Cloth pocket, trouser pocket and waist. The authors collect data on 100 subjects, including eight activities such as falling, running, jumping, walking, walking quickly, step walking, walking upstairs, and walking downstairs. Moreover, HAR applications can involve multiple sensors of different types. To account for all these different types of sensors and activities, Grzeszick et al. [176] proposed a multi-branch CNN architecture. A multi-branch design adopts a parallel structure that trains separate kernels for each IMU sensor and concatenates the output of branches at a late stage, after which one or more fully connected layers are applied on the flattened feature representation before feeding into the final output layer. For instance, a CNN-IMU architecture contains m parallel branches, one per IMU. Each branch contains seven layers, then the outputs of each branch are concatenated and fed into a fully connected and a softmax output layer. Gao et al. [177] has introduced a novel dual attention module including channel and temporal attention to improving the representation learning ability of a CNN model. Their method has outperformed regular CNN considerably on a number of public datasets such as PAMAP2 [91], WISDM [81], UNIMIB SHAR [93], and Opportunity [90]. Another advantage of DL is that the features learned in one domain can be easily generalized or transferred to other domains. The same human activities performed by different individuals can have drastically different sensor readings. To address this challenge, Matsui et al. [163] adapted their activity recognition to each individual by adding a few hidden layers and customizing the weights using a small amount of individual data. They were able to show a 3% improvement in recognition performance. Recurrent Neural Network (RNN) Initially, the idea of using temporal information was proposed in 1991 [178] to recognize a finger alphabet consisting of 42 symbols and in 1995 [179] to classify 66 different hand shapes with about 98% accuracy. Since then, the recurrent neural network (RNN) with time series as input has been widely applied to classify human activities or estimate hand gestures [180,181,182,183,184,185,186,187]. Unlike feed-forward neural networks, an RNN processes the input data in a recurrent behavior. Equivalent to a directed graph, RNN exhibits dynamic behaviors and possesses the capability of modelling temporal and sequential relationships due to a hidden layer with recurrent connections. A typical structure for an RNN is shown in Figure 7 with the current input, x t , and previous hidden state, h t−1 . The network generates the current hidden state, h t , and output, y t , is as follows: h t = (W h h t−1 + U h x t + b h ) y t = (W y h t + b y )(1) where W h , U h , and W y are the weights for the hidden-to-hidden recurrent connection, input-to-hidden connection, and hidden-to-output connection, respectively. b h and b y are bias terms for the hidden and output states, respectively. Furthermore, each node is associated with an element-wise non-linearity function as an activation function such as the sigmoid, hyperbolic tangent (tanh), or rectified linear unit (ReLU). In addition, many researchers have undertaken extensive work to improve the performance of RNN models in the context of human activity recognition and have proposed various models based on RNNs, including Independently RNN (IndRNN) [188], Continuous Time RNN (CTRNN) [189], Personalized RNN (PerRNN) [190], Colliding Bodies Optimization RNN (CBO-RNN) [191]. Unlike previous models with one-dimension time-series input, Lv et al. [192] builds a CNN + RNN model with stacked multisensor data in each channel for fusion before feeding into the CNN layer. Ketykó et al. [193] uses an RNN to address the domain adaptation problem caused by intra-session, sensor placement, and intra-subject variances. HAR improves with longer context information and longer temporal intervals. However, this may result in vanishing or exploding gradient problems while backpropagating gradients [194]. In an effort to address these challenges, long short-term memory (LSTM)-based RNNs [195], and Gated Recurrent Units (GRUs) [196] are introduced to model temporal sequences and their broad dependencies. The GRU introduces a reset and update gate to control the flow of inputs to a cell [197,198,199,200,201]. The LSTM has been shown capable of memorizing and modelling the long-term dependency in data. Therefore, LSTMs have taken a dominant role in time-series and textual data analysis. It has made substantial contributions to human activity recognition, speech recognition, handwriting recognition, natural language processing, video analysis, etc. As illustrated in Figure 7 [202], a LSTM cell is composed of: (1) input gate, i t , for controlling flow of new information; (2) forget gate, f t , setting whether to forget content according to internal state; (3) output gate, o t , controlling output information flow; (4) input modulation gate, g t , as main input; (5) internal state, c t , dictates cell internal recurrence; (6) hidden state, h t , contains information from samples encountered within the context window previously. The relationship between these variables are listed as Equation (2) [202].              i t = σ (b i + U i x t + W i h t−1 ) f t = σ (b f + U f x t + W f x t−1 ) o t = σ (b o + U o x t + W o h t−1 ) g t = σ (b g + U g x t + W g h t−1 ) c t = f t c t−1 + g t i t h t = tanh (c t ) o t(2) As shown in Figure 8, the input time series data is segmented into windows and fed into the LSTM model. For each time step, the model computes class prediction scores, which are then merged via late-fusion and used to calculate class membership probabilities through the softmax layer. Previous studies have shown that LSTMs have high performance in wearable HAR [199,202,203]. Researchers in [204] rigorously examine the impact of hyperparameters in LSTM with the fANOVA framework across three representative datasets, containing movement data captured by wearable sensors. The authors assessed thousands of settings with random hyperparameters and provided guidelines for practitioners seeking to apply deep learning to their own problem scenarios [204]. Bidirectional LSTMs, having both past and future recurrent connections, were used in [205,206] to classify activities. Researchers have also explored other architectures involving LSTMs to improve benchmarks on HAR datasets. Residual networks possess the advantage that they are much easier to train as the addition operator enables gradients to pass through more directly. Residual connections do not impede gradients and could help to refine the output of layers. For example, [200] proposes a harmonic loss function and [207] combines LSTM with batch normalization to achieve 92% accuracy with raw accelerometer and gyroscope data. Ref. [208] proposes a hybrid CNN and LSTM model Deep Learning in Human Activity Recognition with Wearable Sensors: A Review on Advances Figure 7: Schematic diagram of an RNN node and LSTM cell [202]. Left: RNN node where h t−1 is the previous hidden state, x t is the current input sample data, h t is the current hidden state, y t is the current output, and is the activation function. Right: LSTM cell with internal recurrence c t and outer recurrence h t . (DeepConvLSTM) for activity recognition using multimodal wearable sensor data. DeepConvLSTM performed significantly better in distinguishing closely-related activities, such as "Open/Close Door" and "Open/Close Drawer". Moreover, Multitask LSTM is developed in [209] to first extract features with shared weight, and then classify activities and estimate intensity in separate branches. Qin et al. proposed a deep-learning algorithm that combines CNN and LSTM networks [210]. They achieved 98.1% accuracy on the SHL transportation mode classification dataset with CNNextracted and hand-crafted features as input. Similarly, other researchers [211,212,213,214,215,216,217,218,219] have also developed the CNN-LSTM model in various application scenarios by taking advantage of the feature extraction ability of CNN and the time-series data reasoning ability of LSTM. Interestingly, utilizing CNN and LSTM combined model, researchers in [219] attempt to eliminate sampling rate variability, missing data, and misaligned data timestamps with data augmentation when using multiple on-body sensors. Researchers in [220] explored the placement effect of motion sensors and discovered that the chest position is ideal for physical activity identification. Raw IMU and EMG time series data are commonly used as inputs to RNNs [193,221,222,223,224,225]. A number of major datasets used to train and evaluate RNN models have been created, including the Sussex-Huawei Locomotion-Transportation (SHL) [188,198], PAMAP2 [192,226] and Opporunity [203]. In addition to raw time series data [199], Besides raw time series data, custom features are also commonly used as inputs to RNNs. [197] showed that training an RNN with raw data and with simple custom features yielded similar performance for gesture recognition (96.89% vs 93.38%). However, long time series may have many sources of noise and irrelevant information. The concept of attention mechanism was proposed in the domain of neural machine translation to address the problem of RNNs being unable to remember long-term relationships. The attention module mimics human visual attention to building direct mappings between the words/phrases that represent the same meaning in two languages. It eliminates the interference from unrelated parts of the input when predicting the output. This is similar to what we as humans perform when we translate a sentence or see a picture for the first time; we tend to focus on the most prominent and central parts of the picture. An RNN encoder attention module is centred around a vector of importance weights. The weight vector is computed with a trainable feedforward network and is combined with RNN outputs at all the time steps through the dot product. The feedforward network takes all the RNN immediate outputs as input to learn the weights for each time step. [201] utilizes attention in combination with a 1D CNN Gated Recurrent Units (GRUs), achieving HAR performances of 96.5% ± 1.0%, 93.1% ± 2.2%, and 89.3% ± 1.3% on Heterogeneous [86], Skoda [103], and PAMAP2 [91] datasets, respectively. [226] applies temporal attention and sensor attention into LSTM to improve the overall activity recognition accuracy by adaptively focusing on important time windows and sensor modalities. In recent years, block-based modularized DL networks have been gaining traction. Some examples are GoogLeNet with an Inception module and Resnet with residual blocks. The HAR community is also actively exploring the application of block-based networks. In [227], the authors have used GoogLeNet's Inception module combined with a GRU layer to build a HAR model. The proposed model was showed performance improvements on three public datasets (Opportunity, PAMAP2 and Smartphones datasets). Qian et al. [228] developed the model with SM M AR in a statistical module to learn all orders of moments statistics as features, LSTM in a spatial module to learn correlations among sensors placements, and LSTM + CNN in a temporal module to learn temporal sequence dependencies along the time scale. Deep Reinforcement Learning (DRL) AE, DBN, CNN, and RNN fall within the realm of supervised or unsupervised learning. Reinforcement learning is another paradigm where an agent attempts to learn optimal policies for making decisions in an environment. At each time step, the agent takes an action and then receives a reward from the environment. The state of the environment accordingly changes with the action made by the agent. The goal of the agent is to learn the (near) optimal policy (or probability of action, state pairs) through the interaction with the environment in order to maximize a cumulative long-term reward. The two entities-agent and environment-and the three key elements-action, state and reward-collectively form the paradigm of RL. The structure of RL is shown in Figure 9. Figure 9: A typical structure of a reinforcement learning network [229]. In the domain of HAR, [230] uses DRL to predict arm movements with 98.33% accuracy. Ref. [231] developed a reinforcement learning model for imitating the walking pattern of a lower-limb amputee on a musculoskeletal model. The system showed 98.02% locomotion mode recognition accuracy. Having a high locomotion recognition accuracy is critical because it helps lower-limb amputees prevent secondary impairments during rehabilitation. In [232], Bhat et al. propose a HAR online learning framework that takes advantage of reinforcement learning utilizing a policy gradient algorithm for faster convergence achieving 97.7% in recognizing six activities. Deep Learning in Human Activity Recognition with Wearable Sensors: A Review on Advances Generative Adversarial Network (GAN) Originally proposed to generate credible fake images that resemble the images in the training set, GAN is a type of deep generative model, which is able to create new samples after learning from real data [233]. It comprises two networks, the generator (G) and the discriminator (D), competing against each other in a zero-sum game framework as shown in Figure 10. During the training phase, the generator takes as input a random vector z and transforms z ∈ R n to plausible synthetic samplesx to challenge the discriminator to differentiate between original samples x and fake samplesx. In this process, the generator strives to make the output probability D(G(z)) approach one, in contrast with the discriminator, which tries to make the function's output probability as close to zero as possible. The two adversarial rivals are optimized by finding the Nash equilibrium of the game in a zero-sum game setting, which means the adversarial rivals' gains would be maintained regardless of what strategies are selected. However, it is not theoretically guaranteed that GAN zero-sum games reach Nash Equilibria [234]. GAN model has shown remarkable performance in generating synthetic data with high quality and rich details [235,236]. In the field of HAR, GAN has been applied as a semi-supervised learning approach to deal with unlabeled or partially labelled data for improving performance by learning representations from the unlabeled data, which later will be utilized by the network to generalize to the unseen data distribution [237]. Afterwards, GAN has shown the ability to generate balanced and realistic synthetic sensor data. Wang et al. [262] utilized GANs with a customized network to generate synthetic data from the public HAR dataset HASC2010corpus [238]. Similarly, Alharbi et al. [239] assessed synthetic data with CNN or LSTM models as a generator. In two public datasets, Sussex-Huawei Locomotion (SHL) and Smoking Activity Dataset (SAD), the discriminator was built with CNN layers, and the results demonstrated synthetic data with high quality and diversity with two public datasets. Moreover, by oversampling and adding synthetic sensor data into the training, researchers augmented and alleviated the originally imbalanced training set to achieve better performance. In [240,241], they generated verisimilar data of different activities, and Shi et al. [242] used the Boulic kinematic model, which aims to capture the three-dimensional positioning trend to synthesize personified walking data. Due to the ability to generate new data, GAN has been widely applied in transfer learning in HAR to help with the dramatic performance drop when the pre-trained model are tested against unseen data from new users. In transfer learning techniques, the learned knowledge from the source domain (subject) is transferred to the target domain to decrease the lack of performance of the models within the target domain. Moreover, [243] is an attempt that utilized GAN to perform cross-subject transfer learning for HAR since collecting data for each new user was infeasible. With the same idea, cross-subject transfer learning based on GAN outperformed those without GAN on Opportunity benchmark dataset in [243] and outperformed unsupervised learning on UCI and USC-HAD dataset [244]. Even more, transfer learning under conditions of cross-body, cross-user, and cross-sensor has been demonstrated superior performance in [245]. However, much more effort is needed in generating verisimilar data to alleviate the burden and cost of collecting sufficient user data. Additionally, it is typically challenging to obtain well-trained GAN models owing to the wide variability in amplitude, frequency, and period of the signals obtained from different types of activities. Hybrid Models As an advancement of machine learning models, researchers take advantage of different methods and propose hybrid models. The combination of CNN and LSTM endows the model capability of extracting local features as well as longterm dependencies in sequential data, especially for HAR time series data. For example, Challa et al. [246] proposed a hybrid of CNN and bidirectional long short-term memory (BiLSTM). The accuracy on UCI-HAR, WISDM [81], and PAMAP2 [91] datasets achieved 96.37%, 96.05%, and 94.29%, respectively. Dua et al. [247] proposed a model with CNN combined with GRU and obtained an accuracy of 96.20%, 97.21%, and 95.27% on UCI-HAR, WISDM [81], and PAMAP2 [91] datasets, respectively. In order to have a straightforward view of the functionality of hybrid models, we list several papers with CNN only, LSTM only, CNN + GRU, and CNN + LSTM in Tables 3 and 4. In addition, Zhang et al. [296] proposed to combine reinforcement learning and LSTM model to improve the adaptability of different kinds of sensors, including EEG (EID dataset), RFID (RSSI dataset) [248], and wearable IMU (PAMAP2 dataset) [91]. Ref. [249] employed CNN for feature extraction and a reinforced selective attention model to automatically choose the most characteristic information from multiple channels. Summary and Selection of Suitable Methods Since the last decade, DL methods have gradually dominated a number of artificial intelligence areas, including sensor-based human activity recognition, due to its automatic feature extraction capability, strong expressive power, and the high performance rendered. When a sufficient amount of data are available, we are becoming prone to turn to DL methods. With all these types of available DL approaches discussed above, we need to get a full understanding of the pros and cons of these approaches in order to select the appropriate approach wisely. To this end, we briefly analyze the characteristics of each approach and attempt to give readers high-level guidance on how to choose the DL approach according to the needs and requirements. The most salient characteristic of auto-encoder is that it does not require any annotation. Therefore, it is widely adopted in the paradigm of unsupervised learning. Due to its exceptional capability in dimension reduction and noise suppression, it is often leveraged to extract low-dimensional feature representation from raw input. However, auto-encoders may not necessarily learn the correct and relevant characteristics of the problem at hand. There is also generally little insight that can be gained for sensor-based auto-encoders, making it difficult to know which parameters to adjust during training. Deep belief networks are a generative model generally used for solving unsupervised tasks by learning low-dimensional Deep Learning in Human Activity Recognition with Wearable Sensors: A Review on Advances features. Today, DBNs have been less often chosen compared with other DL approaches and are rarely used due to the tedious training process and increased training difficulty with DBN when the network goes deeper [7]. CNN architecture is powerful to extract hierarchical features owing to its layer-by-layer hierarchical structure. When compared with other approaches like RNN and GAN, CNN is relatively easy to implement. Besides, as one of the most studied DL approaches in image processing and computer vision, there is a large range of CNN variants existing that we can choose from to transfer to sensor-based HAR applications. When sensor data are represented as two-dimensional input, we can directly start with pre-trained models on a large image dataset (e.g., ImageNet) to fasten the convergence process and achieve better performance. Therefore, adapting the CNN approach enjoys a higher degree of flexibility in the available network architecture (e.g., GoogLeNet, MobileNet, ResNet, etc) than other DL approaches. However, CNN architecture has the requirement of fixed-sized input, in contrast to RNN, which accepts flexible input size. In addition, compared with unsupervised learning methods such as auto-encoder and DBN, a large number of annotated data are required, which usually demands expensive labelling resources and human effort to prepare the dataset. The biggest advantage of RNN and LSTMs is that they can model time series data (nearly all sensor data) and temporal relationships very well. Additionally, RNN and LSTMs can accept flexible input data size. The factors that prevent RNN and LSTMs from becoming the de facto method in DL-based HAR is that they are difficult to train in multiple aspects. They require a long training time and are very susceptible to diminishing/exploding gradients. It is also difficult to train them to efficiently model long time series. GAN, as a generative model, can be used as a data augmentation method. Because it has a strong expressive capability to learn and imitate the latent data distribution of the targeted data, it outperforms traditional data augmentation methods [36]. Owing to its inherent data augmentation ability, GAN has the advantage of alleviating data demands at the beginning. However, GAN is often considered as hard to train because it alternatively trains a generator and a discriminator. Many variants of GAN and special training techniques have been proposed to tackle the converging issue [255,256,257]. Reinforcement learning is a relatively new area that is being explored for select areas in HAR, such as modelling muscle and arm movements [230,231]. Reinforcement learning is a type of unsupervised learning because it does not require explicit labels. Additionally, due to its online nature, reinforcement learning agents can be trained online while deployed in a real system. However, reinforcement learning agents are often difficult and time-consuming to train. Additionally, in the realm of DL-based HAR, the reward of the agent has to be given by a human, as in the case of [230,232]. In other words, even though people do not have to give explicit labels, humans are still required to provide something akin to a label (the reward) to train the agent. When starting to choose a DL approach, we have a list of factors to consider, including the complexity of the target problem, dataset size, the availability and size of annotation, data quality, available computing resource, as well as the requirement of training time. Firstly, we have to evaluate and examine the problem complexity to decide upon promising venues of machine learning methods. For example, if the problem is simple enough to resolve with the provided sensor modality, it's very likely that manual feature engineering and traditional machine learning method can provide satisfying results thus no DL method is needed. Secondly, before we choose the routine of DL, we would like to make sure the dataset size is sufficient to support a DL method. The lack of a sufficiently large corpus of labelled high-quality data is a major reason why DL methods cannot produce an expected result. Normally, when training a DL model with a limited dataset size, the model will be prone to overfitting, and the generalizability will be sacrificed, thus using a very deep network may not be a good choice. One option is to go for a shallow neural network or a traditional ML approach. Another option is to utilize specific algorithms to make the most out of the data. To be specific, data augmentation methods such as GAN can be readily implemented. Thirdly, another determining factor is the availability and size of annotation. When there is a large corpus of unlabeled sensor data at hand, a semi-supervised learning scheme is a promising direction one could consider, which will be discussed later in this work. Besides the availability of sensor data, the data quality also influences the network design. If the sensor is vulnerable to environmental noise, inducing a small SNR, some type of denoising structure (e.g., denoising auto-encoder) and increasing depth of the model can be considered to increase the noise-resiliency of the DL model. At last, a full evaluation of available computing resources and expected model training time cannot be more important for developers and researchers to choose a suitable DL approach. Challenges and Opportunities Though HAR has seen rapid growth, there are still a number of challenges that, if addressed, could further improve the status quo, leading to increased adoption of novel HAR techniques in existing and future wearables. In this section, we discuss these challenges and opportunities in HAR. Note that the issues discussed here are applicable to general HAR, Challenges in Data Acquisition Data is the cornerstone of artificial intelligence. Models only perform as well as the quality of the training data. To build generalizable models, careful attention should be paid to data collection, ensuring the participants are representative of the population of interest. Moreover, determining a sufficient training dataset size is important in HAR. Currently, there is no well-defined method for determining the sample size of training data. However, showing the convergence of the error rate as a function of training data size is one approach shown by Yang et al. [258]. Acquiring a massive amount of high-quality data at a low cost is critical in every domain. In HAR, collecting raw data is labor-intensive considering a large number of different wearables. Therefore, proposing and developing innovative approaches to augmenting data with high quality is imperative for the growth of HAR research. The Need for More Data Data collection requires a considerable amount of effort in HAR. Particularly when researchers propose their original hardware, it is inevitable to collect data on users. Data augmentation is commonly used to generate synthetic training data when there is a data shortage. Synthetic noise is applied to real data to obtain new training samples. In general, using the dataset augmented with synthetic training samples yields higher classification accuracy when compared to using the original dataset [36,56,259,260]. Giorgi et al. augmented their dataset by varying each signal sample with translation drawn from a small uniform distribution and showed improvements in accuracy using this augmented dataset [56]. Ismail Fawaz et al. [260] utilized Dynamic Time Warping to augment data and tested on UCR archive [105]. Deep learning methods are also used to augment the datasets to improve performance [261,262,263]. Alzantot et al. [263] and Wang et al. [262] employed GAN to synthesize sensor data using existing sensor data. Ramponi et al. [261] designed a conditional GAN-based framework to generate new irregularly-sampled time series to augment unbalanced data sets. Several works extracted 3D motion information from videos and transferred the knowledge to synthesize virtual on-body IMU sensor data [264,265]. In this way, they realized cross-modal IMU sensor data generation using traditional computer vision and graphics methods. Opportunity: We have listed some of the most recent works focusing on cross-modal sensor data synthesis. However, few researchers (if any) used a deep generative model to build a video-sensor multi-modal system. If we take a broader view, many works are using cross-modal deep generative models (such as GAN) in data synthesis, such as from video to audio [266], from text to image and vice versa [267,268]. Therefore, taking advantage of the cutting-edge deep generative models may contribute to addressing the wearable sensor data scarcity issue [269]. Another avenue of research is to utilize transfer learning, borrowing well-trained models from domains with high performing classifiers (i.e., images), and adapting them using a few samples of sensor data. Data Quality and Missing Data The quality of models is highly dependent on the quality of the training data. Many real-world collection scenarios introduce different sources of noise that degrade data quality, such as electromagnetic interference or uncertainty in task scheduling for devices that perform sampling [270]. In addition to improving hardware systems, multiple algorithms have been proposed to clean or impute poor-quality data. Data imputation is one of the most common methods to replace poor quality data or fill in missing data when sampling rates fluctuate greatly. For example, Cao et al. introduced a bi-directional recurrent neural network to impute time series data on the UCI localization dataset [271]. Luo et al. utilized a GAN to infer missing time series data [272]. Saeed et al. proposed an adversarial autoencoder (AAE) framework to perform data imputation [132]. Opportunity: To address this challenge, more research into automated methods for evaluating and quantifying the quality is needed to identify better, remove, and/or correct for poor quality data. Additionally, it has been experimentally shown that deep neural networks have the ability to learn well even if trained with noisy data, given that the networks are large enough and the dataset is large enough [273]. This motivates the need for HAR researchers to focus on other areas of importance, such as how to deploy larger models in real systems efficiently (Section 6.4) and generate more data (Section 6.1.1), which could potentially aid in solving this problem. Privacy Protection The privacy issue has become a concern among users [13]. In general, the more inference potential a sensor has, the less willing a person is to agree to its data collection. Multiple works have proposed privacy preservation methods while classifying human activities, including the replacement auto-encoder, the guardian, estimator, and neutralizer (GEN) architecture [128], and the anonymizing autoencoder [129]. For example, replacement auto-encoders learn to replace features of time-series data that correspond to sensitive inferences with values that correspond to non-sensitive inferences. Ultimately, these works obfuscate features that can identify the individual while preserving features common to each activity or movement. Federated learning is a trending approach to resolve privacy issues in learning problems [274,275,276,277]. It can enable the collaborative learning of a global model without the need to expose users' raw data. Xiao et al. [278] realized a federated averaging method combined with a perceptive extraction network to improve the performance of the federated learning system. Tu et al. [279] designed a dynamic layer sharing scheme, which assisted the merging of local models to speed up the model convergence and achieved dynamic aggregation of models. Bettini et al. [280] presented a personalized semi-supervised federated learning method that built a global activity model and leveraged transfer learning for user personalization. Besides, Gudur and Perepu [281] implemented on-device federated learning using model distillation update and so-called weighted α-updates strategies to resolve model heterogeneities on a resource-limited embedded system (Raspberry Pi), which proved its effectiveness and efficiency. Opportunity: Blockchain is a new hot topic around the world. Blockchain, as a peer-to-peer network without the need for centralized authority, has been explored to facilitate the privacy-preserving data collection and sharing [282,283,284,285]. The combination of federated learning and blockchain is also a potential solution towards privacy protection [286] and is currently still in its very early stage. More collaboration between ubiquitous computing community and networking community should be encouraged to prosper in-depth research in novel directions. Challenges in Label Acquisition Labelled data is crucial for deep supervised learning. Image and audio data is generally easy to label by visual or aural confirmation. However, labelling human activities by looking at time series from HAR sensors is difficult or even impossible. Therefore, label acquisition for HAR sensors generally requires additional sensing sources to provide video or audio data to determine the ground truth, making label acquisition for HAR more labor-intensive. Moreover, accurate time synchronization between wearables and video/audio devices is challenging because different devices are equipped with independent (and often drifting) clocks. Several attempts have been made to address this issue, such as SyncWISE [287] and [288]. Two areas that require more research by the DL-HAR community are shortage in labelled data and difficulty in obtaining data from real-world scenarios. Shortage of Labeled Data As annotating large quantities of data is expensive, there have been great efforts to develop various methods to reduce the need for annotation, including data augmentation, semi-supervised learning, weakly supervised learning, and active learning to overcome this challenge. Semi-supervised learning utilizes both labelled data and unlabeled data to learn more generalizable feature representations. Zeng et al. presented two semi-supervised CNN methods that utilize unlabeled data during training: The convolutional encoder-decoder and the convolutional ladder network [289] and showed an 18% higher F1-score using the convolutional ladder network on the ActiTracker dataset. Dmitrijs demonstrated on the SHL dataset, with a CNN and AAE architecture, that semi-supervised learning on unlabeled data could achieve high accuracy [134]. Chen et al. proposed an encoder-decoder-based method that reduces distribution discrepancies between labelled and unlabeled data that arise due to differences in biology and behavior from different people while preserving the inherent similarities of different people performing the same task [290]. Active learning is a special type of semi-supervised learning that selectively chooses unlabeled data based on an objective function that selects data with low prediction confidence for a human annotator to label. Recently, researchers have tried to combine DL approaches with active learning to benefit from establishing labels on the fly while leveraging the extraordinary classification capability of DL. Gudur et al. utilized active learning by combining a CNN with Bayesian techniques to represent model uncertainties (B-CNN) [291]. Bettini et al. combined active learning and federated learning to proactively annotate the unlabeled sensor data and build personalized models in order to cope with data scarcity problem [280]. Opportunity: Though active learning has demonstrated that fewer labels are needed to build an effective deep neural network model, a real-world study with time-cost analysis would better demonstrate the benefits of active learning. Moreover, given the many existing labelled datasets, another area of opportunity is developing methods that leverage characteristics of labelled datasets to generate labels for unlabeled datasets such as transfer learning or pseudo-label method [292]. Issues of In-the-field Dataset Traditionally, HAR research has been conducted primarily in lab. Recently, HAR research has been moving towards in-field experiments. Unlike in-lab settings, where the ground truth can be captured by surveillance cameras, in-field experiments may have subjects moving around in daily life, where static camera deployment is not sufficient any more. Alharbi et al. used wearable cameras placed at the wrist, chest, and shoulder to record subject's activities as they moved around outside of a lab setting [293] and studied the feasibility of wearable cameras. Opportunity: More research in leveraging human-in-the-loop to provide in-field labelling is required to generate more robust datasets for in situ activities. Besides, one possible solution is to utilize existing in-the-field human activity video datasets and cross-modal deep generative models. If high-fidelity synthetic wearable sensor data can be generated from the available real-world video datasets (such as Stanford-ECM dataset [294]) or online video corpus, it may help alleviate the in-the-field data scarcity issue. Additionally, there are opportunities for semi-supervised learning methods that leverage the sparse labels provided by humans-in-the-loop to generate high-quality labels for the rest of the dataset. Challenges in Modeling In this section, we discuss the challenges and opportunities in the modelling process in several aspects, including data segmentation, semantically complex activity recognition, model generalizability, as well as model robustness. Data Segmentation As discussed in [295], many methods segment time series using traditional static sliding window methods. A static time window may either be too large, capturing more than necessary to detect certain activities, or too small and not capturing enough series to detect long movements. Recently, researchers have been looking to segment time series data more optimally. Zhang et al. used reinforcement learning to find more optimal activity segments to boost HAR performance [296]. Qian et al. [297] proposed weakly-supervised sensor-based activity segmentation and recognition method. Opportunity: More experimentation and research into dynamic activity segments or methods that leverage both short term and long term features (i.e., wavelets) are needed to create robust models at all timescales. While neural networks such as RNNs and LSTMs can model time series data with flexible time scales and automatically learn relevant features, their inherent issues such as exploding/vanishing gradients and training difficulty, make widespread adoption difficult. As such, more research into other methods that account for these issues is necessary. Semantically Complex Activity Recognition Current HAR methods achieve high performance for simple activities such as running. However, complex activities such as eating, which can involve a variety of movements, remain difficult. To tackle this challenge, Kyritsis et al. break down complex gestures into a series of simpler (atomic) gestures that, when combined, form the complex gesture [298]. Liu et al. propose a hierarchical architecture that constructs high-level human activities from low-level activities [299]. Peng et al. proposes AROMA, a complex human activity recognition method that leverages deep multi-task learning to learn simple activities that make up more complex movements [300]. Opportunity: Though hierarchical methods have been introduced for various complex tasks, there are still opportunities for improvements. Additionally, novel black-box approaches to complex task recognition, where individual steps in complex actions are automatically learned and accounted for rather than specifically identified or labelled by designers, have yet to be fully explored. Such a paradigm is perfectly suitable for deep learning because neural networks function on a similar principle. Besides, graph neural network can also be explored to model the hierarchical structure of simple-to-complex human activities [301]. Model Generalizability A model has high generalizability when it performs well on data that it has never seen before. Overfitting occurs when it performs well on training data but poorly on new data. Recently, many efforts have been put into improving the generalizability of models in HAR [86,302,303]. Most research on generalizability in HAR has been focused on creating models that can generalize to a larger population, which often requires a large amount of data and high model complexity. In scenarios where high model complexity and data are not bottlenecks, DL-based HAR generally outperforms and generalize better than other types of methods. In scenarios where data or model complexity is limited, DL-based methods must utilize available data more efficiently or adapt to the specific scenario online. For instance, Siirtola and Röning propose an online incremental learning approach that continuously adapts the model with the user's individual data as it comes in [304]. Qian et al. [305] introduce Generalizable Independent Latent Excitation (GILE), which greatly enhances the cross-person generalization capability of the model. Opportunity: An avenue of generalizability that has yet to be fully explored are new training methods that can adapt and learn predictors across multiple environments, such as invariant risk minimization [306] or federated learning methods [307]. Incorporating Deep Learning in Human Activity Recognition with Wearable Sensors: A Review on Advances these areas into DL-based HAR could not only improve the generalizability of HAR models but accomplish this in a model-agnostic way. Model Robustness A key issue that the community is paying increasing attention to is model robustness and reliability [308,309]. One common way to improve robustness is to leverage the benefits of multiple types of sensors together to create multi-sensory systems [296,310,311,312,313,314]. Huynh-The et al. [311] has proposed an architecture called DeepFusionHAR to incorporate the handcrafted features and deep learning extracted features from multiple sensors to detect daily life and sports activities. Hanif et al. [312] proposed a multi-sensory approach for basic and complex human activity recognition that uses built-in sensors from smartphones and smartwatches to classify 20 complex actions and five basic actions. Pires et al. [313] demonstrated a mobile application on a multi-sensor mobile platform for daily living activity classification using a combination of accelerometer, gyroscope, magnetometer, microphone, and GPS. Multi-sensory networks in some cases are integrated with attention modules to learn the most representative and discriminative sensor modality to distinguish human activities [296]. Opportunity: While there are works that utilize multiple sensors to improve robustness, they require users to wear or have access to all of the sensors they utilize. An exciting new direction is to create generalized frameworks that can adaptively utilize data from whatever sensors happen to be available, such as a smart home intelligence system [315]. For this direction, deep learning methods seem more suitable than classical machine learning methods because neural networks can be more easily tuned and adapted to different domains (i.e., different sensors) than rigid classical models, just by tuning weights or by mixing and matching different layers or embeddings. Creating such systems would not only greatly improve the practicality of HAR-based systems but would also contribute significantly to general artificial intelligence. Challenges in Model Deployment There are several works focusing on deploying deep-learning-based HAR on mobile platforms. Lane et al. [316] proposes a SOC-based architecture for optimizing deep neural network inference, while Lane et al. [317] and Cao et al. [318] utilize the smartphone's digital signal processor (DSP) and mobile GPU to improve inference time and reduce power consumption. Yao et al. [319] propose a lightweight CNN and RNN-based system that accounts for noisy sensor readings from smartphones and automatically learns local and global features between sensor windows to improve performance. The second class of works focus on reducing the complexity of neural networks so that they can run on resource-limited mobile platforms. Bhattacharya and Lane [320] reduces the amount of computation required at each layer by encoding layers into a lower-dimensional space. Edel and Köppe [321] reduces computation by utilizing binary weights rather than fixed-point or floating weights. Emerging trends in deploying neural networks include offloading computation onto application-specific integrated circuits (ASIC) or lower power consumption microcontrollers. Bhat et al. [322], Wang et al. [323] developed custom integrated circuits and hardware accelerators that perform the entire HAR pipeline with significantly lower power consumption than mobile or GPU-based platforms. The downside to ASICs is that they cannot be reconfigured for other types of tasks. Islam and Nirjon [324] present an architecture for embedded systems that dynamically schedules DNN inference tasks to improve inference time and accuracy. Opportunity: Though there are works that explore the deployment of DNNs practical systems, more research is needed for society to fully benefit from the advances in DNNs for HAR. Many of the works discussed leverage a single platform (i.e., either a smartphone or ASIC), but there are still many opportunities for improving the practical use of HAR by exploring intelligent ways to partition computation across the cloud, mobile platforms, and other edge devices. DNN-based HAR systems can largely benefit by incorporating methodologies proposed by works such as [325,326,327,328,329,330,331,332], that carefully partition computation and data across multiple devices and the cloud. Lane et al. performed a small-scale exploration into the performance of DNNs for HAR applications on mobile platforms in various configurations, including utilizing the phone's CPU and DSP and offloading computation onto remote devices [333]. This work demonstrates that mobile devices running DNN inference can scale gracefully across different compute resources available to the mobile platform and also supports the need for more research into optimal strategies for partitioning DNN inference across mobile and edge systems to improve latency, reduce power consumption, and increase the complexity of the DNNs serviceable to wearable platforms. Conclusions Human activity recognition in wearables has provided us with many conveniences and avenues to monitor and improve our life quality. AI and ML have played a vital role in enabling HAR in wearables. In recent years, DL has pushed the boundary of wearables-based HAR, bringing activity recognition performance to an all-time high. In this paper, we provided our answers to the three research questions we proposed in Section 2. We firstly gave an overall picture of the real-life applications, mainstream sensors, and popular public datasets of HAR. Then we gave a review of the advances of the deep learning approaches used in the field of wearable HAR and provided guidelines and insights about how to choose an appropriate DL approach after comparing the advantages and disadvantages of them. At last, we discussed the current road blockers in three aspects-data-wise, label-wise, and model-wise-for each of which we provide potential opportunities. We further identify the open challenges and finally provide suggestions for future avenues of research in this field. By categorizing and summarizing existing works that apply DL approaches to wearable sensor-based HAR, we aim to provide new engineers and researchers entering this field an overall picture of the existing research work and remaining challenges. We would also like to benefit experienced researchers by analyzing and discussing the developing trends, major barriers, cutting-edge frontiers, and potential future directions. Figure 2 : 2Consort diagram outlining how we selected the final papers we included in this work. Figure 3 : 3Taxonomy of Deep Learning-based Human Activity Recognition with Wearables. Figure 4 : 4Placement of inertial sensors in different datasets: WISDOM; ActRecTut; UCI-HAR; SHO; PAMAP2; and Opportunity. Figure 2 . 2Illustration of an autoencoder network. Figure 8 : 8The structure of LSTM and bi-directional LSTM model[204]. (a). LSTM network hidden layers containing LSTM cells and a final softmax layer at the top. (b) bi-directional LSTM network with two parallel tracks in both future (green) and past (red) directions. Figure 10 : 10The structure of generative adversarial network. There are basically two kinds of sensors: Physical sensors and physiological sensors.Physical sensors include Inertial Measurement Unit (IMU), piezoelectric sensor, GPS, wearable camera, etc. Some exemplary physiological sensors are electromyography (EMG) and photoplethysmography (PPG), just to name a few. In terms of the applications of HAR systems, we categorized them into healthcare, fitness& lifestyle, and Human Computer Interaction (HCI). Regarding the DL algorithm, we introduce six approaches, including autoencoder (AE), Deep Belief Network (DBN), Convolutional Neural Network, Recurrent Neural Network (including Long Short-Term Memory (LSTM) and Gated Recurrent Units (GRUs)), Generative Adversarial Network (GAN), and Deep Reinforcement Deep Learning in Human Activity Recognition with Wearable Sensors: A Review on Advances Paper topic filtering (n=7268) Google Scholar using keywords 1 (n=8400) Paper novelty filtering (n=870) Paper language filtering (n=709) Paper relevance selection (n=704) -Not using wearable non-visual sensors (n=2194) -Not using deep learning (n=2031) -Not activity recognition (n=2173) -Remove review/survey papers (n=52) -No technical contribution (n=109) -Not in English (n=5) -Remove duplicates (n=1132) Papers included (n=176) -Of top 25% relevance et al. accentuated the advancements in deep learning models by proposing a taxonomy of generative, discriminative, and hybrid methods along with further explorations for the advantages and Deep Learning in Human Activity Recognition with Wearable Sensors: A Review on AdvancesDeep Learning-based Human Activity Recognition with Wearales Application Sensor DL Approach Challenge Physical Healthcare Autoencoder (AE) Deep Belief Network (DBN) Convolutional Neural Network (CNN) Recurrent Neural Network (RNN) Generative Adversarial Network (GAN) Deep Reinforcement Learning (DRL) Hybrid Models Fitness & Lifestyle Table 1 : 1Major Public Datasets for Wearable-based HAR.Dataset Application Sensor # Classes Spl. Rate Citations/yr WISDM [81] Locomotion 3D Acc. 6 20 Hz 217 ActRecTut [100] Hand gestures 9D IMU 12 32 Hz 153 UCR(UEA)-TSC [105, 106] 9 datasets (e.g., uWave [107]) Vary Vary Vary 107 UCI-HAR [82] Locomotion Smartphone 9D IMU 6 50 Hz 78 Ubicomp 08 [83] Home activities Proximity sensors 8 N/A 69 SHO [84] Locomotion Smartphone 9D IMU 7 50 Hz 52 UTD-MHAD1/2 [85] Locomotion & activities 3D Acc. & 3D Gyro. 27 50 Hz 39 HHAR [86] Locomotion 3D Acc. 6 50-200 Hz 37 Daily & Sports Activities [87] Locomotion 9D IMU 19 25 Hz 37 MHEALTH [88, 89] Locomotion & gesture 9D IMU & ECG 12 50 Hz 33 Opportunity [90] Locomotion & gesture 9D IMU 16 50 Hz 32 PAMAP2 [91] Locomotion & activities 9D IMU & HR monitor 18 100 Hz 32 Daphnet [104] Freezing of gait 3D Acc. 2 64 Hz 30 SHL [108] Locomotion & transportation 9D IMU 8 100 Hz 23 SARD [92] Locomotion 9D IMU & GPS 6 50 Hz 22 Skoda Checkpoint [103] Assembly-line activities 3D Acc. 11 98 Hz 21 UniMiB SHAR [93] Locomotion & gesture 9D IMU 12 N/A 20 USC-HAD [94] Locomotion 3D ACC. & 3D Gyro. 12 100 Hz 20 ExtraSensory [95] Locomotion & activities 9D IMU & GPS 10 25-40 Hz 13 HASC [96] Locomotion Smartphone 9D IMU 6 100 Hz 11 Actitracker [97] Locomotion 9D IMU & GPS 5 N/A 6 FIC [101] Feeding gestures 3D Acc. 6 20 Hz 5 WHARF [98] Locomotion Smartphone 9D IMU 16 50 Hz 4 Most of the datasets listed above are publicly available. The University of California Riverside-Time Series Classification (UCR-TSC) archive is a collection of datasets collected from various sensing modalities [109]. The UCR-TSC archive was first released and included 16 datasets, growing to 85 datasets by 2015 and 128 by October 2018. Recently, researchers from the University of East Anglia have collaborated with UCR to generate a new collection of datasets, which includes nine categories of HAR: BasicMotions , Cricket, Epilepsy, ERing, Handwriting, Libras, NATOPS, RacketSports, and UWaveGestureLibrary Table 3 : 3Comparison of models on UCI-HAR dataset.Model F1-Score (%) Accuracy (%) CNN [250] 92.93 92.71 Res-LSTM [251] 91.50 91.60 Stacked-LSTM [252] - 93.13 CNN-LSTM [215] - 92.13 Bidir-LSTM [253] - 92.67 Residual-BiLSTM [251] 93.5 93.6 LSTM-CNN [211] - 95.78 CNN-GRU [247] - 96.20 CNN-GRU [246] 94.54 94.58 CNN-LSTM [246] 94.76 94.80 CNN-BiLSTM [246] 96.31 96.37 Table 4 : 4Comparison of models on PAMAP2 dataset. Model F1-Score (%) Accuracy (%) CNN[250] 91.16 91.00 BiLSTM [253] 89.40 89.52 LSTM-F [204] 92.90 - COND-CNN [254] - 94.01 CNN-GRU [247] - 95.27 CNN-GRU [246] 93.16 93.20 CNN-LSTM [246] 92.77 92.81 CNN-BiLSTM [246] 94.27 94.29 Deep Learning in Human Activity Recognition with Wearable Sensors: A Review on Advances AcknowledgmentsSpecial thanks to Haik Kalamtarian and Krystina Neuman for their valuable feedback. About One-in-five Americans Use a Smart Watch or Fitness Tracker. E A Vogels, 10Vogels, E.A. About One-in-five Americans Use a Smart Watch or Fitness Tracker. Available online: https://www.pewresearch.org/fact-tank/2020/01/09/ about-one-in-five-americans-use-a-smart-watch-or-fitness-tracker/ (accessed on 10 Feb 2022). End-Use Industry (Consumer Electronics, Healthcare, Enterprise and Industrial. M Research, Wearable Devices Market by Product Type (Smartwatch, Earwear, Eyewear, and others). 10Connectivity Medium, and Region-Global Forecast to 2025Research, M. 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[ "Electrokinetic instability in the sharp interface limit: The perpendicular electric field case", "Electrokinetic instability in the sharp interface limit: The perpendicular electric field case" ]
[ "H.-Y Hsu \nDepartment of Mechanical Engineering\nNorthwestern University\n2145 Sheridan Road60208EvanstonILUSA\n", "Neelesh A Patankar \nDepartment of Mechanical Engineering\nNorthwestern University\n2145 Sheridan Road60208EvanstonILUSA\n" ]
[ "Department of Mechanical Engineering\nNorthwestern University\n2145 Sheridan Road60208EvanstonILUSA", "Department of Mechanical Engineering\nNorthwestern University\n2145 Sheridan Road60208EvanstonILUSA" ]
[]
In this paper, instability at an interface between two miscible liquids with identical mechanical properties but different electrical conductivities is analyzed in the presence of an electric field that is perpendicular to the interface. A parallel electric field case was considered in a previous work[1]. A sharp Eulerian interface is considered between the two miscible liquids. Linear stability analysis leads to an analytic solution for the critical condition of instability. The mechanism of instability is analyzed. Key differences between the perpendicular and parallel electric field cases are discussed. The effect of a microchannel geometry is studied and the relevant non-dimensional parameters are identified.
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[ "https://export.arxiv.org/pdf/1402.1776v1.pdf" ]
119,204,630
1402.1776
6f3ae5556bfe671bef55995f3c5b5574d28ec892
Electrokinetic instability in the sharp interface limit: The perpendicular electric field case H.-Y Hsu Department of Mechanical Engineering Northwestern University 2145 Sheridan Road60208EvanstonILUSA Neelesh A Patankar Department of Mechanical Engineering Northwestern University 2145 Sheridan Road60208EvanstonILUSA Electrokinetic instability in the sharp interface limit: The perpendicular electric field case In this paper, instability at an interface between two miscible liquids with identical mechanical properties but different electrical conductivities is analyzed in the presence of an electric field that is perpendicular to the interface. A parallel electric field case was considered in a previous work[1]. A sharp Eulerian interface is considered between the two miscible liquids. Linear stability analysis leads to an analytic solution for the critical condition of instability. The mechanism of instability is analyzed. Key differences between the perpendicular and parallel electric field cases are discussed. The effect of a microchannel geometry is studied and the relevant non-dimensional parameters are identified. Introduction Instabilities at an interface between two miscible liquids with identical mechanical properties but different electrical conductivities, in the presence of externally applied electric fields, have been studied by Santiago and co-workers [2,3,4,5,6,7,8]. These are instabilities in strong electrolytes and have been termed electrokinetic instabilities. Instabilities with leaky dielectrics have also been considered in literature [9,10]. In this paper, electrokinetic instabilities are considered where the applied electric field is perpendicular to the interface between two strong electrolytes. This case is expected to be more unstable compared to the parallel electric field case [5]. In prior analytic work [3,4,5,6,7,11,8], a diffuse interface was considered between the two miscible electrolytes. Following the approach in our earlier work [1], we show that the correct behavior in the perpendicular electric field configuration can be also obtained with a sharp Eulerian interface between the two miscible electrolytes. A diffuse interface is not required. The assumption of a sharp interface leads to an easier analytic problem, which results in a compact non-dimensional parameter that determines the unstable behavior of the system. In the above problem, the unstable behavior is quantitatively influenced by the thickness of the diffuse interface between the two liquids. However, the sharp interface case, which corresponds to an experiment where the electric field is applied before the interface has diffused significantly, is an important limiting case. Although the approach used in this paper is same as that used by Patankar [1], the difference in orientation of the applied electric field brings out different parametric behavior, e.g., the critical condition for instability. The difference in the parametric dependence cannot be intuitively deduced (especially the dependence on electrical conductivities) and is difficult to obtain simply based on experimental data. In the following sections, the governing equations will be presented first. Instabilities in two geometric configurations -an infinite domain and a microchannel geometry -will be studied. Infinite domain 2.1 Problem formulation The interface is an Eulerian surface which is defined with respect to the base state. The electrical conductivity in the base state changes sharply at this Eulerian interface. An external electric field E is applied in the y-direction, which is perpendicular to the interface. Liquid a is above the interface (positive values of y), and liquid b is below it (negative values of y). In the first configuration considered here, the domain is infinite. The governing equations for this problem can be summarized as follows [1]: ε∇ 2 φ = −ρ b , ∇ · (σE) = 0, Dσ Dt = D σ ∇ 2 σ, ∇ · u = 0, ρ ∂u ∂t + ρ(u · ∇)u = −∇p + µ∇ 2 u + ρ b E,                  (1) where ε is the permittivity, φ is the electric potential, ρ b is the bulk charge density in the liquids, σ is the electrical conductivity which is different in liquids a and b, E is the electrical field, D σ is the diffusion coefficient for the electrical conductivity, u is the velocity field, ρ is the density, p is the dynamic pressure (gravity is balanced by the hydrostatic component), and µ is the viscosity. Liquids a and b are assumed to be strong electrolytes which implies that the electrical conductivities are high. A binary electrolyte is considered [1]. The jump conditions, at the Eulerian interface defined above, are summarized below [1] ρ s = εE · n,(2)σE · n = 0,(3)σu − D σ ∇σ · n = 0,(4)u · n = 0,(5) n · τ − ρuu · n = 0, and t · τ − ρuu · n = 0, where τ = −pI + µ(∇u + ∇u T ) + εEE − ε 2 E · EI, and denotes the value of the variables at the interface in liquid a minus the value in liquid b. n is a unit normal to the interface pointing into liquid a. Semi-infinite domains are considered here in both liquids a and b. The sharp Eulerian interface is located at the center of the domain (y = 0). The material properties ρ, µ, and ε are assumed to be same and constant in both liquids a and b. Only the electrical conductivities are considered to be different in liquids a and b. Since the domain is unbounded and symmetric with respect to the z-direction, the problem is considered to be two-dimensional in the x-y plane. It is assumed that there is no electroosmotic flow in the base state. This assumption is discussed further in the Discussion section. The base solution is given by u 0 = 0, ρ b0 = 0, σ 0 = σ a 0 in liquid a, σ 0 = σ b 0 in liquid b, E 0 a = I 0 σ a 0 j, E 0 b = I 0 σ b 0 j, ρ s0 = ε 0 I 0 ∆σ 0 σ a 0 σ b 0 , p a 0 = εI 2 0 2(σ a 0 ) 2 , p b 0 = εI 2 0 2(σ b 0 ) 2 ,                                               (7) where ρ b0 is the bulk free charge per unit volume inside the fluid, ρ s0 is the charge per unit area at the interface, ∆σ 0 = σ a 0 − σ b 0 , and subscript 0 denotes the variables in the base state. I 0 is a constant current in y-direction. Perturbations are superimposed on the base solution. The conductivity profile in the base state will diffuse with time. However, an approximation is introduced by assuming that the conductivity profile is "frozen" with a sharp jump at the interface. This is not a fully consistent approximation but it is found to be reasonable when the time scale of the instability is short [5,3]. After linearization, the governing equations, under the assumption of a "frozen" base state, for the perturbations are ∇ · u = 0, ε∇ 2 φ = −ρ b , σ 0 ∇ 2 φ − I 0 σ 0 ∂σ ∂y = 0, ∂σ ∂t = D σ ∇ 2 σ , ρ ∂u ∂t = −∇p + µ∇ 2 u + ρ b E 0 ,                         (8) where superscript denotes perturbations. The dimensional form of the perturbations is given by                 u p φ ρ b σ v int                 =                 u(y) p(y) φ(y) ρ b (y) σ(y) v int                 e −ikx+st ,(9) where v int is the y velocity component at the interface, k (a real number) is the wave number of the perturbation, s (a complex number) is the growth rate, and u, p, φ, ρ b , and σ are the amplitudes of the perturbations. An instability is implied by a positive real part of s. The governing equations are non-dimensionalized by using the following scales Length → H, φ → I 0 H σ m , E → I 0 σ m , ρ b → εI 0 Hσ m , p → εI 2 0 σ 2 m σ → σ m (= (σ a 0 + σ b 0 ) 2 ), V elocity → HεI 2 0 σ 2 m µ , time → µσ 2 m εI 2 0 .(10) The length scale is H = 1/k. The velocity scale is based on the balance between the viscous and electrical forces in the momentum equation. After non-dimensionalization, the perturbation equations become ∇ · u = 0, ∇ 2 φ = −ρ b , σ 2 0N ∇ 2 φ − ∂σ ∂y = 0, P e ∂σ ∂t = ∇ 2 σ , Re ∂u ∂t = −∇p + ∇ 2 u + ρ b σ ON j.                           (11) where same symbols have been retained for the non-dimensional variables. The nondimensional parameters in the governing equations are Re = ρεI 2 0 µ 2 σ 2 m k 2 , P e = εI 2 0 µD σ σ 2 m k 2 , σ 0N = 2σ 0 σ a 0 + σ b 0 ,               (12) where Re is the Reynold's number, P e is the Peclet number which is a ratio of convection and diffusion terms, and σ 0N is the non-dimensionalized value of σ 0 in each liquid. In the non-dimensional form of Equation (9), we put k = 1 since the length is nondimensionalized by 1/k, and s will be understood to be non-dimensionalized by the inverse of the time scale. Inserting the non-dimensional form of Equation (9) into Equation (11) and simplifying we get (D 2 − 1 − P es)σ = 0, (D 2 − 1)φ = Dσ σ 2 ON , (D 2 − 1 − Res)(D 2 − 1)υ = − Dσ σ 3 ON , u = −iDυ, P = (D 2 − 1 − Res)Dv,                           (13) where s is the non-dimensional growth rate, D is the derivative with respect to y, and u, v are the x, y components of velocity, respectively. The solution of Equation (13) should approach zero as y → ±∞ in liquids a and b, respectively. This gives the following solutions for liquids a and b σ a = A a e −λy , φ a = B a e −y + Dσ a (σ a 0N ) 2 (P es) , v a = C a e −y + D a e −qy − Dσ a (P es − Res)(P es)(σ a 0N ) 3 ,              (14) σ b = A b e λy , φ b = B b e y + Dσ b (σ b 0N ) 2 (P es) , v b = C b e y + D b e qy − Dσ b (P es − Res)(P es)(σ b 0N ) 3 ,             (15) where λ and q are positive and are given by λ = √ 1 + P es, q = √ 1 + Res.     (16) Superscripts a or b denote liquids a or b, respectively. A, B, C and D's with superscript are constants. The linearized jump conditions for the electrical conductivity at the interface are given by σ = 0, Dσ = P ev int ∆σ ON ,      (17) where v int = v a = v b = (v a + v b ) 2 at y = 0 is the amplitude of the non-dimensional perturbation velocity at the interface, and ∆σ ON = σ a ON − σ b ON . The jump conditions in Equation (17) follow by assuming that there is no self-sharpening mechanism that creates discontinuities in the electrical conductivity. This is reasonable, since the diffusive behavior of σ is important in this problem [12]. The second jump condition in Equation (17) follows from Equation (4). Using these conditions, we get the solution for A a and A b as A a = A b = − P ev int ∆σ ON 2λ .(18) To solve for B a and B b , we use the interface jump conditions for the electric potential: φ = 0, σ ON Dφ + σ y=0 ∆σ 0N σ a 0N σ b 0N = 0.     (19) The first jump condition in Equation (19) follows by assuming that no double layers are formed at the interface. This is reasonable since the interface is not insulating and it is assumed that the current carrying species can pass from one side of the interface to the other [12]. The second jump condition in Equation (19) follows from Equation (3). Thus, B a and B b are B a = P ev int σ 2 0N 4σ a 0N σ b 0N (λ 2 − 1)λ − P ev int σ 0N (σ a2 0N + σ b2 0N ) 4(λ 2 − 1)σ a 0N σ b2 0N ,(20)B b = P ev int σ 2 0N 4σ a 0N σ b 0N (λ 2 − 1)λ + P ev int σ 0N (σ a2 0N + σ b2 0N ) 4(λ 2 − 1)σ a2 0N σ b 0N .(21) The linearized jump conditions for the velocity (from Equation 5) and the stress (from Equation 6) at y = 0 give v = 0, u = 0, − p + 2Dv − Dφ σ 0N = 0, D 2 v + v − φ σ 0N = 0.                   (22) Note that the jump condition for u is due to the no-slip condition. This is reasonable since no double-layers are formed at the interface. Inserting the solutions for velocity in Equations (14) and (15) into Equation (22), we get five homogeneous equations for five unknowns: C a , C b , D a , D b , and v int :             1 −1 1 −1 f 1 1 1 q q f 2 1 1 q 3 q 3 f 3 1 −1 q q 2 f 4 1 1 1 1 f 5                         C a C b D a D b v int             =             0 0 0 0 0             (23) f 1 = − P eΣ(Γ − 2) 2(λ 2 − 1)(λ 2 − q 2 ) Γ − 3 Γ + 1 ,(24)f 2 = λP eΣΓ 2(λ 2 − 1)(λ 2 − q 2 ) ,(25)f 3 = λ 3 P eΓ 2(λ 2 − 1)(λ 2 − q 2 ) + P eΓ 2(λ 2 − 1) ( 1 λ + (Γ − 1) − λΓ),(26)f 4 = λ 2 P eΣ(Γ − 2) 2(λ 2 − 1)(λ 2 − q 2 ) Γ + 1 Γ − 3 + P eΓ 2(λ 2 − 1) Γ − 3 Γ + 1 + P eΣ(Γ − 1) 2 2(λ 2 − 1) (Γ + 1)(Γ − 3) − P eΣ(Γ − 2) 2(λ 2 − 1) Γ + 1 Γ − 3 ,(27)f 5 = P eΣΓ 2(λ 2 − 1)(λ 2 − q 2 ) − 2. (28) where Σ = ∆σ 2 0N σ a2 0N σ b2 0N and Γ = σ a 0N σ b 0N + σ b 0N σ a 0N + 1. The dispersion equation is obtained by setting the determinant of the matrix to zero. The critical condition for instability The dispersion equation obtained from manipulations in Mathematica is − (4Γ(1 + √ 1 + ST )((1 + S) 3 2 √ 1 + ST + √ 1 + S(1 + ST ) + (1 + S) √ 1 + ST (1 + √ 1 + ST ))) +P Σ ( √ 1 + S − Γ √ 1 + S + √ 1 + ST ) = 0,(29) where S = P es, P Σ = P eΣΓ, and T = Re P e . The maximum growth rate for this problem is the largest root of the dispersion equation. It is verified that the maximum growth rate is positive and real. When the system is marginally stable i.e. when the maximum growth rate S = 0, then P Σ is found to be independent of T and is a function of Γ. The marginal stability condition gives the critical condition for the onset of instability. It is given by P cri Σ = 32Γ Γ − 2 , ⇒ ( εE mean 2 D σ µk 2 ) cri = 32 (Γ − 2)Σ ,       (30) where E mean = I 0 σ m . Figure (1) shows a plot of P cri Σ as a function of Γ. The mechanism of instability An approach similar to that in our earlier work [1] is followed. Streamlines spanning one wavelength are plotted for a typical unstable mode in Figure (2). The parameters are A perturbation in the interfacial velocity, v int = cos(x), leads to a perturbation in the electrical conductivity due to the electrohydrodynamic coupling in Equation (17), and Equations (14), (15) and (18): σ a = − P e∆σ ON 2λ e −λy cos(x), σ b = − P e∆σ ON 2λ e λy cos(x).(31) It follows from Equation (31) that a region of lower electrical conductivity is formed when v int is maximum, and higher electrical conductivity is formed when v int is minimum. This is depicted in Figure ( ρ a b = − P e∆σ ON 2σ a2 ON e −λy cos(x), ρ b b = P e∆σ ON 2σ b2 ON e λy cos(x).(32) This leads to an asymmetric bulk charge distribution in the domain as seen in Figure (3). The consequent electrical body force in the fluid gives rise to a cellular flow that reinforces the initial perturbation in velocity and causes instability. Comparison with the parallel electric field case It has been reported that when an electric field is applied perpendicular to the interface the system is more unstable compared to the parallel electric field case [2]. We consider this issue by comparing the critical condition for instability for these cases. The following critical condition for instability of the parallel electric field case is given in our previous work [1]: P cri Σ = 32, ⇒ ( εE 0 2 D σ µk 2 ) cri = 32 Σ ,      (33) where Σ = ∆σ 2 0N σ a 0N σ b 0N . Comparison between Equations (30) and (33) implies that the perpendicular electric field case is more unstable compared to the parallel electric field case. This is discussed below. Consider parameters: σ a /σ b = 10 , µ = 0.001kg/ms, ε = 6.9 × 10 −10 C/V m, ρ = 1000kg/m 3 , and D σ = 10 −9 m 2 /s. We express Σ's in terms of σ a and σ b rather than σ a 0N and σ b 0N in both parallel and perpendicular electric field cases, Thus Σ = (σ a − σ b ) 2 σ a σ b = 8.1.(34) Using equation (33), we get ( εE 0 2 D σ µk 2 ) cri = 32 8.1 ≈ 3.95.(35) For the perpendicular electric field case, we have Σ = ( (σ a ) 2 − (σ b ) 2 2σ a σ b ) 2 = 24.5025, Γ = σ a σ b + σ b σ a + 1 = 11.1.     (36) Using equation (30), we get Comparing Equations (35) and (37), we see that the electric field needed to achieve the instability in the parallel case is larger than the perpendicular electric field case. This difference is due to the difference in the flow pattern in the unstable modes. The parallel electric field case has a different cellular flow pattern in the unstable mode (see [1]). Two pairs of counter rotating vortices are produced in the parallel electric field case. This flow pattern is much less asymmetric and gives rise to weaker flows. In the perpendicular electric field case the asymmetry is much stronger primarily due to the current flowing perpendicular to the interface in the base state. This results in stronger destabilizing forces in the perpendicular electric field case thus making it more unstable. We consider a single parameter above simply for the purpose of comparing parallel and perpendicular electric field cases at typical conditions that are known. Otherwise, the critical conditions for the two cases (parallel and perpendicular) allow comparison over the entire parameter range which shows a similar trend that the perpendicular electric field case is more unstable. 3 Shallow channel Problem formulation Next we apply the linear stability analysis to the case of a shallow microchannel geometry ( Figure 4) that is typical in microfluidic devices [2,5]. The objective is to understand the influence of the device geometry on the instability. An external electrical field applied perpendicular to the interface (y-direction) between two liquids a and b in a domain that is unbounded in the x direction. It is assumed that there is no electroosmotic flow, and there is no charges in the base state. The base state is the same as that in Equation (7) and the perturbation equations are given by Equation (8). The governing equations are non-dimensionalized by using the following scales x, y → H, z → d, φ → I 0 H σ m , E → I 0 σ m , ρ b → εI 0 Hσ m , p → εI 2 0 σ 2 m , σ → σ m (= (σ a 0 + σ b 0 ) 2 ), V elocity → Hβ 2 εI 2 0 σ 2 m µ , time → µσ 2 m β 2 εI 2 0 .(38) where β = d/H << 1 for a shallow channel. For a shallow channel, β → 0 in the governing equations which is the Hele-Shaw limit. This leads to φ = φ (x, y), σ = σ (x, y), and ρ b = ρ b (x, y). It is assumed that ∂φ /∂z = ∂σ /∂z = 0 at the top and bottom walls [1]. In the Hele-Shaw limit the velocity component in the vertical direction is zero, the pressure p = p (x, y), and the horizontal velocity in the x − y plane is of the form u (x, y, z) = 1.5(1 − z 2 )u m (x, y) [5,8,1], where the variables are non-dimensional. As discussed earlier [1] u m is the perturbation velocity at a given location that is averaged with respect to the z direction. The governing equations become [5,8,1] ∇ H · u m = 0, ∇ 2 H φ = −ρ b , σ 2 0N ∇ 2 H φ − ∂σ ∂y = 0, P e ∂σ ∂t = ∇ 2 H σ, Reβ 2 ∂u m ∂t = −∇p + β 2 ∇ 2 H u m − 3u m + ρ b σ ON j,                           (39) where ∇ H denotes the gradient in the x − y plane. All variables carry same meaning as before unless specified otherwise. The non-dimensional parameters in this case are given by Re = β 2 H 2 ρεI 2 0 µ 2 σ 2 m , P e = β 2 H 2 εI 2 0 µD σ σ 2 m , σ 0N = 2σ 0 σ a 0 + σ b 0 .               (40) As discussed earlier [1], the terms involving β 2 in the last of equation (39) should be dropped in the Hele-Shaw limit. However, those terms may be retained to approximately capture the viscous effects due to the flow in the x−y plane [5,8]. Matching of the "inner" solution in the thin viscous layers near the vertical walls with the "outer" Hele-Shaw solution is necessary to obtain a formal solution. Such an analysis will not be considered here. Instead, an approximate approach based on Equation (39) will be considered [5,8]. This will also facilitate comparison with the parallel electric field case considered earlier [1]. The novelty of our effort is the use of a sharp interface approach in the linear stability analysis. Assuming perturbations of the form given by Equation (9) the governing equations become (D 2 − k 2 − P es)σ = 0, (D 2 − k 2 )φ = Dσ σ 2 ON , (D 2 − k 2 − Res − 3 β 2 )(D 2 − k 2 )v m − k 2 Dσ β 2 σ 3 ON , u m = − iDv m k , p = (β 2 (D 2 − k 2 ) − β 2 s − 3)Dv m k 2 .                           (41) Solutions of the governing equations are given by σ a = A a sinh λy + B a cosh λy, φ a = C a sinh λy + D a cosh λy + Dσ a (σ a 0N ) 2 (λ 2 − k 2 ) , v a = E a sinh λy + F a cosh λy + G a sinh λy + H a cosh λy −k 2 Dσ a β 2 (σ a 0N ) 3 (λ 2 − q 2 )(λ 2 − k 2 ) ,                    σ b = A b sinh λy + B b cosh λy, φ b = C b sinh λy + D b cosh λy + Dσ b (σ b 0N ) 2 (λ 2 − k 2 ) , v b = E b sinh λy + F b cosh λy + G b sinh λy + H b cosh λy −k 2 Dσ b β 2 (σ b 0N ) 3 (λ 2 − q 2 )(λ 2 − k 2 ) .                     A a = −A b = − P ev int ∆σ 0N 2λ , B a = B b = − P ev int ∆σ 0N 2λ tanh λ ,      where ∆σ 0N = σ a 0N − σ b 0N , and v int = v a m = v b m = v a m + v b m 2 at y = 0 is the y component of velocity at the interface. The interface conditions for the electrical potential are φ = 0, σ 0N Dφ + σ y=0 ∆σ 0N σ + 0N σ − 0N      at y=0. Since there are electrodes at y = ±1, there is no perturbation of the electric potential at those boundaries. This implies φ = 0 at y = ±1. Using these interface and boundary conditions we solve for C a , C b , D a , and D b to obtain give the maximum growth rate: C a = ∆σ 0N 4σ a 0N σ b 0N k λ 2 − k 2 ( −1 tanh λ ) P ev int ∆σ 0N λ + Pev int ∆σ 0N σ b 0N 4(λ 2 − k 2 ) tanh k σ a2 0N + σ b2 0N σ a2 0N σ b2 0N , C b = ∆σ 0N 4σ a 0N σ b 0N k λ 2 − k 2 ( 1 tanh λ ) P ev int ∆σ 0N λ + Pev int ∆σ 0N σ a 0N 4(λ 2 − k 2 ) tanh k σ a2 0N + σ b2 0N σ a2 0N σ b2 0N , D a = − tanh kC a , D b = tanh kC b .                      The              sinh k cosh k sinh q cosh q 0 k cosh k k sinh k q cosh q q sinh q − k 2 λ 2 P Σ 2λ sinh λ(λ 2 − q 2 )(λ 2 − k 2 ) q 0 q 0 − k 2 λ 2 P Σ 2λ tanh λ(λ 2 − q 2 )(λ 2 − k 2 ) k 3 0 q 3 0 f (k, λ, q, Γ, P Σ ) 0 1 0 1 ( k 2 P Σ 2(λ 2 − k 2 )(λ 2 − q 2 ) − 2 )                           E s F t G s H t v int             =             0 0 0 0 0            (42) In Equation (42), E s = E a − E b and F t = F a + F b . G s and H t are defined similarly. P Σ = P eΣΓ/β 2 , where Σ = ∆σ 2 0N σ a2 0N σ b2 0N and Γ = σ a 0N σ b 0N + σ b 0N σ a 0N + 1. The dispersion equation is once again obtained by setting the determinant of the matrix, above, equal to zero. Results The dispersion equation gives P Σ as a function of λ, k, q, and Γ. P Σ = −(32Γλ(−k 2 + λ 2 )q(k 2 − q 2 )(λ 2 − q 2 )(kcoshqsinhk − qcoshksinhq)) /(k 2 ( 1 2 (−1 + Γ)kλq(λ 2 − q 2 )cothk(31 + coshk 2 − 16coshkcoshq + sinhk 2 ) +8cothλ(q(−Γλ 2 q 2 + k 2 ((−2 + Γ)λ 2 + 2q 2 )) − q(−Γλ 2 q 2 + k 2 ((−2 + Γ)λ 2 + 2q 2 )) coshkcoshq + k(k 2 ((−1 + Γ)λ 2 + q 2 ) + q 2 ((−1 + Γ)λ 2 + q 2 ))sinhksinhq) + λ(−8qcoshq(−(−1 + Γ)k(λ 2 − q 2 )cschk + Γλ(k 2 − q 2 )cschλ + k(Γk 2 + λ 2 − Γλ 2 − q 2 )sinhk + coshk(8Γλq(k 2 − q 2 )cschλ + (−1 + Γ)kq(λ 2 − q 2 )sinhk − 8(q 2 ((−1 + Γ)λ 2 + q 2 ) + k 2 ((−1 + Γ)λ 2 + (1 − 2Γ)q 2 ) )sinhq)))). (43) P Σ can also be expressed as a function of k, Pes, (Res + 3 β 2 ), and Γ i.e. P Σ = f (λ, k, q, Γ) = g(k, Pes, (Res + 3 β 2 ), Γ). For marginal stability at low Re we take Re → 0 and s → 0. In this case P Σ depends on Γ, k, and β i.e. P Σ = f (Γ, k, β).(45) The trends of P Σ will be considered next. We will use parameters corresponding to typical experimental values [5]: d = 5.5µm, H = 78µm, ρ = 1000kg/m 3 , µ = 0.001kg/ms, ε = 6.9 × 10 −10 C/V m, D σ = 10 −9 m 2 /s, and Figure (5) shows the marginal stability curve of P Σ vs.the wavenumber k obtained from the dispersion equation discussed above. It is seen that there is a critical value of the P Σ (or correspondingly I 0 /σ m ) below which the system is stable. This is consistent with the threshold type behavior seen in experiments [2]. Figure (5) shows that the system becomes unstable at P crit Σ = 40211.4. This implies a critical value of (I 0 /σ m ) crit = 0.06kV /cm. Typical values in experiments are 0.1 − 1kV /cm [2,5]. This suggests that an instability should be observed under experimental conditions, which is consistent with the data [2]. Figure (5) also shows that at P crit Σ = 40211.4 the unstable wave corresponds to k = 13.6 which implies a wavelength of the instability that is 0.23 times the channel width. Typical wavelengths of the instability are reported to be of the order of the device width [2]. Boy and Storey [11] presented an instability analysis for the same configuration as that considered in this work with the only difference being that they considered a diffuse interface, in the base state, that was 0.2 times the channel width. In our work, we consider the limiting case of a sharp interface. Figure (5) shows a comparison of the marginal stability curve from Boy and Storey [11] and the present analysis. It is seen that in the perpendicular electric field case the diffusion at the interface can alter the onset of instability significantly. The sharp interface limit sets a lower bound on the critical condition. It is noted that the electric Rayleigh number Ra in the plot of Boy and Storey [11] is related to the parameter P Σ in this work according to the following relation: P Σ = RaΣΓ. This relation is used to re-plot the data of Boy and Storey [11] in terms of Finally, we compare the cases where the electric field is applied parallel or perpendicular to the liquid-liquid interface. Using the same parameters as those used in Figure (5), it was found that the critical electric field for the onset of instability for the parallel electric field case is 0.08kV /cm [1]. Comparing this with the value of 0.06kV /cm computed above for the perpendicular electric field case, it is implied that the perpendicular electric field case is more unstable. Although we have compared these numbers at the experimentally relevant condition, this trend continues at other parameters. Discussion Some comments pertaining to the problem formulation and future experiments are in order. These issues are discussed below. In this work we have assumed that there is no electroosmotic flow. Similar assumption has been made in the past by, e.g., Boy and Storey [11]. While electroosmotic flow can affect the instability, it has been reported in prior work that the influence is little when the ratio of electroosmotic to electroviscous velocity is small (see Lin et al. [3] and Chen et al. [5]). Instabilities caused by electroosmotic slip velocity have been studied by others (see Boy and Storey [11] for references). As noted by Boy and Storey [11], such instabilities do not rely upon bulk conductivity gradient and therefore occur by a different mechanism than the one considered in this work. We have used a constant current condition in the base state with respect to which a linear stability analysis is done. The electrodes have a Dirichlet boundary condition which is similar to that used in prior analytic work [3,4,5,6,7,11,8]. Our work does not consider the charging of the double layer at the electrodes. Boy and Storey [11] note that "the double layer capacitance acts as a high-pass RC filter on the electric field in the bulk. Double layer charging simply adds another mechanism to reduce the instability." Thus, our work helps establish a baseline with respect to which the effects of electrode charging can be studied in the future. The analysis presented here is strictly valid only when the time scale of the growth rate is shorter than the diffusion time scale. Yet, this analysis is useful to understand the nature of the instability and its domain of unstable behavior. This has been discussed in our previous work (Patankar [1]), where thresholds for validity have been shown. Similar discussion is also reported by Boy and Storey [11]. In our analysis, the infinite domain case is considered as a reference case which would be relevant when the channel is very wide. In microfluidic scenarios, this may not be applicable. Hence, we have considered the shallow channel configuration, experiments for which can be set up according to the problem definition in the paper. This is no different from the analyses presented by Santiago and co-workers [3,4,5,6,7,11,8]. El Moctar et al. [13] have reported similar experiments but they have a square cross section channel instead of a shallow channel. Thus, direct comparison with their data is not feasible. Conclusion In this paper the instability at the interface between two miscible liquids with identical mechanical properties but different electrical conductivities was analyzed in the presence of a perpendicular electric field. Linear stability analysis was done by considering a sharp interface between adjacent liquids in the base state. This approach enabled an analytic solution for the critical condition of the electrokinetic instability. It was seen that the instability depends on a non-dimensional parameter P Σ defined in the Equation (30). The mechanism of instability was analyzed. It was found that the electrohydrodynamic coupling due to the interface condition for the electrical conductivity and the electrical body force in the fluid equations led to the instability. The perpendicular electric field case is more unstable compared to the parallel electric field case. The reason for this is the greater asymmetry in the perpendicular field case that results in larger destabilizing electrohydrodynamic force. The effect of a microchannel geometry was studied and the relevant parameters were found to be P Σ , β, and Γ as defined in the paper. The analysis captured the threshold type behavior for the onset of instability. It showed that larger conductivity ratio has a destabilizing effect, while the shallow nature of the channel has a stabilizing effect on the instability. Our approach provides a theoretical estimate and scaling for the desired parameters. Figure 1 : 1The marginal stability curve (P cri Σ vs. Γ) identifying the critical condition for the onset of instability. The region above the curve indicates unstable conditions. Figure 2 : 2Streamlines of an unstable mode in an infinite domain where the applied electric field is perpendicular to the interface between the two liquids. 'c' denotes clockwise rotation of the fluid and 'cc' denotes counterclockwise rotation. T = 0.05, S = 25, P Σ = 683.3914, Figure 3 : 33) with vertical bold arrows at locations of maximum and minimum velocities. Perturbations in the electrical conductivity leads to a perturbation A typical unstable flow cell is depicted. Higher conductivity fluid is in the upper half and the lower conductivity fluid is in the lower half. Interfacial velocity perturbations are shown by bold arrows. Positive and negative signs show the locations of high and low values of the perturbed electrical conductivity. Bold signs indicate that there is greater bulk charge in the lower conductivity fluid. in the bulk charge density (Equation 11) according to ρ b = Figure 4 : 4The shallow channel geometry. a or b denote liquids a or b, respectively. A, B, C, D, E, F, G and H are constants. λ and q are positive and are given by Now we must use the boundary and interface conditions to solve for the constants in the solution. The linearized jump conditions for electrical conductivity at the interface are σ = 0, Dσ = P ev int σ 0N . The linearized boundary conditions for electrical conductivity at the side walls (i.e. at y = ±1) are given by Dσ = 0. The interface conditions together with the boundary conditions give the following solution for the constants interface conditions for velocity are Dv m = 0 (which follows from u m = 0) and v m = 0 at y = 0. The velocity boundary conditions are v m = 0 and Dv m = 0 (i.e. u m = 0) at y = ±1. The stress conditions at the interface are − p + 2β 2 Dv − Dφ σ 0N = 0 and D 2 v + v − φ σ 0N = 0. Using these conditions together with Equations (42) and (42) we get eight equations for the constants E, F , G and H in liquids a and b. Only four of those equations and the equation v int = = 24.5. The only free variables are I 0 /σ m and k. In this case, P Σ is therefore the non-dimensional parameter that represents the variation of I 0 /σ m which is an average measure of the external electric field. P Σ in Figure (5). The critical value for the onset of instability P crit Σ identified in Figure (5) depends onthe the conductivity ratio of the two liquids (i.e. on Γ) and also on the channel height to width ratio (i.e. on β). This is considered next. Figure 5 : 5The marginal stability curve of P Σ vs. k for β=0.07 and Γ=11.1. Figure 6 : 6Plot of P cri Σ vs. Γ indicating critical conditions for the onset of instability at β=0.07. Figure 7 : 7Plot of k vs. Γ at critical conditions for the onset of instability at β=0.07. Figure 8 : 8Plot of P cri Σ vs. β indicating critical conditions for the onset of instability at Γ=11.1. Figure 9 : 9Plot of k vs. β at critical conditions for the onset of instability at Γ=11.1. Figure ( 6 6) shows P cri Σ vs. Γ indicating the critical condition for the onset of instability at β = 0.07. As expected, when Γ increases the critical value of P Σ decreases. It implies that the system is more unstable with a larger conductivity ratio between the two liquids.The wavenumbers at the critical condition for instability, in this case, are plotted in Figure(7). Figure ( 8 8) shows P cri Σ vs. β indicating the critical condition for the onset of instability at Γ = 11.1. As β decreases, i.e. as the channel becomes more shallow, the critical value of P Σ increases. It implies that the shallow nature of the channel has a stabilizing effect on the instability. The wavenumbers at the critical condition for instability, in this case, are plotted inFigure (9). Electrokinetic instability: The sharp interface limit. Na Patankar, PHYSICS OF FLUIDS. 23114101NA Patankar. Electrokinetic instability: The sharp interface limit. PHYSICS OF FLUIDS, 23(1):014101, JAN 2011. Electrokinetic instability micromixing. Mh Oddy, J C Santiago, Mikkelsen, ANALYTICAL CHEMISTRY. 7324MH Oddy, JG Santiago, and JC Mikkelsen. Electrokinetic instability micromixing. ANALYTICAL CHEMISTRY, 73(24):5822-5832, DEC 15 2001. Instability of electrokinetic microchannel flows with conductivity gradients. H Lin, Storey, C H Oddy, J G Chen, Santiago, PHYSICS OF FLUIDS. 166H Lin, BD Storey, MH Oddy, CH Chen, and JG Santiago. Instability of electrokinetic microchannel flows with conductivity gradients. PHYSICS OF FLUIDS, 16(6):1922- 1935, JUN 2004. Multiple-species model for electrokinetic instability. M H Oddy, Santiago, PHYSICS OF FLUIDS. 17664108MH Oddy and JG Santiago. Multiple-species model for electrokinetic instability. PHYSICS OF FLUIDS, 17(6):064108, JUN 2005. Convective and absolute electrokinetic instability with conductivity gradients. Ch Chen, Lin, J G Sk Lele, Santiago, JOURNAL OF FLUID MECHANICS. 524CH Chen, H Lin, SK Lele, and JG Santiago. Convective and absolute electroki- netic instability with conductivity gradients. JOURNAL OF FLUID MECHANICS, 524:263-303, FEB 10 2005. Electrokinetic instabilities in thin microchannels. Bd Storey, H Tilley, J G Lin, Santiago, PHYSICS OF FLUIDS. 17118103BD Storey, BS Tilley, H Lin, and JG Santiago. Electrokinetic instabilities in thin microchannels. PHYSICS OF FLUIDS, 17(1):018103, JAN 2005. Convective instability of electrokinetic flows in a crossshaped microchannel. J D Posner, Santiago, JOURNAL OF FLUID MECHANICS. 555JD Posner and JG Santiago. Convective instability of electrokinetic flows in a cross- shaped microchannel. JOURNAL OF FLUID MECHANICS, 555:1-42, MAY 25 2006. A depth-averaged electrokinetic flow model for shallow microchannels. H Lin, J G Storey, Santiago, JOURNAL OF FLUID MECHANICS. 608H Lin, BD Storey, and JG Santiago. A depth-averaged electrokinetic flow model for shallow microchannels. JOURNAL OF FLUID MECHANICS, 608:43-70, AUG 10 2008. Electric field effect on a two-fluid interface instability in channel flow for fast electric times. A Uguz, O Ozen, N Aubry, PHYSICS OF FLUIDS. 20331702A. Kerem Uguz, O. Ozen, and N. Aubry. Electric field effect on a two-fluid in- terface instability in channel flow for fast electric times. PHYSICS OF FLUIDS, 20(3):031702, MAR 2008. Quantifying the linear stability of a flowing electrified two-fluid layer in a channel for fast electric times for normal and parallel electric fields. A Uguz, N Aubry, PHYSICS OF FLUIDS. 20992103A. Kerem Uguz and N. Aubry. Quantifying the linear stability of a flowing electrified two-fluid layer in a channel for fast electric times for normal and parallel electric fields. PHYSICS OF FLUIDS, 20(9):092103, SEP 2008. Electrohydrodynamic instabilities in microchannels with time periodic forcing. D A Boy, B D Storey, PHYSICS REVIEW E. 7626304DA Boy and Storey BD. Electrohydrodynamic instabilities in microchannels with time periodic forcing. PHYSICS REVIEW E, 76:026304, 2007. J R Melcher, Continuum Electromechanics. MIT PressJ. R. Melcher. Continuum Electromechanics. MIT Press, 1981. Electro-hydrodynamic micro-fluidic mixer. A O El Moctar, Aubry , N , Batton J , LAB CHIP. 3El Moctar AO, Aubry N, and Batton J. Electro-hydrodynamic micro-fluidic mixer. LAB CHIP, 3:273-280, 2003.
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[ "Identifying Differences In Diagnostic Skills Between Physics Students: Developing A Rubric", "Identifying Differences In Diagnostic Skills Between Physics Students: Developing A Rubric" ]
[ "A Mason \nDepartment of Physics and Astronomy\nUniversity of Pittsburgh\n15213PittsburghPAUSA\n", "E Cohen \nDepartment of Science Teaching\nWeizmann Institute of Science\nRehovotIsrael\n", "E Yerushalmi \nDepartment of Science Teaching\nWeizmann Institute of Science\nRehovotIsrael\n", "C Singh \nDepartment of Physics and Astronomy\nUniversity of Pittsburgh\n15213PittsburghPAUSA\n" ]
[ "Department of Physics and Astronomy\nUniversity of Pittsburgh\n15213PittsburghPAUSA", "Department of Science Teaching\nWeizmann Institute of Science\nRehovotIsrael", "Department of Science Teaching\nWeizmann Institute of Science\nRehovotIsrael", "Department of Physics and Astronomy\nUniversity of Pittsburgh\n15213PittsburghPAUSA" ]
[]
Expert problem solvers are characterized by continuous evaluation of their progress towards a solution. One characteristic of expertise is self-diagnosis directed towards elaboration of the solvers' conceptual understanding, knowledge organization or strategic approach. "Self-diagnosis tasks" aim at fostering diagnostic behavior by explicitly requiring students to present diagnosis as part of the activity of reviewing their problem solutions. We have been investigating how introductory physics students perform in such tasks. Developing a robust rubric is essential for objective evaluation of students' self-diagnosis skills. We discuss the development of a grading rubric that takes into account introductory physics students' content knowledge as well as analysis, planning and presentation skills. Using this rubric, we have found the inter-rater reliability to be better than 80%. The rubric can easily be adapted to other problems, as will be discussed in a companion paper.
10.1063/1.3021239
[ "https://export.arxiv.org/pdf/1603.03103v1.pdf" ]
60,665,503
1603.03103
d9b840392cb001e0b941dc041108868f749da4c3
Identifying Differences In Diagnostic Skills Between Physics Students: Developing A Rubric A Mason Department of Physics and Astronomy University of Pittsburgh 15213PittsburghPAUSA E Cohen Department of Science Teaching Weizmann Institute of Science RehovotIsrael E Yerushalmi Department of Science Teaching Weizmann Institute of Science RehovotIsrael C Singh Department of Physics and Astronomy University of Pittsburgh 15213PittsburghPAUSA Identifying Differences In Diagnostic Skills Between Physics Students: Developing A Rubric problem solvingreflectionalternative assessmentself-diagnosis PACS: 0140gb0140Ha Expert problem solvers are characterized by continuous evaluation of their progress towards a solution. One characteristic of expertise is self-diagnosis directed towards elaboration of the solvers' conceptual understanding, knowledge organization or strategic approach. "Self-diagnosis tasks" aim at fostering diagnostic behavior by explicitly requiring students to present diagnosis as part of the activity of reviewing their problem solutions. We have been investigating how introductory physics students perform in such tasks. Developing a robust rubric is essential for objective evaluation of students' self-diagnosis skills. We discuss the development of a grading rubric that takes into account introductory physics students' content knowledge as well as analysis, planning and presentation skills. Using this rubric, we have found the inter-rater reliability to be better than 80%. The rubric can easily be adapted to other problems, as will be discussed in a companion paper. INTRODUCTION Bereiter & Scardamalia (1989) [1] argue in favor of intentional learning, namely a cognitive activity that has learning as a goal. Accordingly, instruction should make sure that students will not only focus their attention on the task, but also on learning from it. In the context of a problem solving process, solvers use self-monitoring questions to elaborate the solution between successive trials. Yet, while self-monitoring is directed mainly towards arriving at a solution, it might also involve self-diagnosis directed towards more general learning goals such as elaboration of the solver's conceptual understanding [2]. We report the analysis of data obtained in a research study focused on self-diagnosis in the context of an abundant activity in physics problem solving: reviewing the solution that the learner has composed in order to improve it or learn from it. The pertinent questions are: 1) what learning outcomes result from this activity, and 2) how can instruction enhance the learning outcomes? We constructed an alternative-assessment task in which students are required to present a diagnosis (namely, identifying where they went wrong, and explaining the nature of the mistakes) as part of the activity of reviewing their quiz solutions. We shall call these tasks "self-diagnosis tasks". Analysis of data requires a robust method of grading that would allow assessing students' solutions as well as students' self-diagnosis. In particular, one would like to assess the solvers' conceptual understanding as it is reflected in their self-diagnosis. To this end, there are two approaches [3]. The 1 st approach maps the student statements to a representation of an expert "ideal" knowledge representation, i.e., what correct ideas needed to solve the problem are reflected in student's solution and diagnosis. The 2 nd approach attempts to describe the novice knowledge per se, i.e., what ideas the student believes are needed to solve the problem are reflected in his/her solution and diagnosis. The scoring rubric developed was aligned with both approaches with the intent of using it to help score student subjects in the continuing study on selfdiagnosis [4,5]. The rubric was designed to comply with common standards of objectivity and reproducibility, i.e., validity as determined by four experts in physics education who perceive it as measuring an appropriate performance of the solution and self diagnosis and a high inter-rater reliability. This paper details the final version of this rubric and its potential use. A companion paper will describe an analysis of student self-diagnosis for which the rubric was used, thus providing more detail. INITIAL CONSIDERATIONS Expert "ideal" knowledge representation There were two main factors we desired to measure with this rubric. The first consideration is the student's use and application of physical principles. The student must invoke appropriate principles as well as apply them correctly in order to solve the problem. The second consideration is the student's presentation of a strategic problem-solving approach [6]. We are interested in evaluating if the student presented a helpful description of the problem's situation in terms FIGURE 1A. Sample student B14's quiz from the selfdiagnosis study. The circled numbers 1 and 2 are references from the student's self-diagnosis as labeled in figure 1B, and the circled number 14 is the code number for the student. FIGURE 1B. Sample student B14's self-diagnosis from the self-diagnosis study. The other groups did not receive this worksheet but instead wrote their diagnosis elsewhere. of physics concepts and principles, e.g., if a diagram is drawn to help visualize the problem. In addition, it is of interest to see if the student constructed a good plan for solving the problem with regard to the target quantity and intermediate problem steps needed to obtain this quantity. Finally, we would like to evaluate if the student checked the reasonability of his or her answer once it is obtained so as to make sure he or she did the problem correctly. These considerations are generic, meaning that a rubric may be designed with a set of general guidelines in place. However, it should also be possible for specific attributes of the problem to be added to or removed from the rubric as befits the problem. Sample Problem Used The problem used features a girl of mass m girl riding a rollercoaster that consists of a steep hill of height h 0 followed by a circularly shaped bump with height h f and reflecting a circular radius r, and asks, given that the girl is sitting on a scale on the rollercoaster cart, how much will the scale read at the top of the circular bump? In order to solve this problem, a student will need to understand that the target variable is the normal force the scale exerts on the girl. To calculate the normal force at point B, the student will have to invoke Newton's 2 nd Law: a m F r r ⋅ = Σ . He will need to find as intermediate variables the net force (sum of the force of gravity and the normal force) and the acceleration at this point. To calculate the acceleration, he will have to invoke the expression for centripetal acceleration: a g = v B 2 /R. The intermediate variable, the speed of the cart at the top of the circular bump, can be found using the law of conservation of mechanical energy, PE i + KE i = PE f + KE f , between point of departure and the top of the bump (which is justified because all forces doing work are conservative forces). A complete description of the problem and its solution is available in Ref. [4]. Figures 1A and 1B THE RESULTING RUBRIC Specific Knowledge: Categories and Subcategories Table 1 represents the rubric that was ultimately developed. The three columns for each student represent the three ways in which the student's work is evaluated for a self-diagnosis experiment: from left to right, they represent the researcher diagnosis of the student quiz solution (RDS), the student's selfdiagnosis of his/her solution (SDS), and the researcher's judgment of this student's self-diagnosis (RSD). For each column, the students are evaluated by a series of criteria represented by the rubric's rows. The rows of the rubric are divided into three main categories: physical principles (hereafter referred to as "physics" for the sake of brevity), problem solving presentation (referred to as "presentation"), and algebra ("math"). The physics category is divided into two subcategories: invoking a physical principle and applying that principle. Each row in each subcategory therefore represents every physical principle that a student will have to invoke and apply to correctly solve the problem. For example, in the problem described here for which the introductory students are evaluated, conservation of energy and a nonequilibrium application of Newton's 2 nd Law in circular motion are both required. Therefore, there will be two rows in the invoking subcategory to evaluate if the student cited these laws, and two corresponding rows in the applying subcategory to evaluate if the student applied them correctly. We unified Newton's 2 nd Law and centripetal acceleration as one subcategory, as those were difficult to differentiate in students' answers. In the applied principle section, we consider how well the student does on grading according to both expert and novice Figures 1A and 1B is graded to serve as an example. Abbreviations: RDS = Researcher diagnosis of solution; SDS = Student's diagnosis of solution; RSD = Researcher judgment of student's diagnosis. In the RDS/RSD "+" is given if a student correctly performs/identifies a mistake defined by some subcategory. A "-" is given if the student incorrectly performs or fails to identify a mistake or identify it incorrectly. If a student is judged to have gotten something partially correct, then the grader may assign ++/-, +/-, or +/--. The term "n/a" is assigned if the student could not reasonably address a subcategory given the prior work done. For example, the sample student correctly invoked conservation of energy on the original quiz and therefore did not address it during self-diagnosis. In the researcher's judgment column, the grader would then state "n/a" and not consider this invoked law in assigning a grade for the researcher's judgment. In the SDS column + and -reflect how the students perceive themselves to be correct/wrong on the quiz solution. An "x" indicates the student did not address a subcategory at all, which is interpreted as the students perceiving themselves to be correct on the quiz solution. General Task representations by noting what specific errors the student makes and evaluating the student's diagnosis of the errors. Also included in the physics category are two other kinds of criteria. The first deals with any justifications that a student should cite for invoking or applying a physical principle. This is determined a priori according to the needs of each problem. In the given case, students are expected to justify invoking conservation of energy in the aforementioned problem as follows: energy is conserved as the only forces doing work on the girl "system" are conservative forces (gravitation). The normal (non-conservative) force is perpendicular to the cart's motion, and hence does no work. There is an additional row in the invoked physics subsection that tracks if the student invoked an inappropriate principle that doesn't apply to the problem, whether legitimate or invalid. The plan/solution category has three different subcategories. Problem description involves anything the student may do to facilitate properly understanding the problem question. This includes drawing visual representations and also listing all given known quantities clearly and appropriately. Planning and solution construction tracks the student's steps, and checks to see if the student has done the following: described the appropriate target variable for which the student is solving, avoided writing down surplus equations, described appropriate intermediate variables to reach the target variable, and explicitly stated some methodology used to take the steps used in the problem. Evaluation involves the student's tracking of his or her own work, usually involving a check of the answer and taking care to write down the proper units. Note that the presentation and algebra subcategories are essentially general and do not have to be changed from problem to problem; only the specific criteria based upon the subcategories need to be changed. The algebra category contains one row that is included to check for mathematical mistakes made during the problem-solving process, for example forgetting a coefficient when rewriting an equation. This row shows whether a student makes a minor math error, or if a student attempts algebraic manipulation to attain a desired quantity. Scoring The scoring reflects researcher priorities. The 1 st approach mentioned in the introduction, in which a researcher assess how student's knowledge compares to the ideal knowledge, would possibly weigh each ideal subcategory as worth 1 point if "+" and worth 0 points if "-". The 2 nd approach, in which a researcher assesses in what detail students are able to identify their mistakes, would possibly also weigh each novice subcategory. Initially, we took the 1 st approach when we scored the quiz solution and the self-diagnosis (RSD). Later on, we took the 2 nd approach for the RSD as it allowed better differentiating between students. Table 2 displays how the marks given in Table 1 are interpreted as an overall score. The overall grade for a physics score can be interpreted as an average of all possible criteria that the student correctly addressed. Eventually, we did not score the "justification" part (row 4), since students' justification of chosen principles was found to occur very rarely. Therefore, we assume that the students did not think of justification as part of the solution procedure, and by extension, the self-diagnosis procedure. Reliability Part of the purpose of this rubric is to assure objectivity in grading. With this in mind, two researchers independently graded ~10 sample students using the rubric for the problem diagrammed in Figures 1A and 1B. They then discussed how criteria should be applied to the students' work objectively. We found that the graders could agree to within at least 80% of each other in grading the rubric. This established a reasonable inter-rater reliability that was consistent in a more thorough analysis of about 200 students. The companion paper will outline the use of this rubric to examine the effect of self-diagnosis of a quiz performance on future exams. Justification for CE (e.g. all forces doing work are conservative since nonconservative forces are perpendicular to the path of motion) -x -Novice knowledge per se 4. inappropriate principle: "-" marked if inappropriate principle is used in student's solution or diagnosis (student used gravitational law here) explicitly stating in words or a generic form the principles used to solve for this intermediate variables (not done by student) respectively represent the quiz and self-diagnosis attempt of a student in the selfdiagnosis studies. This student's work features examples of what a student might do correctly as well as some good examples of what he might have done incorrectly, which the rubric should be able to reflect. TABLE 1 . 1Rubric developed for self-diagnosis study. The student featured in ACKNOWLEDGMENTSThe research for this study was supported by ISF 1283/05 and NSF DUE-0442087. Knowing Learning, and Instruction: Essays in honor of Robert Glaser. C Bereiter, M Scardamalia, Lawrence Erlbaum AssociatesHillsdale, NJC. Bereiter and M. Scardamalia, Knowing Learning, and Instruction: Essays in honor of Robert Glaser, Hillsdale, NJ: Lawrence Erlbaum Associates, 361-392, (1989). . J Larkin, J Mcdermott, D Simon, H Simon, Science. 208J. Larkin, J. McDermott, D. Simon, and H. Simon, Science 208, 1335-1342, (1980). . M Chi, Journal of the Learning Sciences. 63M. Chi, Journal of the Learning Sciences 6(3), 271-315, (1997). . E Yerushalmi, C Singh, B Eylon, Proceedings of the Phys. Ed. Res. Conference. 951AIP Conf. Proc.E. Yerushalmi, C. Singh, and B. Eylon, Proceedings of the Phys. Ed. Res. Conference, Syracuse, NY, AIP Conf. Proc. 951, 27-30, (2007). . C Singh, E Yerushalmi, B Eylon, Proceedings of the Phys. Ed. Res. Conference. 951AIP Conf. Proc.C. Singh, E. Yerushalmi, and B. Eylon, Proceedings of the Phys. Ed. Res. Conference, Syracuse, NY, AIP Conf. Proc. 951, 31-34, (2007). . F Reif, American Journal of Physics. 63F. Reif, American Journal of Physics 63, 17-32, (1995). Rubric scoring for the sample student. Pre: 0.27 SDS Ph: 0.14; Pre: 0.65 RSD Ph: 1; Pre: 0.54Category Grading RDS Ph: 0.17TABLE 2. Rubric scoring for the sample student. Category Grading RDS Ph: 0.17; Pre: 0.27 SDS Ph: 0.14; Pre: 0.65 RSD Ph: 1; Pre: 0.54
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[ "Particles interacting with a vibrating medium: existence of solutions and convergence to the Vlasov-Poisson system", "Particles interacting with a vibrating medium: existence of solutions and convergence to the Vlasov-Poisson system" ]
[ "Stephan De Bièvre \nUMR 8524 -Laboratoire Paul Painlevé\nUniv. Lille\nCNRS\nF-59000Lille\n\nCentre de Recherche INRIA Futurs\nFrance § . & Equipe-Projet MEPHYSTO\nParc Scientifique de la Haute Borne\n40, avenue Halley B.P. 70478\" F-59658 Villeneuve d'Ascq cedexFrance\n", "Thierry Goudon \nLabo. J. A. Dieudonné\nUMR\nInria\nSophia Antipolis Méditerranée Research Centre\nProject COFFEE & Univ. Nice Sophia Antipolis\nCNRS\n7351 Parc ValroseF-06108NiceFrance\n", "Arthur Vavasseur \nLabo. J. A. Dieudonné\nUMR\nInria\nSophia Antipolis Méditerranée Research Centre\nProject COFFEE & Univ. Nice Sophia Antipolis\nCNRS\n7351 Parc ValroseF-06108NiceFrance\n" ]
[ "UMR 8524 -Laboratoire Paul Painlevé\nUniv. Lille\nCNRS\nF-59000Lille", "Centre de Recherche INRIA Futurs\nFrance § . & Equipe-Projet MEPHYSTO\nParc Scientifique de la Haute Borne\n40, avenue Halley B.P. 70478\" F-59658 Villeneuve d'Ascq cedexFrance", "Labo. J. A. Dieudonné\nUMR\nInria\nSophia Antipolis Méditerranée Research Centre\nProject COFFEE & Univ. Nice Sophia Antipolis\nCNRS\n7351 Parc ValroseF-06108NiceFrance", "Labo. J. A. Dieudonné\nUMR\nInria\nSophia Antipolis Méditerranée Research Centre\nProject COFFEE & Univ. Nice Sophia Antipolis\nCNRS\n7351 Parc ValroseF-06108NiceFrance" ]
[]
We are interested in a kinetic equation intended to describe the interaction of particles with their environment. The environment is modeled by a collection of local vibrational degrees of freedom. We establish the existence of weak solutions for a wide class of initial data and external forces. We also identify a relevant regime which allows us to derive, quite surprisingly, the attractive Vlasov-Poisson system from the coupled Vlasov-Wave equations.
10.1137/16m1065306
[ "https://export.arxiv.org/pdf/1603.03575v1.pdf" ]
1,557,949
1603.03575
368429deb29d090a93e074c1e1c6cebb83c33432
Particles interacting with a vibrating medium: existence of solutions and convergence to the Vlasov-Poisson system 11 Mar 2016 March 7, 2022 Stephan De Bièvre UMR 8524 -Laboratoire Paul Painlevé Univ. Lille CNRS F-59000Lille Centre de Recherche INRIA Futurs France § . & Equipe-Projet MEPHYSTO Parc Scientifique de la Haute Borne 40, avenue Halley B.P. 70478" F-59658 Villeneuve d'Ascq cedexFrance Thierry Goudon Labo. J. A. Dieudonné UMR Inria Sophia Antipolis Méditerranée Research Centre Project COFFEE & Univ. Nice Sophia Antipolis CNRS 7351 Parc ValroseF-06108NiceFrance Arthur Vavasseur Labo. J. A. Dieudonné UMR Inria Sophia Antipolis Méditerranée Research Centre Project COFFEE & Univ. Nice Sophia Antipolis CNRS 7351 Parc ValroseF-06108NiceFrance Particles interacting with a vibrating medium: existence of solutions and convergence to the Vlasov-Poisson system 11 Mar 2016 March 7, 2022Vlasov-like equations Interacting particles Inelastic Lorentz gas Math Subject Classification 82C7070F4537K0574A25 We are interested in a kinetic equation intended to describe the interaction of particles with their environment. The environment is modeled by a collection of local vibrational degrees of freedom. We establish the existence of weak solutions for a wide class of initial data and external forces. We also identify a relevant regime which allows us to derive, quite surprisingly, the attractive Vlasov-Poisson system from the coupled Vlasov-Wave equations. Introduction In [8], L. Bruneau and S. De Bièvre introduced a mathematical model intended to describe the interaction of a classical particle with its environment. The environment is modeled by a vibrating scalar field, and the dynamics is governed by energy exchanges between the particle and the field, embodied into a Hamiltonian structure. To be more specific on the model in [8], let us denote by q(t) ∈ R d the position occupied by the particle at time t. The environment is represented by a field (t, x, y) ∈ R × R d × R n → Ψ(t, x, y) ∈ R: it can be thought of as an infinite set of n-dimensional membranes, one for each x ∈ R d . The displacement of the membrane positioned at x ∈ R d is given by y ∈ R n → ψ(t, x, y) ∈ R. The coupling is realized by means of form factor functions x → σ 1 (x) and y → σ 2 (y), which are supposed to be non-negative, infinitely smooth, radially symmetric and compactly supported. Therefore, the dynamic is described by the following set of differential equations     q (t) = −∇V (q(t)) − R d ×R n σ 1 (q(t) − z) σ 2 (y) ∇ x Ψ(t, z, y) dy dz, ∂ 2 tt Ψ(t, x, y) − c 2 ∆ y Ψ(t, x, y) = −σ 2 (y)σ 1 (x − q(t)), x ∈ R d , y ∈ R n .(1) In (1), c > 0 stands for the wave speed in the transverse direction, while q ∈ R d → V (q) ∈ R is a time-independent external potential the particle is subjected to. In [8], the well-posedness theory for (1) is investigated, but the main issue addressed there is the large time behavior of the system. It is shown that the system exhibits dissipative features: under certain circumstances (roughly speaking, n = 3 and c large enough) and for a large class of finite energy initial conditions the particle energy is evacuated in the membranes, and the environment acts with a friction force on the particle. Accordingly, the asymptotic behavior of the particle for large times can be characterized depending on the external force: if V = 0, the particle stops exponentially fast, when V is a confining potential with a minimiser q 0 , then the particle stops at the location q 0 , and for V (q) = −F · q, a limiting velocity V F can be identified. Since then, a series of works has been devoted to further investigation of the asymptotic properties of a family of related models. We refer the reader to [1,10,11,12,25,30] for thorough numerical experiments and analytical studies, that use random walks arguments in particular. The model can be seen as a variation on the Lorentz gas model where one is interested in the free motion of a single point particle in a system of obstacles distributed on a certain lattice. We refer the reader to [4,9,17,19,27] for results and recent overviews on the Lorentz gas problem. Instead of dealing with periodically or randomly distributed hard scatterers as in the Lorentz gas model, here the particle interacts with a vibrational environment, that create the "soft" potential Φ. The asymptotic analysis of the behavior of a particle subjected to an oscillating potential is a further related problem that is also worth mentioning [16,22,24,28]. We wish to revisit the model of [8] in the framework of kinetic equations. Instead of considering a single particle described by its position t → q(t), we work with the particle distribution function in phase space f (t, x, v) ≥ 0, with x ∈ R d , v ∈ R d , the position and velocity variables respectively. This quantity obeys the following Vlasov equation ∂ t f + v · ∇ x f − ∇ x (V + Φ) · ∇ v f = 0, t ≥ 0, x ∈ R d , v ∈ R d .(2) In (2), V stands for the external potential, while Φ is the self-consistent potential describing the interaction with the environment. It is defined by the convolution formula Φ(t, x) = R d ×R n Ψ(t, z, y)σ 1 (x − z)σ 2 (y) dy dz, t ≥ 0, x ∈ R d(3) where the vibrating field Ψ is driven by the following wave equation        ∂ 2 tt Ψ − c 2 ∆ y Ψ (t, x, y) = −σ 2 (y) R d σ 1 (x − z)ρ(t, z) dz, t ≥ 0, x ∈ R d , y ∈ R n , ρ(t, x) = R d f (t, x, v) dv. (4) The system is completed by initial data f (0, x, v) = f 0 (x, v), Ψ(0, x, y) = Ψ 0 (x, y), ∂ t Ψ(0, x, y) = Ψ 1 (x, y). (5) A possible interpretation of the kinetic equation (2) consists in considering the model (1) for a set of N ≫ 1 particles. The definition of the self-consistent potential has to be adapted since all the particles interact with the environment, namely we have, for j ∈ {1, ..., N }           q j (t) = −∇V (q j (t)) − R d ×R n σ 1 (q j (t) − z) σ 2 (y) ∇ x Ψ(t, z, y) dy dz, ∂ 2 tt Ψ(t, x, y) − c 2 ∆ y Ψ(t, x, y) = −σ 2 (y) N k=1 σ 1 (x − q k (t)) . Note that such a many-particle system is not considered in [8]. It is very likely that its asymptotic behavior is much more complicated than with a single particle because, even if the particles do not interact directly, they do so indirectly via their interaction with the membranes. If we now adopt the mean-field rescaling in which Φ → 1 N Φ, then (2) can be obtained as the limit as N goes to ∞ for the empirical measure f N (t, x, v) = 1 N N k=1 δ(x = q k (t), v =q k (t)) of the N −particle system, assuming the convergence of the initial state f N (0, x, v) → f 0 (x, v) in some suitable sense. Such a statement can be rephrased in terms of the convergence of the joint distribution of the N -particle system. This issue will be discussed elsewhere [31] and we refer the reader to the lecture notes [18] and to [20] for further information on the mean-field regimes in statistical physics. In this paper we wish to analyse several aspects of the Vlasov-Wave system (2)- (5). We warn the reader that, despite the similarities in terminology, the model considered here is very different, both mathematically and physically, from the one dealt with in [6], which is a simplified version of the Vlasov-Maxwell system. It is indeed crucial to understand that the wave equation in this paper is set with variables transverse to the physical space: the waves do not propagate at all in the space where the particles move. This leads to very different physical effects; we refer to [8] and references therein for more details on this matter. We add that this paper is less ambitious than [8], since we do not discuss here the large time behavior of the solutions, only their global existence. As mentioned above, since we are dealing with many particles, it is very likely that the question cannot be handled in the same terms as in [8], and that the kinetic model inherits the same technical and conceptual difficulties already mentioned for N > 1 particles. We only mention that a particular stationary solution (with f integrable) has been exhibited in [2], and that this solution is shown to be linearly stable. The paper is organized as follows. Section 2 contains a preliminary and largely informal discussion to set up notation and to establish some estimates on the interaction potential needed in the bulk of the paper. Section 3 establishes the well-posedness of the problem (2)-(5) (Theorem 3.3). We consider a large class of initial data and external potentials with functional arguments which are reminiscient of Dobrushin's analysis of the Vlasov equation [15]. Section 4 is devoted to asymptotic issues which allow us to connect (2)-(5) to Vlasov equations with an attractive self-consistent potential. In particular, up to a suitable rescaling of the form function σ 1 , we can derive this way the attractive Vlasov-Poisson system. This is quite surprising and unexpected in view of the very different physical motivation of the models. Preliminary discussion Throughout the paper, we make the following assumptions on the model parameters and on the initial conditions. First, on the coupling functions σ 1 , σ 2 , we impose:      σ 1 ∈ C ∞ c (R d , R), σ 2 ∈ C ∞ c (R n , R), σ 1 (x) ≥ 0, σ 2 (y) ≥ 0 for any x ∈ R d , y ∈ R n , σ 1 , σ 2 are radially symmetric. (H1) We require that the external potential fulfills V ∈ W 2,∞ loc (R d ), and there exists C ≥ 0 such that V (x) ≥ −C(1 + |x| 2 ) for any x ∈ R d .(H2) This is a rather standard and natural assumption. Note that it ensures global existence when σ 1 = 0 = σ 2 : it then implies that the external potential cannot drive the particle to infinity in finite time. For the initial condition of the vibrating environment, we shall assume Ψ 0 , Ψ 1 ∈ L 2 (R d × R n ).(H3) For the initial particle distribution function, we naturally assume f 0 ≥ 0, f 0 ∈ L 1 (R d × R d ).(H4) For energy considerations, it is also relevant to suppose ∇ y Ψ 0 ∈ L 2 (R d × R n ) and (x, v) → (V (x) + |v| 2 )f 0 (x, v) ∈ L 1 (R d × R d ). (H5) This means that the initial state has finite mass, potential and kinetic energy. Our goal in this section is to rewrite the equations of the coupled system (2)-(5) in an equivalent manner, more suitable for our subsequent analysis. The discussion will be informal, with all computations done for sufficiently smooth solutions. The proper functional framework will be provided in the next section. First, we note that it is clear that (2) preserves the total mass of the particles d dt R d ×R d f (t, x, v) dv dx = 0. In fact, since the field (v, ∇ x V + ∇ x Φ) is divergence-free (with respect to the phase variables (x, v)), any L p norm of the density f is conserved, 1 ≤ p ≤ ∞. Furthermore, the PDEs system (2)-(4) inherits from the Hamiltonian nature of the original equations of motion (1) the following easily checked energy conservation property: d dt 1 2 R d ×R n |∂ t Ψ(t, x, y)| 2 dx dy + c 2 2 R d ×R n |∇ y Ψ(t, x, y)| 2 dx dy + R d ×R d f (t, x, v) |v| 2 2 + V (x) + Φ(t, x) dv dx = 0. As a matter of fact the energy remains finite when the full set of assumptions (H1)-(H5) holds. For the Vlasov-Poisson equation it is well known that the potential can be expressed by means of a convolution formula. Similarly here, the self-consistent potential Φ can be computed explicitly as the image of a certain linear operator acting on the macroscopic density ρ(t, x) = R d f (t, x, v) dv; this follows from the fact that the linear wave equation (4) can be solved explicitly as the sum of the solution of the homogeneous wave equation with the correct initial conditions plus the retarded solution of the inhomogeneous wave equation. To see how this works, we introduce t → p(t) = 1 (2π) n R n sin(c|ξ|t) c|ξ| | σ 2 (ξ)| 2 dξ and Φ 0 (t, x) = 1 (2π) n R n R d σ 1 (x− z) Ψ 0 (z, ξ) cos(c|ξ|t) + Ψ 1 (z, ξ) sin(c|ξ|t) c|ξ| σ 2 (ξ) dz dξ(6) where the symbol · stands for the Fourier transform with respect to the variable y ∈ R n . Note that Φ 0 is the solution of the homogeneous wave equation with the given initial conditions for Ψ. Finally, we define the operator L which associates to a distribution function f : (0, ∞) × R d × R d → R the quantity L(f )(t, x) = t 0 p(t − s) R d Σ(x − z)ρ(s, z) dz ds,(7) where ρ(t, x) = R d f (t, x, v) dv, Σ = σ 1 * x σ 1 . We can then check that the pair (f, Ψ) is a solution of (2)-(4) iff f satisfies ∂ t f + v · ∇ x f = ∇ v f · ∇ x (V + Φ 0 − L(f )) f (0, x, v) = f 0 (x, v)(8) and Ψ is the unique solution of (4). We sketch the computation, which is instructive. Let (f, Ψ) be a solution of (2)-(4). Applying the Fourier transform with respect to the variable y we find      (∂ 2 t + c 2 |ξ| 2 ) Ψ(t, x, ξ) = −(ρ(t, ·) * x σ 1 )(x) σ 2 (ξ), Ψ(0, x, ξ) = Ψ 0 (x, ξ) ∂ t Ψ(0, x, ξ) = Ψ 1 (x, ξ). The solution reads Ψ(t, x, ξ) = − t 0 (ρ(t − s, ·) * σ 1 )(x) σ 2 (ξ) sin(cs|ξ|) c|ξ| ds + Ψ 0 (x, ξ) cos(c|ξ|t) + Ψ 1 (x, ξ) sin(c|ξ|t) c|ξ| .(9) To compute Φ in (3), we use Plancherel's equality: Φ(t, x) = R d ×R n Ψ(t, z, y)σ 1 (x − z)σ 2 (y) dy dz = 1 (2π) n R d ×R n Ψ(t, z, ξ)σ 1 (x − z) σ 2 (ξ) dξ dz = − (σ 1 * σ 1 ) * t 0 ρ(t − s, ·) R n sin(cs|ξ|) c|ξ| | σ 2 (ξ)| 2 (2π) n dξ ds (x) + 1 (2π) n σ 1 * R n Ψ 0 (·, ξ) cos(c|ξ|t) + Ψ 1 (·, ξ) sin(c|ξ|t) c|ξ| σ 2 (ξ) dξ (x) = −L(f )(t, x) + Φ 0 (t, x). Inserting this relation into (2), we arrive at (8). Conversely, let f be a solution of (8) and let Ψ be the unique solution of (4). The same computation then shows that Φ in (3) is given by Φ = Φ 0 − L(f ). Therefore f satisfies (2). The operator L in (7) plays a crucial role in our further analysis. Its precise definition on an appropriate functional space and its basic continuity properties are given in the following Lemma. Lemma 2.1 (Estimates on the interaction potential) For any 0 < T < ∞, the following properties hold: i) L belongs to the space A T of continuous operators on C [0, T ]; W 1,∞ (R d × R d ) ′ with values in C [0, T ]; W 2,∞ (R d ) . Its norm is evaluated as follows: |||L||| A T ≤ σ 1 2 W 3,2 (R d ) σ 2 2 L 2 (R n ) T 2 2 ; ii) L belongs to the space B T of continuous operators on C [0, T ]; W 1,∞ (R d × R d ) ′ with values in C 1 [0, T ]; L ∞ (R d ) . Its norm is evaluated as follows: |||L||| B T ≤ σ 1 2 W 1,2 (R d ) σ 2 2 L 2 (R n ) T + T 2 2 ; iii) Φ 0 satisfies Φ 0 (t, ·) W 2,∞ (R d ) ≤ σ 1 W 2,2 (R d ) σ 2 L 2 (R n ) Ψ 0 L 2 (R n ) + t Ψ 1 L 2 (R n ) , for any 0 ≤ t ≤ T , and, moreover Φ 0 C 1 ([0,T ];L ∞ (R d )) ≤ σ 1 L 2 (R d ) σ 2 W 1,2 (R n ) 2 Ψ 0 L 2 (R n ) + (1 + T ) Ψ 1 L 2 (R n ) . Proof. The last statement is a direct consequence of Hölder and Young inequalities; let us detail the proof of items i) and ii). We associate to f ∈ W 1,∞ (R d × R d ) ′ , the macroscopic density ρ ∈ W 1,∞ (R d ) ′ by the formula: ρ f , χ (W 1,∞ ) ′ ,W 1,∞ (R d ) = f, χ ⊗ 1 v (W 1,∞ ) ′ ,W 1,∞ (R d ×R d ) , ∀χ ∈ W 1,∞ (R d ). Clearly, we have ρ f W 1,∞ (R d ) ′ ≤ f W 1,∞ (R d ×R d ) ′ . For any χ ∈ C ∞ c (R d ), and i ∈ {0, 1, 2} , we can check the following estimates ρ * Σ, ∇ i χ = ρ, ∇ i Σ * χ ≤ ρ W 1,∞ (R d ) ′ ∇ i Σ * χ W 1,∞ (R d ) ≤ f W 1,∞ (R d ×R d ) ′ ∇ i Σ L ∞ (R d ) + ∇ i+1 Σ L ∞ (R d ) χ L 1 (R d ) . Since the dual space of L 1 is L ∞ , for i = 0, we deduce that ρ * Σ L ∞ (R d ) ≤ f W 1,∞ (R d ×R d ) ′ Σ L ∞ (R d ) + ∇Σ L ∞ ≤ σ 1 2 W 1,2 (R d ) f W 1,∞ (R d ×R d ) ′ . Reasoning similarly for i = 1 and i = 2, we obtain ρ * Σ W 2,∞ (R d ) ≤ σ 1 2 W 3,2 (R d ) f W 1,∞ (R d ×R d ) ′ . We now estimate p. Plancherel's inequality yields |p ′ (t)| = 1 (2π) n R n cos(c|ξ|t)| σ 2 (ξ)| 2 dξ ≤ σ 2 2 L 2 (R n ) . Since p(0) = 0, it follows that |p(t)| ≤ σ 2 2 L 2 (R n ) t. Hence, for all 0 ≤ t ≤ T < ∞, we have L(f )(t) W 2,∞ (R d ×R d ) ≤ Σ * ρ L ∞ (0,T ;W 2,∞ (R d )) t 0 |p(t − s)| ds ≤ f C [0,T ]; W 1,∞ (R d ×R d ) ′ σ 1 2 W 3,2 (R d ) σ 2 2 L 2 (R n ) T 2 2 . This proves the estimate in i). That L(f )(t) is continuous as a function of t follows easily from the previous argument. As a further by-product note that L(f )(t) L ∞ ≤ f C [0,T ]; W 1,∞ (R d ×R d ) ′ σ 1 2 W 1,2 (R d ) σ 2 2 L 2 (R n ) T 2 2 holds. Since p(0) = 0, we have ∂ t L(f )(t) = t 0 p ′ (t − s)Σ * ρ(s) ds which gives: ∂ t L(f )(t) L ∞ (R d ×R d ) ≤ f C [0,T ]; W 1,∞ (R d ×R d ) ′ σ 1 2 W 1,2 (R d ) σ 2 2 L 2 (R n ) T. This ends the proof of ii). Existence of solutions The proof of existence of solutions to (8) relies on estimates satisfied by the characteristics curves defined by the following ODE system: Ẋ (t) = ξ(t), ξ(t) = −∇V (X(t)) − ∇Φ(t, X(t)).(10) From now on, we adopt the following notation. The potential Φ being given, we denote by ϕ Φ,t α (x 0 , v 0 ) ∈ R d × R d the solution of (10) which starts from (x 0 , v 0 ) at time t = α: the initial data is ϕ Φ,α α (x 0 , v 0 ) = (x 0 , v 0 ). We use the shorthand notation t → (X(t), ξ(t)) for t → ϕ Φ,t 0 (x 0 , v 0 ) , the solution of (10) with X(0) = x 0 and V (0) = v 0 . Owing to the regularity of V, L and Φ 0 , see Lemma 2.1, the solution of the differential system (10) is indeed well defined for prescribed initial data; this also allows us to establish the following estimates, where characteristics are evaluated both forward and backward. Φ ∈ C 0 ([0, ∞); W 2,∞ (R d )) ∩ C 1 ([0, ∞); L ∞ (R d )). a) There exists a function (N , t, x, v) ∈ [0, ∞) × [0, ∞) × R d × R d → R(N , t, x, v) ∈ [0, ∞), non decreasing with respect to the first two variables, such that the solution t → (X(t), ξ(t)) of (10) with initial data X(0) = x 0 , ξ(0) = v 0 satisfies the following estimate, for any t ∈ R, (X(t), ξ(t)) ∈ B 0, R Φ C 1 ([0,t];L ∞ (R d )) , |t|, x 0 , v 0 ⊂ R d × R d . b) Taking two different potentials Φ 1 and Φ 2 , the following two estimates hold for any t > 0: |(ϕ Φ 1 ,t 0 − ϕ Φ 2 ,t 0 )(x 0 , v 0 )| ≤ t 0 (Φ 1 − Φ 2 )(s) W 1,∞ (R d ) exp t s ∇ 2 (Φ 1 (τ ) + V ) L ∞ (Bτ (x 0 ,v 0 )) dτ ds, |(ϕ Φ 1 ,0 t − ϕ Φ 2 ,0 t )(x, v)| ≤ t 0 (Φ 1 − Φ 2 )(s) W 1,∞ (R d ) exp s 0 ∇ 2 (Φ 1 (τ ) + V ) L ∞ (Bt,τ (x,v)) dτ ds, where we set B τ (x, v) = B 0, R max i=1,2 Φ i C 1 ([0,τ ];L ∞ (R d )) , τ, x, v andB t,τ = B 0, R max i=1,2 Φ i C 1 ([τ,t];L ∞ (R d )) , t − τ, x, v . The proof of the lemma is postponed the end of this section. Given 0 < R 0 < ∞, and Ψ 0 , Ψ 1 satisfying (H3) (they enter into the definition of Φ 0 in (6)), we set r(t, x, v) = R( Φ 0 C 1 ([0,t];L ∞ (R d )) + |||L||| Bt R 0 , t, x, v).(11) Proving uniqueness statements for the wide class of external potentials considered in(H2) requires to strengthen the hypothesis on the initial data. Definition 3.2 Let 0 < T, R 0 < ∞. We say that an integrable function f 0 belongs to the set E R 0 ,T if f 0 ≥ 0 satisfies f 0 L 1 (R d ×R d ) ≤ R 0 and, furthermore,K R 0 ,T (f 0 ) := R d ×R d f 0 (x, v) exp T 0 ∇ 2 V L ∞ (B(0,r(t,x,v))) dt dv dx < ∞. Theorem 3.3 Assume (H1)-(H3). Let 0 < R 0 , T < ∞. Let f 0 ∈ E R 0 ,T . Then, there exists a unique f ∈ C([0, T ]; L 1 (R d × R d )) weak solution of (8). The solution is continuous with respect to the parameters L, Φ 0 and f 0 , respectively in A T ∩ B T , C 1 ([0, ∞); W 2,∞ (R d )) and E R 0 ,T . If f 0 ∈ L 1 (R d × R d ) only, see (H4), then there exists f ∈ C([0, ∞); L 1 (R d × R d )), weak solution of (8). The statement can be rephrased for the original problem (2)- (5). We also establish the conservation of energy. Corollary 3.4 Assume (H1)-(H3). Let 0 < R 0 , T < ∞. Let f 0 ∈ E R 0 ,T . Then, there exists a unique weak solution (f, Ψ) to the system (2)-(5) with f ∈ C([0, T ]; L 1 (R d × R d )) and Ψ ∈ C([0, T ]; L 2 (R d × R n )) . The solution is continuous with respect to the parameters σ 1 , σ 2 , Ψ 0 , Ψ 1 and f 0 in the sets W 3,2 (R d ), L 2 (R n ), L 2 (R d × R n ), L 2 (R d × R n ) and E R 0 ,T , respectively. If f 0 satisfies (H4) only, then there exists a weak solution with f ∈ C([0, ∞); L 1 (R d × R d )) and Ψ ∈ C([0, T ]; L 2 (R d × R n )) . Furthermore, when the initial data satisfies (H5) the total energy 1 2 R d ×R n |∂ t Ψ(t, x, y)| 2 dx dy + c 2 2 R d ×R n |∇ y Ψ(t, x, y)| 2 dx dy + R d ×R d f (t, x, v) |v| 2 2 + V (x) + Φ(t, x) dv dx is conserved. Remark 3.5 Definition 3.2 restricts the set of initial data depending on the growth of the Hessian of the external potential. Of course, any integrable data f 0 with compact support fulfils the criterion in Definition 3.2, and when the potential has at most quadratic growth, any data satisfying (H4) is admissible. As will be clear in the proof, the continuity with respect to the initial data does not involve the L 1 norm only, but the more intricate quantity K R 0 ,T also arises in the analysis. Remark 3.6 The present approach does not need a restriction on the transverse dimension (n ≥ 3 in [8]). The proof can be slightly modified to treat the case of measure-valued initial data f 0 , thus including the results in [8] for a single parti- cle (f 0 (x, v) = δ (x=x 0 ,v=v 0 ) ), and we can consider a set of N > 1 particles as well. The measure-valued solution is then continuous with respect to the initial data in C([0, T ]; (W 1,∞ (R d × R d )) ′ ) . This viewpoint will be further detailed with the discussion of mean-field asymptotics [31]. The proof of Theorem 3.3 relies on a fixed point strategy, the difficulty being to set up the appropriate functional framework. It turns out that it will be convenient to work with the C [0, T ]; (W 1,∞ (R d × R d )) ′ norm. We remind the reader that the dual norm on (W 1,∞ (R d × R d )) ′ is equivalent to the Kantorowich-Rubinstein distance W 1 (f, g) = sup π R 2d ×R 2d |ζ − ζ ′ | dπ(ζ, ζ ′ ) where the supremum is taken over measures π having f and g as marginals, see e. g. [32,Remark 6.5]. This distance appears naturally in the analysis of Vlasov-like systems, as pointed out in [15]. In order to define the fixed point procedure, we introduce the following mapping. For a non negative integrable function f 0 , we denote by Λ f 0 the application which associates to Φ in C([0, ∞); W 2,∞ (R d )) ∩ C 1 ([0, ∞); L ∞ (R d )) the unique solution f of the Liouville equation ∂ t f + v · ∇ x f − ∇ v f · ∇ x (V + Φ) = 0, with initial data f 0 . We shall make use of the following statement, which provides useful estimates. Lemma 3.7 For any f 0 ∈ L 1 (R d × R d ), the application Λ f 0 is continuous on the set C([0, ∞); W 2,∞ (R d )) ∩ C 1 ([0, ∞); L ∞ (R d )) with values in C([0, ∞); L 1 (R d × R d )). Furthermore, we have Λ f 0 (Φ) − Λ g 0 (Φ) L ∞ (0,∞;L 1 (R d ×R d )) = f 0 − g 0 L 1 (R d ×R d ) , for any Φ ∈ C([0, ∞); W 2,∞ (R d )) ∩ C 1 ([0, ∞); L ∞ (R d )). Proof. Let 0 < T < ∞ be fixed once for all. We begin by assuming that f 0 is C 1 and compactly supported. For any 0 ≤ t ≤ T , we have Λ f 0 (Φ)(t) = f 0 • ϕ Φ,0 t , where we remind the reader that ϕ Φ,0 t (x, v) stands for the evaluation at time 0 of the solution of (10) which starts at time t from the state (x, v). Accordingly any L p norm is preserved: Λ f 0 (Φ)(t) L p (R d ×R d ) = f 0 L p (R d ×R d ) holds for any t ≥ 0 and any 1 ≤ p ≤ ∞. By linearity, this immediately proves the continuity estimate with respect to the initial data. To establish the continuity properties with respect to Φ, we first observe, denot- ing Λ f 0 (Φ) = f , that (x, v) ∈ supp(f (t, ·)) iff ϕ Φ,0 t (x, v) ∈ supp(f 0 ), that is (x, v) ∈ ϕ Φ,t 0 (supp(f 0 )). Therefore, by Lemma 3.1, we can find a compact set K T ⊂ R d × R d such that supp(f (t, ·)) ⊂ K T for any 0 ≤ t ≤ T . We are dealing with potentials Φ 1 and Φ 2 in C([0, ∞); W 2,∞ (R d )) ∩ C 1 ([0, ∞); L ∞ (R d )) . We can again find a compact set, still denoted by K T ⊂ R d × R d , such that the support of the associated solutions Λ f 0 (Φ 1 ) and Λ f 0 (Φ 2 ) for any 0 ≤ t ≤ T is contained in K T . We infer that Λ f 0 (Φ 1 )(t) − Λ f 0 (Φ 2 )(t) L 1 (R d ×R d ) = K T |f 0 • ϕ Φ 1 ,0 t − f 0 • ϕ Φ 2 ,0 t | dv dx ≤ f 0 W 1,∞ (R d ×R d ) meas(K T ) sup (x,v)∈K T |ϕ Φ 1 ,0 t (x, v) − ϕ Φ 2 ,0 t (x, v)| holds. As τ ranges over [0, t] ⊂ [0, T ] and (x, v) lies in K T , the backward characteristics ϕ Φ i ,τ t (x, v) still belong to a compact set. We introduce the following quantities R = sup (x,v)∈K T R max i=1,2 Φ i C 1 ([0,T ];L ∞ (R d )) , T, x, v and m T = exp T 0 ∇ 2 Φ 1 (u) L ∞ (R d ) du . For 0 ≤ t ≤ T and any (x, v) ∈ K T , Lemma 3.1-b) yields: |ϕ Φ 1 ,0 t (x, v) − ϕ Φ 2 ,0 t (x, v)| ≤ m T t 0 (Φ 1 − Φ 2 )(s) W 1,∞ (R d ) exp s 0 ∇ 2 V L ∞ (B(0,R)) dτ ds. We conclude with sup (x,v)∈K T |ϕ Φ 1 ,0 t (x, v) − ϕ Φ 2 ,0 t (x, v)| − −−−−−−−−−−−−−−−−−−−−−−−−−−−− → Φ 1 −Φ 2 L ∞ (0,T ;W 2,∞ (R d )) →0 Φ 1 C 1 ([0,T ];L ∞ (R d )) , Φ 2 C 1 ([0,T ];L ∞ (R d )) ≤M 0. (It is important to keep both the C 1 ([0, T ]; L ∞ (R d )) and L ∞ (0, T ; W 2,∞ (R d )) norms of the potentials bounded since these quantities appear in the definition of R and m T .) This proves the asserted continuity of the solution with respect to the potential. By uniform continuity of the flow on the compact set [0, T ] × K T , we obtain the time continuity. Hence the result is proved when the initial data f 0 lies in C 1 c . We finally extend the result for initial data f 0 in L 1 . Those can be approximated by a sequence f k 0 k∈N of functions in C 1 c (R d × R d ). We have Λ f 0 (Φ)(t) − Λ f k 0 (Φ)(t) L 1 (R d ×R d ) = Λ (f 0 −f k 0 ) (Φ)(t) L 1 (R d ×R d ) = f 0 − f k 0 L 1 (R d ×R d ) . Therefore, Λ f 0 is the uniform limit of maps which are continuous with respect to Φ and the time variable. This remark ends the proof. Proof of Theorem 3.3. Existence-uniqueness for initial data in E R 0 .T . We turn to the fixed point reasoning. For f given in C [0, T ]; (W 1,∞ (R d × R d )) ′ , we set T f 0 (f ) = Λ f 0 (Φ 0 − L(f )). It is clear that a fixed point of T f 0 is a solution to (8). Note also that, as a consequence of Lemma 2.1 and Lemma 3.7, T f 0 (f )(t) ∈ L 1 (R d × R d ). More precisely, we know that f → T (f ) is continuous with values in the space C([0, T ]; L 1 (R d × R d )) ⊂ C [0, T ]; (W 1,∞ (R d × R d )) ′ . We shall prove that T admits an iteration which is a contraction on the ball with centre 0 and radius R 0 . Let f 1 and f 2 be two elements of this ball. We denote ϕ Φ i ,t α the flow of (10) with Φ i = Φ 0 − L(f i ): ϕ Φ i ,t α (x 0 , v 0 ) satisfies (10) with (x 0 , v 0 ) as data at time t = α. Let χ be a trial function in W 1,∞ (R d × R d ). We have R d ×R d (T (f 1 )(t, x, v) − T (f 2 )(t, x, v))χ(x, v) dv dx = R d ×R d f 0 • ϕ Φ 1 ,0 t − f 0 • ϕ Φ 2 ,0 t (x, v)χ(x, v) dv dx = R d ×R d f 0 (x, v) χ • ϕ Φ 1 ,t 0 − χ • ϕ Φ 2 ,t 0 (x, v) dv dx ≤ R d ×R d f 0 (x, v) ∇χ ∞ ϕ Φ 1 ,t 0 − ϕ Φ 2 ,t 0 (x, v) dv dx. It follows that T (f 1 )(t)−T (f 2 )(t) (W 1,∞ (R d ×R d )) ′ ≤ R d ×R d f 0 (x, v) ϕ Φ 1 ,t 0 − ϕ Φ 2 ,t 0 (x, v) dv dx. (12) By using Lemma 3.1-b), we obtain ϕ Φ 1 ,t 0 − ϕ Φ 2 ,t 0 (x, v) ≤m T t 0 L(f 1 − f 2 ) L ∞ (0,s;W 2,∞ (R d )) × exp t s ∇ 2 V L ∞ B(0,R( Φ 0 +L(f i ) C 1 ([0,u];L ∞ (R d )) ,u,x 0 ,v 0 )) du ds, where we have used exp T 0 ∇ 2 (Φ 0 (u) − L(f 1 )(u) L ∞ (R d ) du ≤ exp T 0 ∇ 2 Φ 0 (u) L ∞ (R d ) + |||L||| Au f 0 L 1 (R d ×R d ) du =m T . Plugging this estimate into (12) yields T (f 1 )(t) − T (f 2 )(t) (W 1,∞ (R d ×R d )) ′ ≤m T R d ×R d f 0 (x, v) t 0 L(f 1 − f 2 ) L ∞ (0,s;W 2,∞ (R d )) × exp t s ∇ 2 V L ∞ (B(0,r(u,x,v))) du ds dv dx. It recasts as T (f 1 )(t) − T (f 2 )(t) (W 1,∞ ) ′ ≤m ′ T K R 0 ,T t 0 f 1 − f 2 L ∞ 0,s;(W 1,∞ (R d ×R d )) ′ ds withm ′ T =m T × sup 0≤s≤T |||L||| As . By induction, we deduce that T ℓ (f 1 )(t) − T ℓ (f 2 )(t) (W 1,∞ (R d ×R d )) ′ ≤ (tm ′ T K R 0 ,T ) ℓ ℓ! f 1 − f 2 L ∞ 0,T ;(W 1,∞ (R d ×R d )) ′ holds for any ℓ ∈ N and 0 ≤ t ≤ T . Finally, we are led to T ℓ (f 1 )−T ℓ (f 2 ) L ∞ 0,T ;(W 1,∞ (R d ×R d )) ′ ≤ (Tm ′ T K R 0 ,T ) ℓ ℓ! f 1 −f 2 L ∞ 0,T ;(W 1,∞ (R d ×R d )) ′ . This shows that an iteration of T is a contraction. Therefore, there exists a unique fixed point f in C [0, T ]; (W 1,∞ (R d × R d )) ′ . Furthermore, f = T (f ) ∈ C([0, T ]; L 1 (R d × R d )) , and the solution is continuous with respect to the parameters of the system. Note that the continuity estimate involves the quantity in Definition 3.2 which restricts the growth assumption of the initial data. Step 2: Existence for an integrable data We proceed by approximation. Let f 0 be in L 1 (R d × R d ), with f 0 L 1 ≤ R 0 . Then, (x, v) → f k 0 (x, v) = f 0 (x, v)1 √ x 2 +v 2 ≤k lies in E R 0 ,T (with a constant K R 0 ,T which can blow up as k → ∞). The previous step defines f k , solution of (8) with this initial data. Of course we wish to conclude by passing to the limit k → ∞. However, the necessary compactness arguments are not direct and the proof splits into several steps. We start by showing that the sequence f k k∈N is compact in C([0, T ]; M 1 (R d × R d ) − weak − ⋆). Pick χ ∈ C ∞ c (R d × R d ). For any 0 ≤ t ≤ T , we have, on the one hand, R d ×R d f k (t, x, v)χ(x, v) dv dx ≤ f k (t, ·) L 1 (R d ×R d ) χ L ∞ (R d ×R d ) ≤ f k 0 L 1 (R d ×R d ) χ L ∞ (R d ×R d ) ≤ f 0 L 1 (R d ×R d ) χ L ∞ (R d ×R d ) ,(13) and, on the other hand, d dt R d ×R d f k (t, x, v)χ(x, v) dv dx = R d ×R d f k (t, x, v) v · ∇ x χ − ∇ x (V + Φ 0 − L(f )(t)) · ∇ v χ (x, v) dv dx ≤ f 0 L 1 v · ∇ x χ − ∇V · ∇ v χ L ∞ (R d ×R d )) + |||L||| A T f 0 L 1 + Φ 0 L ∞ ([0,T ];W 1,∞ (R d )) ∇ v χ L ∞ . Lemma 2.1 then ensures that the set t → R d ×R d f k (t, x, v)χ(x, v) dv dx, k ∈ N is equibounded and equicontinuous; hence, by virtue of Arzela-Ascoli's theorem it is relatively compact in C([0, T ]). Going back to (13), a simple approximation argument allows us to extend the conclusion to any trial function χ in C 0 (R d × R d ), the space of continuous functions that vanish at infinity. This space is separable; consequently, by a diagonal argument, we can extract a subsequence and find a measure valued function t → df (t) ∈ M 1 (R d × R d ) such that lim k→∞ R d ×R d f k (t, x, v)χ(x, v) dv dx = R d ×R d χ(x, v) df (t) holds uniformly on [0, T ], for any χ ∈ C 0 (R d × R d ). As a matter of fact, we note that df is non negative and for any 0 ≤ t ≤ T it satisfies R d ×R d df (t) ≤ f 0 L 1 (R d ×R d ) . Next, we establish the tightness of the sequence of approximate solutions. Let ǫ > 0 be fixed once for all. We can find M ǫ > 0 such that x 2 +v 2 ≥M 2 ǫ f 0 (x, v) dv dx ≤ ǫ. Let us set A ǫ = sup{r(T, x, v), (x, v) ∈ B(0, M ǫ )} where we remind the reader that r(T, x, v) has been defined in (11): 0 < A ǫ < ∞ is well defined by Lemma 2.1. Let ϕ k,t α stand for the flow associated to the characteristics of the equation satisfied by f k . For any 0 ≤ t ≤ T , we have ϕ k,t 0 (B(0, M ǫ )) ⊂ B(0, A ǫ ) so that ∁ ϕ k,0 t (B(0, A ǫ )) = ϕ k,0 t ∁B(0, A ǫ ) ⊂ ∁B(0, M ǫ ). It follows that ∁B(0,Aǫ) f k (t, x, v) dv dx = ∁B(0,Aǫ) f k 0 (ϕ k,0 t (x, v)) dv dx = ∁ϕ k,0 t (B(0,Aǫ)) f k 0 (x, v) dv dx ≤ ∁B(0,Mǫ) f 0 (x, v) dv dx ≤ ǫ. By a standard approximation, we check that the same estimate is satisfied by the limit f : ∁B(0,Aǫ) df (t) ≤ ǫ. Finally, we justify that f k converges to f in C([0, T ]; (W 1,∞ (R d × R d )) ′ ). Pick χ in W 1,∞ (R d × R d ), with χ W 1,∞ (R d ×R d ) ≤ 1. We introduce a cut-off function θ R as follows: θ R (x, v) = θ(x/R, v/R), θ ∈ C ∞ c (R d × R d ), θ(x, v) = 1 for √ x 2 + v 2 ≤ 1, θ(x) = 0 for x 2 + v 2 ≥ 4, 0 ≤ θ(x) ≤ 1 for any x ∈ R d .(14) Then, we split R d ×R d f k (t, x, v)χ(x, v) dv dx − R d ×R d χ(x, v) df (t) = R d ×R d f k (t, x, v)χθ R (x, v) dv dx − R d ×R d χθ R (x, v) df (t) + R d ×R d f k (t, x, v)χ(1 − θ R )(x, v) dv dx − R d ×R d χ(1 − θ R )(x, v) df (t). Choosing R ≥ A ǫ yields R d ×R d f k (t, x, v)χ(1 − θ R )(x, v) dv dx − R d ×R d χ(1 − θ R )(x, v) df (t) ≤ 2ǫ χ L ∞ (R d ×R d ) . (15) By virtue of the Arzela-Ascoli theorem, W 1,∞ (B(0, 2R)) embeds compactly in C (B(0, 2R)). Thus, we can find a family {χ 1 , ..., ∞ (B(0, 2R))). Therefore, let us write χ mǫ } of functions in W 1,∞ (R d × R d ) such that, for any χ ∈ W 1,∞ (R d × R d ), χ W 1,∞ (R d ×R d ) ≤ 1, there exists an index i ∈ {1, ..., m ǫ } with θ R χ − χ i L ∞ (B(0,2R)) ≤ ǫ (since χθ R lies in a bounded ball of W 1,R d ×R d f k (t, x, v)χθ R (x, v) dv dx − R d ×R d χθ R (x, v) df (t) = R d ×R d f k (t, x, v)χ i (x, v) dv dx − R d ×R d χ i (x, v) df (t)x + R d ×R d f k (t, x, v)(χθ R − χ i )(x, v) dv dx − R d ×R d (χθ R − χ i )(x, v) df (t), where the last two terms can both be dominated by f 0 L 1 (R d ×R d ) ǫ. We thus arrive at R d ×R d f k (t, x, v)χ(x, v) dv dx − R d ×R d χ(x, v) df (t) ≤ 2ǫ( χ L ∞ (R d ×R d ) + f 0 L 1 (R d ×R d ) ) + R d ×R d f k (t, x, v) χ i (x, v) dv dx − R d ×R d χ i (x, v) df (t) ≤ 2ǫ( χ L ∞ (R d ×R d ) + f 0 L 1 (R d ×R d ) ) + sup j∈{1,...,mǫ} R d ×R d f k (t, x, v) χ i (x, v) dv dx − R d ×R d χ i (x, v) df (t) , for any χ ∈ W 1,∞ (R d × R d ), with χ W 1,∞ (R d ×R d ) ≤ 1. The last term can be made smaller than ǫ by choosing k ≥ N ǫ large enough. In other words, we can find N ǫ ∈ N such that sup χ W 1,∞ ≤1 R d ×R d f k (t, x, v)χ(x, v) dv dx − R d ×R d χ(x, v) df (t) ≤ 2ǫ(2 + f 0 L 1 (R d ×R d ) ) holds for any 0 ≤ t ≤ T , and k ≥ N ǫ : f k converges to f in C [0, T ]; (W 1,∞ (R d × R d )) ′ . According to Lemma 3.7, together with Lemma 2.1, it implies that T f 0 (f k ) converges to T f 0 (f ) in C([0, T ]; L 1 (R d × R d )). By definition T f k 0 (f k ) = f k so that f k − T f 0 (f ) C([0,T ];L 1 (R d ×R d )) ≤ T f k 0 (f k ) − T f 0 (f k ) C([0,T ];L 1 (R d ×R d ))) + T f 0 (f k ) − T f 0 (f ) C([0,T ];L 1 (R d ×R d ))) ≤ f k 0 − f 0 L 1 (R d ×R d ) + T f 0 (f k ) − T f 0 (f ) C([0,T ];L 1 (R d ×R d )) ) holds, where we have used Lemma 3.7 again. Letting k go to ∞, we realize that f k also converges to T f 0 (f ) in C([0, T ]; L 1 (R d × R d )). It implies both f = T f 0 (f ) and f ∈ C([0, T ]; L 1 (R d × R d ) ). By definition of T f 0 , f satisfies (8), and it also justifies that f is absolutely continuous with respect to the Lebesgue measure, which ends the proof. Proof of Lemma 3.1. Let (X, ξ) be the solution of (10) with (X(0), X(t)). ξ(0)) = (x 0 , v 0 ). We have d dt V (X(t)) + Φ(t, X(t)) + |ξ(t)| 2 2 = (∂ t Φ)(t, The right hand side is dominated by ∂ t Φ C([0,t];L ∞ (R d )) . With t ≥ 0, integrating this relation yields |ξ(t)| 2 2 ≤ V (x 0 ) + Φ(0, x 0 ) + |v 0 | 2 2 − (V (X(t)) + Φ(t, X(t))) + t ∂ t Φ C([0,t];L ∞ (R d )) . Owing to (H2) we deduce that |ξ(t)| 2 ≤ a(t) + 2C|X(t)| 2 holds with a(t) = 2 V (x 0 ) + Φ(0, x 0 ) + |v 0 | 2 2 + 2t ∂ t Φ C([0,t];L ∞ (R d )) + 2 Φ(t, ·) L ∞ (R d ) + 2C. Next, we simply write d|X(t)| 2 dt (t) = 2X(t) · ξ(t) ≤ X(t) 2 + ξ(t) 2 so that the estimate just obtained on ξ yields |X(t)| 2 ≤ |x 0 | + (1 + 2C) t 0 |X(s)| 2 ds + t 0 a(s) ds. By using the Grönwall lemma we conclude that |X(t)| 2 ≤ |x 0 | 2 e (1+2C)t + t 0 e (1+2C)(t−s) a(s) ds holds. Going back to the velocity, we obtain |ξ(t)| 2 ≤ 2C |x 0 |e (1+2C)t + t 0 e (1+2C)(t−s) a(s) ds + a(t). It concludes the proof of Lemma 3.1-a). Next, let (X 1 , ξ 1 ) and (X 2 , ξ 2 ) be two solutions of (10) with the same initial data (x 0 , v 0 ), but different potentials Φ 1 , Φ 2 . We already know that the two characteristic curves (X i (s), ξ i (s)), for i ∈ {1, 2}, belong to B s (x, v). We have              d ds |X 1 (s) − X 2 (s)| ≤ |ξ 1 (s) − ξ 2 (s)|, d ds |ξ 1 (s) − ξ 2 (s)| ≤ ∇ (Φ 1 (s, ·) − Φ 2 (s, ·)) L ∞ (R d ) +|X 1 (s) − X 2 (s)| ∇ 2 (V + Φ 1 (s, ·)) L ∞ (Bs(x,v)) The Grönwall lemma yields the estimate |(X 1 (t), ξ 1 (t)) − (X 2 (t), ξ 2 (t))| ≤ t 0 (Φ 1 − Φ 2 )(τ, ·) W 1,∞ (R d ) exp t s ∇ 2 (V + Φ 1 (u)) L ∞ (Bu(x,v)) du ds. Finally, we wish to evaluate the backward characteristics, looking at the state at time 0, given the position/velocity pair at time t. Namely we consider ϕ Φ,s t (x, v) for s ≤ t, bearing in mind ϕ Φ,t t (x, v) = (x, v). We set Y ζ (s) = 1 0 0 −1 ϕ Φ,t−s t (x, v). We check that (Y, ζ) satisfies      d ds Y (s) = ζ(s), d ds ζ(s) = −∇V (Y (s)) − ∇Φ(t − s, Y (s)) , Y (0) = x, ζ(0) = v. Changing Φ for Φ(t − ·), this allows us to obtain the same estimates on (Y, ζ) for all s ≥ 0. We conclude by taking s = t. Proof of Corollary 3.4. Theorem 3.3 constructs solutions to (8) in C 0 ([0, ∞); L 1 (R d × R d )). We have now the functional framework necessary to justify the manipulations made in Section 2. For Ψ 0 , Ψ 1 verifying (H3), formula (9) defines a solution Ψ ∈ C([0, ∞); L 2 (R n × R d )) of the wave equation, and finally (f, Ψ) satisfies (2)- (5). Conversely, if f ∈ C 0 ([0, ∞); L 1 (R d × R d )) and Ψ ∈ C([0, ∞); L 2 (R n × R d )) is a solution of the system (2)-(5), then we can rewrite Φ = Φ 0 − L(f ) and f verifies (8). This equivalence justifies the first part of the statement in Corollary 3.4. It only remains to justify the energy conservation. We consider an initial data with finite energy: E 0 = c 2 2 R d ×R n |∇ y Ψ 0 (x, y)| 2 dy dx + 1 2 R d ×R n |Ψ 1 (x, y)| 2 dy dx E vib 0 + R d ×R d f 0 (x, v) |v| 2 2 + V (x) + Φ(0, x) dv dx E part 0 ∈ (−∞, +∞). For the solutions constructed in Theorem 3.3, we have seen that the self-consistent potential remains smooth enough so that the characteristic curves t → (X(t), ξ(t)) are well-defined. Therefore, we can write R d ×R d f (t, x, v) |v| 2 2 + V (x) + Φ(t, x) dv dx = R d ×R d f 0 (x, v) |ξ(t)| 2 2 + V (X(t)) + Φ(t, X(t)) dv dx. For any (t, x, v) we have the following equality d dt V (X(t)) + Φ(t, X(t)) + |ξ(t)| 2 2 = (∂ t Φ)(t, X(t)). Therefore, we get R d ×R d f (t, x, v) |v| 2 2 + V (x) + Φ(t, x) dv dx = E part 0 + R d ×R d f 0 (x, v) t 0 (∂ t Φ)(s, X(s)) ds dv dx = E part 0 + t 0 R d ×R d f (s, x, v)(∂ t Φ)(s, x) dv dx ds = E part 0 + t 0 R d ρ(s, x)(∂ t Φ)(s, x) dx ds. Next, let Ψ be the unique solution of (4) associated to f . We first assume that the initial data Ψ 0 et Ψ 1 are smooth, say in L 2 (R d , H 2 (R n )). Therefore, going back to (9), we can check that Ψ lies in C([0, ∞); L 2 (R d , H 2 (R n ))). Integrations by parts lead to d dt 1 2 R d ×R n |∂ t Ψ(t, x, y)| 2 dy dx + c 2 2 R d ×R n |∇ y Ψ(t, x, y)| 2 dx dy = R d ×R n ∂ t Ψ ∂ 2 t Ψ − c 2 ∆ y Ψ t, x, y) dy dx = − R d ×R n ∂ t Ψ(t, x, y) ρ(t, ·) * x σ 1 (x) σ 2 (y) dy dx = − R d ρ∂ t Φ(t, x) dx. Hence, we obtain 1 2 R d ×R n |∂ t Ψ(t, x, y)| 2 dx dy + c 2 2 R d ×R n |∇ y Ψ(t, x, y)| 2 dx dy = E vib 0 − t 0 R d ρ(s, x)(∂ t Φ)(s, x) dx ds. It proves the energy conservation for such smooth data. We go back to general data with finite energy: Ψ 0 ∈ L 2 (R d , H 1 (R n )) and Ψ 1 ∈ L 2 (R d × R n ). We approximate the data by Ψ k 0 and Ψ k 1 lying in L 2 (R d , H 2 (R n )). Using (9), one sees the associated sequence (Ψ k ) k∈N of solutions to (4) converges to Ψ in C([0, ∞); L 2 (R d , H 1 (R n ))) and C 1 ([0, ∞); L 2 (R d × R n )). This implies one can pass to the limit in the energy conservation. Remark 3. 8 We point out that, whereas energy conservation is an important physical property, it was not used here in the existence proof. In particular, one should notice that it does not provide directly useful a priori estimates on the kinetic energy, since the potential energy associated to the external potential V can be negative and unbounded under our assumptions. In order to deduce a useful estimate the assumptions on the initial data need to be strengthened: in addition to (H5) we suppose M 2 := R d ×R d f 0 (x, v)|x| 2 dv dx < ∞. We set V − (x) = max(−V (x), 0) ≥ 0. Then (H2) implies R d ×R d f (t, x, v)V − (x) dv dx ≤ R d ×R d f (t, x, v)C(1 + |x| 2 ) dv dx ≤ C f 0 L 1 (R d ×R d ) + C R d ×R d f 0 (x, v)|X(t)| 2 dv dx, where X(t) stand for the first (space) component of ϕ t 0 (x, v). Reproducing the estimates of the proof of Lemma 3.1, we get |X(t)| ≤ |x|e √ 2Ct + 1 √ C V (x) + |v| 2 2 + Φ(0, x) 1/2 (e √ 2Ct − 1) + b(t) where b(t) = √ 2 t 0 C + Φ(s, ·)| L ∞ (R d ) + s ∂ t Φ C([0,s];L ∞ (R d )) 1/2 e √ 2C(t−s) ds. It follows that |X(t)| ≤ 9|x| 2 e 2 √ 2Ct + 9 C V (x) + |v| 2 2 + Φ(0, x) (e √ 2Ct − 1) 2 + 9b(t) 2 . Eventually, we find R d ×R d f (t, x, v)V − (x) dv dx ≤ Ce 2 √ 2Ct M 2 +9(e √ 2Ct −1) 2 E 0 +C(9b(t) 2 +1) f 0 L 1 (R d ×R d ) . Therefore the potential energy associated to the external potential cannot be too negative and all terms in the energy balance remain bounded on any finite time interval. Large wave speed asymptotics This section is devoted to the asymptotics of large wave speeds. Namely, we consider the following rescaled version of the system:            ∂ t f ǫ + v · ∇ x f ǫ − ∇ x (V + Φ ǫ ) · ∇ v f ǫ = 0, Φ ǫ (t, x, y) = R n ×R d Ψ ǫ (t, z, y)σ 2 (y)σ 1 (x − z) dz dy, ∂ 2 tt − 1 ǫ ∆ y Ψ ǫ (t, x, y) = − 1 ǫ σ 2 (y) R d ×R d σ 1 (x − z)f (t, z, v) dv dz,(16) completed with suitable initial conditions. We are interested in the behavior of the solutions as ǫ → 0. We shall discuss below the physical meaning of this regime. But, let us first explain on formal grounds what can be expected. As ǫ → 0 the wave equation degenerates to −∆ y Ψ(t, x, y) = −σ 2 (y) σ 1 * x ρ(t, x), ρ(t, x) = R d f (t, x, v) dv. We obtain readily the solution by uncoupling the variables: Ψ(t, x, y) = γ(y) σ 1 * x ρ(t, x) where γ satisfies the mere Poisson equation ∆ y γ = σ 2 . At leading order the potential then becomes Φ(t, x) = −κ Σ * x ρ(t, x), Σ = σ 1 * σ 1 , κ = − R n σ 2 γ dy. Therefore, we guess that the limiting behavior is described by the following Vlasov equation ∂ t f + v · ∇ x f − ∇ x (V + Φ) · ∇ v f = 0. As long as the integration by parts makes sense (we shall see that difficulties in the analysis precisely arise when n ≤ 2), we observe that κ = R n |∇ y γ| 2 dy > 0. It is then tempting to make the form function σ 1 depend on ǫ too, so that Σ resembles the kernel of (−∆ x ). We would arrive at the Vlasov-Poisson system, in the case of attractive forces. We which to justify such asymptotic behavior. Dimensional analysis In (2), f is the density of particles in phase space: it gives a number of particles per unit volume of phase space. Let T, L, V be units for time, space and velocity respectively, and set t ′ = t/T, x ′ = x/L, v ′ = v/V which define dimensionless quantities. Then, we set f ′ (t ′ , x ′ , v ′ ) L −d V −d = f (t, x, v) (or maybe more conveniently f ′ (t ′ , x ′ , v ′ ) dv ′ dx ′ = f (t, x, v) dv dx) . The external and interaction potential, V and Φ, have both the dimension of a velocity squared. We set V (x) = V 2 ext V ′ (x ′ ), Φ(t, x) = W 2 Φ ′ (t ′ , x ′ ), where V ext and W thus have the dimension of a velocity. We switch to the dimensionless equation ∂ t ′ f ′ + VT L v ′ · ∇ x ′ f ′ − T LV V 2 ∇ x ′ V ′ + W V 2 Φ ′ · ∇ v ′ f ′ = 0. The definition of the interaction potential Φ is driven by the product σ 2 (z)σ 1 (x) dx. We scale it as follows σ 2 (z)σ 1 (x) dx = Σ ⋆ L d σ ′ 2 (z ′ )σ ′ 1 (x ′ ) dx ′ . It might help the intuition to think z as a length variable, and thus c has a velocity, but there is not reason to assume such privileged units. Thus, we keep a general approach. For the vibrating field, we set ψ(t, x, z) = Ψ ⋆ ψ ′ (t ′ , x ′ , z ′ ), z ′ = z/ℓ, still with the convention that primed quantities are dimensionless. Accordingly, we obtain W 2 = Σ ⋆ L d Ψ ⋆ ℓ n and the consistent expression of the dimensionless potential Φ ′ (t ′ , x ′ ) = σ ′ 1 (x ′ − y ′ )σ ′ 2 (z ′ )ψ(t ′ , y ′ , z ′ ) dz ′ dy ′ . The wave equation becomes ∂ 2 t ′ t ′ ψ ′ − T 2 c 2 ℓ 2 ∆ z ′ ψ ′ = − T 2 Σ ⋆ L d Ψ ⋆ L −d T 2 Σ ⋆ Ψ ⋆ σ ′ 2 (z ′ ) σ ′ 1 (x ′ − y ′ )f ′ (t ′ , y ′ , v ′ ) dv ′ dy ′ . (17) Note that T 2 Σ ⋆ Ψ ⋆ = Σ ⋆ L d ℓ n Ψ ⋆ T 2 Ψ 2 ⋆ L d ℓ n = W 2 T 2 Ψ 2 ⋆ L d ℓ n . Let us consider the energy balance where the following quantities, all having the homogeneity of a velocity squared, appear: • the kinetic energy of the particles v 2 f dv dx; it scales like V 2 , • the external potential energy V f dv dx; it scales like V 2 ext , • the coupling energy Φf dv dx; it scales like W 2 , • the wave energy which splits into: a) |∂ t ψ| 2 dz dx, which scales like Ψ 2 ⋆ L d ℓ n T 2 , b) c 2 |∇ z ∂ t ψ| 2 dz dx, which scales like c 2 Ψ 2 ⋆ L d ℓ n ℓ 2 . Note that the kinetic energy in a) is ℓ 2 c 2 T 2 times the elastic energy in b). To recap, we have at hand 6 parameters imposed by the model (L, ℓ, c, V ext , W, Σ ⋆ ) and two parameters governed by the initial conditions V and Ψ ⋆ . They allow to define the five energies described above. We turn to the scaling assumptions. It is convenient to think of them by comparing the different time scales involved in the equations. We set ǫ = ℓ cT 2 ≪ 1. If ℓ is the size of the support of the source σ 2 , then this regime means that the time a typical particle needs to cross L (the support of σ 1 ) is much longer than the time the wave needs to cross ℓ (the support of σ 2 ). Next we suppose that the kinetic energy of the particle, the energy of the particle associated to the external potential, the elastic energy of the wave as well as the interaction energy, all have the same strength, which can expressed by setting L T = V = V ext = W = c 2 Ψ 2 ⋆ L d ℓ n−2 . As a consequence, it imposes the following scaling of the coupling constant Ψ ⋆ T 2 Σ ⋆ = ǫ. It also means that the kinetic energy of the wave is small with respect to its elastic energy. Inserting this in (17) yields (16). Statements of the results Throughout this Section, we assume (H1), and we shall strengthen the assumptions (H2)-(H5) as follows (note that since we are dealing with sequences of initial data, it is important to make the estimates uniform with respect to the scaling parameter): the external potential V ∈ W 2,∞ loc (R d ) is non negative, (H7) f 0,ǫ ∈ L 1 (R d × R d ) , with a uniformly bounded norm, and Ψ 0,ǫ , Ψ 1,ǫ ∈ L 2 (R d × R n ) are such that the rescaled initial energy E 0,ǫ = R d ×R d v 2 2 + V + |Φ ǫ | f 0,ǫ dv dx + ǫ 2 R n ×R d |Ψ 1,ǫ | 2 dy dx + 1 2 R n ×R d |∇ y Ψ 0,ǫ | 2 dy dx is uniformly bounded: 0 ≤ sup ǫ>0 E 0,ǫ =Ē 0 < ∞.                      (H8) f 0,ǫ is bounded in L ∞ (R d × R d ),((R d × R d ) − weak)) for any 1 ≤ p < ∞ to f solution of the following Vlasov equation ∂ t f + v · ∇ x f − ∇ x (V +Φ) · ∇ v f = 0, f (0, x, v) = f 0 (x, v),(18)whereΦ = −κΣ * ρ, Σ = σ 1 * x σ 1 , κ = R n | σ 2 (ξ)| 2 (2π) n |ξ| 2 dξ,and f 0 is the weak limit in L p (R d × R d ) of f 0,ǫ . In order to derive the Vlasov-Poisson system from (16), the form function σ 1 need to be appropriately defined and scaled with respect to ǫ. Let θ and δ be two radially symmetric functions in C ∞ c (R d ) verifying: 0 ≤ θ, δ ≤ 1 θ(x) = 1 for |x| ≤ 1, θ(x) = 0 for |x| ≥ 2, R d δ(x) dx = 1. We set θ ǫ (x) = θ( √ ǫx) et δ ǫ (x) = 1 ǫ d/2 δ(x/ √ ǫ) and . Let (f ǫ , Ψ ǫ ) be the associated solution to (16). Then, there exists a subsequence such that σ 1,ǫ = C d δ ǫ * θ ǫ | · | d−1 , with C d = |S d−1 | R d dx |x| d−1 |e 1 − x| d−1 −1/2 .f ǫ converges in C([0, T ]; L p (R 3 × R 3 ) − weak) for any 1 < p < ∞ to f solution of the attractive Vlasov-Poisson equation      ∂ t f + v · ∇ x f − ∇ x (V +Φ) · ∇ v f = 0, ∆Φ = κρ, f (0, x, v) = f 0 (x, v) (19) where f 0 is the weak limit in L p (R 3 × R 3 ) of f 0,ǫ .L p (R d × R d ), 1 ≤ p < ∞, then f ǫ converges to f in C([0, T ]; L p (R d × R d )). Convergence to the Vlasov equation with a smooth convolution kernel Taking into account the rescaling, the analog of (8) for (16) reads ∂ t f ǫ + v · ∇ x f ǫ − ∇ x V + Φ 0,ǫ − 1 ǫ L ǫ (f ǫ ) · ∇ v f ǫ = 0,(20)with Φ 0,ǫ (t, x) = R d ×R n Ψ ǫ (t, z, y)σ 1 (x − z)σ 2 (y) dy dz. where Ψ ǫ stands for the unique solution of the free linear wave equation (in R n ) with wave speed 1/ǫ and initial data Ψ 0,ǫ and Ψ 1,ǫ , and 1 ǫ L ǫ (f ǫ )(t, x) = 1 ǫ R d Σ(x − z) t 0 ρ ǫ (t − s, z) × R n sin(|ξ|s/ √ ǫ) |ξ|/ √ ǫ | σ 2 (ξ)| 2 dξ (2π) n ds dz = Σ * x t/ √ ǫ 0 ρ ǫ (t − s √ ǫ, ·) q(s) ds (x)(21) where we have set q(t) = 1 (2π) n R n sin(t|ξ|) |ξ| | σ 2 (ξ)| 2 dξ. (it is nothing but p(t) as introduced in Section 2 evaluated with c = 1; of course when c = 1 and ǫ = 1, the operators 1 ǫ L ǫ in (21) and L in (7) coincide.) Lemma 4.4 Let n ≥ 3. Then q is integrable over [0, +∞[ with ∞ 0 q(t) dt = 1 (2π) n R n | σ 2 (ξ)| 2 |ξ| 2 dξ := κ > 0. Proof. By virtue of the dominated convergence theorem, t → q(t) is continuous on [0, ∞). Bearing in mind that σ 2 is radially symmetric, integrations by parts yield q(t) = |S n−1 | (2π) n ∞ 0 sin(tr)r n−2 | σ 2 (re 1 )| 2 dr = |S n−1 | (2π) n ∞ 0 cos(tr) t d dr r n−2 | σ 2 (re 1 )| 2 dr = − |S n−1 | (2π) n ∞ 0 sin(tr) t 2 d 2 dr 2 r n−2 | σ 2 (re 1 )| 2 dr. Hence, we can estimate as follows |q(t)| ≤ K t 2 with K = |S n−1 | (2π) n ∞ 0 d 2 du 2 r n−2 | σ 2 (re 1 )| 2 dr < ∞ which proves q ∈ L 1 ([0, ∞)). Next, we compute the integral of q. For M > 0 we get: M 0 q(t) dt = 1 (2π) n R n M 0 sin(t|ξ|) |ξ| dt | σ 2 (ξ)| 2 dξ = 1 (2π) n R n 1 − cos(M |ξ|) |ξ| 2 | σ 2 (ξ)| 2 dξ = κ − |S n−1 | (2π) n ∞ 0 cos(M r)r n−3 | σ 2 (re 1 )| 2 dr = κ − |S n−1 | M (2π) n ∞ 0 sin(M r) d dr r n−3 | σ 2 (re 1 )| 2 dr. We conclude by letting M tend to ∞. Note that κ is infinite for n = 2 since |σ 2 (ξ)| 2 |ξ| 2 ∼ ξ→0 σ 2 2 L 1 (R 2 ) 1 |ξ| 2 does not belong to L 1 (B(0, a)) for any a > 0. We turn to the proof of Theorem 4.1. Of course we have sup ǫ>0 f ǫ (t, ·) L 1 (R d ×R d ) = sup ǫ>0 f 0,ǫ L 1 (R d ×R d ) := M 0 < ∞, and the L p norms f ǫ (t, ·) L p (R d ×R d ) = f 0,ǫ L p (R d ×R d ) are also bounded, for any 1 ≤ p ≤ ∞ by virtue of (H9). Furthermore, the energy conservation yields E ǫ (t) = R d ×R d v 2 2 + V + Φ ǫ f ǫ dv dx + ǫ 2 R n ×R d |∂ t Ψ ǫ | 2 dy dx + 1 2 R n ×R d |∇ y Ψ ǫ | 2 dy dx ≤Ē 0 . Let us set E vib 0,ǫ = ǫ 2 R n ×R d |Ψ 1,ǫ | 2 dy dx + 1 2 R n ×R d |∇ y Ψ 0,ǫ | 2 dy dx. As a consequence of (H1) and (H8), E vib 0,ǫ is bounded uniformly with respect to ǫ. Owing to the standard energy conservation for the free linear wave equation, we observe that ∇ y Ψ ǫ L ∞ (0,∞;L 2 (R d ×R n )) ≤ (2E vib 0,ǫ ) 1/2 . Then Sobolev's embedding (mind the condition n ≥ 3) allows us to deduce the following key estimate on Ψ ǫ : Ψ ǫ L ∞ (R + ;L 2 (R d ;L 2n/(n−2) (R n ))) ≤ C E vib 0,ǫ 1/2 ≤ C Ē 0 1/2(22) Applying Hölder inequalities, we are thus led to: |Φ 0,ǫ (t, x)| ≤ C σ 2 L 2n/(n+2) (R n ) σ 1 L 2 (R d ) Ē 0 1/2 ,(23) and similarly |∇ x Φ 0,ǫ (t, x)| ≤ C σ 2 L 2n/(n+2) (R n ) ∇ x σ 1 L 2 (R d ) Ē 0 1/2 .(24) Concerning the asymptotic behavior, we shall use the following claim. It is not a direct consequence of these estimates and it will be justified later on. Lemma 4.5 Let χ ∈ C ∞ c ([0, ∞) × R d × R d ). Then, we have lim ǫ→0 ∞ 0 R d ×R d f ǫ ∇ x Φ 0,ǫ χ(t, x, v) dv dx dt = 0. The cornerstone of the proof of Theorem 4.1 is the estimate of the self-consistent potential. By virtue of (21), for any 1 ≤ p ≤ ∞ we get 1 ǫ L ǫ (f ǫ )(t, ·) L p (R d ) ≤ Σ L p (R d ) ρ ǫ L ∞ ([0,∞),L 1 (R d )) ∞ 0 |q(s)| ds ≤ Σ L p (R d ) M 0 q L 1 ([0,+∞)) , as well as 1 ǫ ∇ x L ǫ (f ǫ )(t, ·) L p (R d ) ≤ ∇ x Σ L p (R d ) M 0 q L 1 ([0,+∞)) . Let χ ∈ C ∞ c (R d × R d ). We have R d ×R d f ǫ (t, x, v)χ(x, v) dv dx ≤ M 0 χ L ∞ (R d ×R d ) and d dt R d ×R d f ǫ (t, x, v)χ(x, v) dv dx ≤ M 0 v · ∇χ − ∇V · ∇ v χ L ∞ (R d ×R d ) + q L 1 ([0,+∞)) ∇ x Σ L ∞ (R d ) M 2 0 + CM 0 σ 2 L 2n/(n+2) (R n ) ∇ x σ 1 L 2 (R d ) Ē 0 1/2 × ∇ v χ L ∞ (R d ×R d ) . Reproducing arguments detailed in the previous Section, we deduce that we can assume, possibly at the price of extracting a subsequence, that lim ǫ→0 R d ×R d f ǫ (t, x, v)χ(x, v) dv dx = R d ×R d f (t, x, v)χ(x, v) dv dx holds for any χ ∈ L p ′ (R d ×R d ) uniformly on [0, T ], 0 < T < ∞, with f ∈ C([0, T ]; L p (R d × R d ) − weak), 1 < p < ∞, 1/p + 1/p ′ = 1. Next, we establish the tightness of f ǫ ǫ>0 with respect to the velocity variable, which will be necessary to show that the macroscopic density ρ ǫ passes to the limit. Since Φ 0,ǫ and 1 ǫ L ǫ (f ǫ ) are uniformly bounded and V ≥ 0, we infer from the energy conservation the estimate R d ×R d |v| 2 2 f ǫ (t, x, v) dv dx ≤Ē 0 + q L 1 ([0,+∞)) Σ L ∞ (R d ) M 2 0 + CM 0 σ 2 L 2n/(n+2) (R n ) σ 1 L 2 (R d ) Ē 0 1/2 . Hence, we can check that ρ ǫ (t, x) = R d f ǫ (t, x, v) dv dx satisfies lim ǫ→0 R d ρ ǫ (t, x)χ(x) dx = R d ρ(t, x)χ(x) dx(25) for any χ ∈ C 0 (R d ), with ρ(t, x) = R d f (t, x, v) dv. As a matter of fact, we note that (H1) and (25) imply lim ǫ→0 ∇ x Σ * ρ ǫ (t, x) = ∇ x Σ * ρ(t, x) for any (t, x) ∈ [0, T ] × R d .(26) Furthermore, we have |D 2 x (Σ * ρ ǫ )(t, x)| ≤ M 0 Σ W 2,∞ (R d ) , and, by using mass conservation and the Cauchy-Schwarz inquality, |∂ t (∇ x Σ * ρ ǫ )(t, x)| = R d D 2 x Σ(x − y) R d vf ǫ (t, y, v) dv dy ≤ Σ W 2,∞ (R d ) R d ×R d f ǫ dv dx 1/2 R d ×R d v 2 f ǫ dv dx 1/2 ≤ Σ W 2,∞ (R d ) √ 2M 0 Ē 0 + q L 1 ([0,+∞)) Σ L ∞ (R d ) M 2 0 +CM 0 σ 2 L 2n/(n+2) (R n ) σ 1 L 2 (R d ) Ē 0 1/2 1/2 . Therefore convergence (26) holds uniformly on any compact set of [0, ∞) × R d . We turn to examine the convergence of 1 ǫ ∇ x L ǫ (f ǫ ) to κ∇ x Σ * ρ. We have 1 ǫ ∇ x L ǫ (f ǫ )(t, x) − κ∇ x Σ * ρ(t, x) = t/ √ ǫ 0 ∇ x Σ * ρ ǫ (t − s √ ǫ, x)q(s) ds − κ∇ x Σ * ρ(t, x) ≤ t/ √ ǫ 0 ∇ x Σ * ρ ǫ (t − s √ ǫ, x) − ∇ x Σ * ρ(t, x) q(s) ds + ∞ t/ √ ǫ q(s) ds ∇ x Σ * ρ L ∞ ((0,∞)×R d ) ≤ t/ √ ǫ 0 |(∇ x Σ * ρ ǫ − ∇ x Σ * ρ)(t − s √ ǫ, x)| |q(s)| ds + t/ √ ǫ 0 |∇ x Σ * ρ(t − s √ ǫ, x) − ∇ x Σ * ρ(t, x)| |q(s)| ds + ∞ t/ √ ǫ |q(s)| ds ∇ x Σ * ρ L ∞ ((0,∞)×R d ) . Let us denote by I ǫ (t, x), II ǫ (t, x), III ǫ (t), the three terms of the right hand side. Firstly, for any t > 0, III ǫ (t) tends to 0 as ǫ → 0, and it is dominated by κ Σ W 1,∞ (R d ) M 0 . Secondly, for any 0 < T < ∞ and any compact set K ⊂ R d , when (t, x) lies in [0, T ]×K, we can estimate |I ǫ (t, x)| ≤ ∇ x Σ * ρ ǫ − ∇ x Σ * ρ L ∞ ([0,T ]×K) q L 1 ([0,∞) which also goes to 0 as ǫ → 0. Eventually, still considering (t, x) ∈ [0, T ] × K, we write |II ǫ (t, x)| ≤ t/ √ ǫ 0 sup z∈K |∇ x Σ * ρ(t − s √ ǫ, z) − ∇ x Σ * ρ(t, z)| |q(s)| ds. By using the Lebesgue theorem, we justify that it tends to 0 as ǫ → 0 since (t, x) → ∇ x Σ * ρ(t, x) is uniformly continuous over any compact set, the integrand is dominated by 2 Σ W 1,∞ (R d ) M 0 |q(s)|, and q ∈ L 1 ([0, ∞)). Therefore, for any 0 < t < T < ∞ and any compact set K ⊂ R d , sup x∈K 1 ǫ ∇ x L ǫ (f ǫ ) − κ∇ x Σ * ρ (t, x) −−→ ǫ→0 0, and this quantity is bounded uniformly with respect to 0 ≤ t ≤ T < ∞ and ǫ > 0. We go back to the weak formulation of (16). Let χ ∈ C ∞ c ([0, ∞) × R d × R d ). We suppose that supp(χ) ⊂ [0, T ] ×B(0, M ) ×B(0, M ). We have − R d ×R d f 0,ǫ χ(0, x, v) dv dx − ∞ 0 R d ×R d f ǫ ∂ t χ dv dx dt − ∞ 0 R d ×R d f ǫ v · ∇ x χ dv dx dt + ∞ 0 R d ×R d f ǫ ∇ v χ · ∇ x (V + Φ 0,ǫ ) dv dx dt = ∞ 0 R d ×R d f ǫ ∇ x 1 ǫ L ǫ (f ǫ ) · ∇ v χ dv dx dt. Obviously, there is no difficulty with the linear terms of the left hand side. For the non linear term we proceed as follows: ∞ 0 R d ×R d f ǫ ∇ x 1 ǫ L ǫ (f ǫ ) · ∇ v χ dv dx dt − ∞ 0 R d ×R d f κ∇ x Σ * ρ · ∇ v χ dv dx dt = ∞ 0 R d ×R d f ǫ ∇ x 1 ǫ L ǫ (f ǫ ) − κ∇ x Σ * ρ · ∇ v χ dv dx dt + ∞ 0 R d ×R d (f ǫ − f ) κ∇ x Σ * ρ · ∇ v χ dv dx dt. The last term directly passes to the limit. The first integral in the right hand side is dominated by M 0 ∇ v χ L ∞ ([0,∞)×R d ×R d ) T 0 sup y∈B(0,M ) ∇ x 1 ǫ L ǫ (f ǫ ) − κ∇ x Σ * ρ (t, y) dt. We conclude by a mere application of the Lebesgue Theorem. If the initial data f 0,ǫ converge strongly to f 0 in L p (R d × R d ), the nature of the convergence of f ǫ to f can be improved by applying general stability results for transport equations, see [13, Proof of Lemma 4.5 As a matter of fact, the variable x ∈ R d just appears as a parameter for the wave equation, and Υ ǫ (t, x, y) = (σ 1 * Ψ ǫ (t, ·, y))(x) solves the linear wave equation ǫ∂ 2 tt Υ ǫ − ∆ y Υ ǫ = 0, with the data Υ ǫ (0, x, y) = σ 1 * Ψ 0,ǫ (x, y), ∂ t Υ ǫ (0, x, y) = σ 1 * Ψ 1,ǫ (x, y). The parameter x being fixed, we appeal to the Strichartz estimate, see [23,Corollary 1.3] or [29,Theorem 4.2, for the case n = 3], 1 ǫ 1/(2p) ∞ 0 R n |Υ ǫ (t, x, y)| q dy p/q dt 1/p ≤ C E vib 1,ǫ (x) where we set E vib 1,ǫ (x) = ǫ R n |σ 1 * Ψ 1,ǫ (x, y)| 2 dy + R n |σ 1 * ∇ y Ψ 0,ǫ (x, y)| 2 dy. (That 1 ǫ 1/(2p) appears in the inequality can be checked by changing variables and observing that Υ ǫ (t √ ǫ, x, y) satisfies the wave equation with speed equals to 1 and data (σ 1 * Ψ 0ǫ , √ ǫσ 1 * Ψ 1,ǫ ).) This inequality holds for admissible exponents: 2 ≤ p ≤ q ≤ ∞, 1 p + n q = n 2 − 1, 2 p + n − 1 q ≤ n − 1 2 , (p, q, n) = (2, ∞, 3). Observe that R d E vib 1,ǫ (x) dx ≤ σ 1 L 1 (R d ) E vib 0,ǫ ≤ σ 1 L 1 (R d )Ē0 . It follows that R d ∞ 0 R n |Υ ǫ (t, x, y)| q dy p/q dt 2/p dx ≤ C 2 σ 1 L 1 (R d )Ē0 ǫ 1/p −−→ ǫ→0 0. A similar reasoning applies to ∇ x Υ ǫ with ∇ x σ 1 replacing σ 1 . Let χ ∈ C ∞ c ([0, ∞) × R d × R d ). We suppose that supp(χ) ⊂ {0 ≤ t ≤ M, |x| ≤ M, |v| ≤ M } for some 0 < M < ∞. We are left with the task of estimating ∞ 0 R d ×R d f ǫ ∇ x Φ 0,ǫ χ(t, x, v) dv dx dt = ∞ 0 R d R ǫ (t, x)∇ x Φ 0,ǫ (t, x) dx dt where we have set R ǫ (t, x) = R d f ǫ χ(t, x, v) dv. With the standard notation 1/p + 1/p ′ = 1, using Hölder's inequality twice, we get ∞ 0 R d ×R d f ǫ ∇ x Φ 0,ǫ χ(t, x, v) dv dx dt ≤ R d ∞ 0 |R ǫ (t, x)| p ′ dt 2/p ′ dx 1/2 R d ∞ 0 |∇ x Φ 0,ǫ (t, x)| p dt 2/p dx 1/2 . We readily obtain R d ∞ 0 |R ǫ (t, x)| p ′ dt 2/p ′ dx 1/2 ≤ M d+d/2+1/p ′ f ǫ χ L ∞ ((0,∞)×R d ×R d ) ≤ M d+d/2+1/p ′ f 0,ǫ L ∞ (R d ×R d ) χ L ∞ ((0,∞)×R d ×R d ) which is thus bounded uniformly with respect to ǫ > 0. Furthermore, with 1/q +1/q ′ = 1, we have R d ∞ 0 |∇ x Φ 0,ǫ (t, x)| p dt 2/p dx = R d ∞ 0 R n σ 2 (y)∇ x Υ ǫ (t, x, y) dy p dt 2/p dx ≤ σ 2 L q ′ (R d ) R d ∞ 0 R n |∇ x Υ ǫ (t, x, y)| q dy p/q dt 2/p dx which tends to 0 like ǫ 1/p . Convergence to the Vlasov-Poisson system The existence theory for the Vlasov-Poisson system dates back to [3]; an overview of the features of both the repulsive or attractive cases can be found in the lecture notes [5]. The following statements are classical tools of this analysis, that will be useful for our purposes as well. Lemma 4.6 (Interpolation estimates) Let f ∈ L 1 ∩ L ∞ (R d × R d ) be such that |v| m f ∈ L 1 (R d × R d ). Then ρ = R d f dv lies in L (m+d)/d (R d ) with ρ L (d+m)/d (R d ) ≤ C(m, d) f m/(d+m)(R d ) L ∞ |v| m f dv dx d/(d+m) . where C(m, d) = 2|B(0, 1)| m/(m+d) . Lemma 4.7 (Hardy-Littlewood-Sobolev inequality) Let 1 < p, r < ∞ and 0 < λ < d. Assume 1/p + 1/r = 2 − λ/d. There exists a constant C > 0 such that for any f ∈ L p (R d ) and g ∈ L r (R d ) we have R d ×R d f (x)g(y) |x − y| λ dy dx ≤ C f L p (R d ) g L r (R d ) . We refer the reader to [5,Lemma 3.4] and [26,Th. 4.3], respectively, for further details. Next, we check the convergence of the approximate kernel defined by σ 1,ǫ . Lemma 4.8 Let d ≥ 3. For any d/(d − 1) < q < ∞, we have: ∇ C d θ ǫ | · | d−1 * C d θ ǫ | · | d−1 (x) + (d − 2) x |S d−1 ||x| d L q (R d ) −−→ ǫ→0 0. Proof. We remind the reader that the convolution by |x| 1−d is associated to the Fourier transform of the operator with symbol 1/|ξ|, see [26,Th. 5.9]. The convolution of radially symmetric functions is radially symmetric too. For d ≥ 3, we compute as follows 1 | · | d−1 * 1 | · | d−1 (x) = R d dy |y| d−1 |x − y| d−1 = R d |x| d dy |x| d−1 |e 1 − y| d−1 |x| d−1 |y| d−1 = 1 |S d−1 | C 2 d |x| d−2 . Next, the energy conservation becomes E ǫ (t) = ǫ 2 R 3 ×R n |∂ t Ψ ǫ (t, x, y)| 2 dy dx + 1 2 R 3 ×R n |∇ y Ψ ǫ (t, x, y)| 2 dy dx + R 3 ×R 3 f ǫ (t, x, v) |v| 2 2 + V (x) + Φ ǫ (t, x) dv dx = E ǫ (0) ≤Ē 0 . Let us study the coupling term: R 3 ×R 3 f ǫ (t, x, v)Φ ǫ (t, x) dv dx = R 3 ρ ǫ (t, x)Φ ǫ (t, x) dx = S ǫ (t) + T ǫ (t) where we have set S ǫ (t) = − 1 ǫ R 3 ρ ǫ L ǫ (f ǫ )(t, x) dx = − R 3 σ 1,ǫ * σ 1,ǫ * t/ √ ǫ 0 q(s)ρ ǫ (t − s √ ǫ, ·) ds (x)ρ ǫ (t, x) dx = − R 3 σ 1,ǫ * t/ √ ǫ 0 q(s)ρ ǫ (t − s √ ǫ, ·) ds (x) σ 1,ǫ * ρ ǫ (t, x) dx and T ǫ (t) = R 3 ρ ǫ Φ 0,ǫ (t, x) dx, Φ 0,ǫ (t, x) = σ 1,ǫ * R n Ψ ǫ (t, ·, y)σ 2 (y) dy (x). Like in the previous Section, Ψ ǫ stands for the solution of the free linear wave equation with wave speed 1/ǫ and initial data Ψ 0,ǫ and Ψ 1,ǫ . Firstly, we establish a bound for |S ǫ (t)| ≤ q L 1 ([0,∞)) σ 1,ǫ * ρ ǫ 2 L ∞ (0,t;L 2 (R 3 ) ) . However, Lemma 4.7 yields σ 1,ǫ * ρ ǫ L 2 (R 3 ) = C 2 d θ ǫ | · | 2 * δ ǫ * ρ ǫ L 2 (R 3 ) ≤ C ρ ǫ L 6/5 (R 3 ) . Let us set E kin ǫ (t) = R 3 ×R 3 |v| 2 f ǫ (t, x, v) dv dx for the particle kinetic energy. Lemma 4.6 leads to ρ ǫ L 5/3 (R 3 ) ≤ C(2, 3) f ǫ 2/5 L ∞ (R 3 ×R 3 ) E kin ǫ 3/5(28) The Hölder inequality allows us to estimate ρ ǫ L 6/5 (R 3 ) ≤ ρ ǫ 7/12 L 1 (R 3 ) ρ ǫ 5/12 L 5/3 (R 3 ) . Combining these inequalities, we arrive at σ 1,ǫ * ρ ǫ L 2 (R 3 ) ≤ C E kin ǫ 1/4 ,(29) for a certain constant C > 0, which does not depend on ǫ. Therefore, we obtain |S ǫ (t)| ≤ C 2 q L 1 ([0,∞)) E kin ǫ 1/2 L ∞ ([0,t]) . Secondly, we estimate the term involving Φ 0,ǫ : T ǫ (t) = R d ×R N (ρ ǫ * σ 1,ǫ )(t, x) Ψ ǫ (t, x, y)σ 2 (y) dy is dominated by σ 1,ǫ * ρ ǫ L ∞ (0,t;L 2 (R 3 )) Ψ ǫ L ∞ (R + ;L 2 (R d ;L 2n/(n−2) (R n ))) σ 2 L 2n/(n+2) (R n ) . Using (22) and (29), we get |T ǫ (t)| ≤ C ′ E kin ǫ (t) 1/4 E vib 0,ǫ 1/2 where the constant C ′ > 0 does not depend on ǫ. It remains to discuss how (H7)-(H8) implies a uniform estimate on the initial state. Note that S ǫ (0) = 0. Hence, by using (H8), we are led to E vib 0,ǫ + 1 2 E kin ǫ (0) ≤ E ǫ (0) + |T ǫ (0)| ≤Ē 0 + C ′ E kin ǫ (0) 1/4 E vib 0,ǫ 1/2 . It allows us to infer sup 0<ǫ<1 E kin ǫ (0) =Ē kin 0 < ∞, sup 0<ǫ<1 E vib 0,ǫ =Ē vib 0 < ∞. Coming back to the energy conservation, with(H7)-(H8) together with the estimates on T ǫ and S ǫ , we deduce that 1 2 E kin ǫ (t) ≤Ē 0 + C 2 q L 1 ([0,∞)) E kin ǫ 1/2 L ∞ ([0,t]) + C ′ E kin ǫ (t) 1/4 Ē vib 0,ǫ 1/2 , holds, which, in turn, establishes the bound sup 0<ǫ<1, t≥0 E kin ǫ (t) =Ē kin < ∞. Going back to the interpolation inequalities, it follows that ρ ǫ is bounded in L ∞ (0, ∞; L 1 ∩ L 5/3 (R 3 )). Step 2. Passing to the limit. The kinetic equation can be rewritten ∂ t f ǫ + v · ∇ x f ǫ − ∇ x V + Φ 0,ǫ − 1 ǫ L ǫ (f ǫ ) · ∇ v f ǫ = 0. We start by establishing that ∇ v f ǫ · ∇ x Φ 0,ǫ = ∇ v · (f ǫ ∇ x Φ 0,ǫ ) converges to 0 at least in the sense of distributions. Lemma 4.9 Let χ ∈ C ∞ c ([0, ∞) × R d × R d ). Then, we have lim ǫ→0 ∞ 0 R d ×R d f ǫ ∇ x Φ 0,ǫ χ(t, x, v) dv dx dt = 0. Proof. It is convenient to split Φ 0,ǫ (t, x) = R n σ 2 (y)C 3 θ ǫ | · | 2 * δ ǫ * Ψ ǫ (t, x, y) dy = Φ main 0,ǫ (t, x) + Φ rem 0,ǫ (t, x) with Φ main 0,ǫ (t, x) = R n σ 2 (y)C 3 1 | · | 2 * δ ǫ * Ψ ǫ (t, x, y) dy, Φ rem 0,ǫ (t, x) = R n σ 2 (y)C 3 θ ǫ − 1 | · | 2 * δ ǫ * Ψ ǫ (t, x, y) dy, and we remind the reader that Ψ ǫ (t, x, y) is the solution of the free wave equation (ǫ∂ 2 tt − ∆ y ) Ψ ǫ = 0 with initial data (Ψ 0,ǫ , Ψ 1,ǫ ). Accordingly, we are going to study the integral ∞ 0 R d ×R d f ǫ ∇ x Φ 0,ǫ χ(t, x, v) dv dx dt = ∞ 0 R d R ǫ (t, x)(∇ x Φ main 0,ǫ + ∇ x Φ rem 0,ǫ )(t, x) dx dt with R ǫ (t, x) = R d f ǫ χ(t, x, v) dv where χ is a given trial function, supported in {0 ≤ t ≤ M, |x| ≤ M, |v| ≤ M } for some 0 < M < ∞. We observe that ∇ x C 3 θ ǫ − 1 | · | 2 * g = ∇ x θ ǫ | · | 2 − 2(θ ǫ − 1) · | · | 4 * g. Thus, by using (27) with d = 3 and p = 2, we are led to |∇ x Φ rem 0,ǫ (t, x)| ≤ Cǫ 3/4 R d δ ǫ * R n σ 2 (y) Ψ ǫ (t, ·, y) dy (x ′ ) 2 dx ′ 1/2 . However, by (22) we have δ ǫ * R n Ψ ǫ σ 2 (y) dy L ∞ ([0,∞);L 2 (R 3 )) ≤ δ ǫ L 1 (R 3 ) σ 2 L 2n/(n+2) (R n ) sup t≥0 R d Ψ ǫ (t, x, ·) 2 L 2n/(n−2) (R n ) dx 1/2 ≤ C σ 2 L (n+2)/2n (R n ) Ē vib 0 1/2 . It implies that ∇ x Φ rem 0,ǫ (t, x) converges uniformly on (0, ∞)×R d to 0. Since R ǫ is clearly bounded in L 1 ((0, ∞) × R d × R d ), we conclude that ∞ 0 R d R ǫ ∇ x Φ rem 0,ǫ dx dt −−→ ǫ→0 0. We need a more refined estimate to deal with the leading term Φ main 0,ǫ . We begin with ∞ 0 R d R ǫ ∇ x Φ main 0,ǫ dx dt ≤ R d ∞ 0 |R ǫ | p ′ dt 2/p ′ dx 1/2 R d ∞ 0 |∇ x Φ main 0,ǫ | p dt 2/p dx 1/2 . We realize that the components of ∇ x Φ main 0,ǫ are given by the solutions Υ j,ǫ of the wave equation is ξ |ξ| , see [26,Th. 5.9], which implies that the convolution operator g → ∇ x C 3 |x| 2 * g, is an isometry from L 2 (R 3 ) to (L 2 (R 3 )) 3 . Furthermore, we have δ ǫ * g L 2 (R 3 ) ≤ δ ǫ L 1 (R 3 ) g L 2 (R 3 ) = g L 2 (R 3 ) . It follows that ∇ y Υ ǫ (0) L 2 (R 3 x ×R n y ) ≤ ∇ y Ψ 0,ǫ L 2 (R 3 x ×R n y ) , ∂ t Υ ǫ (0) L 2 (R 3 x ×R n y ) ≤ Ψ 1,ǫ L 2 (R 3 x ×R n y ) . Strichartz' estimate then leads to (ǫ∂ 2 t − ∆ y )Υ j,ǫ = 0 with data Υ j,ǫ (0, x, y) = ∂ x j C 3 | · | 2 * δ ǫ * Ψ 0,ǫ (x, y), ∂ t Υ j,ǫ (0, x, y) = ∂ x j C 3 | · | 2 * δ ǫ * Ψ 1,ǫ (x,R d ∞ 0 |∇ x Φ main 0,ǫ | p dt 2/p dx 1/2 ≤ Cǫ 1/(2p) E vib 0,ǫ ≤ Cǫ 1/(2p) Ē vib 0 . Since f ǫ is bounded in L ∞ (0, ∞; L p (R d × R d )) for all 1 ≤ p ≤ ∞, and χ is bounded and compactly supported we conclude that ∞ 0 R d R ǫ ∇ x Φ main 0,ǫ dx dt −−→ ǫ→0 0. (Note that the same argument can be applied to show that ∇ x Φ rem 0,ǫ vanishes faster than what has been obtained with the mere energy estimate.) Next, we study the non linear acceleration term. Let us set ρ ǫ (t, x) = δ ǫ * δ ǫ * t/ √ ǫ 0 ρ ǫ (t − s √ ǫ, x) q(s) ds. It is clear, with Lemma 4.4, that ρ ǫ inherits from ρ ǫ the uniform estimate L ∞ (0, ∞; L 1 ∩ L 5/3 (R 3 )). We also denote E(x) = 1 4π 1 |x| , the elementary solution of the operator −∆ x in R 3 . Note that ∇ x E(x) = − x 4π|x| 3 . Bearing in mind Lemma 4.8, the self-consistent field can be split as follows 1 ǫ ∇ x L ǫ (f ǫ )(t, x) = ∇ x C 3 θ ǫ | · | 2 * C 3 θ ǫ | · | 2 − ∇ x E * ρ ǫ (t, x) + ∇ x E * ρ ǫ (t, x).(30) In the right hand side, the L r norm of the first term is dominated by ρ ǫ L ∞ ([0,∞;L 1 (R 3 )) ... L r (R 3 ) , hence, owing to Lemma Lemma 4.8 it tends to 0 as ǫ → 0 in L ∞ (0, ∞; L r (R 3 )) for any 3/2 < r < ∞. Next, Lemma 4.7 tells us that ∇ x E * ρ ǫ is bounded in L ∞ (0, ∞; L 15/4 (R 3 )). Therefore, adapting the reasoning made in the previous sections, we deduce that we can extract a subsequence, such that, for any trial function χ ∈ L p ′ (R 3 ×R 3 ), 1/p ′ +1/p = 1, 1 < p < ∞, lim ǫ→0 R 3 ×R 3 f ǫ (t, x, v)χ(x, v) dv dx = R 3 ×R 3 f (t, x, v)χ(x, v) dv dx holds uniformly on [0, T ], for any 0 ≤ T < ∞. Since the uniform estimate on the kinetic energy imply the tightness of f ǫ with respect to the velocity variable, we also have lim ǫ→0 R 3 ρ ǫ (t, x, v)ζ(x) dx = R 3 ρ(t, x)ζ(x) dx, ρ(t, x) = R 3 f (t, x, v) dv, uniformly on [0, T ], for any 0 ≤ T < ∞ and any ζ ∈ L q (R 3 ), q ≥ 5/2 or ζ ∈ C 0 (R 3 ). Clearly, for any ζ ∈ C ∞ c (R 3 ), δ ǫ * δ ǫ * ζ converges to ζ in L q (R 3 ), 5/2 ≤ q < ∞, and in C 0 (R 3 ). Therefore R 3 (δ ǫ * δ ǫ * ρ ǫ )(t, x)ζ(x) dx = R 3 ρ ǫ (t, x) (δ ǫ * δ ǫ * ζ)(x) dx −−→ ǫ→0 κ R 3 ρ(t, x) ζ(x) dx uniformly in [0, T ]. Then, we look at the difference R 3 ρ ǫ (t, x)ζ(x) dx − κ R 3 ρ(t, x)ζ(x) dx ≤ t/ √ ǫ 0 R 3 (δ ǫ * δ ǫ * ρ ǫ )(t − √ ǫs, x)ζ(x) dx − R 3 ρ(t − √ ǫs, x)ζ(x) dx |q(s)| ds + t/ √ ǫ 0 R 3 ρ(t − √ ǫs, x)ζ(x) dx − R 3 ρ(t, x)ζ(x) dx |q(s)| ds + ∞ t/ √ ǫ |q(s)| ds R 3 ρ(t, x)ζ(x) dx . Let us denote by I ǫ (t), II ǫ (t) and III ǫ (t) the three integrals in the right hand side. By using Lemma 4.4 and the available estimates, we obtain, for any 0 ≤ t ≤ T < ∞ |I ǫ (t)| ≤ q L 1 ([0,∞)) sup 0≤u≤T R 3 δ ǫ * δ ǫ * ρ ǫ − ρ (u, x)ζ(x) dx −−→ ǫ→0 0, while a direct application of the Lebesgue theorem shows that, for any 0 < t ≤ T < ∞ Therefore, for any ζ ∈ L q (R 3 ), 5/2 ≤ q < ∞ and any ζ ∈ C 0 (R 3 ), lim ǫ→0 R 3 ρ ǫ (t, x)ζ(x) dx = κ R 3 ρ(t, x)ζ(x) dx holds for a. e. t ∈ (0, T ), with the domination R 3 ρ ǫ (t, x)ζ(x) dx ≤ ζ L p ′ (R 3 ) sup ǫ>0, 0≤t≤T ρ ǫ (t, ·) L p (R 3 ) , for any 1 ≤ p ≤ 5/3. In oder to justify that the limit f is a solution of the Vlasov-Poisson equation, the only difficulty relies on the treatment of the non linear acceleration term: NL ǫ (χ) = ∞ 0 R 3 ×R 3 f ǫ ∇ x 1 ǫ L ǫ (f ǫ ) · ∇ v χ dv dx dt where χ is a trial function in χ ∈ C ∞ c ([0, ∞) × R d × R d ). Bearing in mind (30), it is convenient to rewrite NL ǫ (χ) = ∞ 0 R 3 R 3 f ǫ ∇ v χ dv · ∇ x E * ρ ǫ dx dt + R ǫ , lim ǫ→0 R ǫ = 0. Lemma 4.7 implies that ∇ x E * ρ ǫ is bounded in L ∞ (0, T ; L 15/4 (R 3 )). For µ > 0, we introduce the cut-off function θ µ (x) = θ(x/µ). Then we split ∇ x E * ρ ǫ (t, x) = R 3 θ µ (x−y) x − y 4π|x − y| 3 ρ ǫ (t, y) dy+ R 3 1− θ µ (x−y) x − y 4π|x − y| 3 ρ ǫ (t, y) dy. The first term in the right hand side can be made arbitrarily small in L p norm, 1 ≤ p ≤ 5/3, uniformly with respect to ǫ, since it can be dominated by |x−y|≤2µ x − y 4π|x − y| 3 ρ ǫ (t, y) dy L p (R 3 ) ≤ ρ ǫ (t, ·) L p (R 3 ) |x−y|≤2µ dy 4π|x − y| 2 ≤ C µ. In the second term, for fixed x ∈ R 3 and µ, y → 1 − θ µ (x − y) x−y 4π|x−y| 3 | 1 |x−y|≥µ is a continuous function which vanishes as |y| → ∞, so that, for any t > 0, lim ǫ→0 R 3 1 − θ µ (x − y) x − y 4π|x − y| 3 ρ ǫ (t, y) dy = R 3 1 − θ µ (x − y) x − y 4π|x − y| 3 ρ(t, y) dy. By standard arguments of integration theory (see for instance [21,Th. 7.61]), we deduce that (a suitable subsequence of) ∇ x E * ρ ǫ converges to ∇ x E * ρ a. e. and strongly in L p loc ((0, T ) × R 3 ), for any 1 ≤ p < 15/4. On the other hand, R 3 f ǫ ∇ v χ dv is compactly supported and converges to R 3 f ǫ ∇ v χ dv weakly in any L q ((0, T ) × R 3 ). (In fact this convergence, as well as ρ ǫ → ρ can be shown to hold strongly, by applying average lemma techniques, see [14,Th. 5].) We conclude that lim ǫ→0 NL ǫ (χ) = ∞ 0 R 3 R 3 f ∇ v χ dv · ∇ x E * ρ dx dt. It ends the proof of Theorem 4.2. Lemma 3. 1 ( 1Estimates on the characteristic curves) Let V satisfy (H2) and let Theorem 4. 2 2Let d = 3 and n ≥ 3. Assume (H1) and (H7)-(H9) Remark 4. 3 3In Theorem 4.1, if, furthermore, we assume that f 0,ǫ ǫ>0 converge (in the appropriate weak sense) to f 0 , by uniqueness of the solution of the limit equation, the entire sequence f ǫ ǫ>0 converges to f . For Theorem 4.1 and Theorem 4.2, if the initial data converges strongly to f 0 in Th. II.4 & Th. II.5], or [7, Th. VI.1.9]. y), and the space variable x ∈ R 3 has only the role of a parameter. It satisfies the following Strichartz estimate |Υ ǫ (t, x, y)| q dy |∂ t Υ ǫ (0, x, y)| 2 dy + |∇ y Υ ǫ (0, x, y)| 2 dy (for admissible exponents as detailed above). The Fourier transform of x → ∇ x1 ǫ 1/(2p) ∞ 0 R n p/q dt 1/p ≤ C E vib 1,ǫ (x) where E vib 1,ǫ (x) = ǫ R n R n C 3 |x| 2 Differentiating yieldsHence, we can writeLet p > 1. On the one hand, we haveOn the other hand, we getAccordingly, the following estimate holds:where C > 0 depends on p and d only. Finally we remark that 0 ≤ θǫ(x)+1 |x| d−1 ≤ 2 |x| d−1 . 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[ "Configurational entropy and the N * (1440) Roper resonance in QCD", "Configurational entropy and the N * (1440) Roper resonance in QCD" ]
[ "G Karapetyan Id \nCenter of Mathematics\nFederal University of ABC\nSanto André09580-210Brazil\n\nPerimeter Institute for Theoretical Physics\nN2L 2Y5WaterlooOntarioCanada\n" ]
[ "Center of Mathematics\nFederal University of ABC\nSanto André09580-210Brazil", "Perimeter Institute for Theoretical Physics\nN2L 2Y5WaterlooOntarioCanada" ]
[]
The electroexcitation of the N * (1440) Roper resonance, which defines the first radially excited state of the nucleon, is examined within the soft-wall AdS/QCD model. Such excited Fock states are characterized by the leading three-quark component, which determines the main properties of Roper resonance. The differential configurational entropy (DCE) was used in the context of minimal and nonminimal couplings in the nuclear interaction with a gauge vector field for N * (1440) transition. Comparing the main results with the recent data of the CLAS Collaboration at JLab shows a good agreement on the accuracy of the computed data. I. INTRODUCTION Nuclear resonances are an intriguing aspect of the anti-de Sitter (AdS)/quantum chromodynamics correspondence (QCD). Investigation of such phenomena, in particular, the Roper resonance, can shed new light on the mechanism ruling interactions among hadrons, the structure of the nucleons, as well as the fundamental features of the electromagnetic transitions between the nucleon and its resonances. Within the soft-wall AdS/QCD model, one can determine the form factors and helicity amplitudes with high precision at a large range of Q 2 , using the correct power scaling description. The investigation of the nucleon resonances in the AdS/QCD soft wall can model the nucleon-Roper transition, where the Dirac form factor can be successfully determined by the holographic light-front QCD [1-3]. Recently, experiments of the CLAS Collaboration at JLab [4] have shown the most prominent results on the electroexcitation of the low mass resonances. Such resultswere in a comprehensive analysis of data on differential cross sections, longitudinally polarized beam asymmetries, and longitudinal target and beam-target asymmetries for pion electroproduction off the proton. The data obtained can be analyzed using different conceptual approaches. Among both nonrelativistic and relativistic approaches to studying the electro-excitations of nucleon resonances, one can distinguish the potential and hadronic molecular approaches, Dyson-Schwinger equation * Electronic address:
null
[ "https://export.arxiv.org/pdf/2305.05413v1.pdf" ]
258,564,810
2305.05413
9f447fc0f4fdb18b16dc79a645bc3ab40720876b
Configurational entropy and the N * (1440) Roper resonance in QCD G Karapetyan Id Center of Mathematics Federal University of ABC Santo André09580-210Brazil Perimeter Institute for Theoretical Physics N2L 2Y5WaterlooOntarioCanada Configurational entropy and the N * (1440) Roper resonance in QCD The electroexcitation of the N * (1440) Roper resonance, which defines the first radially excited state of the nucleon, is examined within the soft-wall AdS/QCD model. Such excited Fock states are characterized by the leading three-quark component, which determines the main properties of Roper resonance. The differential configurational entropy (DCE) was used in the context of minimal and nonminimal couplings in the nuclear interaction with a gauge vector field for N * (1440) transition. Comparing the main results with the recent data of the CLAS Collaboration at JLab shows a good agreement on the accuracy of the computed data. I. INTRODUCTION Nuclear resonances are an intriguing aspect of the anti-de Sitter (AdS)/quantum chromodynamics correspondence (QCD). Investigation of such phenomena, in particular, the Roper resonance, can shed new light on the mechanism ruling interactions among hadrons, the structure of the nucleons, as well as the fundamental features of the electromagnetic transitions between the nucleon and its resonances. Within the soft-wall AdS/QCD model, one can determine the form factors and helicity amplitudes with high precision at a large range of Q 2 , using the correct power scaling description. The investigation of the nucleon resonances in the AdS/QCD soft wall can model the nucleon-Roper transition, where the Dirac form factor can be successfully determined by the holographic light-front QCD [1-3]. Recently, experiments of the CLAS Collaboration at JLab [4] have shown the most prominent results on the electroexcitation of the low mass resonances. Such resultswere in a comprehensive analysis of data on differential cross sections, longitudinally polarized beam asymmetries, and longitudinal target and beam-target asymmetries for pion electroproduction off the proton. The data obtained can be analyzed using different conceptual approaches. Among both nonrelativistic and relativistic approaches to studying the electro-excitations of nucleon resonances, one can distinguish the potential and hadronic molecular approaches, Dyson-Schwinger equation * Electronic address: framework, or the light-front holographic quantum QCD, [5][6][7][8][9][10][11][12][13][14]. As it has been marked in the above-mentioned studies, the Roper electroproduction requires additional degrees of freedom to be correctly interpreted for a nucleon-scalar σ meson molecular component. Also, in the framework of the light-front holographic QCD, the Dirac form factor for the electromagnetic nucleon-Roper transition was calculated. The dynamically generated resonance N * (1440) suggests the estimation of the electromagnetic helicity form factors, besides valence quark contribution [15]. The electroexcitation of resonances can be studied via the unitary isobar model MAID, which examines the photoproduction and electroproduction, as well as the nucleon resonances [16]. The Roper electroproduction, which comprises in addition of the leading three-quark (3q) state the higher Fock components either, has been investigated within a soft-wall AdS/QCD model [17][18][19][20]. The AdS/QCD model for baryon structure [19] has been successfully used to investigate the nucleon electromagnetic form factors in the Euclidean region of transverse momentum squared, p 2 T , up to 30 GeV 2 . In such a study the higher Fock states, which have been incorporated beyond the 3q state allowed to describe properly the quantitative reproduction of baryons, i. e., their masses, radii, and the electromagnetic form factors at a small value of Q 2 . The Fock states have been limited by the twist dimension τ = 5 and the higher states have been neglected. Such a choice is supported by the fact that at higher values of Q 2 , the estimated contribution of higher Fock states to the hadronic form factors is as (1/Q 2 ) τ−1 . In such cases, the number of adjusting parameters also decreases. The mechanism of the N * (1440) resonance with the determination of the helicity amplitudes at low-Q 2 , takes into account the minimal coupling of the nucleon (with the orbital momentum L = 0) and the N * (1440) (with the orbital momentum L = 1) Fock components with the same twist dimension, and leading twists for the nucleon and the N * (1440) are τ = 3 and τ = 4, respectively. The leading twists of the nucleon and the resonance are linked with the photon through the nonminimal coupling as the minimal one is forbidden by the law of the conservation of a current. To address the N * (1440) transition, the differential configurational entropy (DCE) will be used as a very important criterion to examine some of the most essential phenomenological features of AdS/QCD. The DCE has been successfully applied to study phenomena in QCD [21][22][23][24][25][26][27][28][29][30][31][32][33]. The DCE has been also used for studying different nuclear reactions, which involve high-energy hadrons under extremal circumstances [34][35][36][37][38][39][40]. The latest studies in the such area have shown the useful prospect of DCE techniques to interpret QCD phenomenology [41][42][43][44][45][46], also involving glueballs [47][48][49], charmonium and bottomonium [50], the quark-gluon plasma [51], baryons [52], besides other aspects of information entropy in AdS/QCD [53]. Detailed information about the DCE can be found in Refs. [54][55][56][57][58]. Another set of studies is devoted to DCE phenomena in AdS [59][60][61][62]. Among the latest studies, which incorporate the DCE in the Color Glass Condensate (CGC) settings, one can mention the calculations of the inelastic hadron cross-section in deep hadron-hadron scattering [63,64]. As the hadron cross-section represents the localized function of the probability of nuclear reaction in any spatial configuration of the excited system, we can calculate the critical points of the DCE using the Fourier transform of the cross-section and associated modal fraction [64,65]. This paper is devoted to the investigation of the electromagnetic transitions between the nucleon and its resonances within the soft-wall AdS/QCD. We intend to study the Roper resonance in the negative-parity state N * (1440). We use the formalism suggested in Ref. [66], which uses the adjustable quantum numbers as well as the additional nonminimal terms for the gauge invariance for the form factors and helicity amplitudes momentum dependence. Such an approach allows determining the longitudinal S 1/2 amplitude in the low Q 2 regime. This work is organized as follows: Section II shortly represents the formalism of nucleon resonances in soft-wall AdS/QCD, with special attention to the Roper resonance N * (1440). In Section III, we compute the DCE underlying the electroproduction of the Roper resonance and discuss the main results and analysis. In Section IV we present the summary and outlook of our research. II. FERMION FIELD IN ADS SPACE In this paper, we use the AdS/QCD soft-wall holographic model, which describes the main properties of the electromagnetic structure of the nucleon, such as the analytical power scaling of the elastic nucleon form factors at large momentum transfers [19]. The model also reproduces satisfactorily the experimental data for magnetic moments and electromagnetic radii. In AdS space, one can specify the five-dimensional AdS metric as ds 2 = g M N dx M dx N = η ab exp(2A(z)) dx a dx b = exp(2A(z)) (η µν dx µ dx ν − dz 2 ) ,(1) with η µν = diag(1, −1, −1, −1, −1). In Eq. (1) the indexes a = (µ, z) and b = (ν, z) specify local Lorentz indexes, and g M N and η ab are curved and flat metric tensors, respectively, which are related by the vielbein ε a M (z) = exp(A(z)) δ a M as g M N = ε a M ε b N η ab , with g = |detg M N | = exp(10A(z)), R is the AdS radius, z is the holographic coordinate, and M, N = 0, 1, . . . , 4 are the base manifold space-time indices. In AdS space the calculation approach is restricted by a conformal-invariant metric with warp factor A(z) = log(R/z), and the corresponding AdS/QCD action for the fermion field of twist τ is given by an expression [19]: S τ = d 4 xdz √ g exp(−φ(z)) i=+,−ψ i,τ (x, z) D i (z)ψ i,τ (x, z) ,(2) with covariant derivative D ± (z) = i 2 γ M ↔ ∂ M ∓ (µ + U F (z)) ,(3) where the parameter µ represents the bulk fermion mass associated to the scaling dimension τ (m = µR = τ − 3/2), ψ ±,τ (x, z) is the pair of bulk fermion fields, and in the 4D theory, they are related to the holographic analogs of the left-and right-chirality operators. The dilaton field is assumed in its standard quadratic form, φ(z) = κ 2 z 2 , with κ being a free scale parameter, and γ M = ε M a γ a and γ a = (γ µ , −iγ 5 ) are the five-dimensional Dirac matrices in the chiral representation for the γ µ and γ 5 matrices. The fields ψ τ determines the AdS fermion field for the different scaling dimension: τ = 3, 4, 5, . . .. In the holographic QCD, the scaling dimension of the baryon interpolating operator in the light-front wave, τ = N + L, can be suggested through the scaling dimension of the AdS fermion field, where L = max |L z | is the maximal value of the z-component of the quark orbital angular momentum, and N is the number of partons in the baryon [1]. The effective potential U F (z) = φ(z)/R must satisfy the solutions of the motion equations (EOMs) for the fermionic Kaluza-Klein (KK) modes of left-and right-chirality, and correctly describe the asymptotic behavior of the nucleon electromagnetic form factors at large Q 2 . The P -parity transformation condition implies the fermion masses m and the effective potentials U F (z) for ψ + and ψ − fields must have the opposite signs [19], and the absolute sign of the fermion mass depends on the chirality of the boundary operator [67]. Within the present approach one can choose for the QCD operators O R and O L positive and negative chirality. Thus, the correspondent absolute ± signs have the mass terms of the bulk fields ψ ± . Within the soft-wall AdS/QCD model, one can calculate some of the parameters that regulate wave functions of the interacting hadrons. The first step is to rescale the fermionic fields, using the dilaton, as the following [66] ψ i,τ (x, z) = exp(φ(z)/2)ψ i,τ (x, z).(4) as well as to remove the dilaton field from the overall exponential. It is possible to represent the modified action in the Lorentzian signature via the ψ τ (x, z) field as: S τ = d 4 xdz exp(4A(z)) i=±ψ i,τ (x, z) i / ∂ +γ 5 ∂ z +2A (z)γ 5 −δ i exp(A(z)) R m+φ(z) ψ i,τ (x, z) (5) with ∂ = γ µ ∂ µ , δ ± = ±1 , and the fermion field, ψ i,τ (x, z), must satisfy the equations of motion in the form [19] i ∂ + γ 5 ∂ z + 2A (z)γ 5 ∓ exp(A(z)) R m + φ(z) ψ ±,τ (x, z) = 0 .(6) In the soft-wall AdS/QCD model, the fermion field can be split into the left-(L) and right-chirality (R) components [66]: ψ i,τ (x, z) = ψ L i,τ (x, z) + ψ R i,τ (x, z) ,(7)ψ L/R i,τ (x, z) = 1 ∓ γ 5 2 ψ i,τ (x, z) ,(8) where F L/R i,τ,n (z) are the profile functions, and ψ L/R i,τ (x, z) fields can be represented through the four-dimensional boundary fields in the fermionic KK modes as: ψ L/R i,τ (x, z) = 1 √ 2 n ψ L/R n (x) F L/R i,τ,n (z) .(9) The left-and right-chirality components also compose the Dirac (bi)spinors ψ n (x) = ψ L n (x)+ψ R n (x). The profile functions are linked owing to the four-dimensional P -and C-parity invariance as the following [19], F R ±,τ,n (z) = ±F L ∓,τ,n (z) .(10) It is more convenient to use the alternative definition F R τ,n (z) ≡ F R +,τ,n (z) = F L −,τ,n (z) , F L τ,n (z) ≡ F L +,τ,n (z) = −F R −,τ,n (z) .(11) The AdS/QCD model profile functions F L/R τ,n (z) are the holographic analogs of the nucleon wave function and are defined by the quantum number n and twist dimension τ, which can be considered as the specific partonic content of the nucleon Fock component. The profile functions satisfy the two coupled one-dimensional motion equations [19] in the form: ∂ z ± exp(A) R m + φ(z) + 2A F L/R n,τ (z) = ±M nτ F R/L n,τ (z) .(12) In order to find the solutions for the AdS field profiles in the z direction, one obtains the decoupled EOMs as: ∂ 2 z +4A ∂ z − exp(2A) R 2 (m + φ) 2 ∓ exp(A) R A (m+φ)−φ +4A 2 +2A + M 2 nτ F L/R τ,n (z) = 0 . (13) If we substitute F L/R τ,n (z) = exp(−2A(z)) f L/R τ,n (z), then the Schrödinger-type equations of motion for f L/R τ,n (z) can be expressed as: −∂ 2 z + exp(2A) R 2 (m + φ) 2 ∓ exp(A) R A (m + φ) + φ f L/R τ,n (z) = M 2 nτ f L/R τ,n (z) .(14) Then, substituting A(z) = log(R/z) and φ(z) = κ 2 z 2 , then Eq. (14) yields −∂ 2 z + κ 4 z 2 + 2κ 2 m ∓ 1 2 + m(m ± 1) z 2 f L/R τ,n (z) = M 2 nτ f L/R τ,n (z),(15) with the corresponding notations: f L τ,n (z) = 2γ(n + 1) γ(n + τ) κ τ z τ−1/2 exp(−κ 2 z 2 /2)L τ−1 n (κ 2 z 2 ) , f R τ,n (z) = 2γ(n + 1) γ(n + τ − 1) κ τ−1 z τ−3/2 exp(−κ 2 z 2 /2)L τ−2 n (κ 2 z 2 ),(16) where L τ n (x) are the generalized Laguerre polynomials and the mass spectrum M nτ satisfies Reggelike trajectories given by M 2 nτ = 4κ 2 n + τ − 1 ,(17) for m = τ − 3/2. When we consider the profile function as the nucleon wave function of the nucleon F L/R τ,n (z) = exp(−2A(z)) f L/R τ,n (z) , then for small values of z, a correct scaling relation can be observed for both profile functions for the twist τ: F L τ,n (z) ∼ z τ+3/2 , F R τ,n (z) ∼ z τ+1/2 ,(18) and such a process vanishes during confinement at large values of z. Within the soft-wall AdS/QCD model, bulk fields can play the role of the holographic analog of the nucleon resonance, ψ N ±,τ (x, z), for the nucleon in the ground state with the quantum number n = 0, and holographic analog of the Roper resonance, ψ R ±,τ (x, z), in the first radially excited state with n = 1 [66]. They can be determined in the following form: ψ N ±,τ (x, z) = 1 √ 2 ψ L 0 (x) F L/R τ,0 (z) ± ψ R 0 (x) F R/L τ,0 (z) ,(19)ψ R ±,τ (x, z) = 1 √ 2 ψ L 1 (x) F L/R τ,1 (z) ± ψ R 1 (x) F R/L τ,1 (z) .(20) We consider AdS bulk fields as the products of boundary fermionic fields with spin 1/2, whose profiles depend on the holographic variable. Once the nucleon and Roper resonances are defined, one can represent the free actions at the fixed twist dimension, τ, as: S B τ = d 4 xdz exp(4A(z))L B ,(21) where L B τ = i=±ψ B i,τ (x, z) i ∂ + γ 5 ∂ z + 2A (z)γ 5 − δ i exp(A(z)) R m + φ(z) ψ B i,τ (x, z) ,(22) with the notation for nucleon and Roper resonances B = N , R, respectively. Summing the bulk action, S B τ , over τ values with adjustable coefficients c B τ allows us to consider the higher Fock states for the nucleon and Roper resonances in the form: S B = τ c B τ S B τ ,(23) where c B τ (B = N , R) are adjustable parameters. It is equivalent to expressing the effective Lagrangian L B = τ c B τ L B τ ,(24) weighted by the coefficients c B τ regulating the nucleon and Roper resonances. Eq. (24) will play a prominent role on determining the DCE underlying the Roper resonance N * (1440) in the next section, pointing to the values of c B τ which correspond to higher configurational stability. According to the electromagnetic gauge invariance, one uses the normalization of the kinetic termψ n (x)i ∂ψ n (x) of the boundary fermionic field as τ c B τ = 1. The masses for the nucleon and Roper can be expressed as [19]: M N = 2κ τ c N τ √ τ − 1 , M R = 2κ τ c R τ √ τ .(25) The four-dimensional actions for the fermion field ψ n (x) = ψ L n (x) + ψ R n (x) in n = 0 nucleon state and n = 1 state for Roper resonance can be obtained by the integration procedure over the holographic coordinate z, yielding S B 4D = d 4 x L B 4D ,(26) where L B 4D =ψ 0 (x) i ∂ − M N ψ 0 (x) +ψ 1 (x) i ∂ − M R ψ 1 (x) .(27) One can see that the expression for the effective actions in 4D is obtained using the bulk fields c N 5 = 1 − c N 3 − c N 4 = −0.41 .(29) In order to reproduce the Roper mass M exp R = 1440 MeV, we use the set of parameters [66]: c R 3 = 0.78 , c R 4 = −0.16 , c R 5 = 1 − c R 3 − c R 4 = 0.38 .(30) One can conclude that the value of mass for both resonances is determined by the contribution of the 3q Fock component. III. DIFFERENTIAL CONFIGURATIONAL ENTROPY AND THE ELECTRO-EXCITATION OF THE ROPER RESONANCE The main procedure to obtain the DCE assumes a probability distribution in the form of the localized energy density operator, which represents the temporal T 00 ( r) component of the energymomentum tensor, for r = (x 1 , . . . , x k ) ∈ R k . The Fourier transform of the energy density operator can be obtained as [57] T 00 ( q) = 1 (2π) k/2 R k T 00 ( r)e −i q· r d k x,(31) where T 00 is the q is the spatial part of the 4-momentum q. One can define the modal fraction as [54] T 00 ( q) = |T 00 ( q)| 2 R k |T 00 (q)| 2 d k q .(32) The modal fraction (32) reflects the relative weight transferred by each momentum wave mode q. The DCE allows us to compute the energy density, which involves the complete information about the nuclear system described by the weight of each wave mode [57]: DCE T 00 = − R k T ⊗ 00 (q) ln (T ⊗ 00 (q)) d k q,(33) with T ⊗ 00 ( q) = T 00 ( q) T max 00 ( q) ,(34) and T max 00 ( q) designating the maximum value of the energy density in the momentum space R k . Here we use the nat (natural unit of information) as the DCE unit since it defines the amount of information of encoding in the probability distribution function with the equally likely outcomes in the interval [0, e]. At the given p = 1 in (31) -(33), one can calculate the DCE for the closed nuclear system over all the holographic coordinate z and due to the Kaluza-Klein splitting. Using the Lagrangian density (24), the energy density operator can be obtained as the temporal component of the energymomentum tensor: T 00 = 2 √ −g   ∂( √ −gL) ∂g 00 − ∂ ∂x ∂( √ −gL) ∂ ∂g 00 ∂x   .(35) This procedure allows us to obtain the Fourier transform of the energy density operator, the modal fraction, and the DCE, respectively using Eqs. (31) - (33). The Lagrangian densities can be thought os as being functions of the parameters c N 3 , c N 4 and c R 3 , c R 4 in (28,30). Their phenomenological values in (28,30) were derived via fitting procedure to the helicity amplitudes of the N * (1440) resonance [19]. Instead of using the values of c N 3 , c N 4 and c R 3 , c R 4 in (28,30), these values will be derived out of the global minima of the DCE, which indicates the maximum configurational stability of the nuclear system and the dominant state occupied by the nuclear physical system. The minimal value of the DCE means that the nuclear system has a definite, well-localized energy density with small uncertainty. In order to calculate the main properties of the N * (1440) for the nucleon/Roper resonances, we fix two parameters, κ = 383 MeV and the Roper resonance mass, M exp R = 1440 MeV. Therefore, we can determine two free adjustable parameters, c N 3 , and c N 4 ; c R 3 and c R 4 , respectively in Eqs. (28,30), considering the other two complementary parameters fixed. Such a procedure allows us to obtain the global minimum of the DCE (S c ), hence, determining the most dominant state occupied by the nuclear quantum system. We represent the results of the calculations of the DCE using at c N 3 gmin = 1.264, c N 4 gmin = 0.169, (37) match the data in Eq. (28) from the AdS/QCD soft-wall model in Refs. [19,66] with an accuracy of 1.1% and 5.6%, respectively. In order to run over all possible values of adjustable parameters, corresponding to the experimental nucleon masses and the Roper mass, the range of calculation for c N 3 , c N 4 has been chosen wide enough. Now, the calculations of the DCE using Eqs. at the point of the parameter space c R 3 gmin = 0.803, c R 4 gmin = −0.171,(39) matching the data from the AdS/QCD soft-wall model in Ref. [19,66], given by Eq. (30), with an accuracy of 2.9% and 6.8%, respectively. The fitted parameters (c R 3 , c R 4 ) completely define the parameter space, in which the Roper resonance acquires the configurational stability of a higher level at the global minimum, which is also the most dominant state among all possible quantum states. Our data, corresponding to the global minimum for the nucleon resonance DCE min (c N 3 , c N 4 ) = 767.503 nat, at the point c N 3 gmin = 1.264 and c N 4 gmin = 0.169, are in good agreement with the ones from Ref. [66] within 1.1% and 5.6%, respectively, and for the Roper resonance the value of DCE min (c R 3 , c R 4 ) = 17.639 nat, at the point c R 3 gmin = 0.803 and c R 4 gmin = −0.171, are in good agreement with the corresponding data from Ref. [66] within 0.57% and 2.43%, respectively. The obtained parameters imply that the calculations via DCE show the natural choice of the main properties of the resonance as the optimal tool for describing the localized nuclear system, which points to the most stable configuration of the system. It should be stressed out also that in such a state the nuclear system can be represented as a set of data at a high compression rate of information through the wave modes describing the spatial complexity of the system. Our calculation shows that behind the range in the plot in Figs. 1 and 2, there are six another local minima (lmin) for nucleon resonance. The first one has DCE given by DCE c N 3 lmin1 , c N 4 lmin1 = 662.012 nat,(40) at the point of the parameter space given by c N 3 lmin1 = 0.125, c N 4 lmin1 = 0.476.(41) The second local minimum is given by DCE c N 3 lmin2 , c N 4 lmin2 = 878.379 nat,(42) at c N 3 lmin2 = 2.896, c N 4 lmin2 = −3.429.(43) The third local minimum, DCE c N 3 lmin3 , c N 4 lmin3 = 890.056 nat,(44) at c N 3 lmin3 = 0.011, c N 4 lmin3 = −3.608(45) Finally, the sixth local minimum of the DCE for the nucleon has DCE DCE c N 3 lmin6 , c N 4 lmin6 = 945.093 nat,(50) at the point of the parameter space, c N 3 lmin6 = 0.705, c N 4 lmin6 = 2.247.(51) For the Roper resonance, there are two local minima of the DCE. The first one, DCE c R 3 lmin1 , c R 4 lmin1 = 18.248 nat,(52) at c R 3 lmin1 = 1.938, c R 4 lmin1 = −1.762;(53) is more stable than the second local minimum at c R 3 lmin2 = 0.192, c R 4 lmin2 = −2.987.(54) with DCE given by DCE c R 3 lmin2 , c R 4 lmin2 = 21.750 nat,(55) One can see also the maximum in Fig. 1 for the nucleon resonance with the value of DCE c N 3 gmax , c N 4 gmax = 1208.634 nat,(56) at the global maximum c N 3 gmax = 1.324, c N 4 gmax = −4.187,(57) and five local maxima. The first local maximum, at c N 3 lmax1 = 0.273, c N 4 lmax1 = −1.934;(58) in the (c N 3 , c N 4 )-parameter space has DCE given by DCE c N 3 lmax1 , c N 4 lmax1 = 1012.744 nat.(59) It is slightly less stable than the second local maximum, c N 3 lmax2 = 0.811, c N 4 lmax2 = −0.847,(60) whose associated DCE is given by DCE c N 3 lmax2 , c N 4 lmax2 = 1006.134 nat.(61) Besides, the third local maximum c N 3 lmax3 = 0.825, c N 4 lmax3 = −2.950;(62) is marginally less stable than the first and second local maxima (58,60), since its DCE, vaguely lower than the respective (59, 61), reads DCE c N 3 lmax3 , c N 4 lmax3 = 1005.634 nat.(63) Lastly, there is the DCE c N 3 lmax4 , c N 4 lmax4 = 1008.927 nat,(64) at the fourth local maximum c N 3 lmax4 = 2.765, c N 4 lmax4 = −0.852(65) and DCE c N 3 lmax5 , c N 4 lmax5 = 1007.109 nat, at the fifth local maximum, c N 3 lmax5 = 2.721, c N 4 lmax5 = −2.942.(67) There is one local maximum for the Roper resonance DCE c R 3 lmax1 , c R 4 lmax1 = 30.841 nat,(68) at c R 3 lmax1 = 0.817, c R 4 lmax1 = −2.250.(69) wherein the nuclear system is most unstable, from the configurational entropic point of view. Our data confirmed the suggestion that any limited number of degrees of freedom can define the most dominant state(s) of the nuclear configuration upon the thermodynamic equilibrium. We intend to continue our study of DCE in the context of the seminal Refs. [68][69][70]. IV. CONCLUSIONS The theory of a soft-wall AdS/QCD model has been used to determine the electro-excitation of N * (1440) Roper resonance. In particular, the excited Fock states are characterized by the leading three-quark (3q) component. Such a state allows for determining the main properties of Roper resonance. Within a soft-wall AdS/QCD, the gauge-invariant includes a coupling of two fermion AdS fields with the same twist-dimension for the nucleon and the N * (1440) resonance. To obtain the adjustable parameters, which determine the Lagrangian density of the interacted nucleons, as well as the gauge vector field in the N * (1440) resonances, the concept of DCE has been used at small Q 2 . Both for the nucleon and the Roper resonance, the DCE has a global minimum that determines the parameters respectively in Eqs. (28,30) with good accuracy. Such points in the parameter space present the higher compression rate of the information of the nuclear system, defining the condition of stability of the nuclear configuration, with a lower value of the uncertainty. The global minima of the DCE, obtained for the corresponding value at the parameter space, confirm the data (28,30) for the transversal and longitudinal helicity amplitudes of the N * (1440) resonance [19,66], which indicates that using the DCE approach is an appropriate tool for the study the nuclear reactions at high energy domain. The calculation of the parameters within a soft-wall AdS/QCD using the DCE approach determines the natural choice for the excited nuclear configuration and can be successfully applied to other nuclear composite systems. propagating in AdS. The additional dimension of the conventional hadrons is related to the baryon mass M B . For both, the nucleon and the Roper resonances, the model suggests the contribution of Fock states with twist τ = 3, 4, and 5. Using the following values for the parameters κ, obtain the data for the nucleon/proton mass M exp N = 938.27 MeV. The parameter c N 5 related parameters c N 3 and c N 4 in the form: Eqs. (31) -(33) in Figs. 1 and 2 as a function of the free parameters c N 3 and c N 4 . FIG. 1: DCE (S c ) as a function of c N 3 and c N 4 . There is a global minimum DCE min (c N 3 , c N 4 ) = 767.503 nat, for c N 3 = 1.264 and c N 4 = 0.169, respectively within an accuracy of 1.1% and 5.6%, comparing to data from the AdS/QCD soft-wall model in Refs. [19, 66].InFig. 2the contour plots show the DCE as a function of parameters c N 3 , c N 4 for the nucleon, respectively. The contour plot inFig. 2illustrates the DCE by slices of isentropic domains. FIG. 2 : 2Contour plot of the DCE as a function of the parameters c N 3 and c N 4 for the nucleon resonance. As one can see from Fig. 2 for the nucleon resonance, the dark center of the yellow domain of closed curves indicates the point (37), which matches the global minimum of the DCE (36). The obtained results for the nucleon resonance show that the global minimum (gmin) DCE c N 3 gmin , c N 4 gmin = 767.503 nat, (31) -(33) are depicted in Figs. 3 and 4 as a function of the free parameters c R 3 and c R 4 , where the prevailing quantum states are reached through the global minimum of the DCE. FIG. 3: DCE (S c ) as a function of c R 3 and c R 4 . There is a global minimum DCE min (c R 3 , c R 4 ) = 17.639 nat, for c R 3 = 0.803 and c R 4 = −0.171, respectively within an accuracy of 2.9% and 6.8%, comparing to data from the AdS/QCD soft-wall model in Refs. [19, 66]. For the Roper resonance, Fig. 3 reveals the DCE global minimum DCE c R 3 gmin , c R 4 gmin = 17.639 nat, FIG. 4 : 4Contour plot of the DCE as a function of the parameters c R 3 and c R 4 for the Roper resonance. As one can see from Fig. 4 for the Roper resonance, the dark center of the blue domain of closed curves, in the first quadrant in Fig. 4 for the Roper resonance, indicates the point (39), which matches the global minimum of the DCE (38). As one can note, the (c N , R 3 , c N , R 4 )-parameter space can be divided into configurational isentropic subsectors. In Figs. 2 and 4 the contour plot consists of the closed curves joining points of the parameter space corresponding to the equal value of the DCE with the gradient of the DCE being orthogonal to the contour lines of the configurational isentropic curves. Comparison of Figs. 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[ "Some remarks on Petty projection of log-concave functions", "Some remarks on Petty projection of log-concave functions", "Some remarks on Petty projection of log-concave functions", "Some remarks on Petty projection of log-concave functions" ]
[ "Leticia Alves Da Silva ", "Bernardo González Merino ", "Rafael Villa ", "Leticia Alves Da Silva ", "Bernardo González Merino ", "Rafael Villa " ]
[]
[]
In this note we study the Petty projection of a log-concave function, which has been recently introduced in [9]. The aim of this note is to report a mistake in Theorem 5.2 of[9]and to give correct new inequalities involving this new notion.
10.1007/s12220-023-01322-w
[ "https://export.arxiv.org/pdf/1906.08183v3.pdf" ]
195,068,943
1906.08183
6a09708442009e4ba97d8f520f1538ec477c2cf4
Some remarks on Petty projection of log-concave functions May 29, 2023 Leticia Alves Da Silva Bernardo González Merino Rafael Villa Some remarks on Petty projection of log-concave functions May 29, 2023arXiv:1906.08183v3 [math.FA] 26 May 2023 In this note we study the Petty projection of a log-concave function, which has been recently introduced in [9]. The aim of this note is to report a mistake in Theorem 5.2 of[9]and to give correct new inequalities involving this new notion. Introduction Let K ⊂ R n be a convex body, i.e., a convex and compact set with non-empty interior, whose boundary is denoted by ∂K. Moreover, let K n be the set of all convex bodies in R n . For these and most of the forthcoming definitions and ideas on Convex Geometry, we recommend the books [14] and [3]. If the origin is an interior point of K, the polar body K • of K is K • = {x ∈ R n : x, y ≤ 1 for any y ∈ K}, which is also a convex body with the origin in its interior. A convex body K ∈ K n is uniquely defined by its support function, defined by h K (x) = sup{ x, y : y ∈ K}. For C ⊂ R n of affine dimension k ∈ {1, . . . , n}, we denote by vol k (C) its volume measured inside the affine hull of C, and moreover we also write vol(C) = vol n (C). The mixed volume of two convex bodies K (n − 1 times) and L can be defined by V 1 (K, L) = 1 n lim ε→0 + vol(K + εL) − vol(K) ε . There is a unique finite measure S(K, ·) on the unit euclidean sphere S n−1 , called the surface area of K, so that V 1 (K, L) = 1 n S n−1 h L (v) dS(K, v), (cf. [11]). When K has a C 2 boundary ∂K with positive curvature, the density of S(K, ·) with respect to the Lebesgue measure on S n−1 is the reciprocal of the Gauss curvature of ∂K. For any x ∈ R n we let x ⊥ be the (n − 1)-subspace orthogonal to x, and let P x ⊥ C be the orthogonal projection of C onto x ⊥ . Then the projection body ΠK of K ∈ K n is the centrally symmetric convex body given by its support function h ΠK (u) = vol n−1 (P u ⊥ K), for every u ∈ S n−1 . Using standard properties of the mixed volume V 1 (·, ·) (cf. [14]) it is easy to see that h ΠK (u) = 1 2 S n−1 | u, v |dS(K, v).(2) In fact, S n−1 | u, v |dS(K, v) = S n−1 h Lu (v) dS(K, v), where L u = [−u, u] = {tu : t ∈ [−1, 1]}. Using Fubini's formula, vol(K + εL u ) − vol(K) = 2ε P u ⊥ K dx ′ = 2ε vol n−1 (P u ⊥ K). Then nV 1 (K, L u ) = lim ε→0 + vol(K + εL u ) − vol(K) ε = 2 vol n−1 (P u ⊥ K) = 2h ΠK (u). Finally, the polar projection body Π • K is the polar body of ΠK. A function f : R n → [0, ∞) is log-concave if log f is concave, i.e., if f ((1 − λ)x + λy) ≥ f (x) 1−λ f (y) λ , for every x, y ∈ R n , λ ∈ (0, 1). Then f = e −ϕ for a convex function ϕ : R n → [−∞, ∞). Moreover, let F(R n ) = {f log-concave with f ∈ L 1 (R n )}. Two typical embeddings of all convex bodies onto the set F(R n ) are given by the mappings that identify K either with the characteristic function χ K (x) = e −I ∞ K (x) or the exponential gauge e − x K of K, where In order to define the polar function of a function f = e −ϕ as a log-concave function, it is natural to search for a transformation T between convex functions so that f • = e −T ϕ . Since (K • ) • = K and if K 1 ⊂ K 2 then K • 1 ⊃ K • 2 , we have to ask for T to verify T 2 to be the identity, and if ϕ 1 ≤ ϕ 2 , then T ϕ 1 ≥ T ϕ 2 . From [5], these properties characterize the Legendre transform, so T ϕ = ϕ * . As a consequence, for any log-concave f : R n → [0, +∞) with f = e −ϕ , its polar function f • is defined by f • = e −ϕ * (cf. [5]). With this definition, if f ∈ (R n ) with 0 ∈ int(suppf ), and f • (x M ) = f ∞ for some x M ∈ R n , then f • ∈ F(R n ) too (see Theorem 4.3 below). Note that f • = e −h f . To define the analogue definition of Πf for a log-concave f , firstly defined in [9], we take into account the equality for a convex body K S n−1 | u, v | dS(K, v) = R n | ∇χ K (x), u | dx, for u ∈ S n−1 (see [16], or Proposition 2.2 iii) ). We may now generalize (2) to define the Petty projection function Πf of f given its support function h Πf (u) = 1 2 R n | ∇f (x), u | dx, (see [9]). Note that, by the chain rule, if f = e −ϕ , then ∇f = −f ∇ϕ, and the previous definition admits the form h Πf (u) = 1 2 supp f | ∇ϕ(x), u |f (x) dx. In particular, for any f ∈ F(R n ), the polar projection function is given by Π • f = (Πf ) • . Properties and main result The main result here serves as a correction to [9, Thm. 5.2] and introduces a lower bound for the integral of Π • f . Let us denote by B n 2 the n-dimensional Euclidean unit ball, S n−1 its boundary, and let ω n = vol(B n 2 ) be its volume. Moreover, let |x| = x 2 1 + · · · + x n n be the Euclidean norm for every x = (x 1 , . . . , x n ) ∈ R n . Theorem 2.1. Let f ∈ F(R n ). Then R n |∇f (x)|dx n R n Π • f (z)dz ≥ ω n n! nω n ω n−1 n . Moreover, equality holds if there exists g : [0, ∞) → [0, ∞), g ∈ F(R 1 ), such that f (x) = g(|x|) for every x ∈ R n . In the next proposition we collect some useful computations needed in this paper, refereed to characteristic functions and exponential gauges of convex bodies. Proposition 2.2. Let K ∈ K n . Then we have that i) h χ K = h K . ii) h e − · K = I ∞ K • . iii) h Πχ K = h ΠK . iv) Πχ K = e −I ∞ ΠK = χ ΠK . v) Π • χ K = e −h ΠK . vi) h Πe − · K = (n − 1)!h ΠK . vii) R n e − x K dx = n!vol(K) = n! R n χ K (x)dx. viii) R n Π • (e − · K )(x)dx = n!Γ(n) −n vol(Π • K) = Γ(n) −n R n Π • (χ K )(x)dx. Let us observe that Theorem 5.2 in [9] is not correct. Indeed, using Proposition 2.2, one can verify that if f = χ tB n 2 , for some t > 0, then the Theorem 5.2 in [9] becomes (1 − log t) n−1 ≤ C(n), for some constant C(n) > 0 only depending on the dimension n. Later on, we will discuss how to correct those bounds for the integral of the Petty projection function. One can also bound from above the term |∇f | by means of an entropic function under some extra assumptions; indeed, if aχ B n 2 ≤ f for some f ∈ F(R n ) and a > 0, then R n |∇f (x)|dx ≤ n R n f (y)dy + R n f (z) log f (z) a f ∞ dz,(4) with equality if f = aχ B n 2 (cf. [1], see also [6]). The quantity vol(K) n−1 vol(Π • K) is an affine invariant, its maximum value is provided by Petty's projection inequality [12], with equality if and only if K is an ellipsoid, and its minimum value is given by Zhang's inequality [15], with equality if and only if K is a simplex: 2n n n n ≤ vol(K) n−1 vol(Π • K) ≤ ω n ω n−1 n .(5) For any f ∈ F(R n ), f = e −ϕ , let Π b f be the Petty projection body of f , which is the convex body whose support function is given by h Π b f (y) = supp f | ∇ϕ(x), y |f (x)dx, i.e. h Π b f = 2h Πf . In order to avoid future confusion, here we have changed the original name also given by Fang and Zhou [9] (they used the name Πf , and we insert the subindex b to stress that it is a body). Its polar Π • b f was firstly introduced in [1], and here once more we change the old naming Π • f by Π • b f , and it is the unit ball of the norm given by x Π • b f = R n | ∇f (x), y |dx. Since f = e −ϕ , due to ∇f (x) = −e −ϕ(x) ∇ϕ(x), we have that Π • b f = (Π b f ) • , as one may expect. The right-hand side of (5) was extended to functional settings by Zhang [16] and it is also known as the affine Sobolev inequality, whereas the left-hand side of (5) was recently extended to log-concave functions in [2], 2 −n n! f −n−1 1 R n R n min{f (x), f (y)}dxdy ≤ vol(Π • b f ) ≤ ω n 2ω n−1 n f −n n n−1 .(6) Moreover, equality holds on the right-hand side if and only if f f ∞ = χ AB n 2 , for any regular A ∈ R n×n , and on the left hand side if and only if f f ∞ = e − · S , for any simplex S ∈ K n , with 0 ∈ S. One can immediately verify that Π • f (y) = e −h Πf (y) = e − 1 2 h Π b f (y) = e − 1 2 y Π • b f ,(7) and thus, (6) can be used to give optimal bounds of the integral of Π • f for any f ∈ F(R n ). Proposition 2.3. Let f ∈ F(R n ). Then R n Π • f (x)dx = 2 n n!vol(Π • b f ). After writing this note, Fang and Zhou have told us in personal communication that they also noticed their mistake; however, it seems that they have amended it replacing it by the right-hand side of (6) and using Proposition 2.3. Proofs We start this section by proving Proposition 2.2. Proof of Proposition 2.2. i) See (3). ii) It is a direct consequence of · * K = I ∞ K • . iii) Here we use a similar argument to the one exhibited in [16, §4]. Let us denote by d(x, A) the Euclidean distance from a point x ∈ R n to a set A ⊂ R n . Let ε > 0 and define f ε (x) = 0 if d(x, K) ≥ ε 1 − d(x,K) ε if d(x, K) < ε , If d(x, K) > 0 for some x ∈ R n , then there exists a unique x ′ ∈ ∂K such that d(x, K) = |x − x ′ |. Let ν(x ′ ) = x−x ′ |x−x ′ | be the outer normal of K at x ′ and let D ε = {x ∈ R n : 0 < d(x, K) < ε}. Then ∇f ε (x) = −ε −1 ν(x ′ ) if x ∈ D ε 0 otherwise , from which 1 2 R n | ∇f ε (x), y |dx = 1 2 Dε | ε −1 ν(x ′ ), y |dx = ε −1 2 Dε | ν(x ′ ), y |dx. When ε → 0, we have that ε −1 2 Dε | ν(x ′ ), y |dx → 1 2 ∂K | ν(x ′ ), y |dσ(∂K, x ′ ), where dσ(∂K, ·) is the surface area element of ∂K. Since lim ε→0 f ε = χ K and by (2) we can conclude that h Πχ K (y) = lim ε→0 1 2 R n | ∇f ε (x), y |dx = 1 2 ∂K | ν(x ′ ), y |dσ(∂K, x ′ ) = 1 2 S n−1 | u, y |dS(K, u) = h ΠK (y). iv) Since by definition Πχ K (x) = e −h * Πχ K (x) , by iii) we can conclude that vii) By definition, R n χ K (x)dx = vol(K). Second, Πχ K (x) = e −h * ΠK (x) = e −I ∞ ΠK (x) = χ ΠK (x). v) Using iii) we have that Π • χ K (x) = e −h Πχ K (x) = e −h ΠK (x) ,R n e − x K dx = 1 0 vol({x ∈ R n : e − x K ≥ t})dt = 1 0 vol({x ∈ R n : x K ≤ − log t})dt = vol(K) 1 0 log 1 t n dt = vol(K) ∞ 0 s n e −s ds = Γ(n + 1)vol(K). viii) On the one hand, using v) and vii) we immediately get that R n Π • χ K (x)dx = R n e −h ΠK (x) dx = R n e − x Π • K (x) dx = n!vol(Π • K). On the other hand, using vi) and vii) and the 1-homogeneity of the support function we can conclude that R n Π • (e − · K )(x)dx = R n e −h Πe − · K (x) dx = R n e −Γ(n)h ΠK (x) dx = Γ(n) −n R n e −h ΠK (x) dx = Γ(n) −n n!vol(Π • K). We now prove Theorem 2.1. The main ingredients of it are the integration by polar coordinates and the Jensen inequality [3], which states that if (X, Σ, µ) is a probability space, then for any convex function ϕ : R → R and any µ-integrable function f : X → R, we have that ϕ X f (x)dµ(x) ≤ X ϕ • f (x)dµ(x), and moreover, equality holds if and only if either ϕ is affine or f is independent of x. One can compare the proof below to the one in [9, Thm. 5.2], where we have detected mistakes in (5.17) at the change of variables and at the application of Jensen inequality. Proof of Theorem 2.1. Let f = e −ϕ . Since ∇f (x) = −f (x)∇ϕ(x) and using polar coordinates, we can write R n Π • f (z)dz = R n e − 1 2 R n | ∇f (x),z |dx dz = nω n ∞ 0 S n−1 e − r 2 R n | ∇f (x),u |dx r n−1 dµ(u)dr, where µ is the uniform probability measure in S n−1 . Since e x is convex, Jensen inequality implies that exp S n−1 − r 2 R n | ∇f (x), u |dxdµ(u) ≤ S n−1 exp − r 2 R n | ∇f (x), u |dx dµ(u). Using Fubini and using the fact that S n−1 | v, u |dµ(u) = 2ω n−1 nω n |v|, for any v ∈ R n , then We can thus conclude that R n Π • f (z)dz ≥ nω n ∞ 0 e − r 2 R n S n−1 | ∇f (x),u |dµ(u)dx r n−1 dr = nω n ∞ 0 e − r 2 R nR n Π • f (z)dz ≥ n!ω n ω n−1 nω n ∇f 1 −n = n!ω n ω n−1 nω n −n R n |∇f (x)|dx −n . In the equality case, there must be equality in the inequality above. Hence, by Jensen's equality case, we must have that R n | ∇f (x), u |dx is independent of u ∈ S n−1 . In particular, if f (x) = g(|x|) for some g : [0, ∞) → [0, ∞) log-concave and every x ∈ R n , as desired. A geometrical consequence of Theorem 2.1 is the following result (cf. [16]), which relates the surface area measure S(K) of a K ∈ K n with the volume vol(Π • K), and can be also obtained by Hölder inequality in (5). Corollary 3.1. Let K ∈ K n . Then S(K) n vol(Π • K) ≥ ω n nω n ω n−1 n . Moreover, equality holds if K = B n 2 . Proof. Let us particularize Theorem 2.1 taking f (x) = e − x K . If we denote by dσ(∂K, ·) the surface area element of K, then R n |∇f (x)|dx = R n e −||x|| K |∇||x|| K |dx = ∞ 0 t∂K e −||x|| K |∇||x|| K | · |∇||x|| K | −1 dσ(t∂K, x)dt = ∞ 0 t∂K e −||x|| K dσ(t∂K, x)dt = ∞ 0 ∂K e −t t n−1 dσ(∂K, y)dt = ∞ 0 e −t t n−1 dt ∂K dσ(∂K, y) = ∞ 0 e −t t n−1 dtS(K) = Γ(n)S(K). This, together with viii) in Proposition 2.2, imply that n! (n − 1)! n vol(Π • K) = R n Π • (e − · K )(x)dx ≥ n!ω n ω n−1 nω n −n R n |∇(e − · K )(x)|dx −n = n!ω n ω n−1 nω n −n (n − 1)! −n S(K) −n , as desired. Equality holds if e − x K is independent of x ∈ S n−1 , for instance, if K = B n 2 . Now we show Proposition 2.3. Proof of Proposition 2.3. As a consequence of (7) and vii) in Proposition 2.2, we obtain that R n Π • f (x)dx = R n e −h Πf (x) dx = R n e − 1 2 x Π • b f dx = 2 n R n e − x Π • b f dx = 2 n n!vol(Π • b f ). Integrability of log-concave functions In this section we characterize the integrability of log-concave functions in terms of the value of f over all possible rays. Other characterizations of the integrability of log-concave functions were given in [8]. Before stating the next result, we would like to remember that for any log-concave function f : R n → [0, ∞), the function g(x) =    f (x) if x ∈ int(suppf ), lim sup y→x,y∈int(suppf ) f (y) if x ∈ ∂ suppf, 0 otherwise is log-concave, continuous on its support supp g = supp f = {x ∈ R n : f (x) > 0}, and has g = f (see [7,Lem. 2.1]). Thus, we can always replace f by g and hence we can always extend continuously f to its support. (1) f is integrable. (2) Either f = 0 or there exists no ray R = x 0 + R + u, x 0 , u ∈ R n , u = 0, for which f | R = c > 0. ∇f 1 r r n−1 dr. Letting t = ω n−1 nωn ∇f 1 r = ar, then dt = adr and ∞ 0 e −ar r n−1 dr = ∞ 0 e −t t n−1 a 1−n a −1 dt = a −n Γ(n). Let f : R n → [0, ∞) be log-concave with f (x M ) = f ∞ for some x M ∈ R n .Then the following are equivalent: Proof. We first prove (1) implies(2). Let us suppose that f > 0. Hence, the function f is non-zero in an open ball B(x 0 , r), for some x 0 ∈ R n and r > 0. By continuity of f in B(x 0 , r), let us suppose that f (x) ≥ α, for every x ∈ B(x 0 , r) and for some α > 0. Moreover, for the sake of contradiction, let us suppose that there exists a ray R = y 0 + R + u, y 0 , u ∈ R n , u = 0, t * t f (0) 1− t * t . Acknowledgements. We would like to thank Julián Haddad for his valuable comments and pointers, and to the University of Sevilla for hosting Leticia Alves da Silva during two months, in which this work has been done. She also thanks the support of the IFMG -Campus Bambuí while conducting this work.such that f | R = c > 0. Let z 0 ∈ U = B(x 0 , r) ∩ (x 0 + u ⊥ ) and t > 0. Let us observe thatFurthermore, we have thatand thus thatthus showing that f is not integrable, a contradiction. We now show (2) implies (1). If f = 0, then f is integrable. Let us suppose that f > 0. After a suitable translation, let us assume that f (0) = f ∞ . For every u ∈ S n−1 , since f is not constant on R + u, then there exists s u > 0 such that f (s u u) < f ∞ . Now, using that f is continuous implies that t u = inf{s > 0 : f (su) < f ∞ } fulfills f (t u u) = f ∞ , for every u ∈ S n−1 . We now show that if {t u : u ∈ S n−1 } is unbounded, we arrive at a contradiction. Indeed, in that case let {u k } ⊂ S n−1 be such that t u k → ∞ as k → ∞. Since S n−1 is compact, there exists a subsequence (which we can suppose w.l.o.g. to be the sequence itself) converging u k → u 0 ∈ S n−1 . For every t > 0, thenHence let t * > t u be such that f (t * u) ≤ c < f ∞ for every u ∈ S n−1 and some c > 0. Observe that for every t ≥ t * and every u ∈ S n−1 we have thatThus, integrating in polar coordinates, we get thatwhere µ is the uniform probability measure in S n−1 . Since the first integral is finite as 0 < c/f (0) < 1 and the second one is finite as f is bounded and B(0, t * ) is bounded too, hence we obtain that f is integrable.Proof. Let us suppose that f = e −ϕ . We show now that ϕ * (and thus f • ) is not constant over any ray y 0 + R + u, for any y 0 , u ∈ R n , u = 0, thus concluding by Lemma 4.1 that f • is integrable. Since ϕ is continuous and 0 ∈ int(suppf ), let us suppose that ϕ(x) ≤ C whenever |x| ≤ δ for some C, δ > 0. Let us consider y 0 + tu, t > 0. Then ϕ * (y 0 + tu) = sup z ( y 0 + tu, z − ϕ(z)) ≥ δ|y 0 + tu| − ϕ δ y 0 + tu |y 0 + tu| ≥ δ|y 0 + tu| − C,thus showing that ϕ * is not constant over any ray y 0 + R + u, as desired.Remark 4.4. The integrability of f • can be easily deduced by using some Blaschke-Santaló functional inequality f f • ≤ (2π) n but only for certain particular translations of f (for instance, when the Santaló point of f is the origin, see[4]). However, the comment in [10, Rmk. 2] is not correct (where the authors said that "All of our results hold, with the same proofs, for log-concave functions that reach their maximum at the origin"), since f • is not necessarily integrable if f (0) = f ∞ . For instance, lettingthus having that f • (x) = e 0 = 1 if x < 0, and hence f • would not be integrable. John's ellipsoid and the integral ratio of a log-concave function. D Alonso-Gutiérrez, B Merino, C H Jiménez, R Villa, J. Geom. Anal. 282D. Alonso-Gutiérrez, B. González Merino, C. H. Jiménez, and R. Villa. John's ellipsoid and the integral ratio of a log-concave function. J. Geom. Anal., 28(2):1182-1201, 2018. Zhang's inequality for log-concave functions. D Alonso-Gutiérrez, B G Merino, J Bernués, Geometric Aspects of Functional Analysis. Klartag B., Milman E.ChamSpringerD. Alonso-Gutiérrez, B. G. Merino, and J. Bernués. Zhang's inequality for log-concave functions. In: Klartag B., Milman E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2256. Springer, Cham, 2020. Asymptotic Geometric Analysis, Part I. S Artstein-Avidan, A Giannopoulos, V D Milman, American Mathematical Soc. 202S. Artstein-Avidan, A. Giannopoulos, and V. D. Milman. Asymptotic Geometric Anal- ysis, Part I, volume 202. American Mathematical Soc., 2015. The Santaló point of a function, and a functional form of the Santaló inequality. S Artstein-Avidan, B Klartag, V Milman, Mathematika. 511-2S. Artstein-Avidan, B. Klartag, and V. Milman. The Santaló point of a function, and a functional form of the Santaló inequality. Mathematika, 51(1-2):33-48, 2004. The concept of duality in convex analysis and the characterization of the Legendre transform. S Artstein-Avidan, V D Milman, Ann. of Math. 1692S. Artstein-Avidan and V. D. Milman. The concept of duality in convex analysis and the characterization of the Legendre transform. Ann. of Math., 169(2):661-674, 2009. Reverse Brunn-Minkowski and reverse entropy power inequalities for convex measures. S Bobkov, M Madiman, J. Funct. Anal. 2627S. Bobkov and M. Madiman. Reverse Brunn-Minkowski and reverse entropy power inequalities for convex measures. J. Funct. Anal., 262(7):3309-3339, 2012. Functional inequalities related to the rogers-shephard inequality. A Colesanti, Mathematika. 531A. Colesanti. Functional inequalities related to the rogers-shephard inequality. Mathe- matika, 53(1):81-101, 2006. Moment measures. D Cordero-Erausquin, B Klartag, J. Funct. Anal. 26812D. Cordero-Erausquin and B. Klartag. Moment measures. J. Funct. Anal., 268(12):3834- 3866, 2015. LYZ ellipsoid and Petty projection body for log-concave functions. N Fang, J Zhou, Adv. Math. 340N. Fang and J. Zhou. LYZ ellipsoid and Petty projection body for log-concave functions. Adv. Math., 340:914-959, 2018. Geometry of log-concave functions and measures. B Klartag, V D Milman, Geom. Dedicata. 1121B. Klartag and V. D. Milman. Geometry of log-concave functions and measures. Geom. Dedicata, 112(1):169-182, 2005. The Brunn-Minkowski-Firey Theory I: Mixed volumes and Minkowski Problem. E Lutwak, J. Differential Geom. 381E. Lutwak. The Brunn-Minkowski-Firey Theory I: Mixed volumes and Minkowski Prob- lem. J. Differential Geom., 38(1):131-150, 1993. Isoperimetric problems. C M Petty, Proceedings of the Conference on Convexity and Combinatorial Geometry. the Conference on Convexity and Combinatorial Geometry1C. M. Petty. Isoperimetric problems. Proceedings of the Conference on Convexity and Combinatorial Geometry, 1:26-41, 1971. On the mean width of log-concave functions. L Rotem, In Geom. Funct. Anal. SpringerL. Rotem. On the mean width of log-concave functions. In Geom. Funct. Anal., pages 355-372. Springer, 2012. R Schneider, Convex bodies: the Brunn-Minkowski Theory. Number 151. Cambridge university pressR. Schneider. Convex bodies: the Brunn-Minkowski Theory. Number 151. Cambridge university press, 2014. Restricted chord projection and affine inequalities. G Zhang, Geom. Dedicata. 392G. Zhang. Restricted chord projection and affine inequalities. Geom. Dedicata, 39(2):213-222, 1991. The affine Sobolev inequality. G Zhang, Medeiros -km 05. CP 05 -Bambuí -MG -CEP:38900-000J. Differ. Geom. 531IFMG Campus Bambuí. Brazil E-mail address L. Alves da Silva: [email protected]. Zhang. The affine Sobolev inequality. J. Differ. Geom., 53(1):183-202, 1999. IFMG Campus Bambuí, Instituto Federal de Minas Gerais, -Faz, Varginha -, Rodovia Bambuí/Medeiros -km 05. CP 05 -Bambuí -MG -CEP:38900-000, Brazil E-mail address L. Alves da Silva: [email protected]
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[ "Discussion of \"Vintage factor analysis with Varimax performs statistical inference\" *", "Discussion of \"Vintage factor analysis with Varimax performs statistical inference\" *" ]
[ "Rungang Han \nDepartment of Statistical Science\nDuke University\n\n", "Anru R Zhang \nDepartments of Biostatistics & Bioinformatics\nComputer Science, Mathematics, and Statistical Science\nDuke University\n\n" ]
[ "Department of Statistical Science\nDuke University\n", "Departments of Biostatistics & Bioinformatics\nComputer Science, Mathematics, and Statistical Science\nDuke University\n" ]
[]
We wholeheartedly congratulate Drs. Rohe and Zeng for their insightful paper[2]on vintage factor analysis with Varimax rotation. Varimax rotation is a basic scheme to simplify the expression of a particular subspace and is included in build-in standard packages stats in R and PROC FACTOR statement in SAS. Drs. Rohe and Zeng nicely show that the principal component analysis with Varimax rotation actually performs statistical inference for the explainable factors.Drs. Rohe and Zeng suggested leptokurtosis as a key identifiability condition for Varimax rotation; the number of factors often increases as the data dimension and sample size grow. It is thus natural to ask whether Varimax works with vanishing leptokurtosis and/or a growing number of factors. This note discusses when Varimax recovers the subspace rotation in such high-dimensional regimes. As a first step, we assume the factor matrix Z ∈ R n×k includes a collection of i.i.d. centered random variables satisfyingWe also assume Z ij 's are sub-Gaussian such that E exp (λZ ij ) ≤ e cλ 2 , ∀λ ∈ R for some constant c > 0. LetẐ ∈ R p×r be the observed factors generated aŝwhere R * is an unknown k-dimensional orthogonal matrix that represents the rotation to be recovered. Due to vanishing mean of Z, we focus on the following centered Varimax:
10.1093/jrsssb/qkad056
[ "https://arxiv.org/pdf/2205.10151v1.pdf" ]
248,965,191
2205.10151
468a62aeca32d437950ff6fd39c3d8e4da849d51
Discussion of "Vintage factor analysis with Varimax performs statistical inference" * 20 May 2022 (May 23, 2022) Rungang Han Department of Statistical Science Duke University Anru R Zhang Departments of Biostatistics & Bioinformatics Computer Science, Mathematics, and Statistical Science Duke University Discussion of "Vintage factor analysis with Varimax performs statistical inference" * 20 May 2022 (May 23, 2022) We wholeheartedly congratulate Drs. Rohe and Zeng for their insightful paper[2]on vintage factor analysis with Varimax rotation. Varimax rotation is a basic scheme to simplify the expression of a particular subspace and is included in build-in standard packages stats in R and PROC FACTOR statement in SAS. Drs. Rohe and Zeng nicely show that the principal component analysis with Varimax rotation actually performs statistical inference for the explainable factors.Drs. Rohe and Zeng suggested leptokurtosis as a key identifiability condition for Varimax rotation; the number of factors often increases as the data dimension and sample size grow. It is thus natural to ask whether Varimax works with vanishing leptokurtosis and/or a growing number of factors. This note discusses when Varimax recovers the subspace rotation in such high-dimensional regimes. As a first step, we assume the factor matrix Z ∈ R n×k includes a collection of i.i.d. centered random variables satisfyingWe also assume Z ij 's are sub-Gaussian such that E exp (λZ ij ) ≤ e cλ 2 , ∀λ ∈ R for some constant c > 0. LetẐ ∈ R p×r be the observed factors generated aŝwhere R * is an unknown k-dimensional orthogonal matrix that represents the rotation to be recovered. Due to vanishing mean of Z, we focus on the following centered Varimax: We wholeheartedly congratulate Drs. Rohe and Zeng for their insightful paper [2] on vintage factor analysis with Varimax rotation. Varimax rotation is a basic scheme to simplify the expression of a particular subspace and is included in build-in standard packages stats in R and PROC FACTOR statement in SAS. Drs. Rohe and Zeng nicely show that the principal component analysis with Varimax rotation actually performs statistical inference for the explainable factors. Drs. Rohe and Zeng suggested leptokurtosis as a key identifiability condition for Varimax rotation; the number of factors often increases as the data dimension and sample size grow. It is thus natural to ask whether Varimax works with vanishing leptokurtosis and/or a growing number of factors. This note discusses when Varimax recovers the subspace rotation in such high-dimensional regimes. As a first step, we assume the factor matrix Z ∈ R n×k includes a collection of i.i.d. centered random variables satisfying EZ ij = EZ 3 ij = 0, EZ 2 ij = 1, EZ 4 ij = κ > 3.(1) We also assume Z ij 's are sub-Gaussian such that E exp (λZ ij ) ≤ e cλ 2 , ∀λ ∈ R for some constant c > 0. LetẐ ∈ R p×r be the observed factors generated aŝ Z = ZR * , where R * is an unknown k-dimensional orthogonal matrix that represents the rotation to be recovered. Due to vanishing mean of Z, we focus on the following centered Varimax: R = arg max R∈O(k) n i=1 k j=1 (ẐR ⊤ ) ij 4 , where O(k) is the set of k-by-k orthogonal matrices. Consider the following error metric: dist(R * ,R) = min P ∈P(k) R − P R * F R * F = k −1/2 min P ∈P(k) R R * ⊤ − P F , where P(k) = O(k) ∩ {0, ±1} k 2 is the set of orthogonal matrices that allow for column reordering and sign changes as defined in Eqn. (12) in [2]. The following theorem characterizes the conditions of n, k, κ under which Varimax works or fails. Theorem 1. Let δ ∈ (0, 1/2] be any fixed value and κ ≤ C 0 for some universal constant C 0 > 3. • If n max k log n (κ−3) 2 , k 2 log 2 n κ−3 , then lim sup n→∞ P dist(R * ,R) ≥ δ = 0; • if k < n k 2 , then lim inf n,k→∞ P dist(R * ,R) ≥ δ = 1. If the kurtosis κ > 3 is a constant, Theorem 1 suggests that a necessary and nearly sufficient condition for consistent rotation recovery is n k 2 -this bound is tight up to log 2 (n). Theorem 1 also shows when the leptokurtosis of factors is insignificant, i.e., κ → 3 + , a sufficient sample size to ensure rotation recovery by Varimax is max{k log(n)/(κ− 3) 2 , k 2 log 2 (n)/(κ − 3)}, while it is unclear if this bound is sharp. It is of future interest to investigate the tight condition that guarantees the consistency of Varimax. Proof of Theorem 1. For any random matrix Z and fixed a ∈ R k , define the following stochastic functions v(Z; a) = 1 n n i=1   k j=1 a j Z ij   4 .(2) Note that the Varimax object function can be then written as v(Ẑ; R ⊤ ) = k j=1 v(Ẑ; R j: ) where R j: if the jth row of R. Our proof relies on the following lemmas. Lemma 1. Suppose A ∈ O(k) and min P ∈P(k) A − P F > t for some t > 0. Then, k i=1   k j=1 A 4 ij − 1   ≤ − 1 16 t 2 . Lemma 2. Suppose Z is a n-by-k random matrix with i.i.d. mean-zero, unit-variance and sub-Gaussian entries. Then, with probability at least 1 − n −ck − n −4 , sup a: a 2 ≤1 v(Z; a) − Ev(Z; a) k log n n + k 2 log 2 n n . Lemma 3. Suppose X 1 , . . . , X p are independent random variables with mean zero, variance one, and sub-Gaussian tails. Then there exists constants C, c > 0 such that P p i=1 (X 4 i − EX 4 i ) ≥ C √ pt + Ct 2 ≤ 2 exp(−ct). We start by proving the sufficient condition. Define δ := C 0 √ κ − 3 max k 2 log 2 n n 1/2 , k log n n 1/4 ,(3) for some positive constant C 0 to be specified later. It suffices to show that with high probability, for any R ∈ O(k) such that min P ∈P(k) R * R ⊤ − P F > δk 1/2 , R cannot be the solution of Varimax. By Lemma 2, we know with probability at least 1 − o(1/n), sup a: a 2 ≤1 v(Z; a) − Ev(Z; a) ≤ C k log n n + k 2 log 2 n n ≤ c(κ − 3),(4) where c is some small positive constant. In the meantime, Define A = RR * ⊤ = (a 1 , a 2 , . . . , a k ), where a j is the j-th orthogonal column of A. Note that the Varimax loss function can be written as v(Ẑ; R) = k j=1 v(Z; a j ). By calculating its expectation, we have: Ev(Ẑ; R) = E k j=1 v(Z; a j ) = 3k + (κ − 3) k i=1 k j=1 a 4 ij = κk + (κ − 3) k i=1   k j=1 a 4 ij − 1   Lemma 1 ≤ κk − (κ − 3)δ 2 16 k.(5) Meanwhile, note that Ev(Ẑ, R * ) = κk.(6)Then we have v(Ẑ; R * ) − v(Ẑ; R) = Ev(Ẑ; R * ) − Ev(Ẑ; R) + v(Ẑ; R * ) − E(Ẑ; R * ) − v(Ẑ; R) − E(Ẑ; R) (5),(6) ≥ (κ − 3)δ 2 16 k − 2k sup a: a ≤1 |v(Z; a) − Ev(Z; a)| (3),(4) > 0. Therefore, R is not a solution of varimax. In other words, givenR = arg max R∈O(k) v(Ẑ, R), one must have min P ∈P(k) R * R ⊤ − P F < δk 1/2 , which completes the proof of sufficient condition. Next, we prove the necessary condition. Let Z 1 ∈ R ⌈k/2⌉×k be the first ⌈k/2⌉ rows of the factor matrix Z. Consider the QR-decomposition of Z ⊤ 1 : Z ⊤ 1 = U 1 R 1 , where U 1 is a k-by-⌈k/2⌉ orthogonal matrix and R 1 is an ⌈k/2⌉-by-⌈k/2⌉ upper triangular matrix. Let U 1⊥ ∈ O k,k−⌈k/2⌉ be the perpendicular subspace of U 1 (such that U ⊤ 1 U 1⊥ = 0), and denote A = [U 1 U 1⊥ ] ⊤ R * . Then, (ẐA ⊤ ) [1:⌈k/2⌉,:] = Z 1 [U 1 U 1⊥ ] = [R ⊤ 1 O] and it follows that v(Ẑ, A) = 1 n n i=1 k j=1 (ẐA ⊤ ) ij 4 ≥ 1 n       ⌈k/2⌉ j=1 (R 1 ) 4 jj D 1 + n i=k+1 k j=1 (ẐA ⊤ ) ij 4 D 2       .(7) Now we establish the probabilistic bounds for D 1 and D 2 . • D 1 . By random matrix theory [3], we know that with probability at least 1 − e −ck that σ ⌈k/2⌉ (Z 1 ) ≥ √ k − ⌈k/2⌉ − c √ k ≥ √ 0.1k.(8) Note that R 1 shares the same non-zeros singular values with X 1 . Thus it follows that ⌈k/2⌉ j=1 (R 1 ) 2 jj = | det(R 1 )| 2 = ⌈k/2⌉ j=1 σ 2 j (Z 1 ) (8) ≥ (0.1k) ⌈k/2⌉ . Then by Inequality of arithmetic and geometric means, D 1 = ⌈k/2⌉ j=1 (R 1 ) 4 jj ≥ ⌈k/2⌉   ⌈k/2⌉ j=1 (R 1 ) 4 jj   1/⌈k/2⌉ ≥ 0.005k 3 .(9) • D 2 . Denote B := R * A ⊤ and we can rewrite D 2 = n i=k+1 k j=1 (ẐA ⊤ ) ij 4 = n i=k+1 k j=1 ((ZB) ij ) 4 = k j=1 n i=k+1 k l=1 B lj Z il 4 . Then, conditioning on A (or B), we have E[D 2 |A] = E   k j=1 n i=k+1 k l=1 B lj Z il 4 B   = k j=1 (n − k)E   k k=1 B lj Z 1l 4 B   = (n − k)   3k + (κ − 3) k l=1 k j=1 B 4 lj   ≥ 3(n − k)k. On the other hand, for each fixed j ∈ [k], k l=1 B lj Z il n i=k+1 are i.i.d. mean-0, variance-1 sub-Gaussian random variables (since B is independent of the last n − k rows of Z). By Lemma 3, we have P   n i=k+1 k l=1 B lj Z il 4 − E   n i=k+1 k l=1 B lj Z il 4 A   < −C( √ nt + t 2 ) A   ≤ 2e −ct . Taking t = c 0 k for some small constant c 0 and applying union bound for each j ∈ [k], we know that P D 2 − E[D 2 |A] < −c 0 √ nk 3 + k 3 A ≤ 2ke −ck and it follows that P D 2 < 3(n − k)k − c 0 √ nk 3 + k 3 A ≤ 2ke −ck . Integrate over the density of A and we obtain P D 2 < 3(n − k)k − c 0 √ nk 3 + k 3 ≤ 2ke −ck ≤ e −c ′ k .(10) In conclusion, we proved that with probability at least 1 − e −ck , D 1 ≥ 0.005k 3 , D 2 ≥ 3(n − k)k − c 0 √ nk 3 + k 3 .(11) Since n ≤ ck 2 , by taking c 0 to be sufficiently small, we have v(Ẑ, A) ≥ 1 n 0.004k 3 + 3nk . Next we show that for any R such that dist(R, R * ) ≤ δ, v(Ẑ, R) is upper bounded by the right-hand side of (12) with high probability. Note that v(Ẑ; R) = Ev(Ẑ; R * ) + (v(Ẑ, R * ) − Ev(Ẑ; R * )) + v(Ẑ, R) − v(Ẑ, R * ) = κk + v(Ẑ; R * ) − Ev(Ẑ; R * ) + sup R:dist(R,R * )≤δ v(Ẑ, R) − v(Ẑ, R * ) .(13) Following the same argument as (10), we know that with probability at least 1 − e −ck that v(Ẑ; R * ) − Ev(Ẑ, R * ) ≤ c 0 n √ nk 3 + k 3 . In the meantime, sup R:dist(R,R * )≤δ v(Ẑ, R) − v(Ẑ, R * ) = sup R∈O(k): R −I F ≤δ √ k v(Z;R) − v(Z; I) ≤ sup R∈O(k): R −I F ≤δ √ k k i=1 v(Z;R [i,:] ) − v(Z; e i ) ,(14) where e i is the i-th canonical basis in R k . Let δ i := R [i,:] − e i 2 . Then, by the proof of (20) in Lemma 2, for each i ∈ [k], we have v(Z;R i,: ) − v(Z;ẽ i ) ≤ 6δ i max j,l Z 4 jl ≤ Cδ i log 2 n(15) hold with probability at least 1 − n −4 . Combining (14) and (15), we obtain sup R:dist(R,R * )≤δ v(Ẑ, R) − v(Ẑ, R * ) ≤ C log 2 n k i=1 δ i ≤ C log 2 n k k i=1 δ 2 i 1/2 = C √ k log 2 n R − I F ≤ Cδk log 2 n.(16) Thus, when n k 2 log 2 n , we have sup R:dist(R,R * )≤δ v(Ẑ, R) (13),(16) ≤ κk + c 0 √ nk 3 + k 3 n + δk log 2 n < 0.004k 3 + 3nk n (12) ≤ v(Ẑ, A). Here, the second inequality is obtained by choosing c 0 to be sufficiently small and δ = 1/2. This suggests that the solution of Varimax functionR must satisfy: dist(R, R * ) ≥ δ = 1/2. Proof of Lemma 1. Let P * = arg min P ∈O(k) A − P 2 F . Without loss of generality, we can assume P * = I and A 11 ≥ A 22 ≥ . . . ≥ A kk ≥ 0. This suggests that t 2 < k i=1   (1 − A ii ) 2 + j =i A 2 ij   = 2 k i=1 (1 − A ii ) ≤ 2 k i=1 (1 − A 2 ii ).(17) It suffices to show that for each i ∈ [k], k j=1 A 4 ij − 1 ≤ − 1 8 (1 − A 2 ii ). Now consider the following two situations for each i ∈ [k]: • A 2 ii ≥ 1/16. k j=1 A 4 ij − 1 ≤ A 4 ii +   j =i A 2 ij   2 − 1 = A 4 ii + 1 − A 2 ii 2 − 1 = 2A 2 ii A 2 ii − 1 ≤ − 1 8 (1 − A 2 jj ). • A 2 ii < 1/16. We claim that max j A 2 ij < 3/4. Suppose there exists l ∈ [k]/i such that A 2 il ≥ 3/4. Since I = arg min P ∈P(k) A − P 2 F , we must have A − I 2 F ≤ A − D (i,l) 2 F ,(18) where D (i,l) is the permutation matrix such that D (i,l) il = sgn(A il ), D (i,l) li = sgn(A lj ) and D (i,l) jj = 1 for j ∈ [k]/{i, l}. This yields (A ll − 1) 2 + (A ii − 1) 2 + A 2 il + A 2 li ≤ (|A li | − 1) 2 + (|A il | − 1) 2 + A 2 ii + A 2 ll , which is equivalent to |A li | + |A il | ≤ A ll + A ii . Then it follows that A ll ≥ |A il | − A ii ≥ 2 √ 3 − 1 4 . Consequently, we get 1 ≥ A 2 ll + A 2 il ≥ 13 − 4 √ 3 16 + 3 4 > 1. This contradiction shows that we must have max j A 2 ij < 3/4. Therefore, k j=1 A 4 ij − 1 ≤ max j A 2 ij k j=1 A 2 ij − 1 = − 1 4 ≤ − 1 4 (1 − A 2 ii ). The proof is finished by combining the two situations. Proof of Lemma 2. We first construct an ε-net N = x (1) , . . . , x (N ) ⊂ S k−1 such that for any a ∈ S k−1 , there exists some i a ∈ [N ] with a − x (ia) ≤ ε. By [3], we can choose such an ε-net with N ≤ (3/ε) k . Then, by triangle inequality, sup a∈S k−1 |v(Z; a) − Ev(Z; a)| ≤ sup a∈S k−1 Ev(Z; x (ia) ) − Ev(Z; a) + v(Z; a) − v(Z; x (ia) ) + v(Z; x (ia) ) − Ev(Z; x (ia) ) ≤ sup a∈S k−1 Ev(Z; x (ia) ) − Ev(Z; a) D 1 + sup a∈S k−1 v(Z; a) − v(Z; x (ia) ) D 2 + max i∈N v(Z; x (i) ) − Ev(Z; x (i) ) D 3 . We specify ε = 1/n and N ≤ (3n) k . Next we obtain deterministic bounds for D 1 and D 2 and a probabilistic bound for D 3 . • sup a∈S k−1 Ev(Z; x (ia) ) − Ev(Z; a) . Note that for any fixed a ∈ S k−1 , one can calculate that Ev(Z; a) = E   k j=1 a j X 1j   4 = (κ − 3) k j=1 a 4 j + 3. Therefore, Ev(Z; x (ia) ) − Ev(Z; a) = (κ − 3) k j=1 a 4 j − (x (ia) j ) 4 ≤(κ − 3) max j a 2 j + (x (ia) j ) 2 · k j=1 a 2 j − (x (ia) j ) 2 ≤2(κ − 3) k j=1 a 2 j − (x (ia) j ) 2 .(19)Denote δ = a − b for any b ∈ S k−1 . Note that k j=1 |a j − b j | = k j=1 b 2 j + δ 2 j + 2b j δ j − b 2 j ≤ δ 2 2 + 2 k j=1 |b j δ j | ≤ δ 2 2 + 2 δ 2 , where we use Cauchy-Schwardz inequality and the fact b = 1 to obtain the last inequality. Now we replace b with x (ia) , then D 1 can be further bounded as D 1 ≤ 2(κ − 3) k j=1 a 2 j − (x (ia) j ) 2 ≤ 2(κ − 3) max j |a j + x ia j | · j j=1 a j − x (ia) j ≤ 12ε(κ − 3) ≤ C n . (20) • sup a∈S k−1 v(Z; a) − v(Z; x (ia) ) . For any a, b ∈ S k−1 , |v(Z; a) − v(Z; b)| = 1 n n i=1     k j=1 a j Z ij   4 −   k j=1 b j Z ij   4   ≤ 1 n max i   k j=1 a j Z ij   2 +   k j=1 b j Z ij   2 · n i=1   k j=1 a j Z ij   2 −   k j=1 b j Z ij   2 ≤ 2 n   max i k j=1 Z 2 ij   · max i k j=1 (a j + b j )Z ij · n i=1 k j=1 |(a j − b j )Z ij | ≤ 4k 3/2 (max i,j |Z ij |) 4 · k j=1 |a j − b j | ≤ 4k 2 (max i,k |Z ik |) 4 · k j=1 (a j − b j ) 2 ≤ 4k 2 (max i,j |X ij |) 4 ε ≤ 4k 2 (max i,j |X ij |) 4 n . • max i∈N v(Z; x (i) ) − Ev(Z; x (i) ) . By Lemma 3, for fixed x (l) , since Y (l) i := k j=1 x (l) j X ij are i.i.d. mean-zero, variance one, sub-Gaussian random variables, we have P v(Z; x (j) ) − Ev(Z; x (j) ) ≥ C t n + t 2 n ≤ e −ct . Taking t = Ck log n and applying union bounds on x (l) for each l ∈ [N ], it follows that P D 3 ≤ C k log n n + k 2 log 2 n n ≥ 1 − e −ck log n . In the meantime, by the tail bound of sub-Gaussian Extreme values, we have P max i,k |X i,k | ≤ 10 log(nk) ≤ 1 − 2 π log n n −4 . Combining (20), (19), (21) and (22), we know that with probability at least 1 − n −ck − n −4 that sup a∈S k−1 |v(Z; a) − Ev(Z; a)| k log n n + k 2 log 2 n n . Now the proof is finished. Proof of Lemma 3. Note that {X 4 i } n i=1 are sub-Weibull(1/2) random variables and the result simply comes from the concentration developed by [1, Theorem 3.1]. Moving beyond sub-gaussianity in high-dimensional statistics: Applications in covariance estimation and linear regression. Arun Kumar Kuchibhotla, Abhishek Chakrabortty, arXiv:1804.02605arXiv preprintArun Kumar Kuchibhotla and Abhishek Chakrabortty. Moving beyond sub-gaussianity in high-dimensional statistics: Applications in covariance estimation and linear regres- sion. arXiv preprint arXiv:1804.02605, 2018. 10 Vintage factor analysis with varimax performs statistical inference. Karl Rohe, Muzhe Zeng, Journal of the Royal Statistical Association, Series B. to appear, 2022. 1, 2Karl Rohe and Muzhe Zeng. Vintage factor analysis with varimax performs statistical inference. Journal of the Royal Statistical Association, Series B, to appear, 2022. 1, 2 Introduction to the non-asymptotic analysis of random matrices. Roman Vershynin, arXiv:1011.30274arXiv preprintRoman Vershynin. Introduction to the non-asymptotic analysis of random matrices. arXiv preprint arXiv:1011.3027, 2010. 4, 8
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[ "A note on classes of subgraphs of locally finite graphs", "A note on classes of subgraphs of locally finite graphs" ]
[ "Florian Lehner " ]
[]
[]
We investigate the question how 'small' a graph can be, if it contains all members of a given class of locally finite graphs as subgraphs or induced subgraphs. More precisely, we give necessary and sufficient conditions for the existence of a connected, locally finite graph H containing all elements of a graph class G. These conditions imply that such a graph H exists for the class G d consisting of all graphs with maximum degree < d which raises the question whether in this case H can be chosen to have bounded maximum degree. We show that this is not the case, thereby answering a question recently posed by Huynh et al.
10.1016/j.jctb.2023.02.001
[ "https://arxiv.org/pdf/2205.12824v1.pdf" ]
249,063,136
2205.12824
f1f22fad281625a299ab0e457f2afa52b10fe021
A note on classes of subgraphs of locally finite graphs May 2022 Florian Lehner A note on classes of subgraphs of locally finite graphs May 2022 We investigate the question how 'small' a graph can be, if it contains all members of a given class of locally finite graphs as subgraphs or induced subgraphs. More precisely, we give necessary and sufficient conditions for the existence of a connected, locally finite graph H containing all elements of a graph class G. These conditions imply that such a graph H exists for the class G d consisting of all graphs with maximum degree < d which raises the question whether in this case H can be chosen to have bounded maximum degree. We show that this is not the case, thereby answering a question recently posed by Huynh et al. Introduction Given a graph class G, we call a graph G ∈ G (strongly) universal for G, if it contains every graph in G as an (induced) subgraph. This concept was probably first studied by Rado who showed in [10] that the set of all countable graphs contains a strongly universal element. Numerous other results about the existence or non-existence of universal elements in various graph classes can be found in the literature. Graph classes defined by excluded subgraphs are particularly well studied, especially when the excluded subgraphs are trees or cycles, see for instance [1,2,3,5,7,8,9]. In this short note we are interested in the question whether there is a locally finite (that is, all vertices have finite degree) graph containing all members of a given graph class as (induced) subgraphs. Let us say that a graph H contains a graph class G, if H contains every G ∈ G as a subgraph, and that H strongly contains G, if H contains every G ∈ G as an induced subgraph. Our first main result states that the existence of a locally finite graph H which (strongly) contains G can be determined by looking at the set of balls of finite radius appearing in members of G. To state the theorem, we need to set up some notation. Given a graph class G, let where B G (v, r) and B G ′ (v ′ , r) are considered the same if there is an isomorphism between them which maps v to v ′ . Define a tree structure T (G) on B by connecting B G (v, r − 1) and B G (v, r) by an edge for every G ∈ G, v ∈ V (G), and r ∈ N. Let us call G closed, if for every G / ∈ G there is some v ∈ V (G) and r ∈ N such that B G (v, r) / ∈ B. We note that many interesting graph classes are closed. For instance it is easy to see that any graph class defined by forbidden (induced) subgraphs of finite diameter is closed, and the same is true for graph classes defined by finite forbidden minors or topological minors. With the above notation, and denoting by T ∞ the tree in which every vertex has countably infinite degree, we have the following characterisation of closed graph classes which admit a locally finite graph containing them. Theorem 1.1. Let G be a closed class of connected, locally finite graphs. The following three statements are equivalent. 1. There is a connected, locally finite graph which strongly contains G. 2. There is a connected, locally finite graph which contains G. T (G) does not contain a subdivision of T ∞ . Closedness of graph classes has a topological interpretation which adds another interesting facet to the above theorem: the three equivalent conditions are met if and only if the set G • = {(G, v) | G ∈ G, v ∈ V (G)}, is σ-compact with respect to a natural topology on the set of rooted graphs. We also note that a similar result holds, if we drop the requirement that G is closed (see Theorem 3.1), but the third statement has to be replaced by a more technical condition. The second main result of this paper concerns graphs not containing some fixed star. Denote by G d the class of all connected graphs which do not contain a star with d leaves as a subgraph. Equivalently, G d is the class of all graphs in which all vertex degrees are strictly less than d. We also allow d = ∞ and let G ∞ denote the class of connected, locally finite graphs. The class G ∞ does not contain a universal element by an argument attributed to de Bruijn in [10] (or by Theorem 1.1), and the same is true for G d , d ≥ 4 as shown in [9]. However, it follows from Theorem 1.1 that there is a graph H ∈ G ∞ (strongly) containing G d for d < ∞. We show that this is best possible, thereby answering a question recently posed by Huynh et al. in [6,Section 7]. Theorem 1.2. If a countable graph H contains G 4 , then it necessarily has unbounded vertex degrees. Besides the result itself, the proof method may be of interest. Many results about nonexistence of certain countable universal graphs are based on the fact that partitioning an uncountable set into countably many parts yields at least one uncountable part. The key idea in our proof can be seen as a quantitative version of this infinite pigeonhole principle: if the uncountable set is endowed with a non-null measure and the parts are measurable, then at least one of the parts must have positive measure. Preliminaries Throughout this short note, all graphs are assumed to be connected and simple unless explicitly stated otherwise. Moreover we assume all graphs to be countable in order to avoid set-theoretical subtleties which could arise due to the fact that the class of all graphs is not a set. As usual, denote by V (G) and E(G) the vertex and edge set of a graph G respectively. An embedding of a graph G into a graph H is a map ι : V (G) → V (H) which preserves adjacency, that is, uv ∈ E(G) =⇒ ι(u)ι(v) ∈ E(H). A strong embedding is an embedding ι which additionally preserves non-adjacency, that is, uv ∈ E(G) ⇐⇒ ι(u)ι(v) ∈ E(H). Note that G is a subgraph of H if and only if there is an embedding of G into H, and G is an induced subgraph of H if and only if there is a strong embedding of G into H. A graph class G is a set of graphs such that G ∈ G whenever G is isomorphic to some G ′ ∈ G. We say that a graph H (strongly) contains a graph class G, if every G ∈ G has a (strong) embedding into H. A graph G ∈ G which (strongly) contains G is called (strongly) universal for G. A rooted graph is a pair (G, v) where G is a graph and v ∈ V (G). All of the above definitions carry over to rooted graphs, the only difference is that an embedding of a rooted graph (G, v) into a rooted graph (H, w) must always map the root v of G to the root w of H. For a graph class G, we define the associated rooted graph class by G • = {(G, v) | G ∈ G, v ∈ V (G)}, in other words, G • is the class of all rooted graphs whose underlying graph lies in G. Any class of rooted graphs can be endowed with a metric by letting d((G, v), (G ′ , v ′ )) = exp(− max{r | B G (v, r) is isomorphic to B G ′ (v ′ , r)}), where B G (v, r) as usual denotes the ball in G with radius r and centre v. A class G of locally finite graphs is closed if G • is a closed subset of the class G • ∞ of all locally finite, rooted graphs with respect to the topology induced by the metric given above. We remark that this definition of closedness is equivalent to the one given in the introduction. We slightly abuse notation, and extend the definition of the tree T (G) given in the introduction to rooted graph classes as follows. The vertex set of T (G) is B(G) := {B G (v, r) | (G, v) ∈ G, r ∈ N 0 } where B G (v, r) and B G ′ (v ′ , r) are considered the same if there is an isomorphism between them which maps v to v ′ . Note that isomorphic balls with different radii are considered different, for instance given a finite rooted graph (G, v) the balls B G (v, r) are distinct elements of B(G) although they are isomorphic for all but finitely many r. Connect B G (v, r − 1) and B G (v, r) by an edge for every (G, v) ∈ G and r ∈ N. For a class G of (unrooted) graphs, the definition of T (G) from the introduction coincides with the tree T (G • ) corresponding to the associated rooted graph class. We treat T (G) as a rooted tree with root B 0 = B G (v, 0); note that the definition of B 0 does not depend on the specific choice of (G, v) ∈ G because B G (v, 0) is the trivial one vertex graph for every choice of (G, v). We call B ∈ B(G) an ancestor of B ′ ∈ B(G) and B ′ a descendant of B, if B lies on the B 0 -B ′ -path in T (G). Note that in this case there is (G, v) ∈ G and i ≤ j such that B is isomorphic to B G (v, i) and B ′ is isomorphic to B G (v, j). The tree T (G) is closely linked to the topology of G induced by the metric defined above. Note that any element (G, v) of a rooted graph class G corresponds to a one-way infinite path (B 0 , B 1 , B 2 , B 3 , . . . ) in T (G) where each B i is isomorphic to B G (v, i); if we didn't treat isomorphic balls of different radii as different objects, paths corresponding to graphs of finite diameter would be finite. Mapping each graph to the corresponding path gives a homeomorphism between G and a subset of the end space (see [4] for an introduction) of T (G) and the results given in the remainder of this section can be seen as straightforward consequences of this homeomorphism. We provide proofs of these results for the convenience of the reader. The first result concerns the closure of a graph class G. Note that from an infinite path p = (B 0 , B 1 , B 2 , B 3 , . . . ) starting at the root of T (G) we can construct a rooted graph (G p , v p ) as follows: Find (G i , v i ) ∈ G with B i = B G i (v i , i), take a disjoint union of all B G i (v i , i) and identify B G i−1 (v i−1 , i − 1) and B G i (v i , i − 1) via an automorphism-such an automorphism must exist, otherwise there would not be an edge connecting B i−1 to B i . Let v p be the vertex obtained from identifying all centres v i . Lemma 2.1. Let G be a class of countable rooted graphs. A rooted graph (G, v) is in the closure G of G if and only if it is of the form (G p , v p ) for some infinite path p in T (G) as above. Proof. If p is an infinite path as above, then (G p , v p ) satisfies B Gp (v p , i) = B i . Since there is a graph (G i , v i ) ∈ G such that B i = B G i (v i , i) and hence d((G p , v p ), (G i , v i )) ≤ exp(−i) for every i we conclude that (G p , v p ) ∈ G. Conversely, if (G, v) is contained in the closure, then for any i ∈ N there must be a graph ( G i , v i ) ∈ G such that d((G, v), (G i , v i )) ≤ exp(−i), and the sequence of B G i (v i , i) gives the desired infinite path in T (G). The second result provides a characterisation of compact and σ-compact graph classes; recall that a topological space is called σ-compact if it can be covered by countably many compact subsets, and let T ∞ denote the tree in which every vertex has countably infinite degree. Proof. For the first statement, note that if T (G) is not locally finite, then there is some i ∈ N such that there are infinitely many non-isomorphic B G (v, i) with (G, v) ∈ G. Hence there is an infinite cover of G whose elements are pairwise disjoint open balls of radius exp(−i + 1). This cover clearly has no finite sub-cover, hence G is not compact. For the converse implication, recall that a metric space is compact if and only if it is sequentially compact. Let (G i , v i ) i∈N be a sequence of graphs in G. Since T (G) is locally finite there are only finitely many non-isomorphic k-balls in B(G) for every k. Hence, we can inductively find subsequences , k) and B G j (v j , k) are isomorphic for any pair i, j ∈ I k . Pick an increasing function f (k) : (G i , v i ) i∈I k with N ⊇ I 1 ⊇ I 2 ⊇ . . . such that B G i (v iN → N such that f (k) ∈ I k for every k. The sequence B G f (k) (v f (k) , k) forms an infinite path p in T (G). Since G is closed, the rooted graph (G p , v p ) defined by this path lies in G by Lemma 2. 1. Moreover d((G f (k) , v f (k) ), (G p , v p )) ≤ exp(−k), so (G f (k) , v f (k) ) k∈N is a convergent subsequence of (G i , v i ) i∈N . Now let us turn to the second statement of the lemma. Let G be σ-compact and pick a cover G = i∈N H i where each H i is compact. Note that T (H i ) is the subtree of T (G) induced by B(H i ), and that that this subtree is locally finite because H i is compact. Assume for a contradiction that T (G) contains a subdivision of T ∞ . Call a vertex B a branch point, if it corresponds to a vertex of the subdivision of T ∞ . Note that from every branch point B we can find infinitely many edge disjoint paths to other branch points, and that all but one of these branch points are descendants of B. Since T (H i ) is locally finite, the first edge of at least one of these paths does not lie in T (H i ). Connectedness of T (H i ) implies that the other endpoint of this path is a descendant of B which does not lie in B(H i ). Let B 1 be an arbitrary branch point, and for every i ∈ N let B i+1 be a branch point which is a descendant of B i and does not lie in B(H i ). Let p be an infinite path starting at B 0 and passing through every B i . The graph (G p , v p ) is contained in G by Lemma 2.1, but it is not contained in any H i since there is some r i such that B Gp (v p , r i ) is isomorphic to B i / ∈ B(H i ). Now assume that T (G) does not contain a subdivision of T ∞ . We construct a decomposition of G into countably many compact sets by transfinite induction. In step α of the construction, assume that we have defined compact sets H β ⊆ G for every β < α (for α = 0 this is vacuously true) and let T α = T (G \ β<α H β ). If T α has no vertex B such that the subtree induced by all descendants of B is locally finite, then T α and thus also T (G) contains a subdivision of T ∞ . Hence we can pick such a vertex B α and let H α be the set consisting of all (G, v) ∈ G which correspond to an infinite path in T α which passes through B α . The tree T (H α ) is the union of all these infinite paths in T α , it is locally finite because of our choice of B α , and hence G α is compact. Note that no graph in (G \ β<α H β ) \ H α = G \ β<α+1 H β contains a ball isomorphic to B α , hence B α is not a vertex of any T γ for any γ > α. The class G only contains locally finite graphs, so all graphs in B(G) are finite and thus T (G) has only countably many vertices. Hence the above procedure will terminate after some countable number of steps thereby yielding the desired composition of G. Graph classes with locally finite universal graphs In this section we investigate conditions for the existence of a locally finite graph containing a given graph class and prove Theorem 1.1. We start by giving a more technical condition which is true even if the graph class in question is not closed. Recall that G ∞ denotes the class of connected, locally finite graphs, and consequently G • ∞ denotes the class of rooted connected, locally finite graphs. Theorem 3.1. Let G ⊆ G ∞ . The following are equivalent. 1. There is H ∈ G ∞ which strongly contains G. 2. There is H ∈ G ∞ which contains G. 3. We can decompose G • as a countable union i∈N H i such that T (H i ) is locally finite for every i ∈ N. Note that the classes H i in statement 3 are classes of rooted graphs. Indeed, the theorem becomes false if we decompose G instead of G • : even a class G consisting of a single locally finite graph can contain infinitely many non-isomorphic balls of radius 1. Proof. Condition 1 trivially implies condition 2. For the implication 2 =⇒ 3 let H ∈ G ∞ be a graph which contains G and let (v i ) i∈N be an enumeration of the vertices of H. For each vertex v i let H ′ i be the class of all rooted graphs (G, v i ) where G is a subgraph of H that contains v i . Since H is locally finite there are only finitely many subgraphs of diameter at most r containing any fixed v i , and thus T (H ′ i ) is locally finite for every i. Since H contains G we know that G • ⊆ i∈N H ′ i . Letting H i = G • ∩ H ′ i gives the desired decomposition; T (H i ) is locally finite because it is a subtree of T (H ′ i ) . For the final implication 3 =⇒ 1, we first note that it suffices to find for each i a rooted graph (H i , v i ) ∈ G • ∞ which strongly contains H i . From the disjoint union of these graphs, we can construct a connected, locally finite graph H by connecting v i to v i+1 every i ∈ N. This graph H strongly contains G because any strong embedding of a graph (G, v) ∈ H i into (H i , v i ) induces a strong embedding of H into G. For the construction of the graphs (H i , v i ) we denote by B i the set {B G (v, r) | (G, v) ∈ H i , r ∈ N}. As before, two balls B G (v, r) and B G ′ (v ′ , r) are considered the same if there is an isomorphism between them which maps v to v ′ . For B = B G (v, r) ∈ B we call B G (v, r − 1) the interior B • of B; note that this does not depend on the specific choice of (G, v) ∈ H i . For each B ∈ B we fix a strong embedding ι B : B • → B which maps the centre of B • to the centre of B. Clearly such an embedding always exists; for instance we can pick some (G, v) such that B = B G (v, r) and take the restriction of the identity map to B G (v, r − 1). Now define a graph H i as follows. Start with a disjoint union of all B ∈ B i , and for each B ∈ B i identify every v ∈ V (B • ) with ι B (v). Clearly, the central vertices of all balls in B i are identified to a single vertex of H i , this vertex v i is the designated root of r) is connected and all copies of root vertices of balls are identified. It follows from the definition of the maps ι B that any pair of vertices which is identified has the same distance from the centres of the respective balls. Consequently the vertex set of B H i (v, r) consists of the vertices of r) is finite because G ⊆ G ∞ , and there are only finitely many non-isomorphic balls of radius r since otherwise T (H i ) would not be locally finite. Thus H i is locally finite. (H i , v i ). The rooted graph (H i , v i ) is connected because each B = B G (v,all B G (v, r) for (G, v) ∈ H i . Each such B G (v, It only remains to show that any (G, v) ∈ H i has a strong embedding into (H i , v i ). For this purpose, first note that if a vertex u ∈ B is identified with a vertex u ′ ∈ B ′ a vertex v ∈ B is identified with a vertex v ′ ∈ B ′ , then uv ∈ E(B) ⇐⇒ u ′ v ′ ∈ E(B ′ ). In particular, since B G (v, r) is isomorphic to some B ∈ B i , there is a strong embedding ι r : B G (v, r) → (H i , v i ) for every r. We can construct the desired strong embedding ι : (G, v) → (H i , v i ) from these strong embeddings ι r by a standard compactness argument. Since B H i (v, r) is finite for every r, there are only finitely many ways to embed B G (v, r) into H i . Hence there is an infinite subset I 1 ⊆ N such that the restrictions of ι i and ι j to B G (v, 1) coincide for any pair i, j ∈ I 1 . Inductive application of this argument gives a set I r ⊆ I r−1 for every r ≥ 2 such that the restrictions of ι i and ι j to B G (v, r) coincide for any pair i, j ∈ I r . For x ∈ V (G) whose distance from v is at most r, we pick any i ∈ I r define ι(x) = ι i (x). We note that this does not depend on the specific choice of r or i because ι i (x) = ι j (x) for any x ∈ B G (v, r) and any pair i, j ∈ I r = s≥r I s . Since all ι i are strong embeddings, so is ι. Corollary 3.2. For every d ∈ N there is a graph H ∈ G ∞ strongly contains G d . Proof. There are only finitely many non-isomorphic graphs of diameter at most r in which every vertex has degree less than d. Hence the tree T (G d ) = T (G • d ) is locally finite and Theorem 3.1 concludes the proof. Corollary 3.3. There is a graph H ∈ G ∞ which strongly contains G = d∈N G d . Proof. Decompose G • = d∈N G • d , note that T (G • d ) is locally finite, and apply Theorem 3.1. Theorem 3.1 can also be used to show that graph classes whose members have bounded growth admit a locally finite graph containing them. For example, recall that a graph G is said to have polynomial growth if there is some polynomial P and a vertex v ∈ v(G) such that B G (v, r) contains at most P (r) vertices for every r ∈ N. Corollary 3.4. There is a graph H ∈ G ∞ which strongly contains the class G of all graphs of polynomial growth. Proof. Denote by G(a, b) the class of all rooted graphs (G, v) such that B G (v, r) contains at most a · r b vertices for every r. It is not hard to see that G • = a,b∈N G(a, b). There are only finitely many non-isomorphic connected graphs on at most a · r b vertices, hence T (G(a, b)) is locally finite and we can apply Theorem 3.1. Polynomial growth in the above corollary can of course be replaced by any growth bound, as long as there is a countable set (f i ) i∈N of functions such that for every allowed growth function g there is an i such that g(x) ≤ f i (x) for every x ∈ N. While the above corollaries demonstrate that Theorem 3.1 is useful for showing that there is a graph H ∈ G ∞ containing a certain graph class, the theorem is not quite as useful for showing the non-existence of such a H. For closed graphs the following theorem (which implies Theorem 1.1) gives a condition which is easier to falsify. Theorem 3.5. Let G ⊆ G ∞ be a closed graph class. The following are equivalent. 1. There is H ∈ G ∞ which contains G. 2. G • is σ-compact. 3. T (G) contains no subdivision of T ∞ . Proof. Statements 2 and 3 are equivalent by Lemma 2.2, hence it suffices to show the equivalence of 1 and 2. For the implication 1 =⇒ 2 we first note that by Theorem 3.1 (3) we can decompose G • into countably many sets H i such that T (H i ) is locally finite. Since G • is closed, this implies that G • = i∈N H i , where H i denotes the closure of H i in H • ∞ . Lemma 2.2 implies that G • is σ-compact as claimed. For the converse implication 2 =⇒ 1 let G • = i∈N H i where every H i is compact. By Lemma 2.2, T (H i ) is locally finite for every i and Theorem 3.1 (3) finishes the proof. As a corollary to the above theorem we obtain a result due to de Brujin mentioned in the introduction. Proof. Note that G ∞ is closed. Every ball of radius n in a locally finite graph can be extended in infinitely many ways to a ball of radius n + 1 in a locally finite graph. Hence T (G ∞ ) = T ∞ and by Theorem 3.5 there is no H ∈ G ∞ containing G ∞ . Similar arguments can be made for many other graph classes. For instance, the exact same proof also shows the following result. Corollary 3.7. There is no graph H ∈ G ∞ containing the class of all locally finite, connected, planar graphs. Graphs with bounded degrees By Corollary 3.2, there is a graph H ∈ G ∞ containing G d . On the other hand, it is known (see for instance [9]) that there is no H ∈ G d containing G d . This raises the natural question whether there is any D < ∞ such that there is a graph H ∈ G D containing G d . Theorem 1.2 from the introduction states that such a D does not even exist for G 4 ; the remainder of this section is dedicated to the proof of this theorem. vertex v and let ǫ = P(S(v)) > 0. Since S(v) ⊆ S(v, i) we have ǫ ≤ P(S(v, i)) = P   σ∈S 2 i S(v, i) ∩ S σ   = σ∈S 2 i P(S(v, i) ∩ S σ ) for every i ∈ N. Each summand on the right hand side is bounded above by P(S σ ) = 1 (2 i )! , hence at least ǫ(2 i )! of the summands must be non-zero. We claim that n(v, i) is at least half as large as the number of non-zero summands in the above sum. To this end, it suffices to show that G s (i) and G t (i) are isomorphic if and only if the restrictions of s and t to [2 i ] either coincide or are inverses of one another. Note that the path −2 i+1 , −2 i+1 + 1, . . . 2 i+1 is the unique spanning path of G s (i) with no two consecutive vertices of degree 3. Any isomorphism from G s (i) to G t (i) must map this path to its counterpart in G t (i), and there are precisely two ways of doing this. We have thus shown that ǫ 2 · (2 i )! ≤ n(v, i) ≤ ∆ 5·2 i for arbitrary i. For large enough i this yields a contradiction since the left hand side asymptotically grows faster than the right hand side. Lemma 2. 2 . 2Let G ⊆ G ∞ be a closed class of rooted graphs. 1. G is compact if and only if T (G) is locally finite 2. G is σ-compact if and only if T (G) does not contain a subdivision of T ∞ . Corollary 3.6. G ∞ contains no universal element. Proof of Theorem 1.2. Assume that there is a countable graph H with maximum degree ∆ containing G 4 .We define a subfamily of G 4 as follows. Let [n] = {1, 2, . . . , n} and [m, n] = {m, m + 1, . . . , n}. Let S be the set of all bijective functions s :For each s ∈ S let G s be the graph with vertex set Z and edges connecting n to n + 1 for every n ∈ Z as well as edges connecting 2n to −2s(n) for every n ∈ N; the reason we only connect even vertices is to get rid of unwanted automorphisms. Denote by G s (i) the subgraph of G s induced by [−2 i+1 , 2 i+1 ]. For every vertex v ∈ V (H), let S(v, i) be the set of all s ∈ S such that there is an embedding of G s (i) into H mapping 0 to v.Let n(v, i) the number of non-isomorphic G s (i) with s ∈ S(v, i). Note that for any embedding of G s (i) which maps 0 to v there are at most ∆ 2 i+1 possibilities to embed the path induced by [0, 2 i+1 ], and similarly for the path induced by [−2 i+1 , 0]. Once the paths are embedded, the image of 2n has fewer than ∆ possible neighbours in the embedding of the path induced by [−2 i+1 , 0]. Hence any embedding of the two paths extends to at most ∆ 2 i embeddings of graphs G s (i) (some of which may be isomorphic). In total, this shows that n(v, i) ≤ ∆ 5·2 i for any v ∈ V (H) and every i ∈ N.Define the set S(v) := i∈N S(v, i). We will show that at least one of the sets S(v) must be large enough to derive a contradiction to the above bound. Clearly, if there is an embedding of G s into H, then s is contained in S(v) where v is the image of 0 under this embedding. Since by assumption every G s is a subgraph of H, this implies that S = v∈V (H) S(v). It immediately follows that one of the S(v) must be uncountable. Unfortunately, this is not sufficient for our purpose as we need some control over the sets S(v, i); this is achieved by endowing S with a probability measure.Note that any s ∈ S bijectively maps [2 i−1 + 1, 2 i ] to itself. If we let s i be the permutation that s induces on [2 i−1 + 1, 2 i ], then the map s → (s i ) i∈N is a bijection between S and i∈N S 2 i where S n denotes the symmetric group on n elements. For σ ∈ S 2 i , we let S σ = {s ∈ S | s i = σ}. It is well known (for instance by the Kolmogorov extension theorem) that there is a probability measure on i∈N S 2 i whose projection on every factor is the uniform probability measure. By the above bijection, this defines a probability measure P on S such that P(S σ ) = 1 (2 i )! for every σ ∈ S 2 i . The sets S(v, i) can be written as finite unions of intersections of finitely many S σ , hence they are measurable with respect to P, and thus the same is true for the sets S(v). By subadditivity of the probability measure we have Universal graphs with a forbidden subtree. G Cherlin, S Shelah, J. Comb. Theory, Ser. B. 973G. Cherlin and S. Shelah. Universal graphs with a forbidden subtree. J. Comb. Theory, Ser. B, 97(3):293-333, 2007. Universal graphs with a forbidden subgraph: block path solidity. G Cherlin, S Shelah, Combinatorica. 363G. Cherlin and S. Shelah. Universal graphs with a forbidden subgraph: block path solidity. Combinatorica, 36(3):249-264, 2016. Graphs omitting a finite set of cycles. G Cherlin, N Shi, J. Graph Theory. 213G. Cherlin and N. Shi. Graphs omitting a finite set of cycles. J. Graph Theory, 21(3):351-355, 1996. Locally finite graphs with ends: A topological approach. I: Basic theory. R Diestel, Discrete Math. 31115R. Diestel. Locally finite graphs with ends: A topological approach. I: Basic theory. Discrete Math., 311(15):1423-1447, 2011. On the existence of countable universal graphs. Z Füredi, P Komjáth, J. Graph Theory. 251Z. Füredi and P. Komjáth. On the existence of countable universal graphs. J. Graph Theory, 25(1):53-58, 1997. Universality in minor-closed graph classes. T Huynh, B Mohar, R Šámal, C Thomassen, D R Wood, arXiv:2109.00327PreprintT. Huynh, B. Mohar, R.Šámal, C. Thomassen, and D. R. Wood. Universality in minor-closed graph classes. 2021. Preprint, arXiv:2109.00327. Some remarks on universal graphs. P Komjáth, Discrete Math. 1991-3P. Komjáth. Some remarks on universal graphs. Discrete Math., 199(1-3):259-265, 1999. Some universal graphs. P Komjáth, A H Mekler, J Pach, Isr. J. Math. 642P. Komjáth, A. H. Mekler, and J. Pach. Some universal graphs. Isr. J. Math., 64(2):158-168, 1988. Universal graphs without large bipartite subgraphs. P Komjáth, J Pach, Mathematika. 31P. Komjáth and J. Pach. Universal graphs without large bipartite subgraphs. Math- ematika, 31:282-290, 1984. Universal graphs and universal functions. R Rado, Acta Arith. 9R. Rado. Universal graphs and universal functions. Acta Arith., 9:331-340, 1964.
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[ "Minimal constraints for Maximum Caliber analysis of dissipative steady state systems", "Minimal constraints for Maximum Caliber analysis of dissipative steady state systems" ]
[ "Luca Agozzino \nLaufer Center for Physical and Quantitative Biology\nStony Brook University\n\n\nDepartment of Physics and Astronomy\nStony Brook University\n\n", "Ken Dill \nLaufer Center for Physical and Quantitative Biology\nStony Brook University\n\n\nDepartment of Physics and Astronomy\nStony Brook University\n\n\nDepartment of Chemistry\nStony Brook University\n\n" ]
[ "Laufer Center for Physical and Quantitative Biology\nStony Brook University\n", "Department of Physics and Astronomy\nStony Brook University\n", "Laufer Center for Physical and Quantitative Biology\nStony Brook University\n", "Department of Physics and Astronomy\nStony Brook University\n", "Department of Chemistry\nStony Brook University\n" ]
[]
Maximum Caliber (Max Cal) is purported to be a general variational principle for Non-Equilibrium Statistical Physics (NESP). But recently, Jack and Evans and Maes have raised concerns about how Max Cal handles dissipative processes. Here, we show that the problem does not lie in Max Cal; the problem is in the use of insufficient constraints. We also present an exactly solvable single-particle model of dissipation, valid far from equilibrium, and its solution by Maximum Caliber. The model illustrates how the influx and efflux of work and heat into a flowing system alters the distribution of trajectories. Maximum Caliber is a viable principle for dissipative systems.
10.1103/physreve.100.010105
[ "https://export.arxiv.org/pdf/1904.11426v2.pdf" ]
195,798,682
1904.11426
cbe107853cbfe704be8373f0e9016f853f019773
Minimal constraints for Maximum Caliber analysis of dissipative steady state systems 3 Jul 2019 Luca Agozzino Laufer Center for Physical and Quantitative Biology Stony Brook University Department of Physics and Astronomy Stony Brook University Ken Dill Laufer Center for Physical and Quantitative Biology Stony Brook University Department of Physics and Astronomy Stony Brook University Department of Chemistry Stony Brook University Minimal constraints for Maximum Caliber analysis of dissipative steady state systems 3 Jul 2019 Maximum Caliber (Max Cal) is purported to be a general variational principle for Non-Equilibrium Statistical Physics (NESP). But recently, Jack and Evans and Maes have raised concerns about how Max Cal handles dissipative processes. Here, we show that the problem does not lie in Max Cal; the problem is in the use of insufficient constraints. We also present an exactly solvable single-particle model of dissipation, valid far from equilibrium, and its solution by Maximum Caliber. The model illustrates how the influx and efflux of work and heat into a flowing system alters the distribution of trajectories. Maximum Caliber is a viable principle for dissipative systems. Since the seminal work of Clausius and Boltzmann in the nineteenth century, predicting material equilibria has been based on the concept of entropy and its maximization. There has been a search for a more general variational principle that could also apply to nonequilibria, especially in the far-from-equilibrium regime. A good candidate has been the Principle of Maximum Caliber (Max Cal) [1][2][3][4][5][6], which is a Maximum-Entropy-like principle for inferring distributions over pathways and rate distributions of kinetic processes. Recently, concerns have been raised about whether Maximum Caliber handles dissipation properly. We address those here, and show that Max Cal can handle dissipation properly when given appropriate constraints. Maximum Caliber is a method of inference about probability distributions over pathways or trajectories, in contrast to Maximum Entropy which infers distributions over microstates. Max Cal begins with a model of the accessible trajectories, X = {ξ(t 0 ), ξ(t 1 ), ξ(t 2 )...} of values ξ at different times t. Max Cal infers the probability p(X) of observing trajectory X within trajectory space {X} by maximizing the path entropy S = − X p(X) log p(X) g(X) ,(1) where the function g(X) is some reference/prior distribution in the absence of constraints. Now, in the simple situation of non-dissipative dynamics of a single dynamical quantity J(X), for which the average, J = dXp(X)J(X)(2) is known, the trajectory populations are obtained using the method of Lagrange Multipliers [5,6]. DISSIPATIVE DYNAMICS REQUIRES MORE CONSTRAINTS Two recent papers [7,8] assert that Max Cal will fail in some cases. Jack and Evans (JE) [7] show that applying Max Cal with a single constraint to dissipative systems leads to the apparently inconsistent result of having no dissipation; Maes (M) [8] asserts that problems whenever Max Cal is applied in cases of a time-symmetric component in one of the constraints. Here we clarify that these are not problems of the Principle of Maximum Caliber; these are problems of application of incomplete or incorrect constraints. We first address the JE situation. Consider a dissipative system where a current J(X) flows in conjunction with some finite amount of work δw(X) done on the system and a heat flow δq(X) out. Assume that the statistical ensemble of all trajectories contains also those trajectories that are related to each other through a time-reversal transformation T and a space-reflection transformation P (it can also refer to reflection along only one of the physical coordinates [7]). For the right choice of current-generating force, the resulting current will always be antisymmetric under both time reversal and space-reflection transformations, so we assume that the forces acting on the system are of this type (an example is a shear stress, which generates a current with such property). As a consequence, under a combined PT transformation, the current will be identical to the untransformed current [7]. Now consider the exchange of heat and work between the system and the external bath. This will be antisymmetric under time reversal. Running time backwards would reverse all three: the flow, the work, and the heat along the trajectory. But, it will be invariant under space reflection. No matter whether a force drives a current in a forward or backward direction along a trajectory, an identical amount of heat will be dissipated. This is not an assertion of reversibility of heat transfer; that would violate the Second Law of Thermodynamics. We are considering here only a single non-equilibrium trajectory, not arXiv:1904.11426v2 [cond-mat.stat-mech] 3 Jul 2019 a Second Law average over all trajectories. Rather, it just means that if a trajectory has heat flowing into the system, its time-reversed trajectory has heat flowing out. As an example, consider a particle with mass m sliding on a surface with friction coefficient φ and initial velocity v. The total energy dissipated through the process of slowing down until stopping is equal to the total kinetic energy E k = 1/2mv 2 of the particle, which will increase the temperature of the surface by ∆T = E k /C, where C is the surface's heat capacity. The time-reversed process would be the following: heating up the surface by exactly ∆T and wait for the thermal energy to spontaneously transform back into kinetic energy, accelerating the particle back to velocity v. This reverse process is extremely unlikely. We illustrate a calculation of this probability in Max Cal below. In general, for a dissipative system, a trajectory X will have some current flow J(X), at the same time as work δw(X) performed on it, some heat dissipation δq(X) out of it. In this case, the PT-reversed trajectory, PTX, would have heat δq(PTX) = −δq(X) going into the system and work δw(PTX) = −δw(X) done on the external environment, because a space reflection transformation does not change the heat/work flow, but time reversal does. The probability of the transformed trajectory PTX should be much lower than of the untransformed trajectory X for macroscopic currents, although we know from fluctuation theorems that for very small currents they can become comparable [9,10]. The result below agrees with such predictions. For a dissipative steady state (DSS) the internal energy is unchanging with time, ∆U = δw + δq, because in the steady state, the heat out must equal the work in 1 . The argument of Jack and Evans is straightforward [7]. First, they correctly note that if the only constraint is on J(X) , (eq. 2), then maximizing the Caliber (i.e. the path entropy subject to the constraint) gives the following probability of trajectory X : p(X) = e µJ(X) Z(µ) ,(3) where Z(µ) = e µJ(X) is the sum of weights over all paths. Second, since the flux is PT invariant, substitution of J(X) = J(PTX) into Eq 3 gives the result that the probabilities must be PT invariant, p(PTX) = p(X).(4) JE argue that such systems are not dissipative, because δq = 0, which they show as follows: 1 Note, our convention is that energy going into the system is defined as positive. δq = dXp(X)δq(X) = 1/2 dX (p(X)δq(X) + p(PTX)δq(PTX)) = 1/2 dXp(X)(δq(X) − δq(X)) = 0,(5) where the second line is obtained by considering that the Jacobian of a PT transformation equals 1. JE conclude from this that Maximum Caliber cannot handle systems, such as a sheared fluid, that are dissipative. On the contrary, we show below that the problem above is the use of only a single constraint, namely J(X) . This misses the essentiality of the coupling of the flow J inside the system to the work and heat flows into and out of the system. The latter require additional constraints. THE NUMBER OF CONSTRAINTS MUST AT LEAST EQUAL THE NUMBER OF INDEPENDENT FLOW VARIABLES To illuminate the problem, consider the corresponding situation in equilibrium thermodynamics. The equilibrium entropy can be expressed as a function S = S(U, V, N ) of three independent extensive variables -energy, volume and particle number. If all three independent variables are free to change in a process, you cannot adequately specify the state of the system with only a single Lagrange multiplier, say the pressure p; you must also specify the temperature T and chemical potential µ. You need a Lagrange multiplier for every independent variable. In dissipative dynamical systems too, there are multiple independent variables. You can specify an average flow rate J , but dissipative systems also entail heat and work flows in and out, and those can affect the trajectory distribution. For example, you can achieve a given average particle flow rate in multiple ways, such as increasing the work done on the particle in a medium of increasing viscosity that dissipates more heat. Predicting the trajectory distribution in dissipative systems requires knowing the heat and work rates, not just the particle flow rate. 2 For example, consider particles flowing along the axis of a tube, with an average current of J(X) = J. That particle flow can be independent of the rate of work flow J w (X) and heat flow J q (X) into and out of the tube. Some situations will reduce these 3 variables to fewer; other situations will not. First, consider any steady-state flow, dissipative or not. By definition, the total internal energy will be unchanging with time, ∆U = 0. So, it follows from the First Law that δq = −δw. Thus, in steady-state flows, the heat current must equal the work current, J q (X) = − J w (X)(7) where our convention is that current flows into the system are defined as positive. Now, in a non-dissipative steady state (nDSS), we have J q (X) = − J w (X) = 0, leaving us only one independent variable, J. However, in a dissipative steady state (DSS), energy must continuously enter the system in order to sustain the current J, so now we have 3 constraints, J(X) = J (8) J w (X) + J q (X) = 0 (9) 1 2 J w (X) − J q (X) = J E(10) where J E is the energy influx rate. We note two points here. First, if were to use only a single constraint J for a DSS, as in the JE argument, it is tantamount to setting J E = 0 above, thus effectively asserting that heat and work flow in and out are both zero, and thus that the system is, by definition, not dissipative. Second, near equilibrium and for non-steady states, dissipation J q is proportional to the current J , so in that case a single constraint can be sufficient to describe the system [11]. Therefore, for steady states, arbitrarily far from equilibrium, only non-dissipative systems can be described when only a single constraint, J , is specified. For DSS situations, with constraint eqs 10) above, the expression for Caliber is: C = − dXp(X) ln p(X) (11) −α dXp(X) − 1 (12) −µ dXp(X)J(X) − J (13) −ν dXp(X)J w (X) − J E (14) −λ dXp(X)J q (X) + J E(15) where we chose here, for simplicity, to define each current individually instead of constraining the sum and the difference. Maximizing Caliber gives the trajectory probabilities as p(X) = e µJ(X)+νJw(X)+λJq(X) Z(µ, ν, λ)(16) where Z = e µJ(X)+νJw(X)+λJq(X) . This Max Cal formulation shows that reverse trajectories in dissipative processes are unlikely for large currents. Using the PT transformation, we can calculate the relative probability that a system would absorb heat from the environment (and produce work): p(PTX) p(X) = e −2(νJw(X)+λJq(X)) .(17) This fluctuation relation shows that 'wrong-way' paths, which take up heat in dissipative flows, become exponentially improbable with increasing current, as they should. If the only constraint here were on J , as in JE, then J q = J w = 0 and wrong-way flows would be predicted to be much more probable. The Max Cal procedure gives the distribution of all the trajectories. On the one hand, it uses as an input constraint, the heat uptake J q (X) averaged over all the trajectories: δq = J q ∆t = ∆t dXp(X)J q (X).(18) On the other hand, Max Cal then gives as a prediction the higher moments, such as the mean-square fluctuations of the heat: In this section, we illustrate with a concrete model. Consider one particle moving inside a 1D conduit. The particle is in contact with an external thermal bath with which it can exchange heat. The particle can also interact with a conveyor belt that performs work from outside to boost the particle's velocity; see Fig. 1. δq 2 = ∆t 2 dXJ 2 q (X)p(X).(19) A trajectory X is a series of N steps, each one of which takes time ∆t. In each time step, the particle experiences one of three possibilities: (i) it increases or decreases its velocity by ∆v, by collision with the belt, (ii) it increases or decreases its velocity by ∆u by exchanging heat with the bath, or (iii) it undergoes no change in velocity in that T q(X) < l a t e x i t s h a 1 _ b a s e 6 4 = " H w b o Z P Y j M y e N G 0 2 T 5 D q 4 b z G G 6 Y k = " > A A A B 6 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B a h X k q i g h 4 L X j x W M G 2 h D W W z 3 b R L d z d x d y O U 0 L / g x Y M i X v 1 D 3 v w 3 b t o c t P X B w O O 9 G W b m h Q l n 2 r j u t 1 N a W 9 / Y 3 C p v V 3 Z 2 9 / Y P q o d H b R 2 n i l C f x D x W 3 R B r y p m k v m G G 0 2 6 i K B Y h p 5 1 w c p v 7 n S e q N I v l g 5 k m N B B 4 J F n E C D a 5 9 F j v n g + q N b f h z o F W i V e Q G h R o D a p f / W F M U k G l I R x r 3 f P c x A Q Z V o Y R T m e V f q p p g s k E j 2 j P U o k F 1 U E 2 v 3 W G z q w y R F G s b E m D 5 u r v i Q w L r a c i t J 0 C m 7 F e 9 n L x P 6 + X m u g m y J h M U k M l W S y K U o 5 M j P L H 0 Z A p S g y f W o K J Y v Z W R M Z Y Y W J s P B U b g r f 8 8 i p p X z S 8 y 4 Z 7 f 1 V r N o s 4 y n A C p 1 A H D 6 6 h C X f Q A h 8 I j O E Z X u H N E c 6 L 8 + 5 8 L F p L T j F z D H / g f P 4 A T J y N v A = = < / l a t e x i t > w(X) < l a t e x i t s h a 1 _ b a s e 6 4 = " z l p + + L R n F W N S l X o X k D s F p m l N V m M = " > A A A B 6 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B a h X k q i g h 4 L X j x W M G 2 h D W W z 3 b R L d z d h d 6 O U 0 L / g x Y M i X v 1 D 3 v w 3 b t o c t P X B w O O 9 G W b m h Q l n 2 r j u t 1 N a W 9 / Y 3 C p v V 3 Z 2 9 / Y P q o d H b R 2 n i l C f x D x W 3 R B r y p m k v m G G 0 2 6 i K B Y h p 5 1 w c p v 7 n U e q N I v l g 5 k m N B B 4 J F n E C D a 5 9 F T v n g + q N b f h z o F W i V e Q G h R o D a p f / W F M U k G l I R x r 3 f P c x A Q Z V o Y R T m e V f q p p g s k E j 2 j P U o k F 1 U E 2 v 3 W G z q w y R F G s b E m D 5 u r v i Q w L r a c i t J 0 C m 7 F e 9 n L x P 6 + X m u g m y J h M U k M l W S y K U o 5 M j P L H 0 Z A p S g y f W o K J Y v Z W R M Z Y Y W J s P B U b g r f 8 8 i p p X z S 8 y 4 Z 7 f 1 V r N o s 4 y n A C p 1 A H D 6 6 h C X f Q A h 8 I j O E Z X u H N E c 6 L 8 + 5 8 L F p L T j F z D H / g f P 4 A V c a N w g = = < / l a t e x i t > v(X) < l a t e x i t s h a 1 _ b a s e 6 4 = " Z Z N 2 e r j F p D / C / J m H Z D k G 1 Y T d 0 3 s = " > A A A B + H i c b V D L S g M x F L 3 j s 9 Z H R 1 2 6 C R a h b s q M C r o s u n F Z w T 6 g H U o m T d v Q T D I k m U I d + i V u X C j i 1 k 9 x 5 9 + Y a W e h r Q c C h 3 P u 4 d 6 c M O Z M G 8 / 7 d t b W N z a 3 t g s 7 x d 2 9 / Y O S e 3 j U 1 D J R h D a I 5 F K 1 Q 6 w p Z 4 I 2 D D O c t m N F c R R y 2 g r H d 5 n f m l C l m R S P Z h r T I M J D w Q a M Y G O l n l v q S m t n 6 X Q y q 7 T P e 2 7 Z q 3 p z o F X i 5 6 Q M O e o 9 9 6 v b l y S J q D C E Y 6 0 7 v h e b I M X K M M L p r N h N N I 0 x G e M h 7 V g q c E R 1 k M 4 P n 6 E z q / T R Q C r 7 h E F z 9 X c i x Z H W 0 y i 0 k x E 2 I 7 3 s Z e J / X i c x g 5 s g Z S J O D B V k s W i Q c G Q k y l p A f a Y o M X x q C S a K 2 V s R G W G F i b F d F W 0 J / v K X V 0 n z o u p f V r 2 H q 3 L t N q + j A C d w C h X w 4 R p q c A 9 1 a A C B B J 7 h F d 6 c J + f F e X c + F q N r T p 4 5 h j 9 w P n 8 A t D e T G g = = < / l a t e x i t > FIG. 1. A particle in a dissipative system. The particle can receive energy from the belt or from the thermal bath, but it can also transmit energy to the belt by hitting it or to the thermal bath, by friction on the walls of the conduit. time step. A full trajectory is a string of such states: up, up, stay, up, down, up, .... for example. The quantities ∆v and ∆u are not limited to a fixed value, but can be anything within a given range. The trajectory for a given particle has three identifying quantities. The average velocity of the particle along the trajectory v(X), the work done on the particle by the belt w(X) and the heat absorbed by the particle from the thermal bath q(X). As a convenient convention, we take both w(X) and q(X) to be positive when the energy flows from the external environment to the particle, so for the work, this convention is the opposite with respect to the one used in thermodynamics. Note that for the average velocity of a given trajectory v(X) we have used the overbar symbol to distinguish it from a trajectory-ensemble average; v(X) is just the average velocity maintained by the particle in a specific trajectory, whereas we would use the symbol v(X) ≡ p(X)v(X) to refer to the trajectory-ensemble average, hence averaged over all the possible trajectories. This allows us to enforce some minimal constraints which identify a DSS without ambiguity. The constraints are the following: w(X) = E in (20) q(X) = −E in (21) v(X) = V(22) where E in is the average work input (or negative heat output). The particle starts at time t = 0 with velocity v 0 . So, a given trajectory X can be specified by an initial velocity and a sequence of changes in velocities: X = {v 0 , ξ 1 , ξ 2 , ..., ξ N −1 }(23) where ξ j = ∆v j or ∆u j , where j is an index of the time step, depending on which processes occurred along the given trajectory. Now, Maximum Caliber gives the probability of a given DSS trajectory as p(X) = p(v 0 , ξ 1 , ξ 2 , ..., ξ N −1 ) = e νw(X)+λq(X)+µv(X) Z (24) All the functions w(X), q(X) and v(X) can be expressed in terms of the particular sequence of velocity changes in trajectory X. Now, under a PT transformation, each trajectory function is transformed as follows: w(PTX)= −w(X) (25) q(PTX) = −q(X) (26) v(PTX) = v(X).(27) This is because both heat and work are invariant under space reflection, but are anti-symmetric under time reversal. Therefore, the ratio between the PT-transformed and untransformed trajectory is p(PTX) p(X) = e −2[νw(X)+λq(X)](28) which does not equal 1, except in the non-dissipative case that the trajectory does not involve any energy exchange 3 . For a general N -step process, the functional form is too complex for analytical solution, due to the non-linear relation between velocity and kinetic energy: the change in velocity at time step n will depend upon all the changes in velocity at time steps n − 1, n − 2, ..., 0. The partition function can be calculated numerically in that case, and the values of the Lagrange multipliers can be tuned to make sure that constraint averages are satisfied. In the next section we will show how to solve the problem analytically in an even simpler case. Simplified trajectories with only 3 time steps Now, we can obtain a closed-form expression if we further simplify the model above to just 3 total time steps. Any trajectory is now described by the vector X = {v 0 , ξ 1 , ξ 2 }(29) where v 0 is the initial velocity of the particle (first step), ξ 1 is the change in velocity in the second step and ξ 2 is the change in velocity in the third step. At steps 2 and 3, the velocity can either remain the same (ξ i = 0) or change by interaction with the moving belt (ξ i = ∆v i ) or change by heat exchange (ξ i = ∆u i ) (see Supporting Material for details). Again, the functional form of the probability is given by Eq. 24, but now just for the short-trajectories of Eq. 29. The Max Cal dynamical partition function is obtained by computing the following sum over all the small number of trajectories X: Z = X e νw(X)+λq(X)+µv(X)(30) In order to correctly express the form of the sum in Eq. 30 we take into account the fact that at every step we are assuming that only one type of velocity change is possible, either heat driven or work driven. We can compute the change in particle velocity that is due to work or heat exchange. In the Supporting Material, we calculate the sum in Eq. 30 and solve Eqs. 20-22, to obtain the following values of the Lagrange multipliers: µ 3η V (31) ν 2E in (32) λ − 2E in(33) where η = V 2 /V 2 max and = ∆V Q /∆V W . V max is the maximum velocity that the conduit can withstand, ∆V Q is the maximum change in velocity due to heat exchange and ∆V W the one due to heat exchange. In order to obtain this result, we have assumed that the measured velocity V is much smaller than the maximum rate V max , so η << 1. We also assumed that the maximum change in velocity due to work is much larger than the one due to heat, because work is always directed in a specific direction, so this means << 1. Such assumptions, although not necessary to solve the problem, make it easier to obtain an analytical expression for the Lagrange multipliers. The trajectory probabilities in this 3-step model are: p(X) = 1 Z exp w(X) − q(X) 2E in + 3η v(X) V(34) Eq 34 computes the probability of any pathway X for fixed values of the two observables, E in and V . Using 3 constraints (blue plane) it is able to capture the difference between trajectories with different energy sources, which is not possible when only one constraint is used (orange plane). The coloured dots show the difference of the probability of three trajectory with the same average velocity but different energy source, as depicted in Fig. 3. depend on the source of energy into the system (heat or work). The blue plane shows two things: (1) how trajectories that have a higher speed and in the same direction of the average V become more populated, and (2) the trajectories become more populated when more work flows in and heat is dissipated (w(X) − q(X) > 0), and become less populated as more energy flows in to the system from the external bath, producing work (w(X) − q(X) < 0). From Eq 34, we can readily compute the ratio of probabilities for PT-reversal: p(PTX) p(X) e −2 w(X) − q(X) E in(35) Eq 35 shows that for a given amount of energy that is put into the system, a trajectory that has a large dissipative current is more likely than the PT-reversed, nondissipative one. Eq 35 also correctly predicts that when energy exchange is small, the probability of a wrong-way flow is comparable to a right-way flow. In this way, Max Cal captures the difference between trajectories having the same average velocity but caused by very different processes. In Fig 3 we show three examples of trajectories, each with the same average velocity v but with different values of w and q. The first trajectory corresponds to the process in which the particle is hit twice by the belt; in the second the particle is hit first by the belt and then it receives energy from the thermal bath; in the third, the particle receives energy from the bath twice. If the process were non-dissipative, the three trajectories would have the same probability, but in this dissipative case, Max Cal shows how the probabilities are different (Fig 2). t 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " d 2 z v i 6 q r Y 2 Z j N Z P Z Y i P k v a J n 5 y A = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k o M e C F 4 8 V T V t o Q 9 l s N + 3 S z S b s T o Q S + h O 8 e F D E q 7 / I m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 M J X C o O t + O 6 W N z a 3 t n f J u Z W / / 4 P C o e n z S N k m m G f d Z I h P d D a n h U i j u o 0 D J u 6 n m N A 4 l 7 4 S T 2 7 n f e e L a i E Q 9 4 j T l Q U x H S k S C U b T S A w 7 c Q b X m 1 t 0 F y D r x C l K D A q 1 B 9 a s / T F g W c 4 V M U m N 6 n p t i k F O N g k k + q / Q z w 1 P K J n T E e 5 Y q G n M T 5 I t T Z + T C K k M S J d q W Q r J Q f 0 / k N D Z m G o e 2 M 6 Y 4 N q v e X P z P 6 2 U Y 3 Q S 5 U G m G X L H l o i i T B B M y / 5 s M h e Y M 5 d Q S y r S w t x I 2 p p o y t O l U b A j e 6 s v r p H 1 V 9 9 y 6 d 9 + o N R t F H G U 4 g 3 O 4 B A + u o Q l 3 0 A I f G I z g G V 7 h z Z H O i / P u f C x b S 0 4 x c w p / 4 H z + A P / P j Y 0 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " d 2 z v i 6 q r Y 2 Z j N Z P Z Y i P k v a J n 5 y A = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k o M e C F 4 8 V T V t o Q 9 l s N + 3 S z S b s T o Q S + h O 8 e F D E q 7 / I m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 M J X C o O t + O 6 W N z a 3 t n f J u Z W / / 4 P C o e n z S N k m m G f d Z I h P d D a n h U i j u o 0 D J u 6 n m N A 4 l 7 4 S T 2 7 n f e e L a i E Q 9 4 j T l Q U x H S k S C U b T S A w 7 c Q b X m 1 t 0 F y D r x C l K D A q 1 B 9 a s / T F g W c 4 V M U m N 6 n p t i k F O N g k k + q / Q z w 1 P K J n T E e 5 Y q G n M T 5 I t T Z + T C K k M S J d q W Q r J Q f 0 / k N D Z m G o e 2 M 6 Y 4 N q v e X P z P 6 2 U Y 3 Q S 5 U G m G X L H l o i i T B B M y / 5 s M h e Y M 5 d Q S y r S w t x I 2 p p o y t O l U b A j e 6 s v r p H 1 V 9 9 y 6 d 9 + o N R t F H G U 4 g 3 O 4 B A + u o Q l 3 0 A I f G I z g G V 7 h z Z H O i / P u f C x b S 0 4 x c w p / 4 H z + A P / P j Y 0 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " d 2 z v i 6 q r Y 2 Z j N Z P Z Y i P k v a J n 5 y A = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k o M e C F 4 8 V T V t o Q 9 l s N + 3 S z S b s T o Q S + h O 8 e F D E q 7 / I m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 M J X C o O t + O 6 W N z a 3 t n f J u Z W / / 4 P C o e n z S N k m m G f d Z I h P d D a n h U i j u o 0 D J u 6 n m N A 4 l 7 4 S T 2 7 n f e e L a i E Q 9 4 j T l Q U x H S k S C U b T S A w 7 c Q b X m 1 t 0 F y D r x C l K D A q 1 B 9 a s / T F g W c 4 V M U m N 6 n p t i k F O N g k k + q / Q z w 1 P K J n T E e 5 Y q G n M T 5 I t T Z + T C K k M S J d q W Q r J Q f 0 / k N D Z m G o e 2 M 6 Y 4 N q v e X P z P 6 2 U Y 3 Q S 5 U G m G X L H l o i i T B B M y / 5 s M h e Y M 5 d Q S y r S w t x I 2 p p o y t O l U b A j e 6 s v r p H 1 V 9 9 y 6 d 9 + o N R t F H G U 4 g 3 O 4 B A + u o Q l 3 0 A I f G I z g G V 7 h z Z H O i / P u f C x b S 0 4 x c w p / 4 H z + A P / P j Y 0 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " d 2 z v i 6 q r Y 2 Z j N Z P Z Y i P k v a J n 5 y A = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k o M e C F 4 8 V T V t o Q 9 l s N + 3 S z S b s T o Q S + h O 8 e F D E q 7 / I m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 M J X C o O t + O 6 W N z a 3 t n f J u Z W / / 4 P C o e n z S N k m m G f d Z I h P d D a n h U i j u o 0 D J u 6 n m N A 4 l 7 4 S T 2 7 n f e e L a i E Q 9 4 j T l Q U x H S k S C U b T S A w 7 c Q b X m 1 t 0 F y D r x C l K D A q 1 B 9 a s / T F g W c 4 V M U m N 6 n p t i k F O N g k k + q / Q z w 1 P K J n T E e 5 Y q G n M T 5 I t T Z + T C K k M S J d q W Q r J Q f 0 / k N D Z m G o e 2 M 6 Y 4 N q v e X P z P 6 2 U Y 3 Q S 5 U G m G X L H l o i i T B B M y / 5 s M h e Y M 5 d Q S y r S w t x I 2 p p o y t O l U b A j e 6 s v r p H 1 V 9 9 y 6 d 9 + o N R t F H G U 4 g 3 O 4 B A + u o Q l 3 0 A I f G I z g G V 7 h z Z H O i / P u f C x b S 0 4 x c w p / 4 H z + A P / P j Y 0 = < / l a t e x i t > t 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " J V D L b t O q d X 8 z J s P 2 w 6 j w g w 1 j V I 4 = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k o M e C F 4 8 V T V t o Q 9 l s N + 3 S z S b s T o Q S + h O 8 e F D E q 7 / I m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 M J X C o O t + O 6 W N z a 3 t n f J u Z W / / 4 P C o e n z S N k m m G f d Z I h P d D a n h U i j u o 0 D J u 6 n m N A 4 l 7 4 S T 2 7 n f e e L a i E Q 9 4 j T l Q U x H S k S C U b T S A w 6 8 Q b X m 1 t 0 F y D r x C l K D A q 1 B 9 a s / T F g W c 4 V M U m N 6 n p t i k F O N g k k + q / Q z w 1 P K J n T E e 5 Y q G n M T 5 I t T Z + T C K k M S J d q W Q r J Q f 0 / k N D Z m G o e 2 M 6 Y 4 N q v e X P z P 6 2 U Y 3 Q S 5 U G m G X L H l o i i T B B M y / 5 s M h e Y M 5 d Q S y r S w t x I 2 p p o y t O l U b A j e 6 s v r p H 1 V 9 9 y 6 d 9 + o N R t F H G U 4 g 3 O 4 B A + u o Q l 3 0 A I f G I z g G V 7 h z Z H O i / P u f C x b S 0 4 x c w p / 4 H z + A A F i j Y 4 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " J V D L b t O q d X 8 z J s P 2 w 6 j w g w 1 j V I 4 = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k o M e C F 4 8 V T V t o Q 9 l s N + 3 S z S b s T o Q S + h O 8 e F D E q 7 / I m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 M J X C o O t + O 6 W N z a 3 t n f J u Z W / / 4 P C o e n z S N k m m G f d Z I h P d D a n h U i j u o 0 D J u 6 n m N A 4 l 7 4 S T 2 7 n f e e L a i E Q 9 4 j T l Q U x H S k S C U b T S A w 6 8 Q b X m 1 t 0 F y D r x C l K D A q 1 B 9 a s / T F g W c 4 V M U m N 6 n p t i k F O N g k k + q / Q z w 1 P K J n T E e 5 Y q G n M T 5 I t T Z + T C K k M S J d q W Q r J Q f 0 / k N D Z m G o e 2 M 6 Y 4 N q v e X P z P 6 2 U Y 3 Q S 5 U G m G X L H l o i i T B B M y / 5 s M h e Y M 5 d Q S y r S w t x I 2 p p o y t O l U b A j e 6 s v r p H 1 V 9 9 y 6 d 9 + o N R t F H G U 4 g 3 O 4 B A + u o Q l 3 0 A I f G I z g G V 7 h z Z H O i / P u f C x b S 0 4 x c w p / 4 H z + A A F i j Y 4 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " J V D L b t O q d X 8 z J s P 2 w 6 j w g w 1 j V I 4 = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k o M e C F 4 8 V T V t o Q 9 l s N + 3 S z S b s T o Q S + h O 8 e F D E q 7 / I m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 M J X C o O t + O 6 W N z a 3 t n f J u Z W / / 4 P C o e n z S N k m m G f d Z I h P d D a n h U i j u o 0 D J u 6 n m N A 4 l 7 4 S T 2 7 n f e e L a i E Q 9 4 j T l Q U x H S k S C U b T S A w 6 8 Q b X m 1 t 0 F y D r x C l K D A q 1 B 9 a s / T F g W c 4 V M U m N 6 n p t i k F O N g k k + q / Q z w 1 P K J n T E e 5 Y q G n M T 5 I t T Z + T C K k M S J d q W Q r J Q f 0 / k N D Z m G o e 2 M 6 Y 4 N q v e X P z P 6 2 U Y 3 Q S 5 U G m G X L H l o i i T B B M y / 5 s M h e Y M 5 d Q S y r S w t x I 2 p p o y t O l U b A j e 6 s v r p H 1 V 9 9 y 6 d 9 + o N R t F H G U 4 g 3 O 4 B A + u o Q l 3 0 A I f G I z g G V 7 h z Z H O i / P u f C x b S 0 4 x c w p / 4 H z + A A F i j Y 4 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " J V D L b t O q d X 8 z J s P 2 w 6 j w g w 1 j V I 4 = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k o M e C F 4 8 V T V t o Q 9 l s N + 3 S z S b s T o Q S + h O 8 e F D E q 7 / I m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 M J X C o O t + O 6 W N z a 3 t n f J u Z W / / 4 P C o e n z S N k m m G f d Z I h P d D a n h U i j u o 0 D J u 6 n m N A 4 l 7 4 S T 2 7 n f e e L a i E Q 9 4 j T l Q U x H S k S C U b T S A w 6 8 Q b X m 1 t 0 F y D r x C l K D A q 1 B 9 a s / T F g W c 4 V M U m N 6 n p t i k F O N g k k + q / Q z w 1 P K J n T E e 5 Y q G n M T 5 I t T Z + T C K k M S J d q W Q r J Q f 0 / k N D Z m G o e 2 M 6 Y 4 N q v e X P z P 6 2 U Y 3 Q S 5 U G m G X L H l o i i T B B M y / 5 s M h e Y M 5 d Q S y r S w t x I 2 p p o y t O l U b A j e 6 s v r p H 1 V 9 9 y 6 d 9 + o N R t F H G U 4 g 3 O 4 B A + u o Q l 3 0 A I f G I z g G V 7 h z Z H O i / P u f C x b S 0 4 x c w p / 4 H z + A A F i j Y 4 = < / l a t e x i t > t 2 < l a t e x i t s h a 1 _ b a s e 6 4 = " r W O W r q K 5 q 4 M r M g M i F n D w 9 X W M P k 4 = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l K o R 4 L X j x W t B / Q h r L Z b t q l m 0 3 Y n Q g l 9 C d 4 8 a C I V 3 + R N / + N 2 z Y H b X 0 w 8 H h v h p l 5 Q S K F Q d f 9 d g p b 2 z u 7 e 8 X 9 0 s H h 0 f F J + f S s Y + J U M 9 5 m s Y x 1 L 6 C G S 6 F 4 G w V K 3 k s 0 p 1 E g e T e Y 3 i 7 8 7 h P X R s T q E W c J 9 y M 6 V i I U j K K V H n B Y G 5 Y r b t V d g m w S L y c V y N E a l r 8 G o 5 i l E V f I J D W m 7 7 k J + h n V K J j k 8 9 I g N T y h b E r H v G + p o h E 3 f r Y 8 d U 6 u r D I i Y a x t K S R L 9 f d E R i N j Z l F g O y O K E 7 P u L c T / v H 6 K 4 Y 2 f C Z W k y B V b L Q p T S T A m i 7 / J S G j O U M 4 s o U w L e y t h E 6 o p Q 5 t O y Y b g r b + 8 S T q 1 q u d W v f t 6 p V n P 4 y j C B V z C N X j Q g C b c Q Q v a w G A M z / A K b 4 5 0 X p x 3 5 2 P V W n D y m X P 4 A + f z B w L m j Y 8 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " r W O W r q K 5 q 4 M r M g M i F n D w 9 X W M P k 4 = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l K o R 4 L X j x W t B / Q h r L Z b t q l m 0 3 Y n Q g l 9 C d 4 8 a C I V 3 + R N / + N 2 z Y H b X 0 w 8 H h v h p l 5 Q S K F Q d f 9 d g p b 2 z u 7 e 8 X 9 0 s H h 0 f F J + f S s Y + J U M 9 5 m s Y x 1 L 6 C G S 6 F 4 G w V K 3 k s 0 p 1 E g e T e Y 3 i 7 8 7 h P X R s T q E W c J 9 y M 6 V i I U j K K V H n B Y G 5 Y r b t V d g m w S L y c V y N E a l r 8 G o 5 i l E V f I J D W m 7 7 k J + h n V K J j k 8 9 I g N T y h b E r H v G + p o h E 3 f r Y 8 d U 6 u r D I i Y a x t K S R L 9 f d E R i N j Z l F g O y O K E 7 P u L c T / v H 6 K 4 Y 2 f C Z W k y B V b L Q p T S T A m i 7 / J S G j O U M 4 s o U w L e y t h E 6 o p Q 5 t O y Y b g r b + 8 S T q 1 q u d W v f t 6 p V n P 4 y j C B V z C N X j Q g C b c Q Q v a w G A M z / A K b 4 5 0 X p x 3 5 2 P V W n D y m X P 4 A + f z B w L m j Y 8 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " r W O W r q K 5 q 4 M r M g M i F n D w 9 X W M P k 4 = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l K o R 4 L X j x W t B / Q h r L Z b t q l m 0 3 Y n Q g l 9 C d 4 8 a C I V 3 + R N / + N 2 z Y H b X 0 w 8 H h v h p l 5 Q S K F Q d f 9 d g p b 2 z u 7 e 8 X 9 0 s H h 0 f F J + f S s Y + J U M 9 5 m s Y x 1 L 6 C G S 6 F 4 G w V K 3 k s 0 p 1 E g e T e Y 3 i 7 8 7 h P X R s T q E W c J 9 y M 6 V i I U j K K V H n B Y G 5 Y r b t V d g m w S L y c V y N E a l r 8 G o 5 i l E V f I J D W m 7 7 k J + h n V K J j k 8 9 I g N T y h b E r H v G + p o h E 3 f r Y 8 d U 6 u r D I i Y a x t K S R L 9 f d E R i N j Z l F g O y O K E 7 P u L c T / v H 6 K 4 Y 2 f C Z W k y B V b L Q p T S T A m i 7 / J S G j O U M 4 s o U w L e y t h E 6 o p Q 5 t O y Y b g r b + 8 S T q 1 q u d W v f t 6 p V n P 4 y j C B V z C N X j Q g C b c Q Q v a w G A M z / A K b 4 5 0 X p x 3 5 2 P V W n D y m X P 4 A + f z B w L m j Y 8 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " r W O W r q K 5 q 4 M r M g M i F n D w 9 X W M P k 4 = " > A A A B 6 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l K o R 4 L X j x W t B / Q h r L Z b t q l m 0 3 Y n Q g l 9 C d 4 8 a C I V 3 + R N / + N 2 z Y H b X 0 w 8 H h v h p l 5 Q S K F Q d f 9 d g p b 2 z u 7 e 8 X 9 0 s H h 0 f F J + f S s Y + J U M 9 5 m s Y x 1 L 6 C G S 6 F 4 G w V K 3 k s 0 p 1 E g e T e Y 3 i 7 8 7 h P X R s T q E W c J 9 y M 6 V i I U j K K V H n B Y G 5 Y r b t V d g m w S L y c V y N E a l r 8 G o 5 i l E V f I J D W m 7 7 k J + h n V K J j k 8 9 I g N T y h b E r H v G + p o h E 3 f r Y 8 d U 6 u r D I i Y a x t K S R L 9 f d E R i N j Z l F g O y O K E 7 P u L c T / v H 6 K 4 Y 2 f C Z W k y B V b L Q p T S T A m i 7 / J S G j O U M 4 s o U w L e y t h E 6 o p Q 5 t O y Y b g r b + 8 S T q 1 q u d W v f t 6 p V n P 4 y j C B V z C N X j Q g C b c Q Q v a w G A M z / A K b 4 5 0 X p x 3 5 2 P V W n D y m X P 4 A + f z B w L m j Y 8 = < / l a t e x i t > v > 0 w > 0 q = 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " O h D B W X y V f B / U y + Z 2 U V D K n r R W g B k = " > A A A B + H i c b V C 7 T s M w F H V 4 l v B o g J H F o k J i q h x A o g u o E g t j k e h D a q L K c Z 3 W q u M E 2 y k q U b + E h Q G E W P k U N v 4 G N 8 0 A L U e 6 V 0 f n 3 C t f n y D h T G m E v q 2 V 1 b X 1 j c 3 S l r 2 9 s 7 t X d v Y P W i p O J a F N E v N Y d g K s K G e C N j X T n H Y S S X E U c N o O R j c z v z 2 m U r F Y 3 O t J Q v 0 I D w Q L G c H a S D 2 n P L 5 G n m c / 5 v 3 h C v W c C q q i H H C Z u A W p g A K N n v P l 9 W O S R l R o w r F S X R c l 2 s + w 1 I x w O r W 9 V N E E k x E e 0 K 6 h A k d U + V l + + B S e G K U P w 1 i a E h r m 6 u + N D E d K T a L A T E Z Y D 9 W i N x P / 8 7 q p D m t + x k S S a i r I / K E w 5 V D H c J Y C 7 D N J i e Y T Q z C R z N w K y R B L T L T J y j Y h u I t f X i a t s 6 p 7 X k V 3 F 5 V 6 r Y i j B I 7 A M T g F L r g E d X A L G q A J C E j B M 3 g F b 9 a T 9 W K 9 W x / z 0 R W r 2 D k E f 2 B 9 / g A b W p F k < / l a t e x i t > v > 0 w > 0 q > 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " g k V w / Y z E Q F U L E y A n Y P q 8 p f t f j 9 c = " > A A A B + H i c b V C 7 T s M w F H V 4 l v B o g J H F o k J i q h x A o l N V i Y W x S P Q h N V H l u E 5 r 1 X G C 7 R S V q F / C w g B C r H w K G 3 + D m 2 a A l i P d q 6 N z 7 p W v T 5 B w p j R C 3 9 b a + s b m 1 n Z p x 9 7 d 2 z 8 o O 4 d H b R W n k t A W i X k s u w F W l D N B W 5 p p T r u J p D g K O O 0 E 4 5 u 5 3 5 l Q q V g s 7 v U 0 o X 6 E h 4 K F j G B t p L 5 T n t S R 5 9 m P e X + o o 7 5 T Q V W U A 6 4 S t y A V U K D Z d 7 6 8 Q U z S i A p N O F a q 5 6 J E + x m W m h F O Z 7 a X K p p g M s Z D 2 j N U 4 I g q P 8 s P n 8 E z o w x g G E t T Q s N c / b 2 R 4 U i p a R S Y y Q j r k V r 2 5 u J / X i / V Y c 3 P m E h S T Q V Z P B S m H O o Y z l O A A y Y p 0 X x q C C a S m V s h G W G J i T Z Z 2 S Y E d / n L q 6 R 9 U X U v q + j u q t K o F X G U w A k 4 B e f A B d e g A W 5 B E 7 Q A A S l 4 B q / g z X q y X q x 3 6 Three possible trajectories for a single particle. Top: The particle interacts in both steps with the conveyor belt, receiving energy as work; Center: The particle first receives work then heat from the thermal bath; Bottom: The particle only receives energy as heat from the thermal bath. In Fig. 2 the corresponding coloured dots with the respective probabilities. THE MAES ARGUMENT AND THE PROPER NUMBER OF CONSTRAINTS Maes argument [8] is a bit more subtle, because it points out the fact that when the only chosen constraints are time-asymmetric currents, the only possible outcome is a system without any dissipation. We agree in principle that this is the case, but this would be a problem of making a poor choice of constraints, and not with Max Cal itself. When there is available knowledge of the system that is being ignored, like work or heat transfer, (or in general what Maes calls a frenetic contribution [8]), it is to be expected that Max Cal will not necessarily be consistent with it. In this case too, the problem is with the choice of constraints, not the Max Cal principle. One further point is that our discussion here considers only the 3 constraints needed for DSS; some different situations may need more or different constraints. CONCLUSIONS We have shown here that Maximum Caliber can handle dissipation properly, but it requires the application of appropriate restraints. In DSS, you need both the mean rate of flow,and also the work performed on the system and the heat that is dissipated. We show this on general grounds, but we also give a specific solvable model of a single particle flow that is subjected to heat and work input and output. This toy model may be useful for studying dissipative flows. FOR DISSIPATIVE STEADY-STATES, MAXIMUM CALIBER REQUIRES AT LEAST 3 CONSTRAINTS. Fig 2 shows an example of trajectory populations as a function of the three properties, v(X), w(X) and q(X) of each trajectory, for fixed values of V , the particle flow velocity, and for fixed energy input E in . The orange plane in Fig 2 shows what you would predict if you knew only the mean flow velocity v(X). The trajectory population would not FIG. 2. The Max Cal probability distribution vs JE. FIG. 3. Three possible trajectories for a single particle. Top: The particle interacts in both steps with the conveyor belt, receiving energy as work; Center: The particle first receives work then heat from the thermal bath; Bottom: The particle only receives energy as heat from the thermal bath. In Fig. 2 the corresponding coloured dots with the respective probabilities. Note, however, that while (U, V, N ) are conserved quantities in the equilibrium metaphor, Jq and Jw are not necessarily conserved in flow situations. In this case, the PT-reversed trajectory must have identical probability, because it is the identical trajectory. ACKNOWLEDGMENTSThis research was supported by the National Science Foundation grants PHY1205881 and MCB1344230 and the Laufer Center for Physical and Quantitative Biology. We wish to thank Jason Wagoner, Purushottam Dixit and Kingshuk Ghosh for the precious insights and discussions. Where do we stand. E Jaynes, E. Jaynes et al., Ed. Levine, RD, Tribus, M., Where do we stand (1979). E Jaynes, H Haken, Springer Series in Synergetics. Haken, H. Berlin, Heidelberg, New YorkSpringerE. Jaynes and H. Haken, Springer Series in Synergetics. Haken, H.(ed.). Berlin, Heidelberg, New York: Springer (1985). . H Haken, Zeitschrift für Physik B Condensed Matter. 63505H. Haken, Zeitschrift für Physik B Condensed Matter 63, 505 (1986). . R C Dewar, Journal of Physics A: Mathematical and General. 38371R. C. Dewar, Journal of Physics A: Mathematical and General 38, L371 (2005). . S Pressé, K Ghosh, J Lee, K A Dill, Reviews of Modern Physics. 851115S. Pressé, K. Ghosh, J. Lee, and K. A. Dill, Reviews of Modern Physics 85, 1115 (2013). . P D Dixit, J Wagoner, C Weistuch, S Pressé, K Ghosh, K A Dill, The Journal of chemical physics. 14810901P. D. Dixit, J. Wagoner, C. Weistuch, S. Pressé, K. Ghosh, and K. A. Dill, The Journal of chemical physics 148, 010901 (2018). . R L Jack, R Evans, Journal of Statistical Mechanics: Theory and Experiment. 93305R. L. Jack and R. Evans, Journal of Statistical Mechan- ics: Theory and Experiment 2016, 093305 (2016). C Maes, Non-dissipative effects in nonequilibrium systems. SpringerC. Maes, Non-dissipative effects in nonequilibrium sys- tems (Springer, 2018). . C Jarzynski, Physical Review Letters. 782690C. Jarzynski, Physical Review Letters 78, 2690 (1997). . G E Crooks, Physical Review E. 602721G. E. Crooks, Physical Review E 60, 2721 (1999). Communication: Maximum caliber is a general variational principle for nonequilibrium statistical mechanics. M J Hazoglou, V Walther, P D Dixit, K A Dill, M. J. Hazoglou, V. Walther, P. D. Dixit, and K. A. Dill, "Communication: Maximum caliber is a general vari- ational principle for nonequilibrium statistical mechan- ics," (2015).
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[ "A modified-Boltzmann approach for modeling the hot QCD medium-induce splitting vertices in the deep LPM region", "A modified-Boltzmann approach for modeling the hot QCD medium-induce splitting vertices in the deep LPM region" ]
[ "Weiyao Ke \nDepartment of Physics\nDuke University\n27708-0305DurhamNC\n", "Yingru Xu \nDepartment of Physics\nDuke University\n27708-0305DurhamNC\n", "Steffen A Bass \nDepartment of Physics\nDuke University\n27708-0305DurhamNC\n" ]
[ "Department of Physics\nDuke University\n27708-0305DurhamNC", "Department of Physics\nDuke University\n27708-0305DurhamNC", "Department of Physics\nDuke University\n27708-0305DurhamNC" ]
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Hard probes produced in perturbative processes are excellent probes for the study of the hot and dense QCD matter created in relativistic heavy-ion collisions. Transport theory, allowing for coupling to an evolving medium with fluctuating initial conditions, has become a powerful tool in this endeavor. However, the implementation of the Landau-Pomeranchuk-Migdal (LPM) effect for medium-induced parton bremsstrahlung and pair production, poses a challenge to semi-classical transport models based on Boltzmann-type transport equations. In this work, we investigate a possible solution to approximate the LPM effect in a "modified Boltzmann transport" approach, including a prescription for the running coupling constant. By fixing a numerical parameter, this approach quantitatively reproduces the rates of medium-induced parton splitting predicted by the next-to-lead-log solution of the AMY equation which is valid in the deep-LPM regime of an infinite medium. We also find qualitative agreement of our implementation with calculations in a finite and expanding medium, but future improvements are needed for added precision at small path length. This work benefits transport model-based studies and the usage of these models in the phenomenological extraction of the jet transport coefficient. arXiv:1810.08177v2 [nucl-th]
10.1103/physrevc.100.064911
[ "https://export.arxiv.org/pdf/1810.08177v2.pdf" ]
195,874,214
1810.08177
30d24d576b9d3de7f1fa723831115e8db5997b51
A modified-Boltzmann approach for modeling the hot QCD medium-induce splitting vertices in the deep LPM region Weiyao Ke Department of Physics Duke University 27708-0305DurhamNC Yingru Xu Department of Physics Duke University 27708-0305DurhamNC Steffen A Bass Department of Physics Duke University 27708-0305DurhamNC A modified-Boltzmann approach for modeling the hot QCD medium-induce splitting vertices in the deep LPM region (Dated: March 2, 2022) Hard probes produced in perturbative processes are excellent probes for the study of the hot and dense QCD matter created in relativistic heavy-ion collisions. Transport theory, allowing for coupling to an evolving medium with fluctuating initial conditions, has become a powerful tool in this endeavor. However, the implementation of the Landau-Pomeranchuk-Migdal (LPM) effect for medium-induced parton bremsstrahlung and pair production, poses a challenge to semi-classical transport models based on Boltzmann-type transport equations. In this work, we investigate a possible solution to approximate the LPM effect in a "modified Boltzmann transport" approach, including a prescription for the running coupling constant. By fixing a numerical parameter, this approach quantitatively reproduces the rates of medium-induced parton splitting predicted by the next-to-lead-log solution of the AMY equation which is valid in the deep-LPM regime of an infinite medium. We also find qualitative agreement of our implementation with calculations in a finite and expanding medium, but future improvements are needed for added precision at small path length. This work benefits transport model-based studies and the usage of these models in the phenomenological extraction of the jet transport coefficient. arXiv:1810.08177v2 [nucl-th] I. INTRODUCTION The study of hard probes in relativistic heavy-ion collisions is moving towards the precision era thanks to upcoming experimental upgrades [1][2][3][4][5] as well as theoretical and computational advances that allow for the calculation of jet propagation in a realistic Quark-Gluon-Plasma (QGP) medium (including event-by-event fluctuating initial conditions and temperature-dependent transport coefficients) . Among the goals for this research is the characterization of the QGP medium in terms of its jet transport coefficientsq. Transport models are powerful tools that hard probes can be coupled to the realistic time evolution of the medium with an event-by-event fluctuating initial condition. However, the numerical implementation of the QCD analog of the Landau-Pomeranchuk-Migdal (LPM) effect, poses a serious challenge to a class of widely used models based on the Boltzmann-type transport equation or those based on Langevin dynamics. In a dense medium, multiple scatterings act coherently within the formation time (τ f ) of the medium-induced parton splitting [6-9, 27, 28]. The resultant bremsstrahlung rate differs from those estimated assuming independent incoherent) collisions. As a result, bremsstrahlung in a medium effectively becomes an n-body to (n + 1)-body process with a finite extended time scale τ f . At high energy, τ f can be much larger than the collisional mean-free-path λ el , and can be comparable to the typical inverse gradient of the macroscopic quantities of the medium created in heavy-ion collisions. These situations are particularly difficult to treat in a Boltzmann transport equation with local and few-body collision terms. To simplify the parton bremsstrahlung while retaining essential qualitative features, different approximations and prescriptions are used to incorporate the LPM effect into transport models [21,[29][30][31][32][33]. These models are then applied to phenomenological studies and used for the extraction of physical parameters such asq. However, insufficient attention has been paid to quantitative comparisons of these approximate LPM implementations to the theoretical baselines. Such a step is essential for the reliable extraction of jet transport properties. In this work, we have developed an approximation scheme for the inclusion of the LPM effect into transport models, aiming for a quantitative comparison between the model and the theory. The resultant technique is hereafter termed as "the modified Boltzmann transport" approach. Here we give a short overview of the approach. The semi-classical transport model propagates the hard partons inside a quark-gluon plasma, under the influence of both elastic collisions with the medium and incoherent parton bremsstrahlung (inelastic processes) from independent collisions. We first focus on the simplest scenario of a hard parton propagating in an infinite medium with a constant temperature. The LPM effect is implemented as a modification to the incoherent inelastic rates, where elastic collisions broaden the transverse momentum between the outgoing hard partons and the incoherent rate is reduced by a factor P . The suppression probability P is obtained from the leading-log (LL) approximation of in-medium parton splitting when the number of coherent collisions is large N coh = τ f /λ el 1 (the deep-LPM region), and P ∝ λ el /τ f . Moreover, a quantitative agreement with the theory can be achieved by introducing a next-to-lead-log (NLL) correction to P . The modified transport model describes the rates for in-medium bremsstrahlung q → q + g, g → g + g, and pair production g → q +q surprisingly well in the deep-LPM region. Next, we apply the model to cases beyond an infinite and static medium, because the realistic medium created in heavy-ion collisions is finite, fluctuating, with a fast dropping temperature profile due to the violent expansion. It is true that the current approach is devel-oped by matching to theoretical calculations in the deep LPM region assuming many coherent collisions, but for a thin medium, the role of the interference pattern between only a few coherent collisions becomes important. In principle, one should resort to other techniques such as the opacity expansion [34,35] for computation in a thin medium. Nevertheless, we do find the current method qualitatively reproduces the theoretical calculations of path-length dependence [36] and the medium expansion rate dependence [37] of the parton bremsstrahlung. Finally, we find it instructive to compare the current method to two other Monte Carlo implementations of the medium-induced radiation processes that have been used in previous studies. We find that subtle differences in the construction of these models can lead to an incorrect implementation of the LPM effect or introduce correlations between sequential bremsstrahlung partons that are beyond the leading-order theory used for the model. We draw attention to the phenomenological consequences of these differences, especially, how these differences affect the interpretation of the extracted physical parameters such as the in-medium coupling and transport coefficient q. This paper is organized as follows. Section II introduces the ingredients of the semi-classical transport model that is to be modified. In section III, we propose the modifications to the aforementioned model in Section II to include the LPM effect. In section IV and appendix B, we provide detailed comparisons between the simulation and the NLL solution in the deep-LPM regime. Effects of a finite and expanding medium are investigated in section V. In section VI, we compare this work to two previously used Monte-Carlo implementations of the LPM effect in transport models. Finally, section VII summarizes and discusses the future applications of the model. II. SEMI-CLASSICAL TRANSPORT APPROACH WITH INCOHERENT RATES In this section, we construct a transport model assuming independent collisions. This type of model sets the basis for our subsequent discussion regarding the inclusion of the LPM effect. Focusing on hard partons, collisions are categorized into elastic (hard particle number conserving) and inelastic processes (hard particle number non-conserving). The inelastic processes are further divided into partonsplitting and parton-fusion contributions. The Boltzmann equation relates the evolution of the hard parton distribution function f H to these collision processes, df H dt = C el [f H ] + C inel [f H ].(1) As a remark, we have omitted denoting the dependence of the collision term on the light parton distribution functions, because we assume them to be thermal equilibrium distribution functions with Boltzmann statistics f 0 (p) = e −p·u/T . T and u are the medium temperature and four-velocity that can be obtained from a medium evolution model. We also neglect the collisions between two or more hard partons and the back reaction from hard parton scattering to the medium due to their small population in actual high energy nuclear collisions. As a result, the above Boltzmann equation is linearized with respect to f H . A linearized Boltzmann equation can be solved using a collision rate approach. First, hard partons are represented by δ-functions in the phase space f H (t, x, p) ≈ δ (3) (p − p(t))δ (3) (x − x(t)) . Each particle travels in a straight line and occasionally its momentum changes drastically due to collisions. This way, one obtains a stochastic solution of the time evolution of f H . The collision probability per unit time (i.e., the collision rate) is R = g i 2E 1 d 3 p 2 2E 2 (2π) 3 f 0 (p 2 )2ŝ i 0 −ŝ dσ i dt dt. (2) For simplicity, we have only shown the case that the hard parton with momentum p 1 collides with one medium parton with momentum p 2 .ŝ,t are the Mandelstam variables of the two-body collisions. dσ i /dt is the differential cross-sections of channel i, including both elastic and inelastic contributions. The in-medium cross-sections dσ i are dominated bŷ t channel processes which diverge like 1/t 2 in the vacuum. In a medium, screening effects regulate the crosssection and leave the physical quantities finite; however, the cross-section becomes complicated and its form depends on the choice of the reference frame. Here, we borrow an elegant solution from [38] to separate the inmedium collisions based on the momentum transfer-q between the hard parton and the medium. The large-q transfer (hard) scattering rate uses cross-sections computed in the vacuum since the medium modification to these hard modes is small. The associated rates are obtained from equation 2 while restricting q to be larger than a switching scale Q cut . The small-q transfer processes are frequent and soft, which allows for a diffusion approximation of its effect on the trajectory of the hard parton, x(t + ∆t) = p E ∆t (3) p(t + ∆t) = p − η D,S p∆t + ξ(t)∆t(4) The effects of the soft-interaction are absorbed into the drag coefficients η D,S and the covariance of the thermal random force ξ, ξ i (t)ξ j (0) = δ(t) p i p j p 2q L,S + δ ij − p i p j p 2 q S 2 .(5) q S andq L,S are the soft transverse and longitudinal momentum broadening coefficients, and the subscript "S" reminds us that these numbers should only contain soft contributions with q < Q cut . The switching scale Q cut is chosen to be greater than the Debye screening mass m D , m 2 D = 4πα s 3 N c + N f 2 T 2 .(6) One of the many advantages of this separation is the avoidance of complicated in-medium propagators, while still achieving a leading order accuracy with a reasonable choice of Q cut at weak coupling as shown in [38]. Moreover, the Lorentz-invariance of the vacuum matrixelements used in large-q processes simplifies the computation in different reference frames. The frame-dependence only appears in the diffusion equation, which is easiest solved in the medium rest frame. a. Elastic processes The two-body matrix-elements in the vacuum that enter the large-q collision rates can be found in the standard literature [39]. In the small-q diffusion sector, the transverse and longitudinal momentum diffusion coefficientsq S ,q S,L in a weakly coupled theory have been calculated in [38] at leading order, q S = Q 2 cut 0 dq 2 α s m 2 D T q 2 (q 2 + m 2 D ) ,(7)q L,S = Q 2 cut 0 dq 2 α s m 2 ∞ T q 2 (q 2 + m 2 ∞ ) .(8) m ∞ is the asymptotic gluon thermal mass m 2 ∞ = m 2 D /2. Finally, the drag coefficient is determined by the Einstein relation between the transport coefficients, η D,S =q L,S 2ET − dq L,S dp 2 −q L,S −q S /2 p 2 .(9) b. Inelastic processes The inelastic collision term is also separated into a large-q 2 ↔ 3 body inelastic collision rate plus an effective 1 ↔ 2 body diffusion-induced parton splitting / fusion rate. The matrix-elements of 2 ↔ 3 body collisions are derived under the limit k 2 , q 2 x(1 − x) √ŝ . Here, x is the light-cone energy fraction of the initial state hard parton carried by one of the final state hard parton, q is the transverse component of q in the center-of-mass frame of the collision, and k is the transverse momentum between the two split hard partons in the final state. A list of these matrixelements can be found in appendix A. Again, the rates are obtained from equation 2 imposing q > Q cut . The diffusion-induced splitting rate uses the restriction q < Q cut in equation 2, and uses the limit q k of the 2 → 3 matrix-elements listed in appendix A, R 1→2 = g i 2E 1 d 3 p 2 f 0 (p 2 ) 2E 2 (2π) 3 2ŝ Q 2 cut 0 q 2 dσ el q 2 dq 2 × dk 2 dx α s P (x) 2π(k 2 + m 2 ∞ ) 2(10) where m ∞ is added to screen the divergence and P (x) is the vacuum splitting function listed in appendix A. Now, one may notice that the first line in equation 10, upon summing over all channels, simply computes the variance of transverse momentum received by the hard parton per unit time from soft interactions below the momentum cut-off Q cut . So we rewrite R 1→2 using the soft transport coefficientsq S as R 1→2 =q S dk 2 dx α s P (x) 2π(k 2 + m 2 ∞ ) 2 ,(11) which is our final expression for the diffusion induced inelastic collision rate for the incoherent transport equation. Finally, the collision rates of the reverse processes 3 → 2 and 2 → 1 processes can be written down by the requirement of detailed balance. c. The transport equation in the incoherent limit Combining all these processes, we summarize the semiclassical transport equation with independent collisions into df dt = D[f ] + C 1↔2 [f ] + C 2↔2 [f ] + C 2↔3 [f ]. (12) The distribution function of the hard parton evolves under the effect of diffusion D, large-q elastic collision C 2↔2 , diffusion induced parton splitting / merging C 1↔2 , and large-q inelastic collisions C 2↔3 . III. MODELING LPM EFFECT BY A MODIFIED TRANSPORT SIMULATION The incoherent transport equation requires: first, the transition time scale of a process is small compared to the mean-free-path, so that multiple-collision contribution to the transition rate is negligible; and second, the transition time scale is small compared to the inversegradient of the system, so that the collision terms in equation 12 depend only on distribution functions at a localized space-time region. Such a semi-classical picture does work for elastic collisions at weak coupling g 1. This is because a medium scattering center is statistically independent from others with distances greater than 1/m D ∼ 1/gT and the mean-free-path λ ∼ 1/g 2 T is larger than 1/m D . For an inelastic collision, consider the splitting of a hard parton "a" with energy E to two hard partons "b" and "c", with "b" carrying an x fraction of the original parton's energy. From uncertainty principle, the formation time τ f of the hard final state is obtained as the inverse of the light-cone energy difference δE between the initial and final states, τ −1 f ∼ δE = k 2 2x(1 − x)E .(13) k is the transverse momentum of "b" relative to the direction-of-motion of "a". For hard and collinear splittings, this formation time can be very large compared to λ and multiple-collision effect becomes important and needs to be resummed into the transition rate. In an infinite and static medium, when the number of multiple collisions N is large (the deep-LPM region), theoretical calculation indicates a qualitative change to the parton radiation pattern comparing to the results obtained in the independent collision picture [28]. First, transverse momentum of the splitting is broadened from multiple collisions; second, the transition rate is reduced from producing O(α s ) radiation every collision to producing O(α s ) radiation every formation time, dR dω ∝ α s ωλ → α s ωτ f .(14) The average inverse formation time can be estimated using equation 13 and the condition k 2 ≈qτ f , because the transport coefficientq = d k 2 /dt quantifies the momentum broadening per unit time. These two conditions lead to, τ −1 f ∼ q 2x(1 − x)E .(15)At weak couplingq ∝ g 4 T 3 and τ f /λ ∼ 2x(1 − x)E/T . The radiative pattern with moderately small x in an infinite medium is changed to, dR dω ∝ α s T 1/2 ω 3/2 λ(16) Therefore, a fundamental modification to the semiclassical equation is necessary once the final-state partons become hard xE, (1 − x)E > T . A. The modification to the semi-classical evolution We start by investigating the leading-order calculation of medium-induced parton splitting in a "brick" medium of constant temperature to identify the modification we need. Here, we quote the reorganized leading-order formula for the probability of a single medium-induced splitting in a brick medium from [7,36], dP a bc dω = ∞ 0 dt g 2 πE P a(0) bc (x) ∞ t dt F (t , t), (17) F (t , t) = Re q ,q iq · q δE C(t ) • K(t , q ; t, q). (18) P a(0) bc is the vacuum splitting function and δE is the lightcone energy difference between initial and final states. The C(t ) operator is a Boltzmann-type collision operator in the momentum space such that, C • f p = q g 2 A(q 2 ) C b + C c − C a 2 (f p − f p−q ) (19) + C a + C c − C b 2 (f p − f p+xq ) + C a + C b − C c 2 f p − f p+(1−x)q , where ω is energy of daugther parton "b". C i is the color factor of each parton and q represents an integration over transverse momentum dq 2 /(2π) 2 . The collision kernel is given by [40], A(q 2 ) = m 2 D T q 2 (m 2 D + q 2 ) .(20) Finally, K(t , q ; t, q) is the propagator of the transverse HamiltonianĤ = δE − iC in the momentum representation. This rather compact result is actually hard to implement in a Boltzmann formulation, because of the double-time integral that comes from the nature of a quantum transition. Moreover, the splitting probability generally dependents on the temperature and flow velocity profiles of the medium, making it a computationally heavy task when coupled to dynamical evolving and fluctuating medium. Our approximation towards a modified Boltzmann transport formulation starts by replacing the effect of the temporal two-point function F (t , t) with a simple ansatz, F (t , t) → 1 N N i=1 b τ i (t) δ(t − t − aτ i (t)).(21) Here the function F (t , t) is approximated by an ensemble of N independent copies of the system a → b + c. These copies are generated according to the incoherent 1 → 2 and 2 → 3 rates as hard parton a propagates. Each copy "i" evolves with the influence of the elastic broadening. Its formation time τ i (t) at time t during the evolution can be computed by equation 13. The delta function imposes that the branching that starts at time t is thought to be formed at time t + aτ f . The additional factor b/τ f accounts for that the branching probability is suppressed compared to the incoherent case. This ansatz of representing the function F (t , t) with information at t and t of an ensemble of particles follows the same spirit of representing the distribution function by an ensemble of particle states. Of course, this is only a crude ansatz for F (t , t), as the latter is actually highly oscillating, and the validity has to be examined by comparing its prediction with the theoretical calculations. Finally, a and b are dimensionless factors whose forms shall be determined in later comparison with theory and will be tuned to achieve an optimal level of agreement to theoretical calculations. With such an ansatz, the medium-induced splitting probability reduces to, dP a bc dω = ∞ 0 dt g 2 P a(0) bc πEλ(t) 1 N N i=1 ∞ t dt bλ(t) τ i (t) δ(t − t − aτ i (t)) = 1 N N i=1 ∞ 0 dt dR incoh (t) dω × abλ(t) τ f (t , t) t =t+aτ f(22) where we have divided and multiplied back an effective mean-free-pathλ(t) = m 2 D /q g so that we may interpret the quantity immediately after the dt integral as the incoherent splitting rate R incoh . The reason for using an effectiveλ is that it can be defined for both scattering and diffusion processes, provided only the screening mass and the transverse momentum diffusion coefficient. From equation 22, it is clear how we can modify the standard Boltzmann transport to include the LPM effect in the deep-LPM region approximately. During the simulation of the incoherent transport (equation 12) of each hard parton, we implement the following modification the parton bremsstrahlung and pair production processes. Suppose a parton splitting process "a → b + c" happens at t = t , 1. Final state partons b and c are not immediately treated as physical objects ("preformed") to the system. A parton a can carry arbitrary numbers of such "preformed" final-state copies. 2. Evolve from t to t + ∆t the "preformed" final state partons in each copy with only elastic processes. While the mother parton a is evolved under the full collision term in equation 12. 3. Recalculate formation time τ f after each time step. The elastic broadening from multiple collisions in step 2, on average, shortens the formation time. 4. Repeat steps 1 to 3 until t − t > aτ f (t) is satisfied. 5. Then, the "performed" final state is considered to become the physical final state with probability p, p = min 1, abλ(t) τ f(23) . In the last step, daughter partons from accepted final state will be treated as independent objects thereafter and be propagated by the full transport equation. Rejected final states are dropped and do not cause any physical effect. A key step is a self-consistent determination of the formation time as described in step 4. In a static medium, it results in the expected scaling of the average inverse formation time in equation 15. This procedure also generalizes to medium with evolving temperature and flow velocity profiles, as the elastic broadening is performed along the trajectory of the hard parton. This iterative procedure was first developed and implemented by [21]. In the following subsection, we shall compare such modification to the leading-log (LL) and the next-toleading-log (NLL) approximation of splitting rates in an infinite medium. We will show that this modification indeed reproduces the qualitative features given by the lead-log results once a is related to the color factors, while the NLL results help to determine the form of b to achieve a quantitative agreement with the theory. Going to an infinitely large medium with an uniform temperature, equation 17 can be cast into its asympototic form known as the AMY formalisim [10,41,42], dR a bc dω = α s d a P a(0) bc (x) Eν a d 2 k (2π) 2 2k · ReF 4x 2 (1 − x) 2 (24) where we have dropped the Bose enhancement and the Pauli blocking factors from the original formula. The vector-valued function F(k; E, x) satisfies the following static equation, 2k = i k 2 F(k) 2x(1 − x)E + g 2 C[F](25) The exact solution can be solved numerically and has already been applied to transport study [13,17], but we shall use approximated semi-analytic solutions obtained in [43]. These approximated solutions were obtained at the leading-log (LL) and next-to-leading-log (NLL)accuracy, and the NLL solution was shown to be a good approximation of the numerical results in the deep LPM regime ω T . At leading-log order, a small-q expansion was performed to obtain a diffusion approximation to the operator C below a certain cut-off q < Q 0 . The resulting leading-log solution is [43], dR a,LL bc dω = α s P a(0) bc πE √ 2 q 3 (x, Q 2 0 ) 2x(1 − x)E(26) Where theq 3 is defined as an effective transport parameter,q 3 (x, Q 2 0 ) = C abc (x) q 2 <Q 2 0 g 2 A(q 2 )q 2 dq 2 (2π) 2 (27) = C abc (x)α s T m 2 D ln 1 + Q 2 0 m 2 D .(28) Note thatq 3 is logarithmically dependent on the cut-off Q 0 . It comes from integrating the perturbative tail of the collision kernel in equation 20. C abc (x) is a processand x-dependent color factor, C abc (x) = C b + C c − C a 2 + x 2 C a + C c − C b 2(29)+ (1 − x) 2 C a + C b − C c 2 . Comparing the leading-log formula to the modified Boltzmann approach in equation 22, the term q 3 /2x(1 − x)E in equation 26 plays the role of the inverse formation time. We can also insert 1/λ ×λ into equation 26 so that it resembles the modified rate in equation 22. However, the effectiveq 3 differs from theq of a single particle, e.g. particle "b" on which we performed the elastic broadening, by an x and color-dependent factor C abc . This can be improved by using a process-and x-dependent a parameter in equation 21, a → a abc (x) = C b C abc (x)(30) Another issue is that the LL solution in equation 26 is still ambiguous up to an unknown cut-off Q 0 througĥ q 3 (x, Q 2 0 ). This Q 0 ambiguity can be improved by going to the NLL order, where the contribution of the collision kernel with q > Q 0 is treated as a perturbation to the LL solution. This "hard" correction to the "multiplesoft" approximation is also recently studied in [44] in the BDMPS framework. In both works, the NLL result is expressed as the LL solution with the unknown Q 0 replaced by Q 1 , Q 2 1 ≈ ωq ≈ ωα s C R m 2 D T ln Q 2 0 m 2 D(31) or one can uses a self-consistent determination of Q 1 as in [43] to eliminated the dependence of Q 0 from the formula, Q 2 1 = 2x(1 − x)Eα s T m 2 D (32) × C b + C c − C a 2 ln 2ξQ 2 1 m 2 D + C a + C c − C b 2 x 2 ln 2ξQ 2 1 x 2 m 2 D + C a + C b − C c 2 (1 − x) 2 ln 2ξQ 2 1 (1 − x) 2 m 2 D 1/2 Therefore, the reasonable choice of the scale inq 3 is of similar order as the transverse momentum of the branching k 2 ≈ 2x(1 − x)Eq. Now, check what this scale is in the original transport approach: the large-Q two-body matrix-element in equation 2 is always integrated up to the maximum transfer bounded by the center-of-mass energy √ŝ of each independent collision. Ignore the sloŵ s-dependence of the elastic cross-section at high-energy and average over the medium parton momentum p 2 in s = (p + p 2 ) 2 over the thermal distribution, then the average Q 2 0 in the transport simulation is ŝ = 6ET . Therefore, we must correct for this difference in scale; otherwise, the ansatz with a naïve choice of an upper limit of the matrix-element integration leads to systematic deviation from the NLL solution in a logarithmic manner. Noticing that the inverse formation time is proportional to ln(1 +Q 2 /m 2 D ), a simple correction can be made using a scale-dependent b parameter in the acceptance probability of the modified transport equation in equation 23, b = 0.75 ln(Q 2 1 ) ln(Q 2 0 ) ,(33) with the NLL-improved scaleQ 2 1 and the native choice of scale in the transport equationQ 2 0 given by, Q 2 1 = 1 + k 2 m 2 D ≈ 1 + τ f,ĩ λ ,(34)Q 2 0 = 1 + 6ET m 2 D .(35) The prefactor 0.75 in equation 33 is the only numerical constant that we have tuned when we compared the modified Boltzmann simulation to the NLL solutions in the next section, and it is the same throughout the rest of the paper. As a remark, this logarithmic correction comes from the integration of the perturbative collision kernel at large-q 2 . Therefore, if one assumes and implements certain collision kernel that vanishes sufficiently fast at large-q, this logarithm factor in b should be dropped. C. Agreement in the Bethe-Heitler region at large-q In the previous subsection, we have shown how the modified Boltzmann transport approach can be matched to the NLL solution in the deep-LPM region where τ f λ el . In this subsection, we demonstrate that the transport equation agrees with the theoretical calculation in another region of phase-space This is the large-q (q > Q cut ) region in the Bethe-Heitler regime τ f λ el . The acceptance probability 23 goes back to unity for this region, and the transport equation is simply the original Boltzmann equation with incoherent rates. Here, we briefly show that the incoherent 2 → 3 rates used in the equation 12 is the same as the rate dR/dω obtained as the leading term in the 1/τ f expansion of the AMY formalism in equations and . When the formation time is very short, one treats 1/τ f = k 2 /2x(1−x)E in equation III C as a large number and solve for the real-part of F by one iteration, d 2 k (2π) 2 2k · ReF 4x 2 (1 − x) 2 E 2 = 2g 2 φ k · C[φ k ],(36) where φ k = k/k 2 . Taking q → q + g as an example, the total 2 → 3 rate in the Boltzmann equation is (usê s ≈ 6ET ), dR q qg dω ∝ i q 2 >Q 2 cut f i (p 2 )dp 3 2 (2π) 3 2p 2 g 4 T q 4 dq 2 dk 2 (37) C F (φ k−q − φ k−xq ) 2 + C F (φ k−q − φ k ) 2 −(2C F − C A ) (φ k−q − φ k−xq ) · (φ k−q − φ k )} where we have used the cross-section from appendix A and the splitting function is not shown explicitly. This transverse momentum integration looks different from the one in equation 36. Summing over the species, colors, and degeneracy of the particle "i", the integration of the momentum p 2 yields the Debye mass with classical statistics, g 4 T q 4 i f i (p 2 )dp 3 2 (2π) 3 2p 2 = 2g 2 m 2 D q 4 .(38) Expanding each squares and changing the integration variable from k to one of k − q, k − xq, k − (1 − x)q for each term, one can show that the integration can be cast into, dR q qg dω ∝ q 2 >Q 2 cut 2g 2 m 2 D q 4 dq 2 dk 2 (39) C A φ k · (φ k − φ k+q ) + C A φ k · φ k − φ k+(1−x)q +(2C F − C A )φ k · (φ k − φ k+xq )} = dk 2 φ k · C[φ k ](40) which agrees with the large-q 2 limit of of the AMY solution in equation 36 when q 2 m 2 D . D. Implement running of αs At the end of this section, we introduce the prescription for a running coupling constant in the modified Boltzmann approach. We used the leading order running coupling constant with n f = 3 and Λ = 0.2 GeV, α s (Q 2 ) = 4π 9 ln (Q 2 /Λ 2 )(41) To avoid the pole when Q approaches the nonperturbative scale, we introduce a cut-off at a medium scale Q med = πT . Therefore the actual coupling constant is α s (max{Q, Q med }). Following the prescription described in [43], the coupling constant associated to the collision kernel C are evaluated at q 2 and the resultingq can be integrated approximately to get the running version of the Eq. 28, q running 3 ≈ 4π 9 g 2 (m 2 D ) − g 2 (Q 2 0 ) 1.27T 3 C abc (x)(42) Where the scale Q 0 is of order m D [E/T ln(E/T )] 1/4 . And the scale of the α s associated to the splitting vertex in equation 26 is chosen at the typical transverse momentum k 2 ∼ 2x(1 − x)Eq 3 . In the modified Boltzmann approach, such a running coupling prescription is straightforward for the elastic part. The transport coefficientsq S andq S,L in Eq. 7 and equation 8 for the diffusion sector should be integrated with α s (q 2 ). The α s associated to the 2 → 2 scattering matrix-elements (including the ones that appear in the 2 → 3 matrix-elements) are evaluated at the t-channel momentum transfer squared. The scale k 2 for the splitting vertex coupling requires an additional treatment, because k comes from the summation over multiple scatterings within the formation time, However, the initial splitting processes are generated using incoherent processes with the coupling evaluated at k 2 (t 0 ). This is on average ω/T times smaller than k 2 (t 0 + τ f ) in the deep LPM region. Therefore the effective coupling after multiple scattering is smaller than the one we used in incoherent calculation and allows for a rejection implementation by modifying the acceptance probability p in equation 23 to k 2 t0+τ f = (k t0 + q 1 + · · · + q n ) 2 .(43)p running = p × α s (k 2 t0+τ f ) α s (k 2 t0 ) .(44) This final step completes the inclusion of running coupling effect in the modified Boltzmann approach. IV. RESULTS In this section, we compare the rate of in-medium bremsstrahlung and pair production from the simulation of the modified Boltzmann approach to the NLL approximation of the AMY equation in an infinite medium. The differential rate dR/dω is shown in Figure 1 for a 1 TeV quark splitting into a gluon and a quark. The temperature of the medium is T = 0.5 GeV and we choose a relatively small coupling constant α s = 0.1. Please also refer to appendix B for a full comparison varying both the FIG. 2. The splitting rate of q → q + g * , g → g + g * , and g → q * +q as a function of the parton energy labeled by the star. The mother parton with E = 1 TeV evolves inside an infinite medium with T = 0.5 GeV. The simulations (thick dashed lines) are compared to the NLL solutions (thin solid lines). parton energy and the coupling constant. The horizontal axis is the radiated gluon energy. In the upper plot, we divided the spectrum into different regions by the gluon thermal mass m ∞ and an estimated Bethe-Heitler energy λ g m 2 D ∼ 2πT . The spectrum at ω < m g is suppressed due to the gluon thermal mass. In the Bethe-Heitler region ω < 2πT , the spectrum scales like ω −1 as given by the incoherent radiation rate. In the deep LPM region T ω < E, the spectrum is dominated by coherent multiple scatterings and scales like ω −3/2 . The power-law fits in each domain are very close to the expected scaling. In the middle plot, we compare the simulation to the NLL solution with self-consistent Q 2 1 from equation 33. The ratio between the two is shown in the bottom plot. There is a good agreement in the LPM region between the simulation and the theory calculation, which we have used as guidance in developing our Monte-Carlo approach. A comparison of the other channels g → g + g and g → q +q to the theory calculations are shown in Figure 2. For the splitting parton energy much greater than temperature ω > 10T , the simulation agrees well with the NLL solution. Again, please also refer to appendix B for comparison varying both E and α s for these two channels. Finally, we compare the running coupling calculation with the theory curve in Fig. 3 for the g → g +g channel. The theory curves (black lines) are obtained combining equation 26 and equation 42. Different line styles correspond to the variation of the Q 0 value around an initial guess m D (E/T ln(E/T )) 1/4 by a factor of 2 above and below. For this 1 TeV parton, the scale Q 0 is actually very large and the running of α s is rather slow, which explains the theory curve is not very sensitive to a factor of 4 change in Q 0 . The simulation was performed using the running coupling prescription described in Section III D. The overall shape of the spectrum in the deep LPM region is again well described by the modified Boltzmann simulation. V. TOWARDS PHENOMENOLOGICAL APPLICATIONS In the previous section, we showed that the modified Boltzmann equation simulation has a good agreement with the theoretical calculation for parton splitting in the LPM regime in an infinite medium. Towards future phenomenological application, we would like to investigate a few more complex scenarios involving a finite and expanding medium. A. A finite medium For calculation in a finite medium, there is an intricate interference pattern near the boundary which requires solving the original equation using a finite medium temperature profile. Or for the case of a thin medium, FIG. 4. Comparison of the path-length dependent rate dR/dω from the simulation using αs = 0.3 to the theoretical calculation for splitting q → q + g [36]. The quark energy is 16 GeV. such effect can be analyzed order by order in the "opacity (L/λ) expansion" [34,35]. One important effect in a finite medium is the path-length dependent radiation rate for L τ f . Considering the formation time of very energetic splitting can be comparable to the size of the QGP fireball, the finite size effect is important for heavyion collision phenomenology, Though the modified Boltzmann approach has been constructed to mimic the case in an infinite medium, it does predict an L-dependent results in a finite medium and we would like to check whether it is significantly different from the theory expectation. The origination of the path-length dependence in the modified Boltzmann approach is that the gluons sampled at t = t 0 are not considered as independent objects until t = t 0 + τ f , meaning the splitting at time t is initiated by an inelastic vertex at t − τ f . In a semi-infinite medium with a step function like temperature profile, T = 0, z < 0 T 0 , z > 0(45) there are no scattering centers at L − τ f < 0 and thus introduces a reduction in the radiation rate for small pathlength. In Figure 4, the q → q + g rate simulated in a semiinfinite medium is compared to the full calculations obtained in [36]. The energy of the quark is 16 GeV, and α s = 0.3. The medium temperature of the left and the right columns are 0.2 GeV and 0.4 GeV respectively. Top and bottom rows show the differential rates for the emit- ted gluon energy ω = 3 GeV and ω = 8 GeV 1 . dR/dω is plotted as a function of the path length L. It is evident that theoretical rates (black solid lines) first grow approximately linearly with L and then bend over to transit to L-independent ones. The simulation describes the large L limit well and also approximately captures the point at which the transition happens. But there are systematic deviations compared to the theory at small path-length. Therefore it will be of great interest to improve to the current simulation approach at small path-length in the future. For example, one possible solution would be using the results from the opacity expansion in the simulation for those splittings that happened close to the boundary and developing matching conditions to the approach we used for the deep LPM regime. B. An expanding medium At the beginning of section III, we have mentioned that the semi-classical transport equation has to be improved when the formation time is comparable to the inverse-gradient of the system. This is indeed an issue for describing hard parton propagation in realistic medium created in heavy-ion collisions. Here, we test the modified transport approach in a simple case that only the temporal change of the medium temperature is considered. It introduces the inverse medium expansion rate as another time scale τ ex . If τ ex τ f , the multiple-collision effect comes from collision centers along the hard parton trajectory in the time-dependent medium, as is done in the modified transport approach. In this work, we shall define τ ex as τ ex = d ln(T 3 ) dτ −1 ,(46) understood as the time over which the localq ∝ T 3 changes notably. For simplicity, parametrize the temperature profile as a power law function of the proper time, T (τ ; ν) 3 = T 3 0 τ 0 τ 2−1/ν ,(47) which mimic the fast-dropping medium temperature at mid-rapidity. The static case is recovered when ν = 1/2 and ν = 1 corresponds to the temperature profile of a Bjorken flow [45]. The resultant τ ex is τ ex = τ 2 − 1/ν .(48) To compare the response of the modified Boltzmann approach to an expanding medium to theoretical calculations, we use the results obtained in the BDMPS framework [37] using the power-law type temperature profile in equation 47. The splitting probability is, dP dω = α s 2πE P q→qg (x)Re τ0+L τ0 dt f t f t f τ0 dt i t i 1 ν 2(49)[I ν−1 (z i )K ν−1 (z f ) − I ν−1 (z f )K ν−1 (z i )] −2 ω=∞ ω , z i,f = 2iν q g (1 − x + C F /C A x 2 ) 2(1 − x)ω τ 0 t i,f τ 0 1/2ν(50) for the q → q + g splitting. For ν = 1/2, this expression reduces to the static BDMPS result [28]. As a remark, the BDMPS calculation considers the multiple-soft limit of the collision kernel and therefore does not include the logarithm that comes from the perturbative tail 1/q 4 . Accordingly, we turn off the large-Q matrix-element scatterings and only retain diffusion plus diffusion-induced radiation components in our simulation. Also, b = 0.75 is used without the logarithmic correction factor in equation 33, and the sameq g = m 2 D C A α s T are input to the theory and the simulation. To suppress other difference in the simulation and the theory, instead of making a direct comparison of the spectrum dP/dω to the BDMPS result, we compare the ratio of the splitting probability in an expanding medium to that of a static medium dP (T = T (τ ; ν))/dω dP (T = T 0 )/dω (51) between simulation and theory to focus on the response to an expanding medium compared to the static case. The medium expansion starts at τ 0 = 0.2 fm/c with T 0 = 1 GeV and stops at τ = 20 fm/c. We take four choices of the expansion rate ν = 1/2, 3/4, 1, 3/2, corresponding to a static medium, a slowly expanding medium, Bjorken flow, and a faster-than-Bjorken expansion respectively. The ratio R ν from both theory and simulation are shown in Fig. 5 for a 100 GeV quark with α s = 0.3. Again, for ω/T 1, the simulation displays the expected decreasing of medium-induced radiation due to the dropping of temperature. In the future, we are looking forward to making a direct comparison to the solution of equation 17 with both varying temperature and adding medium flow effects. VI. COMPARISON WITH TWO OTHER MONTE-CARLO METHODS FOR MEDIUM-INDUCED SPLITTINGS Before we conclude this work, it is beneficial to compare the current implementation of the medium-induced splittings with two other Monte-Carlo approaches. We will also summarize the features and caveats of these two methods for readers reference on this subject. Also, because the other two methods predict very different radiation spectrum, it is not so instructive to compare their spectrum directly. Instead, we compare among these approaches the "energy loss" of a testing quark from gluon radiation, dE dL = E m D ω g dR q qg dω g dω g .(52) As a remark, this definition is not related to the actually parton / jet energy loss in a medium, but only as a simply way to quantify the difference between different methods. We shall see that due to different implementations of the medium-induced radiation, the amount of "energy loss" is very different between these approaches with the same α s . As a result, an extraction of interesting medium properties from experimental data using these models can be biased by the way they treat radiative processes. Therefore, being able to calibrate a model to theoretical calculations as demonstrated in this work is an essential step prior to the comparison to experimental data. A. The approached used in the improved Langevin equation This approach is implemented in the improved Langevin equation [29], using a higher-twist calculation of medium-induced single-gluon emission rate and a prescription for multiple emission in a time-evolution manner. The higher-twist formula is developed in [46,47] and the single-gluon radiation rate is, dN g dxdk 2 dt = α s P (x)q g πk 4 2 1 − cos t − t 0 τ f(53) The rate is time-dependent, coming from the interference between the production of the hard parton at time t 0 and one interaction with the medium at time t. From the second emission, a multiple-radiation is implemented by setting the time t 0 to be the time of the previous emission so that the probability of the next emission starts to accumulate from zero again as time increases [29]. Though it is not immediately clear what this prescription predicts without a simulation, it is possible to get a qualitative understanding by realizing that the typical time separation between two emissions of this model is a time scale ∆t = t − t 0 within which the emission probability is of order one, 1 ∼ t t0 dt 1 xc dx dk 2 dN g (t − t 0 ) dxdk 2 dt .(54) In the soft limit where P (x) ∼ 2/x, τ f ∼ 2xE/k 2 , and perform the time integral first, then the k integral with limits from 0 to xE. The x (with a change of variable to u = xE∆t/2) integral here divergent at 0. Though this is not a problem for calculating energy loss if one also included the absorption processes are required by detailed balance. But to apply the collision rate formalism, one has to render the integral finite with a minimum cut-off x c , 1 ∼ 4α sq ∆t 1 xc dx x dk 2 k 4 1 − sin(∆t/τ f ) ∆t/τ f (55) = α sqg ∆t 3 3u 3 u 3 Ci(u) − 3u 2 Si(u) − u 2 sin(u) (56) +3u − sin(u) − 2u cos(u)) | ∆tE 2 ∆tExc 2 The result can be expanded at small u: 1 3 ln(u) − 0.752 and it quickly decays to 0 at infinity, so a good proxy is to use the small-u expansion but cut-off the upper bound at its zero, 1 ∼ α sq ∆t 3 3 ln 2 x c E∆t ∝ (g 2 T ∆t) 3 ln 2 x c E∆t .(57) One sees that typical time between two emissions in this approach is on the order of 1/g 2 T , which is the same as λ. Putting typical ∆t ∼ λ back to the factor 2(1 − cos(∆t/τ f )), the radiation spectrum is indeed suppressed when τ f λ. However, this suppression is introduced by controlling the correlation between two subsequent emissions, while the LPM suppression actually happens on the level of single emission rate. Moreover, the logarithmic factor in equation 57 depends on the infrared cut-off 2 , therefore the prediction is cut-off dependent, though logarithmically slow. This is because that the prescription changes the second emission rate in the same way no matter how soft the previous emission is. This is, in fact, a feature we have avoided in this work. B. The "blocking radiation" approach Another method [30] will be termed as the "blocking radiation approach". Similar to this work, the splitting is 2 This cut-off is chosen to be ω > πT in [29] also first generated through an incoherent process at time t . It is then followed by a self-consistent determination of formation time with elastic broadening similar to the procedures described in section III. However, different from this work, the "blocking radiation approach" implements the LPM suppression by requiring that no additional radiation is allowed during the formation time of the previous one, while remembering that in our approach, the emitter can have an arbitrary number of independent "pre-formed" final-state copies. Again, this "blocking radiation" approach introduces correlations between subsequent emissions. A closer investigation reveals a bigger problem. In an infinite medium, this approach reduces every τ f /λ inel incoherent emission to one. λ inel is the mean-free-path of incoherent inelastic collisions, and is related to the elastic mean-free-path by λ inel ∝ λ el /α s . It only results in an overall reduction in the radiation spectrum without changing its shape, and the suppression factor λ inel /τ f is off by a power of α s compared to the expected one λ el /τ f . C. Comparison with the modified Boltzmann approach and the analytic results In Figure 6, we show the calculation of "energy loss" defined in 52 of a quark in an "infinitely large" medium. The results presented are normalized by 1/(α 2 s √ ET 3 ) in anticipation of the scaling dE/dL ∝ α 2 s √ ET 3 . For each column, we double the value of α s and for each row, the temperature is increased by 0.2 GeV. Within each subplot, the parton energy varies from 10 GeV to 200 GeV. The three Monte-Carlo methods of medium-induced energy loss are shown in colored lines and NLL AMY results are shown as black lines. As expected, the modified Boltzmann approach (red-dashed lines) which describes the radiation spectrum also reproduces the energy, temperature, and coupling constant dependence of the energy loss. The method used in the improved-Langevin equation (blue-dash-dotted lines) has a similar energy and temperature dependence as the theoretical baseline; however, it systematically deviates from the baseline for different values of the coupling constant in a logarithmic manner. For the "block radiation" approaches, the deviations from the baseline regarding their α s -dependence is completely off, which is not surprising as we have discussed its shortcomings. VII. SUMMARY AND OUTLOOK We have investigated the modification to the semiclassical Boltzmann transport equation to include the LPM effect for parton splitting processes in the deep-LPM region, with the guidance from the leading-log and the next-to-leading-log solutions of the AMY equation. The running coupling effect has also been implemented. The overall level of agreement between the simulated re- ET 3 . The MC implementations of the LPM effect referred to as "modified Boltzmann", "coherence factor", and "blocking radiation" approaches are shown with red-dashed lines, blue-dash-dotted lines, and green-dotted lines respectively. The AMY NLL results are denoted as black boxes. sults and theoretical calculations in the infinite medium limit is promising given the simplicity of this Monte-Carlo procedure. Although it is developed for the deep-LPM regime, this approach captures qualitative features of the path-length dependence of medium-induced splittings and the qualitative change of the spectrum shape in an expanding medium compared to the static case. Future study will focus on improved treatment in a thin medium, and consistent inclusion of the heavy-quark mass effect into the current approach. Being able to calibrate a Monte-Carlo transport model to the theoretical calculation is important. As we have demonstrated in section VI, different modeling of the medium-induced radiation can bias the extraction of the interaction strength between the probe and the medium. The design of this modified Boltzmann transport approach and its systematic comparison to theoretical calculations allow us to reduce and estimate the uncertainty in these implementations. This is instrumental for performing an examination of theoretical assumptions and a more meaningful phenomenological extraction of jet transport properties from future model-to-data comparisons in transport model-based studies. Regarding the large-q 2 → 3 matrix-elements, in previous study [48], we emploied an improved version of the original Gunion-Bertsch cross-section that works under the limits k, q √ s and xq k [49][50][51]. The original Gunion-Bertsch cross-section [49] only works for soft emissions x = k + /p + 1. With the improvements made in [50,51], the agreement with the exact matrix-elements is extended to larger x, but still the splitting function is only reproduced to O(x). In the present study, we relax the condition xq k in the derivation, so that the full leading order vacuum splitting function can be recovered. We summarize the matrix-elements here, |M 2 | g+i→g+g+i = |M 2 | g+i→g+i P g(0) gg D g gg ,(58)|M 2 | g+i→q+q+i = C F d F C A d A |M 2 | g+i→g+i P g(0) qq D g qq , (59) |M 2 | q+i→q+g+i = |M 2 | q+i→q+i P q(0) qg D q qg ,(60) where |M 2 | g+i→g+i , |M 2 | g+i→g+i and |M 2 | q+i→q+g+i are the spin-color averaged two-body collision matrixelements with onlyt-channel contribution. Index i represent a medium quark / anti-quark or a medium luon. The P a(0) bc (x) terms are vacuum splitting functions of parton a to partons b and c. P g(0) gg = g 2 C A 1 + x 4 + (1 − x) 4 x(1 − x) ,(61)P q(0) qg = g 2 C F 1 + (1 − x) 4 x ,(62)P g(0) qq = g 2 N f 2 x 2 + (1 − x) 4 .(63) Finally, the D a bc terms are, D g qq = C A (a − b) 2 + C A (a − b) 2 (64) − C A (a − b) · (a − c), D g qq = C F (a − b) 2 + C F (a − b) 2 (65) − (2C F − C A )(a − b) · (a − b), D q qg = C F (c − a) 2 + C F (c − b) 2 (66) − (2C F − C A )(c − a) · (c − b), with the vectors given by a = k − xq (k − xq) 2 ,(67)b = k − q (k − q) 2 ,(68)c = k k 2 .(69) B. ENERGY AND COUPLING CONSTANT DEPENDENCE OF THE SPLITTING RATE In this appendix, we provide comparisons of splitting rate at different values of energy and coupling constant for the reader's references. Fig. 7, Fig. 8 and Fig. 9 shows the comparison for channels q → q + g, g → g + g and g → q +q respectively. The results are shown as the ratio between the simulations and the NLL solution. Within each figure, the mother parton energy is 10 GeV, 100 GeV, and 1000 GeV from the top to bottom plot. We have used two coupling constants at α s = 0.1 (red solid lines) and α s = 0.3 (blue dashed lines). FIG. 1 . 1The q → q + g splitting rate in an infinite medium from a quark with E = 1 TeV,and a coupling constant αs = 0.1. The top plot shows the simulated spectrum dR/dω (reddashed line) and power law fit (green-dotted and blue-dashdotted lines) in different gluon energy regions, separated by energy scales ωBH ≈ 2πT . The middle plot compares to the simulation to NLL solution to the AMY equation, and the ratio is shown in the bottom plot FIG. 3 . 3Top plot: comparison of modified Boltzmann simulation with the NLL solution with running coupling. Three initial guesses of the The ratio between simulation and theory is shown in the bottom plot. FIG. 5 . 5The ratios of induced splitting rate in expanding medium to that of a stat medium, with expansion parameter ν = 3/4, 1, and 3/2. The analytic results are shown in solid lines and simulations denoted as symbols. The conpling constant αs = 0.3, the expansion starts at τ0 = 0.2 fm/c with an initial temperature T0 = 1 GeV. FIG. 6 . 6Energy loss per unit path lengh dE/dL as a function of energy E, temperature T and coupling constant αs. Each column corresponds to a value of the coupling constant αs = 0.075, 0.15, 0.3, and 0.6 (from left to right). Each row corresponds to a temperature of T = 0.2, 0.4, and 0.6 GeV (from top to bottom). dE/dx is divided by the expected scaling α 2 s √ ACKNOWLEDGMENTS SAB, WK, and YX are supported by the U.S. Department of Energy Grant no. DE-FG02-05ER41367. WK is also supported by NSF grant OAC-1550225. WK would like to thank Florian Senzel, Jean-Francois Paquet and Yacine Mehtar-Tani for helpful discussions. In a practical simulation, the rates are obtained by counting gluons within a finite energy range ω ± 0.5 GeV ATL-COM-PHYS-2012-1116A. Collaboration (ATLAS). A. Collaboration (ATLAS), (2012), ATL-PHYS-PUB- 2012-002, ATL-COM-PHYS-2012-1116. . 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[ "A non-empirical free volume viscosity model for alkane lubricants under severe pressures", "A non-empirical free volume viscosity model for alkane lubricants under severe pressures" ]
[ "Kerstin Falk \nFraunhofer IWM\nWöhlerstr. 1179108FreiburgGermany\n", "Daniele Savio \nFraunhofer IWM\nWöhlerstr. 1179108FreiburgGermany\n", "Michael Moseler \nFraunhofer IWM\nWöhlerstr. 1179108FreiburgGermany\n\nInstitute of Physics\nUniversity of Freiburg\nHerrmann-Herder-Str. 379104FreiburgGermany\n" ]
[ "Fraunhofer IWM\nWöhlerstr. 1179108FreiburgGermany", "Fraunhofer IWM\nWöhlerstr. 1179108FreiburgGermany", "Fraunhofer IWM\nWöhlerstr. 1179108FreiburgGermany", "Institute of Physics\nUniversity of Freiburg\nHerrmann-Herder-Str. 379104FreiburgGermany" ]
[]
Viscosities η and diffusion coefficients Ds of linear and branched alkanes at high pressures P <0.7 GPa and temperatures T =500-600 K are calculated by equilibrium molecular dynamics (EMD). Combining Stokes-Einstein, free volume and random walk concepts results in an accurate viscosity model η(Ds(P, T )) for the considered P and T. All model parameters (hydrodynamic radius, random walk step size and attempt frequency) are defined as microscopic ensemble averages and extracted from EMD simulations rendering η(Ds(P, T )) a parameter-free predictor for lubrication simulations.
10.1103/physrevlett.124.105501
[ "https://export.arxiv.org/pdf/1905.06130v1.pdf" ]
155,089,214
1905.06130
1387e9d33284a412370743e176738d0338119157
A non-empirical free volume viscosity model for alkane lubricants under severe pressures 13 May 2019 Kerstin Falk Fraunhofer IWM Wöhlerstr. 1179108FreiburgGermany Daniele Savio Fraunhofer IWM Wöhlerstr. 1179108FreiburgGermany Michael Moseler Fraunhofer IWM Wöhlerstr. 1179108FreiburgGermany Institute of Physics University of Freiburg Herrmann-Herder-Str. 379104FreiburgGermany A non-empirical free volume viscosity model for alkane lubricants under severe pressures 13 May 2019 Viscosities η and diffusion coefficients Ds of linear and branched alkanes at high pressures P <0.7 GPa and temperatures T =500-600 K are calculated by equilibrium molecular dynamics (EMD). Combining Stokes-Einstein, free volume and random walk concepts results in an accurate viscosity model η(Ds(P, T )) for the considered P and T. All model parameters (hydrodynamic radius, random walk step size and attempt frequency) are defined as microscopic ensemble averages and extracted from EMD simulations rendering η(Ds(P, T )) a parameter-free predictor for lubrication simulations. Knowlegde-based design and optimization of liquid lubricants require a quantitative modelling of their rheological properties under relevant tribological conditions [1]. For instance, lubricants in roller element bearing and gear applications are subject to pressures of the order of GPa [2]. Traditional empirical viscosity models (such as Barus or Roelands equation) fail to describe η(P ) over the relevant pressure range [3] indicating that improved viscosity models for the extreme pressure regime [4,5] should be based on physical insights [6]. A promising approach employs the Stokes-Einstein relation [7] D s = k B T /(nπηR h )(1) that connects the viscosity with the self-diffusion coefficient D s . Here, R h denotes the hydrodynamic radius and n lies between the slip and no-slip hydrodynamics limits, 4 and 6. The applicability of Eq. (1) on a microscopic level has been theoretically motivated and is well established under normal conditions [8,9], also for nonspherical molecules if R h is considered a free parameter. However, a breakdown of Stokes law has been observed in various dense liquids, including molecular glass formers [10,11]. The mechanism of this breakdown is still subject to extensive research, mostly focused on densification obtained by supercooling [12][13][14]. This raises the question whether Eq. (1) remains valid for liquids which are densified by pressurisation instead of cooling. In this letter, EMD simulations are utilized to validate Eq. (1) for linear and branched alkanes (constituents of ordinary lubricant base stock) for a pressure and temperature range representing typical tribological high load applications. We suggest a microscopic defintion of R h = a /π(2) by introducing an EMD averaged molecular cross section a . This represents an important step towards a parameter-free structure-property relationship (SPR) relateing molecular structure with macroscale viscosities. Employing Eq. (1) necessitates an additional SPR for D s . Here, we utilize the free volume (FV) concept [15,16] D s = D 0 exp(−v c /v f ).(3) with the mean FV per molecule v f and the critical volume v c . Note, that since v f is determined by the lubricant density ρ an equation of state ρ(P ) is required to arrive at a pressure dependent viscosity law. Although widely used for soft matter systems [17][18][19], the FV concept is being challenged due to concerns about the relevance of FV compared to energetic effects and about the physical interpretation of the free parameters D 0 and v c [20][21][22][23][24]. We show in the second part of this letter that Eq. (3) can be applied to our alkane lubricants and suggest microscopic definitions of D 0 and v c . The latter relies on the observation that self-diffusion can be considered a random walk of a molecule's center of mass (COM) with step length and attempt frequency determined by EMD simulations. Two linear alkanes and three poly-α-olefines (PAO) (for structures see Fig.1a) are modelled with the allatom optimized potentials for liquid simulations (AA-OPLS) [25]. The EMD of these lubricants is simulated within a constant volume subject to periodic boundary conditions for densities ranging roughly from 470 to 850 kg/m 3 . Time integration is performed employing the LAMMPS software suite [26,27] with timestep 0.5 fs, and a Nosé-Hoover thermostat with relaxation time 0.1 ps [28,29]. Viscosities η are determined via the Green-Kubo formalism and self-diffusion coefficients D s via the mean squared displacement (MSD) [9] (see supp. Figs. S1/2). As shown in Fig. 1b, results for η and D s vary over 3 orders of magnitude and are fully compatible with Eq. (1) (assuming slip boundary conditions n=4 [8,9]). A parameter-free quantitative agreement is achieved by introducing the hydrodynamic radius via Eq. (2) as follows. Since the Stokes drag on macroscopic solid objects with slippery surfaces scales with the object's cross section in the direction of a displacement [30], we calculate a directional molecular cross section a as indicated in Fig. 1c. To each molecule a volume v mol is assigned using a coarse grained hard sphere approach based on a CH X (X = 1, 2, 3) united atom representation [31] (see details in supp. Fig. S3). Then the molecule is displaced over a short distance ε and its effective cross section area is defined by a = ∆V /ε, where ∆V is the newly occupied volume. Finally, the mean cross section is obtained by a = aq(a)da, where q(a) is the probability for the molecule to have a configuration with cross section a. Interestingly, a scales linearly with the molecule size, despite the different morphologies (i.e. number of branches in alkanes -see inset in Fig. 1b). This scaling can be rationalized by a cylindrical shape estimate of long alkane chains, neglecting the contribution of chain ends and knots (supp. Fig. S4). After having established a parameter-free relation between η and D s , we now focus on the diffusive motion of the alkanes. Fig. 2a shows part of a C 10 -trimer COM trajectory with a behaviour which is characteristic of a caging effect. The COM position oscillates within a compact volume due to confinement by the neighboring molecules (snapshots 1,3,5 of Fig. 2a). Elementary diffusion steps (EDS) take place via occasional irreversible translations (indicated by red arrows in snapshots 2,4,6 of Fig. 2a). The FV ansatz leading to Eq. (3) assumes that the probability for an EDS is given by p(v c ) = exp(−v c /v f ). Here, v c is the critical void size in the cage formed by a molecule's neighbours allowing for an irreversible COM jump. Following the simple argument that this void has to accommodate the molecule, the critical volume v c is expected to be of the order of the hard core molecule volume v mol . Note, that p(v c ) depends parametrically on the mean free volume per molecule v f = v − v mol , where v denotes the molar volume [16]. Indeed, the self diffusion coefficient follows the form Eq. (3), as shown in Fig.2b, where lines are best fits for the 600 K data. Fig. 2c displays the dependence of the fit parametersṽ c andD 0 on the size of the molecules. Surprisingly, the critical volume is about 3 times larger than v mol in contradiction to the simple argument stated above. An alternative interpretation of v c is based on the following consideration. To perform an EDS, a molecule needs to move from its cage center to a void in the cage wall. The necessary critical volume for this displacement over the cage size r c is then r c · a with a the molecule's cross section. For constant a and r c , the diffusion process could then be pictured as a random walk with stepsize r c and step frequency 1/∆t. The latter is the product of an attempt frequency 1/τ 0 and the success probability exp(−r c a/v f ) resulting in D s (a, r c ) = r 2 c /(6∆t) = r 2 c /(6τ 0 ) exp(−r c a/v f ). (4) However, both a and r c depend a priori on the molecules' configurations with respect to the direction of the EDS. Fluctuating shapes and distances in molecular fluids result in a probability distribution q(a, r c ) for a and r c that determine the diffusion coefficent D s (a, r c ) in a certain direction and thus the total diffusion coefficient is D s = D s (a, r c )q(a, r c ) da dr c . In the following, we demonstrate by sampling q(a, r c ) over all possible configurations and orientations that the first moments a = aq(a, r c ) da dr c and r c = r c q(a, r c ) da dr c dominate the diffusion process: D s ≈ r c 2 /(6τ 0 ) exp(− r c a /v f ).(5) For a given configuration, Eq.(4) implies a direction dependent stepsize and success probability for an individual EDS. To study this anisotropy we consider an auxiliary system of preferentially oriented alkanes immersed in a bath of unconstrained molecules. This artificial test situation is realized for n-hexadecane at two different densities by applying opposing external forces ±0.05 eV/Å to the head and tail carbon atoms of 2.6% of the molecules (Fig.3a). The resulting preferential orientation leads to a permanently anisotropic cross section a(θ) (Fig.3b) and cage radius r c (θ) (red dots in Fig.3c), where θ denotes the angle between the applied forces and the EDS direction. Here, a(θ) was calculated as previously defined (see Fig. 1c) for a given direction θ. Lacking an unambiguous definition of r c (θ) , a pragmatic estimate was based on the direction dependent radial distribution function g COM (θ, r) of the molecules' COM ( Fig. 3c) via g COM (θ, r c (θ) ) = 1. In the same spirit, the isotropic cage size r c (circle in Fig. 3c) was determined from the isotropic radial distribution function g COM (r) (bold line in Fig. 3c). Note, that this value is close to the result obtained via r c = r c (θ) d cos θ. As expected the preferentially oriented molecules exhibit a pronounced anisotropy in the MSD r 2 (θ) (Fig.3d) and consequently in the diffusion coefficient D s (θ). Interestingly, the mean diffusion coefficient D s = (D s (θ = 0) + 2D s (θ = π/2))/3 is equal to the isotropic diffusion coefficient of the unperturbed molecules, confirming that diffusion is given by an average over all directions with respect to the main axis of a hexadecane molecule. Most importantly however, D s (θ) scales with the respective static structure properties as predicted by Eq.(5) validating the applicability of a FV ansatz on the microscopic level of an EDS (Fig. 3e). Comparing Eq.(3) with (5) leads to D 0 = r c 2 /(6τ 0 ) and v c = r c a . Indeed, applying the above structure evaluation for a (inset in Fig. 1b) and r c (Fig. 4a) to the unperturbed systems of all 5 fluid types reveals that the product v c = r c a agrees well with the fitted critical volumesṽ c (Fig. 4a inset). Both, a and r c are only weakly dependent on density and temperature (supp. Figs. S5/6) and can be conveniently estimated from an EMD simulation for a single ρ and T . The remaining free parameter τ 0 (time between random walk attempts) can be interpreted as the time scale for structure decorrelation in the molecule/cage system. The connection between structural relaxation, diffusion and viscosity is subject of ongoing research [33], and fully unravelling the underlying mechanisms goes far beyond the scope of this work. Nevertheless, as a starting point we consider the time autocorrelation function e(0)·e(t) of the molecules' end-to-end vector orientation e(t) (see Fig. 4b) to quantify the intramolecular structure decorrelation. For long hydrocarbon chains, e(0) · e(t) is well described by a double exponential function [34] e(0) · e(t) = Ce −t/τα + (1 − C)e −(t/τ β ) b with two separate characteristic decay times τ α and τ β . On the one hand, a long time decay is observed on the time scale of the diffusion process τ α ≈ ∆t = r 2 c /(6D s ), ranging from 10 ps -1 ns. On the other hand, τ β is of the order of 1 − 10 ps and is insensitive to the fluid density (within statistical uncertainties, see supp. Fig. S7). This β-relaxation time fits well with the expected attempt frequency for the random walk τ β ≈ τ 0 = r 2 c /(6D 0 ) as illustrated in Fig. 4c, which suggests it as a good measure for the relevant structure decorrelation on short times. The presented results are further validated in a series of scaling tests with modified model parameters for both intra-and intermolecular interactions (supp. Fig. S8). In particular, the strength of nonbonded interactions has little influence, but a scaling of the atomic radii σ LJ results in strong variations of the diffusivity caused by the exponential term in Eq. (5). Moreover, the prefactor D 0 is sensitive to variations of the energy barrier for bond rotation, which influences intramolecular relaxations. All scaling tests are also in quantitative agreement with the Stokes-Einstein relation, supporting our definition of a hydrodynamic radius in Eq. (2). Finally, combining Eqs.(1), (2) and (5) the viscosity can be expressed as a function of density log η(T, ρ) η 0 (T ) = r c a v mol (ρ m /ρ − 1) −1(7) with ρ m = M/v mol the maximum hypothetical density for zero free volume (1 − v f /v = ρ/ρ m ; supp. Fig. S4). Apart from the density, the r.h.s. contains only equilibrium structure properties, namely the molecules' volume v mol , mean cross section a and mean next neighbor distance r c . The temperature dependence of η enters via η 0 (T ) = 1.5k B T τ 0 (T )/(r 2 c π a ). By employing Eq.(7) and identifying τ 0 with τ β a parameter free rescaling of the simulated viscosities can be established (Fig. 4d). This scaling law can also be applied to experimental high T and P viscosity data for n-dodecane [32]. We find a good agreement of the experimental data with our parameter-free viscosity model Eq. (7) (inset of Fig. 4d). To conclude, a combination of basic random walk and FV theory fully describes the self diffusion mechanism of long alkane chains, linear and branched alike, in the high T and P regime. A crucial part of the presented work is the introduction of the mean cross section a and mean cage size r c as novel molecule shape parameters. While a establishes a quantitative link between viscosity and self diffusion via the Stokes-Einstein relation, a and r c allow for a parameter-free density scaling of both transport coefficients. The viscosity model can be directly implemented in density-based Reynolds-solvers [35] and will contribute to a new cutting-edge simulation tool for tribological applications. The proposed shape parameters also opens new possibilities to quantify the role of molecular structure on rheology in anisotropic situations, such as shear thinning [36,37] or in nanometer-thin boundary lubrication films [38]. Our approach might also be useful for other soft materials, such as self assembled membranes [17,18], polymer-solvent systems [19] or adsorbates in nanopores [39]. FIG. 1 . 1Stokes radius of complex molecules. (a) Considered molecular structures: n-dodecane (1, red), n-hexadecane (2, orange), PAO-C10-dimer (3, green), PAO-C10-trimer (4, light blue), and PAO-C10-tetramer (5, dark blue) (b) Shear viscosity η and self diffusion coefficient Ds under extreme pressure (up to 0.7 GPa) and temperature ( / 600/500 K) from EMD simulations: Quantitative agreement with the Stokes-Einstein relation Eq.(1), assuming a slip boundary condition (n=4) and a molecule radius R h = a /π with a the molecule's mean cross section area (inset: mean a , standard deviation). (c) Definition of the configuration dependent cross section area a := ∆V /ε (∆V newly occupied volume after a small virtual displacement ε), and resulting distributions q(a); for comparison, lines show Gaussian distributions. FIG. 2 . 2Dependence of diffusion on free volume. (a) Part of a C10-trimer trajectory within bulk fluid at 0.6 MPa. Light blue: C-atoms; large dark blue: COM position; small violet: all COM positions (every 1 ps during 85 ps). The COM diffusive motion (arrows) can be described by a random walk between caged positions due to the confining presence of the surrounding molecules (not displayed). (b) Self diffusion coefficient Ds vs. inverse of the mean free volume per molecule v f = v − v mol (colors as in Fig. 1, / 600/500 K). Lines are best fits of Ds = D0 exp(−vc/v f ) on 600 K data. (c) Fit resultsD0 andṽc; errorbars: 68%-confidence interval. artificially oriented n-hexadecane molecules in an unconstrained bath. (a) Superimposed configurations of one randomly chosen bath and one oriented molecule (±F applied along θ = 0; π, COM motion subtracted, 1 frame/ns). (b) mean cross section a(θ) for un-/constrained ( / ) molecules, unperturbed value a seeFig.1. (c) θ-dependent COM radial distribution function (shifted for better visibility) with estimate of cage radius rc(θ) (•); same for isotropic COM-RDF of bath molecules (bold line, ). (d) Anisotropic MSD for oriented molecules in least dense system. (e) Anisotropic self diffusion coefficient Ds(θ) (2 different densities , ) normalized with static structure properties according to Eq.(5); direction averaged values Ds = (Ds(0) + 2Ds(π/2))/3, isotropic Ds of bath molecules and of unperturbed systems (×, +, ) are identical. FIG. 4 . 4Scaling of viscosity and self diffusion coefficient with molecule structure properties [Colors as inFig. 1]. (a) COM radial distribution functions (lines, shifted for better visibility) with estimate of rc (•); inset: fit result for the critical volumẽ vc (seeFig. 2) vs. rc a ( a mean cross section area, seeFig. 1). (b) Time autocorrelation function of the C10-dimer endto-end vector orientation e(t) for all considered densities at 500 K (symbols) with fits of Eq. (6) (lines); all other molecules in supp.Fig. S7. (c) Correlation times τ β and τα (full/empty symbols) vs. predicted attempt and waiting time of the random walk diffusion model τ0 = rc 2 /(6D0) and ∆t = rc 2 /(6Ds), respectively; data for τ β shows the mean value from all densities. (d) Scaled viscosity log(η/η0) with η0 = 1.5kBT τ β /( rc 2 π a ) vs. ratio of critical to free volume rc a /v f (no free parameter); inset: Experimental data for n-dodecane at T = 473 K from Ref.[32] ( ) and prediction from simulations (line). The authors gratefully acknowledge funding by the industrial partners of the MikroTribologie Centrum µTC (Karlsruhe, Germany), computing time within project HFR14 at NIC Jülich and useful discussions with L. Joly, S. Kapfer and L. Bocquet. S Bair, High Pressure Rheology for Quantitative Elastohydrodynamics. 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[ "Learning to be adversarially robust and differentially private 1 CONVERGENCE RATES OF THE LOGISTIC LOSS ON A LINEARLY SEPARABLE PROBLEM WITH GRADIENT DESCENT", "Learning to be adversarially robust and differentially private 1 CONVERGENCE RATES OF THE LOGISTIC LOSS ON A LINEARLY SEPARABLE PROBLEM WITH GRADIENT DESCENT" ]
[ "Jamie Hayes ", "Borja Balle ", "M Pawan ", "Kumar Deepmind " ]
[]
[]
We study the difficulties in learning that arise from robust and differentially private optimization. We first study convergence of gradient descent based adversarial training with differential privacy, taking a simple binary classification task on linearly separable data as an illustrative example. We compare the gap between adversarial and nominal risk in both private and non-private settings, showing that the data dimensionality dependent term introduced by private optimization compounds the difficulties of learning a robust model. After this, we discuss what parts of adversarial training and differential privacy hurt optimization, identifying that the size of adversarial perturbation and clipping norm in differential privacy both increase the curvature of the loss landscape, implying poorer generalization performance.
null
[ "https://arxiv.org/pdf/2201.02265v1.pdf" ]
245,828,032
2201.02265
b427dbff2e4a544430cf8b3e125f201bac0b28f2
Learning to be adversarially robust and differentially private 1 CONVERGENCE RATES OF THE LOGISTIC LOSS ON A LINEARLY SEPARABLE PROBLEM WITH GRADIENT DESCENT Jamie Hayes Borja Balle M Pawan Kumar Deepmind Learning to be adversarially robust and differentially private 1 CONVERGENCE RATES OF THE LOGISTIC LOSS ON A LINEARLY SEPARABLE PROBLEM WITH GRADIENT DESCENT We study the difficulties in learning that arise from robust and differentially private optimization. We first study convergence of gradient descent based adversarial training with differential privacy, taking a simple binary classification task on linearly separable data as an illustrative example. We compare the gap between adversarial and nominal risk in both private and non-private settings, showing that the data dimensionality dependent term introduced by private optimization compounds the difficulties of learning a robust model. After this, we discuss what parts of adversarial training and differential privacy hurt optimization, identifying that the size of adversarial perturbation and clipping norm in differential privacy both increase the curvature of the loss landscape, implying poorer generalization performance. CONVERGENCE RATES OF THE LOGISTIC LOSS ON A LINEARLY SEPARABLE PROBLEM WITH GRADIENT DESCENT We begin by analyzing logistic regression as a motivating example. Let ℓ (·, (·, ·)) : W × (X, Y) → R be the logistic loss ℓ ( ; , ) = log(1+ − ). We assume X ⊆ R and ∀( , ) ∈ (X, Y), ∥ ∥ ≤ 1 and ∈ {±1}. We further assume the data is linearly separable with margin , and set = arg max ∥ ∥=1 min ∈ [ ] , the optimal hyperplane that classifies all ( , ) correctly with margin at least . We let ∈ N denote the training step, and ∈ R denote the learning rate. We will see that the difference in rate of convergence between non-robust and robust optimization grows with the dimensionality of the input space X. Following a similar analysis in [6], we compute convergence rates for the loss under gradient descent with and without adversarial training and differential privacy. Without robustness we optimize the loss ( ) = 1 =1 ℓ ( ; , ), while in adversarial training we minimize the adversarial loss ( ) = 1 =1 ℓ ( ; , ), where ℓ ( ) = max ∥ ∥ ≤ log(1 + − ( + ) ) = log(1 + − + ∥ ∥ * ), where is the size of adversarial perturbations. Throughout our analysis we use the ℓ 2 norm on , however we also conduct experiments with the ℓ ∞ norm. Under differential privacy the gradient at each time step is perturbed by Gaussian noise with variancethe privacy guarantees can be computed based on the amount of noise, number of iterations and Lipschitz constant of the loss in a standard way [2]. Table 1 gives the converge rates in each of our four possible optimization settings, with proofs given in appendix A. 1 We observe that optimizing the adversarial loss introduces a dependence on the adversarial budget in the convergence bound, while making the optimizer differentially private introduces terms 2 depending on the dimension and amount of noise. Combining adversarially robust and private optimization introduces the two modifications on the bound almost independently -the only cross-contribution is through a negative term. In general, two factors dominate the rate of convergence in robust and private optimization: (i) the dimensionality of data ; and (ii) the size of the adversarial budget with respect to the size of the margin . We discuss some interesting observations that arise from considering these different optimization settings. Hierarchy of empirical risk bounds. We inspect the hierarchy of empirical nominal and adversarial risk in non-private and private settings. Firstly, empirical nominal risk is always smaller than empirical adversarial risk in either setting. The non-private empirical adversarial risk can be smaller or larger than private empirical nominal risk. The position of overlap depends upon respective hyperparameters and , and data dependent parameters such as dimensionality. These effects are shown in fig. 1 for a simple binary classification task. Step 10 3 10 0 10 3 Gap d=10 1 d=10 3 d=10 5 Non-private Figure 2: Gap between bounds on empirical adversarial and nominal risk as a function as the number of optimization steps. We measure the gap in the non-private setting, and in the private setting for different values of . We use the same parameters as described in fig. 1. The gap between empirical adversarial risk and nominal risk is approximately equal in non-private and private settings for small . Despite the aforementioned hierarchy, it is worth asking: does the gap between nominal and adversarial risk expand as we introduce privacy into the optimization process? The answer is yes, however, arXiv:2201.02265v1 [cs.LG] 6 Jan 2022 ✗ ✗ ( ) 8− 8 ( log ) 2 + 1 + ( 8− 4 ) log( +1 ) ✗ ✓ E[ ( ) ] 8− 8 ( log ) 2 + 1 + 2 + ( 8− 4 ) log( +1 ) + 2 ✓ ✗ ( ) 8− (1+ ) 2 8 − ( log − ) 2 + (1 + ) 2 + 8− (1+ ) 2 4 − 2 log( +1 ) ✓ ✓ E[ ( ) ] 8− (1+ ) 2 8 − ( log − ) 2 + (1 + ) 2 + 2 + 8− (1+ ) 2 4 − 2 log( +1 ) + 2 this gap tends to zero over training given a sufficient number of steps. We measure the gap between adversarial and nominal risk throughout training in either the private or non-private setting. Figure 2 shows that this gap tends to zero as → ∞, and that the gap increases in the private setting as increases. Empirical adversarial risk comparison. It is worth noting that fig. 1 is not an apples-to-apples comparison because we are comparing standard loss ( ) under standard training against robust loss ( ) under adversarial training. We can compare with robust accuracy under standard training by noting that under gradient descent, we can upper bound on the robust loss by ( ) ≤ Hierarchy of convergence on non-linear models. We now evaluate how well our bounds on logistic regression match the empirically observed convergence rates of non-linear models. We train a three layer convolutional neural network on the MNIST dataset, and evaluate canonical measures as introduced previously. Firstly, fig. 4 shows the convergence hierarchy as previously introduced is preserved in practice; nominal training has the lowest loss while private and robust optimization incurs the largest loss over training. Nearly identical observations were made for the test loss. The nominal model has a test set accuracy of 99.03%, the robust model a test set accuracy of 99.20%, the private model a test set accuracy of 96.10%, and the robust and private model a test accuracy of 94.43%. The private models are ( , )-DP, with = 3.8 and = 10 −5 , where individual gradients were clipped to norm one (cf. Abadi et al. [1]). The robust models were trained with ℓ ∞ perturbations of size 0.3 using 40 steps of PGD. All models are trained with standard SGD. Additionally, we train analogous models on CIFAR-10 using six layers with max-pooling and tanh activations, and observe similar descent curves. The private models are ( , )-DP, with = 8 and = 10 −5 , where individual gradients were clipped to norm 0.1. The robust models are trained to be robust to ℓ ∞ perturbations of size 8 /255 using 20 steps of PGD. Gap in practice. We now evaluate the gap between empirical adversarial and nominal risk on this non-linear model. From fig. 2, we would expect that this gap shrinks as → ∞, with the gap being strictly larger in the private setting. This is precisely what we observe in fig. 5. Of course, we can no longer be confident the gap will tend to zero in this non-toy setting. Robustness gap. In fig. 6a and fig. 6c, we empirically evaluate the robustness of each model trained on MNIST and CIFAR-10, using 100 PGD steps. For both robust and non-robust optimized models, the private robustness curve resembles a shifted version of the non-private robustness curve. From fig. 6b, we note the relative improvement in robust accuracy when going from a non-robust model to a robust model is slightly higher in the non-private setting in comparison to the private setting. This implies that learning to be both robust and private is a more difficult learning task, and as a result robust accuracy suffers. We also note, from fig. 6c and fig. 6d, the relative gap in improvement in robust accuracy between private and non-private optimization on CIFAR-10 models is larger than on MNIST. This is to be expected as our previous analysis suggested learning to be robust and private is more difficult for datasets of larger dimensions. THE ROLE OF CLIPPING, NOISE AND ADVERSARIAL BUDGET In the previous section we have seen examples of the increased risk brought about by robust and private optimization, and that theory and practice are in alignment. We now take a deeper look at why private and robust optimization is difficult. In particular, we concretize the connection between the smoothness of the loss landscape, adversarial training and the magnitude of learned parameters. Once again, we use the example of binary logistic regression on separable data. Adversarial training. We again consider binary logistic regression with loss ℓ ( ) = max ∥ ∥ ≤ log(1 + − ( + ) ) = log(1 + − + ∥ ∥ * ). We show in appendix B that: ∇ 2 ℓ ( * ) = 2 ∥ * ∥ − * * ∥ * ∥ 2 ,(1) where * is the optimal solution, perfectly separating data from the two classes. Clearly ∇ 2 ℓ ( * ) is a positive semi-definite matrix with eigenvalues at 0 and 2 ∥ * ∥ -we can take the eigenvector to be * and then ( − * * ∥ * ∥ 2 ) * = 0, and we can also take the eigenvector to be orthogonal to * resulting in: 2 ∥ * ∥ − * * ∥ * ∥ 2 = 2 ∥ * ∥(2) Thus, the curvature of our loss landscape is completely determined by the ratio of adversarial budget to the magnitude of the optimal parameters, and as decreases we achieve smoother solutions. Role of clipping. We define the clipping operation on a function, , as ( , ) = min(1, ∥ ∥ ) , for some ∈ R + . Xie et al. [12] showed that that gradient clipping implies bounded weight parameters. We can assume the optimal parameter takes the form * := ℎ( ) ∥ ∥ , where ℎ( ) is a monotonically increasing function of the gradient clipping parameter, . Note, we also assume we can find the optimum under the clipping norm > 0, and so we can repeat the above analysis. This leaves the smoothness of the loss landscape determined by 2ℎ ( ) . Similar to the purely adversarial setting, this has the intuitive property that decreasing the clipping norm decreases the smoothness of the loss landscape. Role of noise. The landscape sharpness is governed by the ratio of adversarial noise to weight norm. If we perform noisy gradient descent: +1 = − (∇ + ), where ∼ N (0, 2 ), then +1 ∼ N ( − ∇, 2 2 ). We can use Jensen's inequality to upper bound the expected norm of +1 : E[ +1 ] ≤ E[ +1 2 ] 1 2 = √︃ ∥ − ∇∥ 2 + 2(3) This implies the expected norm is larger with an additional Gaussian noise term added to gradients. Empirical validation. We now demonstrate that these effects are observed in practice using multi-class logistic regression on MNIST. We show that smaller adversarial budgets, , and larger clipping norms, , lead to smoother solutions with smaller generalization error. Firstly, we train without differentially private noise; only clipping and adversarial training are activated. This means our models are not differentially private but allows us to show our analysis holds empirically (c.f. fig. 7). In total, we train 2500 multi-class logistic regression on MNIST, sweeping over different training attack budget ( ) and clipping threshold ( ) configurations. Each point in the fig. 7 represents a model trained with a specific ( , ) configuration. After this, we will show that moving to differentially private models has virtually no difference empirically, implying our analysis is useful in these settings (c.f. fig. 8). Again, in total, we train 2500 multi-class logistic regression on MNIST, sweeping over different training attack budget ( ) and differentially private configurations. Each point in the fig. 8 represents a model trained with a specific ( , ) configuration. In fig. 7a, we plot the maximum eigenvalue of the Hessian for different training attack budgets, , and maximum gradient norm, . As expected, the smallest eigenvalues belong to solutions with small and large . For values of close to zero, effective learning becomes increasingly difficult, resulting in non-smooth solutions regardless of the choice of attack budget, . Likewise, for attack budgets close to 0.1, the maximum Hessian eigenvalue remains high even if we increase the clipping threshold, , (i.e. the space of learnable solutions). In fig. 8a, we plot analogous results but for fully differential private models, and observe similar results; small attack budgets, , and less privacy (large ) results in smaller eigenvalues. In fig. 7b and fig. 8b, we fix and , respectively, and visualize the relationship between maximum Hessian eigenvalue and test accuracy as a function of the attack budget . We can immediately note that smaller eigenvalues (that imply smoother solutions) result in smaller generalization error, confirming that smooth solutions are preferable. Unsurprisingly, as increases, the probability of finding a smooth model with high test accuracy decreases. Similarly, in fig. 7c and fig. 8c, we fix the attack budget , and visualize the relationship between maximum Hessian eigenvalue and test accuracy as a function of either or . Again, we observe that less smooth solutions are more likely to have smaller test set accuracy. Decreasing or (more privacy) increase the chances of finding a model with a large maximum eigenvalue and (comparatively) small test set accuracy. We have a theoretical understanding of how clipping and adversarial training affect the learned model. Importantly, the trends 3 fig. 6b and fig. 6d, given an attack budget, , we plot the difference in accuracy between a robust model and a nominal model, in either a non-private or private setting. We refer to this as the improvement of the robust model over the nominal model. observed in fig. 7 are mirrored in fig. 8, implying that our analysis is useful in the fully differentially private and robust setting. DISCUSSION Recent work by Song et al. [10] has empirically shown a tension between privacy and robust learning. In particular, it is shown that six state-of-the-art defense methods designed to reduce the success of adversarial examples increase the risk of membership inference attacks due to overfitting on the training set [9]. The phenomenon of overfitting in adversarially robust optimization has also been observed by Rice et al. [7]. If Song et al. [10] showed that robust models are less private, Tursynbek et al. [11] recently identified the contrapositive relation, empirically showing that differentially private models come at the expense of robustness. Meanwhile, Ghazi et al. [5] shows that the sample complexity of learning both robust and private halfspaces is worse than learning with either property by itself. A CONVERGENCE RATES OF LOGISTIC LOSS ON A LINEARLY SEPARABLE PROBLEM WITH GRADIENT DESCENT Let ℓ (·, (·, ·)) : W × (X, Y) → R be the logistic loss, we assume X ⊆ R and ∀( , ) ∈ (X, Y), ∥ ∥ ≤ 1 and ∈ {±1}. Following a similar analysis in [6], we compute convergence rates for gradient descent with and without adversarial training and differential privacy. We assume the data is linearly separable with margin , and set = arg max ∥ ∥=1 min ∈ [ ] , the optimal hyperplane that classifies all ( , ) correctly with margin at least . No privacy / No robustness. We find convergence rates under the logistic loss ℓ ( ; , ) = log(1 + − ) 2 . The logistic loss has the following first and second derivatives: ∇ℓ ( ) = − − 1 + − (4) ∇ 2 ℓ ( ) = − (1 + − ) 2 (5)(6) Assuming ∥ ∥ ≤ 1, we have ∥∇ℓ ( )∥ ≤ 1 and ∇ 2 ℓ ( ) ≤ 1 4 . We let ( ) = 1 =1 ℓ ( ) , and so ∇ ( ) = 1 =1 − − 1+ − . By Taylor expansion we have: ( +1 ) = ( − ∇ ( )) ≤ ( ) − ∇ ( ) 2 + 2 8 ∇ ( ) 2(7) where is the learning rate at step . For simplicity, we assume gradient descent is performed with a constant learning rate := for every step . Our following analysis will depend on a sufficiently small learning rate, which we specify at the relevant places. Because is smooth (bounded Hessian), we can apply standard gradient descent convergence analysis. For any ∈ R : +1 − 2 = − 2 − 2 ⟨∇ ( ), − ⟩ + 2 ∇ ( ) 2 (8) ≤ − 2 − 2 ( ( ) − ( )) + 2 ∇ ( ) 2 (by convexity of ℓ) (9) ≤ − 2 − 2 ( ( ) − ( )) + 1 − 8 ( ( ) − ( +1 )) (by Taylor expansion)(10)= − 2 + ( 1 − 8 − 2 ) ( ) + 2 ( ) − 1 − 8 ( +1 )(11) If < 4, the contribution of the ( ) term is negative and so: +1 − 2 ≤ − 2 + 2 ( ) − 1 − 8 ( +1 )(12) If we set 0 = 0, then 1 = − =1 − 2 and 1 ≤ 2 ≤ 1 as long as < 2. Under the assumption that ( +1 ) ≤ ( ), summing the above between = 1, ..., , we get: +1 − 2 ≤ 1 − 2 − 1 − 8 ( +1 ) − 2 ( ) (13) =⇒ ( +1 ) ≤ 1 − 8 ( 1 2 + ∥ ∥ 2 ) + 2(1 − 8 ) ( )(14) Since we are free to choose , we set it to := log , then: ( ) = 1 ∑︁ =1 log(1 + − log ) ≤ 1 ∑︁ =1 log(1 + − log ) = log( + 1 )(15) Together with the fact that ∥ ∥ = log , we get the final upper bound: 2 We omit the and terms in ℓ for brevity hereon in. 6 ( +1 ) ≤ 1 − 8 1 + ( log ) 2 + 2(1 − 8 ) log( + 1 ) (16) = 8 − 8 1 + ( log ) 2 + ( 8 − 4 ) log( + 1 )(17) No privacy / robustness. We next find convergence rates under gradient-based adversarial training. Let ℓ ( ) = max ∥ ∥ ≤ log(1 + − ( + ) ) = log(1 + − + ∥ ∥ ). We first find the first and second derivative of this loss. If we let ℎ( ) := − + ∥ ∥ , then ℓ ( ) = log(1 + ℎ( )) and ∇ℓ ( ) = ℎ ′ ( ) 1+ℎ ( ) , where ℎ ′ ( ) = ℎ( ) ( ) if we take ( ) := − + ∥ ∥ . Similarly: ∇ 2 ℓ ( ) = ℎ( ) ( ) ( ) + ℎ( ) 2 ′ ( ) + ℎ( ) ′ ( ) (1 + ℎ( )) 2(18) To bound the above we must bound ′ ( ) = ( ∥ ∥ − ∥ ∥ 3 ). We first note that the problem can be reduced to finding a lower bound to for any > 0 because ∥ ′ ( ) ∥ ≤ 2 ∥ ∥ . If we let 0 = 0, then ℓ is no longer differentiable at zero, so we take the sub-gradient to be: − + ∥ ∥ (− + ) 1 + − + ∥ ∥ ∈ ℓ ( 1 )(19) where is a vector with ∥ ∥ ≤ 1. For example, we can take = 0, and then the sub-derivative becomes − 2 ∈ ℓ ( 1 ), and is bounded above like 1 = 0 2 =1 , and ⟨ 1 , ⟩ ≥ 0 2 . We then take := for > 1 such that 0 2 > . We then note the inner product between −∇ℓ ( ) and is strictly positive: ⟨−∇ℓ ( ), ⟩ ≥ − + ∥ ∥ 1 + − + ∥ ∥ ( − ) > 0(20) and so ⟨ 2 , ⟩ = ⟨ 1 − ∇ ( 1 ), ⟩ = ⟨ 1 , ⟩ + ⟨−∇ ( 1 ), ⟩ ≥ ⟨ 1 , ⟩ ≥(21) By a similar argument for any > 1, ⟨ , ⟩ ≥ . Because is the global minimizer it follows that ≥ . We have a lower bound on for any and so ∥ ′ ( )∥ ≤ 2 . The other term we upper bound is ( ) ( ) = − 2 ∥ ∥ + 2 ∥ ∥ 2 , and since we assume bounded data we can bound this like ( ) ( ) ≤ (1 + ) 2 . Piecing this altogether gives: ∇ 2 ( ) ≤ ℎ( ) ( ) ( ) (1 + ℎ( )) 2 + ℎ( )(1 + ℎ( )) ′ ( ) (1 + ℎ( )) 2 (22) = ℎ( ) ( ) ( ) (1 + ℎ( )) 2 + ℎ( ) ′ ( ) (1 + ℎ( )) (23) ≤ ℎ( ) (1 + ℎ( )) 2 ( ) ( ) + ℎ( ) (1 + ℎ( )) ′ ( )(24)≤ (1 + ) 2 4 + 2(25) Letting := (1+ ) 2 4 + 2 , now that we have a bound on the second derivatives we can again use the Taylor expansion of gradient descent: ( +1 ) = ( − ∇ ( )) (26) ≤ ( ) − ∇ ( ) 2 + 2 2 ∇ ( ) 2(27) Then for any ∈ R , ( +1 ) ≤ 2 − 2 (1 + ) 2 + 2 + ( log − ) 2 + (2 − ) log(1 + 1 ) + 2(42) Importantly, if we compare the non-private case to the private case for either the robust or non-robust setting, we take an equivalent hit in convergence that depends on the variance of noise and dimensionality of data. Empirical adversarial risk comparison. Figure 1 is not an apples-to-apples comparison because we are comparing standard loss ( ) under standard training against robust loss ( ) under adversarial training. We can compare with robust accuracy under standard training by noting that under gradient descent, if 1 ≤ 1, then ≤ 1 + ( − 1), and together with: ℓ ( ) = log(1 + − + ∥ ∥ * ) (43) = log( − ∥ ∥ * + − ) + ∥ ∥ * (44) ≤ log(1 + − ) + ∥ ∥ * (45) = ℓ ( ) + ∥ ∥ * ,(46) we get the following upper bound on the robust loss under gradient descent: ( ) ≤ 8 − 8 1 + ( log ) 2 + ( 8 − 4 ) log( + 1 ) + 1 + ( − 1)(47) B BINARY LOGISTIC REGRESSION WITH CLIPPING AND ADVERSARIAL TRAINING. The following analysis only considers a single input, but can extended to a batch of inputs through simple averaging. We again consider binary logistic regression with loss ℓ ( ) = max ∥ ∥ ≤ log(1 + − ( + ) ) = log(1 + − + ∥ ∥ ). As stated previously the first and second derivatives are given by: ∇ℓ ( ) = ℎ( ) ( ) 1 + ℎ( )(48)∇ 2 ℓ ( ) = ℎ( ) ( ) ( ) + ℎ( ) 2 ′ ( ) + ℎ( ) ′ ( ) (1 + ℎ( )) 2(49) where ℎ( ) := − + ∥ ∥ , ( ) := − + ∥ ∥ , and ′ ( ) = ( ∥ ∥ − ∥ ∥ 3 ). At the optimal * = arg min ℓ ( ), we have − + * ∥ * ∥ = 0 =⇒ * = ∥ * ∥ . The ℎ( ) ( ) ( ) term in the second derivative vanishes at the optimum and ℎ( * ) = 1 and we are left with: ∇ 2 ℓ ( * ) = ′ ( * ) 2 = 2 ∥ * ∥ − * * ∥ * ∥ 2(50) Clearly ∇ 2 ℓ ( * ) is a positive semi-definite matrix with eigenvalues at 0 and 2 ∥ * ∥ -we can take the eigenvector to be * and then ( − * * ∥ * ∥ 2 ) * = 0, and we can also take the eigenvector to be orthogonal to * resulting in: 2 ∥ * ∥ − * * ∥ * ∥ 2 = 2 ∥ * ∥(51) Thus, the curvature of our loss landscape is completely determined by the ratio of adversarial noise to the magnitude of the optimal parameters, and as decreases we achieve smoother solutions. C EXCESS RISK ANALYSIS FOR ROBUST & PRIVATE LEARNING So far we have concentrated our analysis on logistic regression. We now give a simple risk analysis for general convex losses for robust learning with differential privacy. Let ℓ (·, (·, ·)) : W × (X, Y) → R be a loss function with the following properties: -Lipschitz, -smooth, convex and linear in its parameters, ℓ ( , ( , )) = ℓ ( ), ∀ ∈ W, ∥ ∥ ≤ , X ⊆ R , ∀( , ) ∈ (X, Y), ∥ ∥ ≤ 1 and ∈ {±1}. 9 Ongoing work. Appeared in PPML, 2021 Consider two batch updates that differ in their final element: To achieve ( , ) − , we must bound ( ) = max , ′ ( ; , ′ ) as defined in [1], where ( ) = +1 ( || ) is as defined in [4]. This results in the following: ( ) = +1 ( ∥ ) = ( + 1) ∥ − ′ ∥ 2 2 2(54) Under the assumption that ∥ ∥ ≤ 1, ∈ {±1}, and ℓ is L-Lipschitz, we have the following upper bound: − ′ ≤ 2 =⇒ ( ) ≤ 2 ( + 1) 2 2(55) Summing over all iterations gives: ( ) ≤ ∑︁ =1 ( ) ≤ 2 ( + 1) 2 2 ≤ 2 2 2(56) for some constant . Then for some ′ , taking 2 = ′ 2 log( 1 ) 2 will give ( , ) − according to Theorem 2.2 in [1]. Turning to the robust setting, we wish to solve min ∈W max ∥ ∥ ≤ ℓ ( + ) , and in the convex setting the inner-maximization can be given in closed-form as: max ∥ ∥ ≤ ℓ ( + ) = ℓ ( − ∥ ∥ )(57) where 1 + 1 = 1. If we set = ∞, then squared Euclidean distance between and ′ becomes − ′ ≤ ∇ℓ − ′ ∇ℓ ′ ′ − sign( )∇ℓ + sign( )∇ℓ ′ (58) ≤ 2 √ (1 + )(59) This similarly holds for = 2 by noting that that the derivative of ∥ ∥ 2 is of unit length and replacing sign ( ) with this unit length value. Then ( , ) − holds by taking 2 = ′ (1+ ) 2 2 log( 1 ) 2 . First note that if ℓ is -Lipschitz, then E[∥^∥ 2 ] = ∥ ∥ 2 + E[∥^∥ 2 ] ≤ 2 + 2 . Assuming ℓ is -strongly convex, then by Theorem 1 of [8] the bound on excess risk is given by the following: E[ℓ ( ) − ℓ ( * )] ≤ 17( 2 + 2 )(1 + log( ))(60) where for > 0 the per step learning rate satisfies = 1 . For private learning we can plug in 2 = Figure 1 : 1Converge of logistic regression with = 10, = 0.25, = 1.0, = 0.1, = 0.1. log( +1 ) + 1 + ( − 1) (a full derivation can be found in appendix A). We can then plot the robust loss under gradient descent and adversarial training + gradient descent infig. 3, where the benefits of adversarial training are clear to see. Figure 3 : 3Comparison of robust loss under gradient descent with and without adversarial training, using the same parameters as described infig. 1. Figure 4 : 4Training loss of a three-layer CNN on MNIST. Figure 5 : 5Gap in loss between robust and non-robust optimization on MNIST. Figure 6 : 6Robustness comparison of MNIST and CIFAR-10. For ; x, y) (k) (c) The correspondence between maximum eigenvalue and test accuracy for different clipping thresholds ( ). Figure 7 : 7Robust and clipped logistic regression. Each point in the figures represents a different model trained to be robust to perturbations of size with gradients clipped to be smaller than . The correspondence between maximum eigenvalue and test accuracy for different privacy guarantees ( ). Figure 8 : 8Private and robust logistic regression. Each point in the figures represents a different model trained to be robust to perturbations of size and differentially private with . , = ∇ℓ ( ) , ′ = ′ ∇ℓ ( ′ ′ ) ′ , we have = N ( + , 2 ) and = N ( + ′ , 2 ). robust and private learning we can plug in 2 = ′ (1+ ) 2 2 log( 1 ) 2 . Table 1 : 1Convergence rates of empirical nominal and adversarial risk of logistic loss in non-private and private settings.Optimizer Loss Bound Robust Private We stress that we do not claim our bounds are tight. For example, we should be able to improve the bound in the non-private non-robust setting by a factor of log by appealing to Theorem 3.3 in[3]. As long as satisfies < 1, the ( ) contribution is negative and we get:Isolating in the condition < 1 gives < 4( −2 ) (1+ ) 2 which also implies that we require < 2 for our results to hold. Summing over the iterations as before gives:By choosing = log − we again get ℓ ( ) = log(1 + 1 ) and ∥ ∥ = log − , giving the final upper bound:Privacy / No robustness. We next find convergence rates under gradient descent with differential privacy. At each step we add noise; we do not clip gradients as they are already 1-Lipschitz and we assume here the clipping norm is larger than one. Let ∇ ( ) = ∇ ( ) + , where ∼ N (0, 2 ). Throughout the following analysis we find convergence results under expectations taken over the noise added for differential privacy. Our analysis is almost identical to the non-private case, and we sometimes write ∇ or ∇ where the context is clear. Firstly, the Taylor expansion follows:Note that ∇ 2 ≤ 1 4 and ∇ = ∇ + . Although we take expectations over the random noise in following, we omit the notation E for conciseness:Then for any ∈ R , and again taking expectations over the noise,Repeating the same process as in the non-private case we get:Privacy / robustness. We can repeat the analysis as in non-private and robust case, while taking expectations over the random noise to give the upper bound: Deep learning with differential privacy. M Abadi, A Chu, I Goodfellow, H B Mcmahan, I Mironov, K Talwar, L Zhang, Proceedings of the 2016 ACM SIGSAC conference on computer and communications security. the 2016 ACM SIGSAC conference on computer and communications securityAbadi, M., Chu, A., Goodfellow, I., McMahan, H. B., Mironov, I., Talwar, K., and Zhang, L. Deep learning with differential privacy. In Proceedings of the 2016 ACM SIGSAC conference on computer and communications security, pp. 308-318, 2016. Private empirical risk minimization: Efficient algorithms and tight error bounds. R Bassily, A Smith, A Thakurta, IEEE 55th Annual Symposium on Foundations of Computer Science. IEEEBassily, R., Smith, A., and Thakurta, A. Private empirical risk minimization: Efficient algorithms and tight error bounds. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pp. 464-473. IEEE, 2014. S Bubeck, arXiv:1405.4980Convex optimization: Algorithms and complexity. arXiv preprintBubeck, S. Convex optimization: Algorithms and complexity. arXiv preprint arXiv:1405.4980, 2014. Concentrated differential privacy: Simplifications, extensions, and lower bounds. M Bun, T Steinke, Theory of Cryptography Conference. Bun, M. and Steinke, T. Concentrated differential privacy: Simplifications, ex- tensions, and lower bounds. In Theory of Cryptography Conference, pp. 635-658. . Springer, Springer, 2016. Robust and private learning of halfspaces. B Ghazi, R Kumar, P Manurangsi, T Nguyen, International Conference on Artificial Intelligence and Statistics. PMLRGhazi, B., Kumar, R., Manurangsi, P., and Nguyen, T. Robust and private learning of halfspaces. In International Conference on Artificial Intelligence and Statistics, pp. 1603-1611. PMLR, 2021. Inductive bias of gradient descent based adversarial training on separable data. Y Li, E X Fang, H Xu, T Zhao, arXiv:1906.02931arXiv preprintLi, Y., Fang, E. X., Xu, H., and Zhao, T. Inductive bias of gradient descent based adversarial training on separable data. arXiv preprint arXiv:1906.02931, 2019. Overfitting in adversarially robust deep learning. L Rice, E Wong, Z Kolter, International Conference on Machine Learning. PMLRRice, L., Wong, E., and Kolter, Z. Overfitting in adversarially robust deep learning. In International Conference on Machine Learning, pp. 8093-8104. PMLR, 2020. Stochastic gradient descent for non-smooth optimization: Convergence results and optimal averaging schemes. O Shamir, T Zhang, International conference on machine learning. PMLRShamir, O. and Zhang, T. Stochastic gradient descent for non-smooth opti- mization: Convergence results and optimal averaging schemes. In International conference on machine learning, pp. 71-79. PMLR, 2013. Membership inference attacks against machine learning models. R Shokri, M Stronati, C Song, V Shmatikov, 2017 IEEE Symposium on Security and Privacy (SP). IEEEShokri, R., Stronati, M., Song, C., and Shmatikov, V. Membership inference attacks against machine learning models. In 2017 IEEE Symposium on Security and Privacy (SP), pp. 3-18. IEEE, 2017. Privacy risks of securing machine learning models against adversarial examples. L Song, R Shokri, P Mittal, Proceedings of the 2019 ACM SIGSAC Conference on Computer and Communications Security. the 2019 ACM SIGSAC Conference on Computer and Communications SecuritySong, L., Shokri, R., and Mittal, P. Privacy risks of securing machine learning models against adversarial examples. In Proceedings of the 2019 ACM SIGSAC Conference on Computer and Communications Security, pp. 241-257, 2019. N Tursynbek, A Petiushko, I Oseledets, arXiv:2012.07828Robustness threats of differential privacy. arXiv preprintTursynbek, N., Petiushko, A., and Oseledets, I. Robustness threats of differential privacy. arXiv preprint arXiv:2012.07828, 2020. L Xie, K Lin, S Wang, F Wang, J Zhou, arXiv:1802.06739Differentially private generative adversarial network. arXiv preprintXie, L., Lin, K., Wang, S., Wang, F., and Zhou, J. Differentially private generative adversarial network. arXiv preprint arXiv:1802.06739, 2018.
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[ "Akaike's Bayesian information criterion (ABIC) or not ABIC for geophysical inversion", "Akaike's Bayesian information criterion (ABIC) or not ABIC for geophysical inversion", "Akaike's Bayesian information criterion (ABIC) or not ABIC for geophysical inversion", "Akaike's Bayesian information criterion (ABIC) or not ABIC for geophysical inversion" ]
[ "Peiliang Xu [email protected] \nDisaster Prevention Research Institute\nKyoto University\n611-0011UjiKyotoJapan\n", "Peiliang Xu [email protected] \nDisaster Prevention Research Institute\nKyoto University\n611-0011UjiKyotoJapan\n" ]
[ "Disaster Prevention Research Institute\nKyoto University\n611-0011UjiKyotoJapan", "Disaster Prevention Research Institute\nKyoto University\n611-0011UjiKyotoJapan" ]
[]
Akaike's Bayesian information criterion (ABIC) has been widely used in geophysical inversion and beyond. However, little has been done to investigate its statistical aspects. We present an alternative derivation of the marginal distribution of measurements, whose maximization directly leads to the invention of ABIC by Akaike. We show that ABIC is to statistically estimate the variance of measurements and the prior variance by maximizing the marginal distribution of measurements. The determination of the regularization parameter on the basis of ABIC is actually equivalent to estimating the relative weighting factor between the variance of measurements and the prior variance for geophysical inverse problems. We show that if the noise level of measurements is unknown, ABIC tends to produce a substantially biased estimate of the variance of measurements. In particular, since the prior mean is generally unknown but arbitrarily treated as zero in geophysical inversion, ABIC does not produce a reasonable estimate for the prior variance either.
10.1016/j.jfranklin.2021.03.003
[ "https://export.arxiv.org/pdf/1911.06564v1.pdf" ]
208,076,939
1911.06564
14f4852ba1e250585da77fa94532e1c7870be080
Akaike's Bayesian information criterion (ABIC) or not ABIC for geophysical inversion 15 Nov 2019 Peiliang Xu [email protected] Disaster Prevention Research Institute Kyoto University 611-0011UjiKyotoJapan Akaike's Bayesian information criterion (ABIC) or not ABIC for geophysical inversion 15 Nov 2019arXiv:1911.06564v1 [stat.ME]Akaike's Bayesian information criterionmarginal likelihood functionsprior varianceregularization parametervariance of measurements Akaike's Bayesian information criterion (ABIC) has been widely used in geophysical inversion and beyond. However, little has been done to investigate its statistical aspects. We present an alternative derivation of the marginal distribution of measurements, whose maximization directly leads to the invention of ABIC by Akaike. We show that ABIC is to statistically estimate the variance of measurements and the prior variance by maximizing the marginal distribution of measurements. The determination of the regularization parameter on the basis of ABIC is actually equivalent to estimating the relative weighting factor between the variance of measurements and the prior variance for geophysical inverse problems. We show that if the noise level of measurements is unknown, ABIC tends to produce a substantially biased estimate of the variance of measurements. In particular, since the prior mean is generally unknown but arbitrarily treated as zero in geophysical inversion, ABIC does not produce a reasonable estimate for the prior variance either. Introduction A linear inverse ill-posed problem can often be written symbolically as the following linear model: y = Aβ + ǫ,(1) where y is an (n×1) vector of measurements, A is the deterministic coefficient matrix, which is assumed to be theoretically of full column rank but with singular values close to zero, β is a (t × 1) vector of unknown parameters to be estimated, and the random error vector ǫ is of zero mean and variancecovariance matrix Wσ 2 , W is an (n × n) (positive definite) weighting matrix. If the least squares (LS) method is applied to (1), we have the weighted LS solution of β below: β = (A T WA) −1 A T Wy.(2) Althoughβ is unbiased, it can become highly unstable and practically meaningless physically, because the normal matrix A T WA, often denoted by N = A T WA, is of almost zero eigenvalues (see e.g., Phillips 1962;Tikhonov 1963;Tikhonov and Arsenin 1977). To obtain a mathematically and/or physically meaningful solution to inverse ill-posed problems (1), we can either limit ourselves to a sub-space spanned by the vectors corresponding to sufficiently large eigenvalues (see e.g., Xu 1998) or add a positive (semi-)definite matrix, say W β κ, to the normal matrix N such that the addition of both matrices avoids sufficiently small eigenvalues, where W β is positive (semi-)definite and κ is a positive scalar. This latter method is well known either as ridge regression in the statistical literature (see e.g., Hoerl and Kennard 1970;Vinod and Ullah 1981;Xu 1992a;Xu and Rummel 1994) or regularization in the literature of mathematics and applied sciences (see e.g., Tikhonov 1963;Tikhonov and Arsenin 1977), with the solution being given as follows: β r = (A T WA + κW β ) −1 A T Wy,(3) which will be called a regularized solution in this letter, for simplicity but without loss of confusion. Determination of the regularization parameter κ is of crucial importance in regularizing the inverse illposed model (1). If κ is too small, the regularized solutionβ r of (3) may still remain unstable; however, if it is chosen too large,β r will become over-smoothed. In particular, if κ tends positively to infinity, β r will shrink towards zero and the measurements y play no role at all in retrieving the information on β. There are a number of methods to determine the regularization parameter κ, depending on how the regularized solutionβ r of (3) is interpreted. The view points can be either presented in terms of frequentist, under the Bayesian framework, or even intuitively in terms of some norms of residuals of measurements and/or estimates of β. From the frequentist point of view,β r of (3) has been known as a biased estimator of β. As a result, the regularization parameter has been mainly motivated to compromise between noise amplification and bias such as the criterion of mean squared errors (see e.g., Hoerl and Kennard 1970;Vinod and Ullah 1981;Xu 1992a;Xu and Rummel 1994) or predict measurements such as generalized cross-validation and maximum likelihood (see e.g., Golub et al. 1979;Wahba 1985;Xu 2009). Norms of residuals and parameters have played an important role in choosing the regularization parameter. Very often, one can either determine the regularization parameter, given a fixed level of noise for the residual norm (see e.g., Miller 1970;Tikhonov and Arsenin 1977;Morozov 1984) or by finding a balance between some norms of residuals and parameters (see e.g., Hansen and O'Leary DP 1993). The inverse ill-posed model (1) has also been solved under the Bayesian framework (see e.g., Akaike 1980;Tarantola 1987). Assuming that there exists prior information on the parameters β in terms of the first and second (central) moments, namely, E(β) = µ, D(β) = W −1 β σ 2 β ,(4) where the prior vector µ is given, D(·) stands for variance operator, W β is a positive definite matrix and the scalar σ 2 β is given as well. Then applying stochastic inference to inverse problems (1) with prior information (4) results in the following estimator: β i = A T WA σ 2 + W β σ 2 β −1 A T Wy σ 2 + W β µ σ 2 β .(5) If the prior values µ are equal to zero, namely, µ = 0, and denoting κ = σ 2 /σ 2 β , then the stochastic inferenceβ i of (5) formally becomes the regularized or biased solutionβ r of (3). An alternative use of prior information to solve inverse problems (1) is based on full Bayesian inference, which requires the assumption of distributions of both measurements and prior data. As in the case of regularization and/or stochastic inference, proper choice of the regularization parameter is crucial in Bayesian inversion. Akaike (1980) proposed a Bayesian information criterion to determine the regularization parameter, which has since been widely applied in geophysical inversion. However, little has been known about statistical aspects of Akaike's Bayesian information criterion (ABIC) for ill-posed inverse problems. The purpose of this paper is primarily to investigate statistical performances of ABIC. We will provide an alternative representation of the marginal distribution of measurements. We will show that the ABIC determination of regularization parameter is statistically equivalent to estimating the variance σ 2 of measurements and the prior variance σ 2 β . We will also prove in this section that if the prior mean µ is unknown but arbitrarily treated as zero, as almost always in the case of practical geophysical inversion, the estimate of σ 2 will be significantly biased, which will further affect the estimate of σ 2 β and thus further the determination of the regularization parameter. The theoretical results will then be summarized in the concluding remarks. ABIC and its statistical aspects The measurement and prior distributions are usually assumed to be normal and respectively given as follows: f y (y/β, σ 2 ) = det(W) (2π) n/2 σ n exp − 1 2σ 2 (y − Aβ) T W(y − Aβ) ,(6a) and π β (β/σ 2 β ) = det(W β ) (2π) t/2 σ t β exp − 1 2σ 2 β (β − µ) T W β (β − µ) ,(6b) as can be found, for example, in Zellner (1971) and Akaike (1980), where det(·) stands for the determinant of a square matrix. Then the joint distribution density of both y and β is given by f (y, β/σ 2 , σ 2 β ) = f y (y/β, σ 2 )π β (β/σ 2 β ).(7) Bayesian inference on β has been done on the basis of the following posterior distribution of β given the measurements y: π(β/y, σ 2 , σ 2 β ) = f (y, β/σ 2 , σ 2 β ) m(y/σ 2 , σ 2 β ) = f y (y/β, σ 2 )π β (β/σ 2 β ) m(y/σ 2 , σ 2 β ) ,(8) where π(β/y, σ 2 , σ 2 β ) stands for the posterior distribution of β given y, and m(y/σ 2 , σ 2 β ) is the marginal distribution of y, which is defined and given as follows: m(y/σ 2 , σ 2 β ) = f (y, β/σ 2 , σ 2 β )dβ.(9) As a result, one can derive the Bayesian (posterior) estimate, either by maximizing or computing the posterior mean from the posterior distribution π(β/y, σ 2 , σ 2 β ) of (8), which is denoted byβ b and simply written belowβ b = A T WA σ 2 + W β σ 2 β −1 A T Wy σ 2 + W β µ σ 2 β , = (A T WA + κW) −1 (A T Wy + κW β µ),(10) (see e.g., Zellner 1971;Akaike 1980). It is obvious that both stochastic and Bayesian inferences result in the same estimator under the assumption of normal distributions by comparingβ i of (5) withβ b of (10). The Bayesian posterior estimator (10) can also be equivalently rewritten aŝ β b = A T WA σ 2 + W β σ 2 β −1 A T WAβ σ 2 + W β µ σ 2 β ,(11) indicating thatβ b is actually the weighted mean of the LS estimateβ and the prior mean µ. If the prior distribution is too poor to be useful, or more precisely, σ 2 β =⇒ ∞, thenβ b becomesβ. However, for an ill-posed inverse problem, the LS estimateβ can be too poor in accuracy and the prior information will constrainβ b even for a large value of σ 2 β . If µ = 0 in (10), the corresponding Bayesian estimator is then denoted byβ b0 and (10) becomeŝ β b0 = A T WA σ 2 + W β σ 2 β −1 A T Wy σ 2 = (A T WA + κW β ) −1 A T Wy,(12) which essentially turns out to beβ r of (3). Remark 1: For a geophysical inverse problem, the geophysical signals we are seeking from measurements y are not equal to zero. In other words, it is generally not fair/reasonable to assume µ = 0; otherwise, we have no reasons to collect the data y, as explained in the case of satellite gravimetry (Xu and Rummel 1994). By setting µ = 0 for geophysical inverse problems, the Bayesian estimator (12) is biased, as correctly pointed out by Akaike (1980) (see also Xu 1992b). From this point of view, it may become easily understandable why the regularized solutionβ r of (3) has been called a biased estimator from the frequentist point of view. Realizing the importance of choosing the parameter κ, or equivalently σ 2 and σ 2 β , Akaike (1980) proposed a criterion by maximizing the marginal distribution of y, i.e. m(y/σ 2 , σ 2 β ) of (9) to choose κ, which is known as Akaike's Bayesian information criterion or simply ABIC. ABIC has since been widely applied to geophysical inverse problems, as can be seen, for example, in Tamura et al. (1991), Yabuki and Matsu'ura (1992) and Fukuda and Johnson (2008), just to name a few of geophysical applications. Although m(y/σ 2 , σ 2 β ) plays a key role in ABIC, it was simply given in Akaike (1980) without providing any details of its derivation, which has been further basically used verbatim in geophysical applications (see e.g., Tamura et al. 1991;Yabuki and Matsu'ura 1992). Given the normal distribution (6a) of measurements and the normal prior distribution (6b), we have re-derived the marginal distribution m(y/σ 2 , σ 2 β ) of the measurements y, which is simply given as follows: m(y/σ 2 , σ 2 β ) = f (y, β/σ 2 , σ 2 β )dβ = 1 (2π) n/2 det(Σ py ) exp − 1 2 (y − Aµ) T Σ −1 py (y − Aµ) ,(13) where Σ py = W −1 σ 2 + AW −1 β A T σ 2 β .(14) The detailed derivation of (13) is given in appendix A. The univariate version of (13) can be found in Zellner (1971, p.28). The marginal distribution m(y/σ 2 , σ 2 β ) appears to be formally different from the corresponding distribution in Akaike (1980), they are essentially identical. It has become clear that ABIC for geophysical inversion is the same as estimating the two unknown variance components σ 2 and σ 2 β or equivalently, the unknown variance σ 2 and the relative weight κ, by maximizing the marginal distribution (13) of measurements. Mathematically, maximizing m(y/σ 2 , σ 2 β ) of (13) is equivalent to minimizing min: L(σ 2 , σ 2 β ) = ln{det(Σ py )} + (y − Aµ) T Σ −1 py (y − Aµ). Since κ = σ 2 /σ 2 β , the objective function L(σ 2 , σ 2 β ) can also be rewritten in terms of σ 2 and κ as follows: L(σ 2 , κ) = n ln(σ 2 ) + ln{det(E py )} + (y − Aµ) T E −1 py (y − Aµ)/σ 2 ,(16) where E py = W −1 + AW −1 β A T /κ. We should like to note that L(σ 2 , κ) of (16) is essentially the same as the corresponding formula in Section 5 of Akaike (1980). Case 1: both σ 2 and σ 2 β unknown. Differentiating L(σ 2 , κ) of (16) with respect to σ 2 and letting it equal to zero, we have n σ 2 − (y − Aµ) T E −1 py (y − Aµ) ( σ 2 ) 2 = 0, or equivalently, σ 2 = (y − Aµ) T E −1 py (y − Aµ)/n.(17) Given the prior variance σ 2 β or the equivalent relative weight κ, it is rather easy to prove that σ 2 of (17) is an unbiased estimate of σ 2 . Substituting (17) into (16) and neglecting a constant term, we have the ABIC for determining the relative weight (or regularization parameter) κ as follows: L(κ) = n ln{(y − Aµ) T E −1 py (y − Aµ)} + ln{det(E py )}.(18) The likelihood function L(κ) consists of two parts: a positive definite quadratic form of the predicted residuals (y − Aµ) and their cofactor matrix E py . It is easy to prove mathematically that the first part of L(κ) increases with κ, while the second part decreases with the increase of κ. Although µ is generally not equal to zero but unknown in geophysical inversion, it is almost always replaced with a zero vector under the framework of Bayesian geophysical inversion. In this case, by imposing µ = 0 (even if we know it cannot not zero), both formulae (17) and (18) become σ 2 s = y T E −1 py y/n,(19) and L s (κ) = n ln{y T E −1 py y} + ln{det(E py )},(20) respectively. Given a value κ, applying the expectation operator to (19) yields E( σ 2 s ) = E(E −1 py yy T )/n = y T E −1 py y/n + tr{E −1 py W −1 }σ 2 /n,(21) which clearly indicates that σ 2 s of (19) can be significantly biased from its true value σ 2 , depending on the true values of measurements, where y stands for the vector of true values of measurements y. On the other hand, since both L(κ) of (18) and L s (κ) of (20) are different only in whether the predicted residuals (y − Aµ) or the measurements y are used to compute the positive definite quadratic form. Since y are generally larger than the residuals (y − Aµ), the relative weighting factor or regularization parameter κ is expected to be estimated with a bias. As a result, we expect that the prior variance σ 2 β will also be computed with a bias from both σ 2 s and κ. Case 2: σ 2 known but σ 2 β unknown. If σ 2 is given/known but σ 2 β unknown, use of ABIC to determine the regularization parameter is mathematically equivalent to finding the best estimate of the prior variance σ 2 β such that it maximizes the marginal distribution m(y/σ 2 , σ 2 β ) of measurements to most favor the output of the measurements y. It is also equivalent to finding the optimal ABIC estimate of σ 2 β (or κ) that minimizes the following likelihood function: L b (κ) = (y − Aµ) T E −1 py (y − Aµ)/σ 2 + ln{det(E py )}.(22) As in the case of L(κ) in (18), the first term of L b (κ) increases with κ, while its second term with the cofactor matrix E py decreases with the increase of κ. Since the prior mean µ is practically unknown in geophysical inversion, it is almost always treated as zero and the likelihood function L b (κ) of (22) becomes L bs (κ) = y T E −1 py y/σ 2 + ln{det(E py )}. Since y is generally expected to be much larger in size than the predicted residual vector (y − Aµ), the optimal κ from minimizing (23) may tend to be smaller than that from minimizing (22). Thus, arbitrarily assigning the prior mean µ to zero would affect the ABIC estimate of σ 2 β . Concluding remarks Akaike's Bayesian information criterion (ABIC) was proposed by Akaike (1980) and has since been widely applied in geophysical inversion. The ABIC method is to determine the regularization parameter such that the marginal distribution of measurements is maximized. In other words, the ABIC-based regularization parameter is optimally chosen to most favor the output of the collected measurements. However, little has been done to investigate its statistical aspects for geophysical inversion. We have presented an alternative representation of the marginal distribution of measurements, which is the starting point of ABIC. We have shown that ABIC for geophysical inverse problems is statistically equivalent to estimating the variance of measurements and the prior variance by maximizing the marginal distribution of measurements or minimizing the likelihood function of the variance of measurements and the prior variance. If the prior distribution is correct, the regularization parameter actually reflects the relative weighting factor between these two variances. We have proved that if the noise level of measurements is unknown, ABIC tends to produce a substantially biased estimate of the variance of measurements. In particular, since the prior mean is generally unknown but arbitrarily treated as zero in geophysical inversion, ABIC does not produce a reasonable estimate for the prior variance either. In case of a given variance of measurements, the determination of the regularization parameter on the basis of ABIC is mathematically equivalent to estimating the prior variance for geophysical inverse problems. We may also like to note that although ABIC maximizes the marginal distribution of measurements under the Bayesian framework, it does not directly target at constructing a solution of high quality in terms of mean squared errors. Appendix A: the derivation of the marginal distribution m(y/σ 2 , σ 2 β ) of measurements y in(9)Based on the normal distributions (6a) and (6b), we can write the major exponent part of the joint distribution (7), denoted by F (y, β/σ 2 , σ 2 β ), as follows:whereandβ b has been given in (10). We now rewrite F y (y/σ 2 , σ 2 β ) of (25) as follows:On the other hand, since the Bayesian estimatorβ b is derived from the following normal equations:which is equivalent toThus, we haveBy substituting (µ −β b ) of (28) into (26) and after some slight rearrangement, we finally obtainActually, the term (W −1 σ 2 + AW −1 β A T σ 2 β ) in the last line of (29) is exactly equal to the variancecovariance matrix of the random vector (Aβ + ǫ). It is also obvious that the mean of (Aβ + ǫ) is Aµ.Inserting F (y, β/σ 2 , σ 2 β ) of (24) and F y (y/σ 2 , σ 2 β ) of (29) into m(y/σ 2 , σ 2 β ) of (9) and after the integration, we finally obtain the marginal distribution m(y/σ 2 , σ 2 β ) of y as follows:m(y/σ 2 , σ 2 β ) = 1 (2π) n/2 det(Σ py ) exp − 1 2 (y − Aµ) T Σ −1 py (y − Aµ) ,which is the same as (13). Likelihood and the Bayes procedure. H Akaike, Bayesian Statistics. Bernardo JM, de Groot MH, Lindley DV, Smith AFMValencia, SpainUniversity PressAkaike H (1980). Likelihood and the Bayes procedure. in: Bayesian Statistics, Bernardo JM, de Groot MH, Lindley DV, Smith AFM, eds., University Press, Valencia, Spain, pp.1-13. Statistical inference in factor analysis. T W Anderson, H Rubin, Proceedings of the third Berkeley symposium on mathematical statistics and probability. the third Berkeley symposium on mathematical statistics and probability5Anderson TW, Rubin H (1956). Statistical inference in factor analysis. In Proceedings of the third Berkeley symposium on mathematical statistics and probability, Vol.5, pp. 111-150. A fully Bayesian inversion for spatial distribution of fault slip with objective smoothing. J Fukuda, K M Johnson, Bull. seism. Soc. Am. 98Fukuda J, Johnson KM (2008). A fully Bayesian inversion for spatial distribution of fault slip with objective smoothing. Bull. seism. Soc. Am., 98, 1128-1146. Generalized cross-validation as a method for choosing a good ridge parameter. 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[]
[ "Splashes in isotropic media", "Splashes in isotropic media" ]
[ "Eugene B Kolomeisky \nDepartment of Physics\nUniversity of Virginia\nP. O. Box 40071422904-4714CharlottesvilleVirginiaUSA\n" ]
[ "Department of Physics\nUniversity of Virginia\nP. O. Box 40071422904-4714CharlottesvilleVirginiaUSA" ]
[]
The response of a weakly absorbing isotropic medium to a sudden localized perturbation (a "splash") is explained within the framework of linear response theory. In this theory splashes result from the interference of the collective excitations of the medium, with the outcome determined by the interplay between their phase and group velocities as well as the sign of the latter. The salient features of splashes are controlled by the existence of extremal values of the phase and the group velocities: the group velocity gives the expansion rate of the locus of the points where new wavefronts nucleate or existing ones disappear, while the phase velocity determines the large-time expansion rate of a group of wavefronts. If the group velocity is negative in a spectral range and takes on a minimal value within it, then converging wavefronts will be present in the splash. These results are relevant to the studies of several experimentally viable setups, such as a splash on the surface of deep water due to a small pebble or a raindrop, a splash in the two-dimensional electron gas caused by a short voltage pulse applied with the tip of a scanning tunneling microscope, or a bulk splash in superfluid 4 He due to formation of an electron bubble. Specifically, the gross features of a splash in superfluid 4 He are determined by five extremal velocities. Additionally, due to the existence of a negative group velocity spectral range, some of the wavefronts in the superfluid splash are converging.
10.1103/physreve.107.024117
[ "https://export.arxiv.org/pdf/2209.01261v2.pdf" ]
252,089,015
2209.01261
883190d11e54f677cfe42c9737e0cfe009937980
Splashes in isotropic media Eugene B Kolomeisky Department of Physics University of Virginia P. O. Box 40071422904-4714CharlottesvilleVirginiaUSA Splashes in isotropic media (Dated: February 1, 2023) The response of a weakly absorbing isotropic medium to a sudden localized perturbation (a "splash") is explained within the framework of linear response theory. In this theory splashes result from the interference of the collective excitations of the medium, with the outcome determined by the interplay between their phase and group velocities as well as the sign of the latter. The salient features of splashes are controlled by the existence of extremal values of the phase and the group velocities: the group velocity gives the expansion rate of the locus of the points where new wavefronts nucleate or existing ones disappear, while the phase velocity determines the large-time expansion rate of a group of wavefronts. If the group velocity is negative in a spectral range and takes on a minimal value within it, then converging wavefronts will be present in the splash. These results are relevant to the studies of several experimentally viable setups, such as a splash on the surface of deep water due to a small pebble or a raindrop, a splash in the two-dimensional electron gas caused by a short voltage pulse applied with the tip of a scanning tunneling microscope, or a bulk splash in superfluid 4 He due to formation of an electron bubble. Specifically, the gross features of a splash in superfluid 4 He are determined by five extremal velocities. Additionally, due to the existence of a negative group velocity spectral range, some of the wavefronts in the superfluid splash are converging. The response of a weakly absorbing isotropic medium to a sudden localized perturbation (a "splash") is explained within the framework of linear response theory. In this theory splashes result from the interference of the collective excitations of the medium, with the outcome determined by the interplay between their phase and group velocities as well as the sign of the latter. The salient features of splashes are controlled by the existence of extremal values of the phase and the group velocities: the group velocity gives the expansion rate of the locus of the points where new wavefronts nucleate or existing ones disappear, while the phase velocity determines the large-time expansion rate of a group of wavefronts. If the group velocity is negative in a spectral range and takes on a minimal value within it, then converging wavefronts will be present in the splash. These results are relevant to the studies of several experimentally viable setups, such as a splash on the surface of deep water due to a small pebble or a raindrop, a splash in the two-dimensional electron gas caused by a short voltage pulse applied with the tip of a scanning tunneling microscope, or a bulk splash in superfluid 4 He due to formation of an electron bubble. Specifically, the gross features of a splash in superfluid 4 He are determined by five extremal velocities. Additionally, due to the existence of a negative group velocity spectral range, some of the wavefronts in the superfluid splash are converging. I. INTRODUCTION Recent years have seen major advances in imaging of the fluid density n(r, t) and velocity v(r, t) fields in neutral quantum liquids such as superfluid 4 He [1][2][3][4], superfluid 3 He [5], atomic gas superfluids [6], and charged quantum liquids such as electrons in graphene [7] and Cooper-pair liquids in superconductors [8]. These emerging capabilities open a door to visualization of a variety of effects, some of which have already been described [9,10]. A theoretical study of a family of such effects, splashes, is given in this paper. A splash is the response of the medium to a local perturbation of a short duration; it is described by an initial value problem. A familiar example from classical physics is the expanding pattern of annular waves caused by a small rock impacting a surface of calm water [11][12][13]. Analogs of this phenomenon exist in quantum liquids. For instance, a low-energy electron injected inside liquid 4 He triggers quick formation of a bubble around it [14]; the reaction of the superfluid to the formation of the bubble is an example of a three-dimensional splash in a neutral quantum liquid. The surfaces of superconductors and two-dimensional electron systems allow for a large degree of control over the perturbation. Specifically, applying a short voltage pulse with the tip of a scanning tunneling microscope to a graphene sheet [15] can initiate a splash in the two-dimensional sea of Dirac electrons. For any localized disturbance, the response of the medium is small at long times t after the perturbation ceased to operate, and then is described by a linear theory. In such a theory, developed below, splashes are a result of the interference of collective excitations of the medium; the outcome is determined by their frequency spectrum Ω(k) (here k is the wave vector). Previous treatments of splashes focused on the water surface as a medium. Here the relevant excitations are the capillary gravity waves whose dispersion law in the deep water (and incompressible fluid) limit has the form [16] Ω 2 (k) = gk + γ ρ k 3(1) where g is the free-fall acceleration, k = |k| is the wave number, γ is the coefficient of surface tension of water and ρ is the density of water. For γ = 0 the initial value problem has been fully solved by Cauchy and Poisson (CP) [11]. Specifically, the position of the l-th wavefront in the splash r l as measured from the point of impact in the large time limit gt 2 /r l 1 is given by the expression r l = gt 2 8πl (2) whose hallmark is accelerated expansion of the rings. The parameters of the spectrum (1) can be combined to form a spatial scale λ, the capillary length, and a time scale τ , such as λ = γ gρ 1/2 = 0.28 cm, τ = γ g 3 ρ 1/4 = 0.017 s (3) where the numerical values are for water at 20 • C [16]. The CP theory is only valid for wavelengths long compared to the capillary length. When the scales (3) are used as units of length and time (see below), the dispersion law (1) acquires the parameter-free form Ω 2 = k + k 3(4) which means that there is more to the water splash than implied by the CP result (2). Kelvin [17] pioneered a general method for analyzing splashes due to excitations with arbitrary dispersion law, and found that the dynamics of splashes is determined by the interplay between the phase and the group velocities. The CP result (2) has its simple form because the group velocity for gravity waves (γ = 0 in Eq.(1)) is half the phase velocity. For the general dispersion relation, Kelvin argued that interesting phenomena are sure to occur whenever there are extrema of the phase velocity, because then the phase velocity Ω/k and group velocity dΩ/dk ≡ Ω (k) are equal (indeed the condition (Ω/k) = 0 is equivalent to Ω/k = Ω (k)). For the particular case of capillary gravity waves (shown in Figure 1), there is a minimum when k = k c = 1 and v = v c = √ 2 (23 cm/s, physical units). The group velocity Ω also has a minimum ( Figure 1) at [12,13] that a few seconds following the perturbation of a water surface, a quiescent region inside the annular waves is formed. Rayleigh demonstrated [18] that this region expands with the constant velocity v m (6). Outside the quiescent region there are two systems of waves of different wavelength present at the same place. In practice only one system is visible, and Rayleigh conjectured that the other (corresponding to short waves of predominantly capillary origin) is rapidly damped. Rayleigh's conjecture has been justified by Lighthill [19] who also observed that new wavefronts nucleate "from nowhere" at the boundary of the expanding region of calmed water. k m = (2/ √ 3 − 1) 1/2 ≈ 0.39 corresponding to the velocity v m = √ 3 2 √ 3 − 1 1/4 ≈ Le Méhauté [20] additionally argued that the waves in the annular region have a narrow range of wave numbers centered around k m , Figure 1, corresponding to the minimum group velocity v m (6). Below we give a general theory of dynamics of wavefronts in splashes in the weakly absorbing isotropic medium and apply it to various cases. While elaborating on Rayleigh's results [18] regarding the water spash, we expand on Lighthill's observation [19], showing that new wavefronts arise at the inner boundary of the annular region in pairs at equal time intervals. We also support Kelvin's intuition regarding the significance of the minimum phase velocity v c (5): it sets the velocity of expansion of a group of capillary gravity rings in the long-time limit where the conjecture of Ref. [20] fails. This theory also applies to a plasmonic splash in a twodimensional Fermi sea. Here the relevant dispersion law is [21][22][23][24][25]: Ω 2 (k) = gk + u 2 k 2(7) where g (no longer the free-fall acceleration) and the speed of sound u are determined by the equation of state of the electron gas [25]. While the spectra (1) and (7) are the same in the long-wavelength limit, the remaining difference -k 3 versus k 2 contributions -makes the plasmonic splash a simpler version of its water counterpart as discussed below. A final application of the theory involves a splash in bulk superfluid 4 He whose elementary excitations exhibit a spectral region with negative group velocity [26]. A recent analysis of the wake patterns in this system [27] established that these excitations are responsible for features similar to the Kelvin ship wake. Below it will be shown that negative group velocity excitations are responsible for converging wavefronts in superfluid 4 He splashes. II. FORMALISM Regardless of its particular manifestation, splashes are described by linear response theory [26,28,29]. Let us suppose that every particle of the medium is perturbed by an external field of the potential energy U (r, t). Then the operator of the perturbation acting on the whole medium isV (t) = n(r, t)U (r, t)d d x(8) wheren(r, t) is the Heisenberg density operator and d is the space dimensionality (in classical linear water wave theory n(r, t) is the height of the water surface while U (r, t) is the excess pressure [30]). The Fourier transform of the induced density due to the perturbation is given by δn(ω, k) = −α(ω, k)U (ω, k) where α(ω, k) is a generalized susceptibility [26,28,29] and U (ω, k) is the Fourier transform of U (r, t). Inverting the Fourier transform and specifying to the case of a point instantaneous source, U (ω, k) = const, the induced density will be given by δn(r, t) ∝ dωd d k (2π) d+1 α(ω, k)e i(k·r−ωt) .(9) The dynamics of the wavefronts in the splash can be determined by using Kelvin's stationary phase argument [11,17]. At positions and times such as the phase f = k · r − ωt is large in magnitude, the exponential in (9) is highly oscillatory, and contributions of elementary plane waves interfere destructively leaving almost no net result, unless ω = ±Ω(k) (which are the poles of the susceptibility α(ω, k) [26,29]) and have a phase which is stationary with respect to k. Subjecting the phase f = k · r ∓ Ω(k)t(10) to the condition of stationarity ∇ k f = 0 one finds r = ±∇ k Ω · t.(11) Since the phase f is constant along the wavefronts, the last two equations can be solved to determine the wavefront dynamics in a parametric form. In an isotropic medium they become r(k) = Ω Ω k − Ω f, t(k) = ± 1 Ω k − Ω f(12) where the lower sign in the expression for t(k) accounts for the possibility of a negative group velocity. Since r and t are positive, the phase f is determined by the interplay between the phase and the group velocities, as well as by the sign of the latter. Specifically, there are three possibilities: f =      2πl, if Ω > Ω/k − 2πl, if 0 < Ω < Ω/k 2πl, if Ω < 0(13) where l is a positive integer. The dynamics of the wavefronts of the last type is given by Eqs. (12) with the lower sign chosen in the expression for t(k); otherwise, the expression for t(k) with the upper sign should be used. Several conclusions anticipating the gross features of splashes can be deduced from Eqs. (12): (i) When the equation Ω = 0 has real solutions, i. e. the group velocity has an extremum v = v m , the expressions for r(k) and t(k) have simultaneous extrema. Then the equation r = |v m |t gives the locus of the points where new wavefronts nucleate or existing ones disappear. When this happens at a nonzero k = k m which is not an end point of the spectrum, the wavefronts appear or disappear in pairs. Since positions of extrema of t(k) are l-independent, the wavefronts appear (or disappear) at regular time intervals t(k m ) ≡ t m = 2π |v m k m − Ω(k m )| .(14) (ii) In the vicinity of an extremum of the group velocity, the spectrum can be approximated by its Taylor expansion Ω(k) = Ω(k m ) + v m (k − k m ) + Ω (3) (k m ) 3! (k − k m ) 3 (15) Combining it with Eqs. (12) and (13), one can then see that when the group velocity has a minimum ( Ω (3) (k m ) > 0), that is negative, v m < 0, then t(k) has a minimum while r(k) has a maximum at k = k m . The consequence is that pairs of wavefronts nucleating with period t m (14) will be converging toward the center of the splash. (iii) The large t limit is controlled by the points of the spectrum where the phase and the group velocities are equal; these are also extrema of the phase velocity. If this happens at k = k c with finite common velocity v = v c , equation r = v c t gives the asymptotic large t behavior of wavefronts whose wave numbers are close to k = k c . If k c = 0, then v c is the speed of sound u. If k c = 0 and v c = 0 (v c = ∞), then the asymptotic large t expansion of wavefronts is sub-ballistic (super-ballistic). (iv) Eqs. (12) and (13) imply that if r/l and t/l are used as variables to represent splashes, visual complexity of their original r(t) patterns is reduced since all the wavefronts of given family (according to Eq.(13)) "collapse" onto a single curve (or a pair of curves) representing that family. III. APPLICATIONS We now proceed to selected applications of the general results (12), (13) and (14). A. Acoustic spectrum When the excitation spectrum is linear in the wavenumber k, Ω = uk,(16) the phase Ω/k and the group Ω velocities are equal to u for all k. According to Eqs. (12) this is the marginal case. The well-known outcome r = ut then follows from Eq. (11): there is only one wavefront propagating away from the point of disturbance with the speed of sound. B. Gravity waves For gravity waves (γ = 0 in Eq.(1)) the group velocity is always smaller than the phase velocity, thus implying, Eq. (13), that f = −2πl. Then according to Eq.(12) one finds that r = 2πl/k and t = 4πl/ √ gk; combining them recovers the CP result (2). This is an example of a superballistic expansion. C. Capillary waves For capillary waves (g = 0 in Eq.(1)) the group velocity is always larger than the phase velocity, thus implying, Eq.(13), that f = 2πl. Then according to Eq.(12) one finds that r = 6πl/k and t = 4πl/ γk 3 /ρ. Eliminating the wave number k one obtains r l = 3 πγlt 2 2ρ 1/3 ,(17) i.e. the expansion is sub-ballistic, r l ∝ t 2/3 . D. Capillary gravity waves With capillarity included (4), the first two possibilities of Eq.(13) are realized. To understand the dynamics of the wavefronts, in Figure 2 we plotted corresponding t(k) dependences (12) for several values of l (the r(k) dependences are not shown; they are qualitatively the same). We observe that at times smaller than (14), the equation for t(k) (12) only has solutions corresponding to short waves k (l) (t) > k c where f = 2πl. In the t t m limit the effects of gravity are negligible, and evolution of these annular rings follows Eq. (17). As t → ∞ the solutions k (l) (t) asymptotically approach k = k c from above. The corresponding rings expand with the velocity v c (5) corresponding to zero of the denominator in Eqs. (12). The behavior for arbitrary t is displayed in Figure 3 by a series of dashed wavefronts; line styling is coordinated with the k > k c regions in Figures 1 and 2. As in the purely capillary case (17), the number of these wavefronts is infinite and they extend over the whole surface; this is an artifact of the incompressible liquid approximation. t m = 2 3/2 π (1 + 2/ √ 3) 1/4 1 − 1/ √ 3 ≈ At t = t m (18) the equation for t(k) (12) acquires an additional solution k = k m which for t > t m bifurcates into two: k − (t) < k m and k + (t) > k m . As the time progresses, the first of these tends to zero, k − (t → ∞) → 0 where Eqs. (12) reduces to the CP result (2) for r 1 . On the other hand, as t → ∞ the second solution k + (t) asymptotically approaches k = k c from bellow; corresponding ring expands with the constant velocity v c (5). For arbitrary t t m this is shown in Figure 3: "nucleation" at t = t m followed by bifurcation into two branches. At t = 2t m (18) the equation for t(k) (12) acquires yet another solution k = k m that for t > 2t m bifurcates into two. Their evolution repeats what was already found for the first pair of solutions k −,+ (t). Generally, new wavefronts are created periodically in 0.43 s intervals (18) followed by bifurcation into two, one of which, at t large, approaches the CP result (2) while the other expands with the velocity v c = 23 cm/s (5). These wavefronts given by f = −2πl solutions to Eqs. (12), are shown in Figure 3. The shaded grey region of calmed water expands with the constant velocity v g = 18 cm/s (5). In practice it should become clearly defined in a time interval estimated as several t m (18), i.e. several seconds, which explains observations. The greyscale line r = v c t is shown for reference; it separates the two f = ±2πl regimes discussed earlier. Despite their ubiquity, systematic quantitative studies of splashes on deep water in the linear regime have been lacking. Possible reasons for this have already been given by Rayleigh [18]: the short waves of capillary-gravity origin represented in Figure 3 by the "dashed" wavefronts may be rapidly damped, and the length scale for full development of the splash pattern may be inconveniently large. Both of these obstacles can be overcome if instead of water one uses superfluid 4 He. Damping is eliminated in the superfluid state, and the coefficient of surface tension of the superfluid extrapolated to zero temperature, γ = 0.37 erg/cm 2 [31], is about 200 times smaller than that of water. Given the density of the superfluid ρ = 0.145 g/cm 3 [26], the 4 He counterparts of the capillary length and the time scale (3) can be found as λ (He) = 0.051 cm, τ (He) = 0.0072 s. (19) Since the capillary length of water (3) is five times larger, many more wavefronts will be present within the same observation area in the case of the superfluid. Moreover, the velocity unit in the case of superfluid 4 He, λ (He) /τ (He) = 7 cm/s, is about a half of that of water λ/τ = 16 cm/s (3). E. Plasma waves in a two-dimensional electron gas Evaluation of Eq. (7) shows that both the group Ω and the phase Ω/k velocities are monotonically decreasing functions of k which asymptotically approach the speed of sound as k → ∞, leading to the results that v m = u and k m = ∞. Since Ω < Ω/k, Eq. (13) further implies that f = −2πl. The second of Eqs.(12) then becomes t = 4πl(u/g) 1 + g/u 2 k. The consequence is that Eqs. (12) acquire solutions only for t t m = 4πu/g. The first of these is a monotonically increasing function of time shown in Figure 4 length, and u/g, respectively) following a sudden localized perturbation of a two-dimensional electron gas, as described by Eqs. (12), (13) and (7). The region of calmed electron liquid r < ut is shaded grey. the CP result (2) for r 1 for t → ∞. More generally, new wavefronts are created periodically at times t = lt m = 4πlu/g. Their evolution is shown in Figure 4; as t → ∞ they approach the CP result (2). These wavefronts are found at space-time locations r ut. The region of calmed electron liquid r < ut is shaded grey. The annular waves in Figure 4 are counterparts to the accelerating wavefronts found in the water splash, Figure 3. The central difference is that annular waves in the two-dimensional electron gas are created one at a time. F. Elementary excitations in superfluid 4 He The dispersion law Ω(k) of elementary excitations in a superfluid is a non-monotonic function of the wave number k [26]: after an initial linear in k increase (16), the function Ω(k) reaches a maximum at k = k * followed by a "roton" minimum at k = k 0 . Therefore the group velocity is negative and takes on its minimal value in the [k * ; k 0 ] range. Additionally, the spectrum has an end point k = k e where the group velocity vanishes [26]. As a result, the group velocity is positive and takes on its maximal value in the [k 0 ; k e ] range. Dependences of the group and the phase velocities on k are sketched in Figure 5; the phase velocity also has a minimum and a maximum at k finite, and both velocities have simultaneous maxima of magnitude u at k = 0. A color legend is employed phase Ω(k)/k velocities of elementary excitations in superfluid 4 He. The extrema of Ω(k) are located at k = k * , k = k 0 (roton minimum), and k = k e (end point). Hereafter the function Ω = 2k + sin 2πk is employed to mimic the true dispersion law. Color legend is explained in the main text. for the phase velocity curve to distinguish, according to the inequalities (13), five different spectral ranges corresponding to five families of the wavefronts: (i) The [0; k * ] range (blue). Here the group velocity is smaller than the phase velocity. (ii) The [k * ; k 0 ] range (red). Here the group velocity is negative and takes on a minimal value. (iii) The [k 0 ; k c1 ] range (black). Here the group velocity is smaller than the phase velocity. In the superfluid this range of the wave numbers [27] is very narrow; an arrow in Figure 5 points to it. (iv) The [k c1 ; k c2 ] range (green). Here the group velocity is larger than the phase velocity and the former takes on a maximal value. (v) The [k c2 ; k e ] range (magenta). Here the group velocity is smaller than the phase velocity. Employing the empirically known dispersion law [26], the extremal phase and group velocities (extrapolated to zero pressure) characterizing the splash in a superfluid can be estimated as: v m1 = −2 · 10 4 cm/s, v c1 = 5.9 · 10 3 cm/s, v m2 u = 2.4 · 10 4 cm/s, v c2 = 9 · 10 3 cm/s. (20) The velocity v = v c1 known as the Landau critical roton velocity to destroy superfluidity is also a threshold velocity for the generation of a wake pattern behind a small source uniformly moving through the superfluid [27]. To understand the dynamics of the wavefronts, in Figure 6 we sketched the t(k)/l and r(k)/l dependences (12) (12) and (13) based on the behavior of the velocities in Figure 5 using the same legend and colors. Dashed curves of the same color display the r(k)/l dependences. and (13) color coordinated with Figure 5. The resulting splash pattern in the (r/l, t/l) variables consisting of five families of wavefronts is shown in Figure 7: (i) The blue colored (largest slope) wavefronts emerge with period t * = 2π/Ω(k * ) at the center of the splash and expand, asymptotically reaching the speed of sound u (20) for large t. (ii) The magenta colored (second largest slope) wavefronts nucleate at the center of the splash with period t e = 2π/Ω(k e ) (14). As t → ∞, the corresponding solution to the equation for t(k) (Figure 6) approaches k = k c2 , thus implying that the wavefronts expand asymptotically reaching the velocity v = v c2 (20); the greyscale double-dashed line r = v c2 t is shown for reference. (iii) The black colored (smallest slope) wavefronts emerge at the center of the splash with period t 0 = 2π/Ω(k 0 ) and expand, asymptotically reaching the Landau critical roton velocity v = v c1 (20) for large t; the greyscale dashed line r = v c1 t is shown for reference. (iv) The dynamics of the green colored (diverging) wavefronts can be understood via the argument already given in the discussion of the water splash. They nucleate at r finite with period t m2 (14); the locus of these events belongs to the straight line r = v m2 t. Each nucleation event results in a pair of diverging spherical wavefronts. In the large time limit the slower of the two expands with velocity approaching the Landau critical roton velocity v = v c1 (20), while the faster expands with velocity approaching v = v c2 (20). (v) The red colored (converging) wavefronts made by elementary waves of negative group velocity nucleate at r finite with period t m1 (14); the locus of these events wavefronts vs time (scaled by integer factor l to "collapse" wavefronts of given family onto a single curve or a pair of curves) in a superfluid following a sudden localized perturbation, according to Eqs. (12), (13), and the functional dependences of the phase and group velocities depicted in Figure 5. The color legend is coordinated with Figures 5 and 6. Greyscale dashed and double-dashed lines, r = v c1 t and r = v c2 t, respectively, are shown for reference. belongs to the straight line r = |v m1 |t. Since now r(k)/l has a maximum at k = k m1 , each nucleation event results in a pair of converging spherical wavefronts. The faster of these reaches the center of the splash exactly as the blue colored (largest slope) wavefronts nucleate there while the slower one arrives at the center when black colored (smallest slope) wavefronts emerge there. This lends itself to the following interpretation: converging wavefronts made of waves of negative group velocity rebound off the splash center in the form of diverging wavefronts made of waves of positive group velocity. IV. CONCLUSIONS To summarize, we demonstrated that in large time regime dispersion law of relevant collective excitations alone suffices to understand dynamics of wavefronts in splashes in isotropic media. The outcome is determined by the interplay between excitation's phase and group velocities as well as the sign of the latter. The salient features of splashes are controlled by the existence of extremal values of the phase and the group velocities: the group velocity gives the expansion rate of the locus of the points where new wavefronts nucleate or existing ones disappear, while the phase velocity determines the large-time expansion rate of a group of wavefronts. If the group velocity is negative in a spectral range and takes on a minimal value within it, then converging wavefronts will be present in the splash. To illustrate our theory we also carried out several case studies of experimentally relevant setups. Specifically, splashes on water and in two-dimensional electron gas were found to be similar: following a localized perturbation, a quiescent region inside the annular waves forms. This region expands with a constant rate corresponding to the minimum group velocity -the speed of sound for the two-dimensional electron gas or 18 cm/s for water, the conclusion due to Rayleigh [18]. New wavefronts nucleate at the boundary of the quiescent region at regular time intervals (0.43 s for water), in pairs (in water) or one at a time (in a two-dimensional electron gas). When the wavefronts appear in pairs, one of them expands with the minimal phase velocity (23 cm/s for water). The other (water and two-dimensional electron gas) expands asymptotically with a constant acceleration. The gross features of a splash in a superfluid are determined by five extremal velocities. Additionally, due to the existence of a negative group velocity spectral range, some of the wavefronts in superfluid 4 He splash are converging. The existence of converging wavefronts in splashes is not unique to superfluid 4 He. Similar conclusions apply to a dipolar quantum gas whose spectrum also features a roton minimum [32]. More generally, whenever there exists a spectral range where the group velocity as a function of the wavenumber is negative and it takes on a minimal value within this range, then converging wavefronts will be necessarily present in the splash. The first known realistic example of a spectrum featuring excitations of negative group velocity, the optical branch of vibrations in crystals [33], belongs to this category, too. We hope that both the general analysis and sample studies carried out in this work will guide future observations of splashes. V. ACKNOWLEDGEMENTS The author is grateful to J. P. Straley and A. P. Levanyuk for valuable comments. FIG. 1 : 1Group Ω (k) and phase Ω(k)/k velocities versus the wave number k in units of l/τ and 1/l, respectively, Eq.(3), for capillary gravity waves (4). FIG. 2 :FIG. 3 : 23Dependences of t(k), Eqs.(12) and(13), for the capillary gravity waves(4). The legend and line styling are the same as inFigure 1. Numbers next to the curves are values of integer l. Radii of annular capillary gravity wavefronts vs time following a sudden localized perturbation of the water surface, in units of Eq.(3) for several values of l according to Eqs.(4),(12),(13) and(18). The line styling is coordinated withFigures 1 and 2. The greyscale line r = v c t separates annular rings made by short k > k c (dashed) and long k < k c waves. The shaded grey region of calmed water expands with the velocity v g(5). online) Radii of the annular plasma wavefronts vs time (in units of u 2 /g, the screening online) Sketches of the group Ω (k) and online) The t(k)/l and r(k)/l dependences online) Radii of the spherical density Large-scale turbulent flow around a cylinder in counterflow superfluid He (He(ii)). T Zhang, S W Van Sciver, Nat. Phys. 136T. Zhang and S.W. Van Sciver, Large-scale turbulent flow around a cylinder in counterflow superfluid He (He(ii)), Nat. Phys. 1, 36 (2005). Visualization of quantized vortices. 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P. Straley, Screening and plasma oscillations in an electron gas in the hydrodynamic ap- proximation, Phys. Rev. B 96, 165116 (2017). E M Lifshitz, L P Pitaevskii, Statistical Physics. Butterworth-Heinemann2Third edition. III and IXE. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Third edition, Part 2: Volume 9 (Course of Theoretical Physics) (Butterworth-Heinemann, 1980), Chapters III and IX. Negative group velocity and Kelvin-like wake pattern. E B Kolomeisky, J Colen, J P Straley, Phys. Rev. B. 10554509E. B. Kolomeisky, J. Colen and J. P. Straley, Negative group velocity and Kelvin-like wake pattern, Phys. Rev. B 105, 054509 (2022). L D Landau, E M Lifshitz, Statistical Physics. Butterworth-Heinemann1123Third editionL. D. Landau and E. M. Lifshitz, Statistical Physics, Third edition, Part 1: Volume 5 (Course of Theoretical Physics) (Butterworth-Heinemann, 1980), Section 123. D Pines, P Nozières, The Theory of Quantum Liquids. New YorkBenjamin1D.Pines and P. Nozières, The Theory of Quantum Liq- uids, Vol.1 (Benjamin, New York 1966), Chapter 2. Kelvin-Froude wake patterns of a traveling pressure disturbance. J Colen, E B Kolomeisky, Eur. J. Mech. /B Fluids. 85400J. Colen and E. B. Kolomeisky, Kelvin-Froude wake pat- terns of a traveling pressure disturbance, Eur. J. Mech. /B Fluids 85, 400 (2021). Surface Tension of Liquid He 4. K R Atkins, Y Narahara, Phys. Rev. 138437K. R. Atkins and Y. Narahara, Surface Tension of Liquid He 4 , Phys. Rev. 138, A437 (1965). Observation of roton mode population in a dipolar quantum gas. L Chomaz, R M W Van Bijnen, D Petter, G Faraoni, S Baier, J H Becher, M J Mark, F Wächtler, L Santos, F Ferlaino, Nature Physics. 14and references thereinL. Chomaz, R. M. W. van Bijnen, D. Petter, G. Faraoni, S. Baier, J. H. Becher, M. J. Mark, F. Wächtler, L. Santos and F. Ferlaino, Observation of roton mode population in a dipolar quantum gas, Nature Physics 14, 442 (2018), and references therein. A brief history of negative group velocity is outlined in K. T. McDonald, Negative group velocity. 69607A brief history of negative group velocity is outlined in K. T. McDonald, Negative group velocity, Am. J. Phys. 69, 607 (2001).
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[ "arXiv:hep-th/0504227v1 28 Apr 2005 ON DYNAMICS OF STRINGS AND BRANES", "arXiv:hep-th/0504227v1 28 Apr 2005 ON DYNAMICS OF STRINGS AND BRANES" ]
[ "A V Golovnev \nV.A. Fock Institute of Physics Sankt-Petersburg State University\n\n", "L V Prokhorov \nV.A. Fock Institute of Physics Sankt-Petersburg State University\n\n" ]
[ "V.A. Fock Institute of Physics Sankt-Petersburg State University\n", "V.A. Fock Institute of Physics Sankt-Petersburg State University\n" ]
[]
We study Nambu-Goto strings and branes. It is shown that they can be considered as continuous limits of ordered discrete sets of relativistic particles for which the tangential velocities are excluded from the action. The linear in unphysical momenta constraints are found. It allows to derive the evolution operators for the objects under consideration from the "first principles".
10.1007/s10773-006-9087-2
[ "https://arxiv.org/pdf/hep-th/0504227v1.pdf" ]
119,017,816
hep-th/0504227
ec6ccce6dbade125e9653c9451685ea82123bb5e
arXiv:hep-th/0504227v1 28 Apr 2005 ON DYNAMICS OF STRINGS AND BRANES A V Golovnev V.A. Fock Institute of Physics Sankt-Petersburg State University L V Prokhorov V.A. Fock Institute of Physics Sankt-Petersburg State University arXiv:hep-th/0504227v1 28 Apr 2005 ON DYNAMICS OF STRINGS AND BRANES We study Nambu-Goto strings and branes. It is shown that they can be considered as continuous limits of ordered discrete sets of relativistic particles for which the tangential velocities are excluded from the action. The linear in unphysical momenta constraints are found. It allows to derive the evolution operators for the objects under consideration from the "first principles". Introduction. We consider Nambu-Goto strings and branes. We start with the simplest case of one relativistic particle which formally can be considered as a 0-brane. The relativistic invariant form of its action is raparametrization invariant and has a constraint, commonly taken in a quadratic form with the loss of information on the momentum sign. It led to some problems [1,2] in quantum description of free relativistic particles. In section 2 we explain how these problems can be overcome [3,4] using a linear "constraint" containing the sign of the velocity ∂x 0 ∂σ 0 ( ∂t ∂τ in usual notations) and fixing the physical sector of quantum theory by the condition ∂x 0 ∂σ 0 > 0. After that the operator of evolution (and propagator [3]) can be derived from the "first principles" of quantum mechanics. In sections 3 and 4 we prove that p-branes may be considered as continuous limits of discrete sets of relativistic particles and that the p-brane action S = −γ d p+1 σ (−1) p g, g = det ∂x µ ∂σ a ∂x µ ∂σ b ,(1) is a continuous limit of sum of properly modified relativistic particle actions. Here a, b = 0, 1, . . . , p and µ = 0, 1, . . . , n where p + 1 and n + 1 are the brane worldsheet and the bulk space dimentions respectively; g is the induced metric determinant on the worldsheet. Strings correspond to the p = 1 case in (1). The modification is such that particle motions along the brane hypersurface become unphysical. It yields p constraints; the remaining constraint (H = 0) is a consequence of arbitrariness of "time" σ 0 . In sections 5 and 6 we consider dynamics of strings (p = 1) and branes (p > 1). The theories are reparametrization invariant for all p, n > p; each of them has p + 1 constraints. Usualy one of these constraints physicists take in a quadratic form. It causes the same problems as in the case of a single particle. We found that the operator of evolution can be constructed in the same way as in section 2; new difficulties are rather technical than conceptual ones. The solution is also similar to the p = 0 case (a particle). Relativistic particle. Motion of free relativistic particle is defined by the well-known action S = −m 1 − − → v 2 dt, where − → v = d − → x (t) dt . The canonical momentum is − → p ≡ ∂L ∂ − → v = m − → v √ 1− − → v 2 ≡ E p − → v with L being the Lagrangian, and the Hamiltonian is H = E p = m 2 + − → p 2 . We can write down the action in the explicitly relativistic invariant form by parametrization of the world line: x µ = x µ (σ 0 ) (usually one uses τ instead of σ 0 ) with x µ (σ 0 ) = (t(σ 0 ), − → x (σ 0 )), µ = 0, . . . , n. We denoteẋ µ ≡ dx µ dσ 0 , so that − → v =− → x dσ 0 dt and S = −m ẋ µẋ µ dσ 0 . It is the p = 0 case of (1) in which we put m (particle's mass) instead of γ. The vector of canonical momentum is p µ ≡ ∂L ∂ẋ µ = −mẋ µ √ẋ 2 .(2) The obtained theory is invariant under reparametrization group σ 0 → σ 0 = f (σ 0 )), hence its Hamiltonian is zero and by squaring the equation (2) one gets a constraint: p 2 − m 2 = 0.(3) Of course, the information about the sign of p 0 is lost. On the other hand one can prove that any constraint has to be linear in unphysical momenta [3,4]. It was a serious obstruction to the ab initio derivation of relativistic particle propagator (authors of [1,2] used some additional assumptions). The problem was overcome only in [3]. One can obtain the solution in the following way. From the constraint (3) one finds p 0 = ± m 2 + − → p 2 , and it is obvious from (2) that sign p 0 = − signẋ 0 . Combining these facts we have p 0 + E p sign(ẋ 0 ) = 0 (4) with E p ( − → p ) = m 2 + − → p 2 . The Hamiltonian is zero and the total Hamiltonian [5] is H T = v p 0 + E p sign(ẋ 0 ) . Here v is the Lagrange multiplier. Strictly speaking Eq. (4) is not a constraint because it contains velocity (and H T is not a Hamiltonian due to the same reason). But it depends only upon the sign ofẋ, and this fact allows us to formulate the quantum theory. We fix the σ 0 "time arrow" by condition ∂x 0 ∂σ 0 > 0(5) which forces us to admit that v > 0 (because ∆x 0 = v∆σ 0 , see later). We use the evolution operator U ω (x,x) = x| exp(−iωĤ T )|x , ψ(x, σ 0 + ω) = d 4x U ω (x,x)ψ(x, σ 0 ) and the following relation for infinitesimal ω: x| exp(−iωĤ T )|x = d 4 p x| exp(−iωH T (x, p))|p p|x . Integrating over p 0 and (with use of δ-function δ(x 0 −x 0 − ωv)) overx 0 we get for the wave function [4]: ψ(x 0 , − → x ) = d 3 pd 3x (2π) 3 exp i[p i ∆x i − ∆x 0 E p ] ψ(x 0 , − → x ),(6) where ∆x 0 = vω, and we omit the argument σ 0 because all the information on σ 0 is accumulated in x 0 . Using (6) one can get the right Feynman propagator for a relativistic particle [4]. String as an ordered system of relativistic particles. Now we show that any string and brane can be described as a system of particles. More precisely, the action (1) may be regarded as a continuous limit of sum of free relativistic particle actions provided that we take − → v ⊥ instead of − → v , where − → v ⊥ is the part of velocity orthogonal to the constant time hypersurface of the brane worldsheet. We deal here with a kind of indirectly introduced particle interaction. In this section we consider a string (1-brane) and reproduce the proof of our statement by an explicit calculation [6]. We consider N + 1 particles with the position vectors − → x k (x 0 ), k = 0, 1, . . . , N and the action S = −m N k=0 dx 0 1 − − → v 2 k⊥ (x 0 ),(7)in which − → v k = d − → x k dx 0 and − → v k⊥ = − → v k − ( − → v k · ∆ − → x k ) (∆ − → x k ) 2 ∆ − → x k , ∆ − → x k ≡ − → x k+1 − − → x k . In the continuous limit we define kA N → σ 1 , ∆ − → x k A/N → − → k (σ 1 ), m |∆ − → x k | → γ. Here σ 1 ∈ [0, A]; usually one takes A = π, but for our purposes it may be natural to consider A as the string length and |∆ − → x k | = A N , | − → k | = 1. In any case we have S = −γ N k=0 dx 0 ∆l 1 − − → v 2 k⊥ → −γ dx 0 | − → k |dσ 1 1 − − → v 2 ⊥ with − → k = ∂ − → x (x 0 ,σ 1 ) ∂σ 1 , − → v = ∂ − → x (x 0 ,σ 1 ) ∂x 0 and − → v ⊥ = − → v − ( − → v − → k ) k 2 − → k . The string length is L = | − → k |dσ 1 . After that we parametrize the worldsheet x 0 = x 0 (σ 0 , σ 1 ), − → x = − → x (σ 0 , σ 1 ) introducing a new parameter σ 0 (again the stan- dard notations are x 0 t and σ 0 τ ). We haveẋ ≡ ∂x(σ 0 ,σ 1 ) ∂σ 0 = (1, − → v )ẋ 0 , x ′ ≡ ∂x(σ 0 ,σ 1 ) ∂σ 1 = (x 0 ′ , − → k + − → v x 0 ′ ) and S = −γ d 2 σẋ 0 | − → k | 1 − − → v 2 ⊥ . The last expression is equal to the Nambu-Goto action: S = −γ d 2 σ (ẋx ′ ) 2 −ẋ 2 x ′ 2 . Of course, one could start with it and get S = −γ dx 0 dl 1 − − → v ⊥ 2 which is a continuous limit of (7). We can also propose another discrete analogue of the Nambu-Goto string. Let's consider the following action: S = −m N k=0 dσ 0 ẋ µ k⊥ẋ µ k⊥ , whereẋ µ k⊥ is the part ofẋ µ k perpendicular to x µ k+1 − x µ k . The continuous limit is N → ∞, kA N → σ 1 , m |∆s k | → γ with the invariant interval (∆s k ) 2 = (x µ k+1 − x µ k )(x µ k+1 − x µ k ). We haveẋ µ ⊥ =ẋ µ −ẋ ν x ′ ν x ′ρ x ′ ρ x ′µ , ds dσ 1 2 = x ′2 , and S = −γ dτ |ds| ẋ 2 − (ẋx ′ ) 2 x ′ 2 = −γ dσ 0 dσ 1 (ẋx ′ ) 2 −ẋ 2 x ′ 2 . In contrast to the previous paragraph the presented discrete theory has the relativistic invariant form from the very begining but even the sense ofẋ ⊥ depends upon the parametrization of the worldsheet. In the gauge σ 0 = x 0 these two approaches coincide. Brane as an ordered system of relativistic particles. Now we shall prove our statement for p > 1. We consider (N + 1) p particles arranged into some p-dimensional lattice with the position vectors − → x i 1 i 2 ...ip and the action S = −m dx 0 N i 1 =0 · · · N ip=0 1 − − → v 2 i 1 ,...ip⊥ where − → v i 1 ,...ip⊥ is the component of − → v i 1 ...ip perpendicular to − → x i 1 ...i k +1...ip − − → x i 1 ...i k . ..ip for all k. In continuous limit we demand Ai k N → σ k and m ∆V → γ with ∆V being volume of a cell of the lattice. The action takes the form S = −γ dx 0 dV 1 − − → v 2 ⊥ , and we need to prove that S is equal to (1). For the sake of simplicity first we consider some special coordinate system on the brane. Let the brane be parametrized by σ 0 , σ 1 , . . . , σ p , σ i ∈ [0, A] for i = 1, . . . , p. We choose a coordinate system in which σ 0 = x 0 , so that ∂x 0 ∂σ i = 0, and ∂x µ ∂σ i ∂xµ ∂σ j = 0, i = j. It is always possible, locally at least. Let's denote − → k i ≡ ∂ − → x (x 0 ,σ i ) ∂σ i and − → v ≡ ∂ − → x (x 0 ,σ i ) ∂x 0 on the worldsheet. In our coordinate system − → k i is orthogonal to − → k j , i = j and − → v ⊥ = − → v − p i=1 ( − → v − → k i ) k i 2 − → k i . Of course, one can find − → v ⊥ without the orthogonality condition with the use of standard orthogonalization procedure, but in section 6 more simple proof is presented. We have ∂x µ ∂σ 0 = (1, − → v ) and ∂x µ ∂σ i = (0, − → k i ), so the determinant in (1) is det ∂x µ ∂σ a ∂x µ ∂σ b = 1 − − → v 2 − − → v − → k 1 − − → v − → k 2 . . . − − → v − → k p − − → v − → k 1 −k 2 1 0 . . . 0 − − → v − → k 2 0 −k 2 2 . . . 0 . . . . . . . . . . . . . . . − − → v − → k p 0 0 . . . −k 2 p = = p i=1 (−k i 2 )    1 − − → v 2 + p i=1 − → v − → k i 2 k i 2    where a, b = 0, 1, . . . , p. It is not also difficult to see that the volume element at the constant time hypersurface on the brane worldsheet is dV = p i=1 |k i |dσ i . We conclude that the action S = −γ dσ 0 d n σ i det ∂x µ ∂σ a ∂x µ ∂σ b = −γ dx 0 dV 1 − − → v 2 ⊥ .(8) It proves our statement. Dynamics of strings. As it was mentioned in the Introduction, the evolution operator U ω can be constructed for strings and branes in the same way as in section 2 [7]. For p = 1 action (1) is the action of free bosonic string with the Lagrange density [6,[8][9][10][11] L = −γ √ −g = −γ (ẋx ′ ) 2 −ẋ 2 x ′ 2 .(9) We assume thatẋ is timelike and x ′ is spacelike, so that σ 0 can be regarded as a time parameter. In this case the momentum p µ ≡ ∂L ∂ẋ µ = −γ (ẋx ′ )x ′ µ − x ′ 2ẋ µ (ẋx ′ ) 2 −ẋ 2 x ′ 2(10) will also be timelike. One easily gets two constraints p µ x ′ µ = 0,(11)p 2 + γ 2 x ′ 2 = 0.(12) The second one is obtained by squaring the Eq. (10) and hence some information is lost. As in the case of a pointlike particle Eq. (12) yields p 0 = ±E p with E p = − → p 2 − γ 2 x ′ 2 . The sign of p 0 follows from the definition of the momentum. We have p 0 + E p (x, − → p ) sign(y 0 ) = 0, where y µ = (ẋx ′ )x ′ µ − x ′ 2ẋ µ . The vector y is obviously timelike (indeed, y 2 = x ′ 2 g > 0 and y µẋ µ = −g > 0, so that signẋ 0 = sign y 0 ). We get the "constraint" analogous to (4): p 0 + E p (x, − → p ) sign(ẋ 0 ) = 0.(13) The "Hamiltonian" of the theory H = p 0 +E p signẋ 0 is zero, and demanding again signẋ 0 > 0 we get the total Hamiltonian H T = u(p 0 + E p ) + vp µ x ′ µ . We have two unphysical momenta and need to exclude them from E p . Let's exclude p 0 and p 1 (we denote the remaining components by the lower index " > ": p µ = (p 0 , p 1 , p > )). One can find p 1 from (11) if x ′ 1 = 0. In [7] we restricted ourselves to the case x ′ 0 = 0 and had E p (x, p > ) = p > x ′ > x ′ 1 2 + p 2 > − γ 2 x ′ 2 . In general case one can substitute p 1 from (11) to (12) and get a quadratic equation for p 0 : p 0 2 1 − x ′ 0 x ′ 1 2 + 2p 0 (p > x ′ > )x ′ 0 (x ′ 1 ) 2 − p 2 > 1 + x ′ > x ′ 1 2 + γ 2 x ′ 2 = 0. If |x ′ 1 | > |x ′ 0 |, it has two real roots of opposite signs, and we can choose a proper one following (13). Otherwise we have to try to exclude another component of p µ from E p . It's always possible because x ′ is spacelike and | − → x ′ | > |x ′ 0 |. In section 3 we have seen that the unphysical degree of freedom is related to the motion of particles along the string. Now we write down the evolution equation (see section 2) ψ(x) = D n+1x (σ 1 ) x(σ 1 )| exp(−iωĤ T )|x(σ 1 ) ψ(x(σ 1 )) = = D n+1 pD n+1x exp(i[p µ ∆x µ − ωu(p 0 + E p (x, p ⊥ )) − ωvp µ x ′ µ ])ψ(x) = = D n−1 p > D n+1x exp(i[−p > ∆x > − ωuE p (x, p > ) + ωvp > x ′ > ])× × δ(∆x 0 − ωu − ωvx ′ 0 )δ(−∆x 1 + ωvx ′ 1 )ψ(x) . Here Dx and Dp denote differentials in the functional spaces and all the integrals are functional ones (path integrals). δ-Functions determine the Lagrange multipliers ωv = ∆x 1 x ′ 1 and ωu = ∆x 0 − x ′0 ∆x 1 x ′ 1 yielding the final result ψ(x) = D n−1 p > D n−1x > exp(i[−p > ∆x > − − ∆x 0 − x ′ 0 ∆x 1 x ′ 1 E p (x, p > ) + ∆x 1 x ′ 1 p > x ′ > ])ψ(x). If x ′ 0 = 0, Eq. (5) implies u > 0. Dynamics of branes. We turn to the general case of action (1). We denote the σ 0 , σ 1 , . . . , σ p derivatives of x by x ,0 , x ,1 , . . . , x ,p . Again we assume that the vector x µ ,0 is timelike, and vectors x µ ,i are spacelike (here and hereafter in this chapter i, k, l = 1, . . . , p while a, b = 0, . . . , p). Now in the action (1) g(σ) = 1 (p + 1)! ǫ a 0 ...ap ǫ b 0 ...bp x α 0 ,b 0 x α 0 ,a 0 · · · x αp,bp x αp,ap with ǫ being the unit antisymmetric Levi-Civita symbol, and the canonical momentum is p µ = −γ(−1) p p! (−1) p g ǫ 0a 1 ...ap ǫ b 0 ...bp x µ,b 0 x ,b 1 · x ,a 1 . . . x ,bp · x ,ap . Evidently p µ x µ ,i = 0 due to antisymmetry of ǫ, and using the equality ǫ a 0 ...ap g(σ) = ǫ b 0 ...bp x ,b 0 · x ,a 0 . . . x ,bp · x ,ap one obtains p 2 = (−1) p γ 2 ζ(x), ζ(x) = det x µ ,i x µ,p 2 − (−1) p γ 2 ζ(x) = 0.(15) From (15) we have p 0 = ±E p with E p = − → p 2 + (−1) p γ 2 ζ(x). Again, the p 0 sign can be easily found: p µ andẋ µ are timelike and p µẋ µ = −γ (−1) p g < 0, hence sign(p 0 ) = − sign(ẋ 0 ). The result is similar to (4): p 0 + E p sign(ẋ 0 ) = 0, the "Hamiltonian" is equal to zero and the total Hamiltonian H T = u(p 0 + E p (x, − → p )) + v i p µ x µ ,i . Here p+1 momenta are unphysical ones. We assume that det(x i,k ) = 0 (x i,k is an p×p matrix) and exclude momenta p 1 , . . . , p p from E p . Due to (14) we have p i = ([x .,. ] −1 ) il (p 0 x 0,l − p > x >,l ). Here we denoted all the components of p µ with µ > p by the lower index " > ", and ([x .,. ] −1 ) il ≡ d il is a matrix inverse of x l,i . Then (15) turns into quadratic equation p 0 2 1 − d il x 0,l d ik x 0,k + 2p 0 d il (p > x >,l )d ik x 0,k − − d il (p > x >,l )d ik (p > x >,k ) − (−1) p γ 2 ζ(x) = 0. It has two real roots of opposite signs if and only if d il x 0,l d ik x 0,k < 1. The sufficient condition is that the norm of x i,k as a linear operator is greater than the length of p-dimensional vector x 0,l . If x 0,l = 0 the simple answer exists: E p = E p (x, p > ) = p i=1 (p i (x, p > )) 2 + p 2 > + (−1) p γ 2 ζ(x). For the wave function we have path integrals (taking (5) into account) ψ(x) = D n+1 pD n+1x exp(i[p µ ∆x µ − − ωu(p 0 + E p (x, p ⊥ )) − ωv i p µ x µ ,i ])ψ(x) = = D n−p p > D n+1x exp(i[−p > ∆x > − ωuE p (x, p > ) + ωv i p > x >,i ])× × δ(∆x 0 − ωu − ωv i x ′ 0,i ) r l=1 δ(−∆x l + ωv i x l,i )ψ(x). Again δ-functions determine the Lagrange multipliers ωv i = d il ∆x l , ωu = ∆x 0 − ωv i x 0,i and reduce the number of integrals over x: ψ(x) = D n−p p > D n−px > exp(i[−p > ∆x > − − (∆x 0 − d il (∆x l )x 0,i )E p (x, p > ) + d il (∆x l )p > x >,i ])ψ(x). So, we found out the structure of constraints and wrote down the evolution operator in the same way as in section 2. It isn't surprising remembering results of sections 3 and 4. Now we are ready to prove this statement without fixing any special coordinate system. We just need to find out the general formula for − → v ⊥ . Notice that by definition p µ lies in a hyperplane of x µ ,0 and x µ ,i . Then, due to the constraints (14), p µ is proportional to x µ ⊥,0 . We have v µ = ∂x µ (x 0 ,σ i ) ∂x 0 = x µ ,0 x 0 ,0 and hence p µ is also proportional to v µ ⊥ : v µ ⊥ = αp µ . To find the coefficient α we take v µ ⊥ p µ = x µ ⊥,0 pµ x 0 ,0 = −γ √ (−1) p g x 0 ,0 (the last equality is just the Euler's homogeneous function theorem) and p µ p µ = (−1) p γ 2 ζ(x). But v µ ⊥ p µ = αp µ p µ , hence we have v µ ⊥ = − (−1) p g (−1) p γζx 0 ,0 p µ and 1 − − → v 2 ⊥ = v µ ⊥ v µ⊥ = (−1) p gp 2 γ 2 ζ 2 x 0 ,0 2 = g ζx 0 ,0 2 , so dx 0 dV 1 − − → v 2 ⊥ = x 0 ,0 dσ 0 | det x µ ,i x µ,k | d p σ i g ζx 0 ,0 2 = d p+1 σ |g|. We proved (8) and the main statement of sections 3 and 4 for the most general case. k . So, with the loss of information about the sign of p 0 , the constraints are p µ x µ ,i = 0, i = 1, 2, . . . , p;(14) Path integral for a scalar propagator: Preprint HUTP 80-A-003. F Krausz, CambridgeKrausz F. Path integral for a scalar propagator: Preprint HUTP 80-A- 003. Cambridge, 1980. . P P Fiziev, Theor, Math.Phys. 62123Fiziev P.P. Theor.Math.Phys., 62, p. 123 (1985). . L V Prokhorov, A G Nuramatov, Vestnik Leningr. Univ. (Ser. 41886in RussianProkhorov L.V., Nuramatov A.G. Vestnik Leningr. Univ. (Ser. 4) 3 (18), p. 86 (1991). (in Russian) Gamiltonova mekhanika kalibrovochnikh sistem. L V Prokhorov, S V Shabanov, Izd. SPbGU. in RussianProkhorov L.V., Shabanov S.V. Gamiltonova mekhanika kalibrovoch- nikh sistem. Izd. SPbGU, 1997. (in Russian) Lectures on quantum mechanics. P A Dirac, DoverDirac P.A.M. Lectures on quantum mechanics. Dover, 2001. Introduction to the relativistic string theory. B M Barbashov, V V Nesterenko, World ScientificBarbashov B.M., Nesterenko V.V. Introduction to the relativistic string theory. World Scientific, 1990. A V Golovnev, L Prokhorov, Vestnik SPb Univ. (Ser.4). 286in RussianGolovnev A.V., Prokhorov L.V. Vestnik SPb Univ. (Ser.4), 2 (12), p. 86 (2003). (in Russian) Superstring theory, vols. 1 and 2. M B Green, J H Shwartz, E Witten, Cambridge University PressGreen M.B., Shwartz J.H., Witten E. Superstring theory, vols. 1 and 2, Cambridge University Press, 1986. Introduction to superstrings. M Kaku, Springer-VerlagKaku M. Introduction to superstrings. Springer-Verlag, 1988. . T Goto, Prog.Theor.Phys. 461560Goto T. Prog.Theor.Phys, 46, p. 1560 (1971). . O Hara, Prog.Theor.Phys. 461549Hara O. Prog.Theor.Phys, 46, p. 1549 (1971).
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[ "LINEAR CODES OVER Z 4 + uZ 4 : MACWILLIAMS IDENTITIES, PROJECTIONS, AND FORMALLY SELF-DUAL CODES", "LINEAR CODES OVER Z 4 + uZ 4 : MACWILLIAMS IDENTITIES, PROJECTIONS, AND FORMALLY SELF-DUAL CODES" ]
[ "Bahattin Yildiz ", "Suat Karadeniz " ]
[]
[]
Linear codes are considered over the ring Z 4 + uZ 4 , a non-chain extension of Z 4 . Lee weights, Gray maps for these codes are defined and MacWilliams identities for the complete, symmetrized and Lee weight enumerators are proved. Two projections from Z 4 + uZ 4 to the rings Z 4 and F 2 + uF 2 are considered and self-dual codes over Z 4 +uZ 4 are studied in connection with these projections. Finally three constructions are given for formally self-dual codes over Z 4 + uZ 4 and their Z 4 -images together with some good examples of formally self-dual Z 4 -codes obtained through these constructions.
10.1016/j.ffa.2013.12.007
[ "https://arxiv.org/pdf/1307.3047v1.pdf" ]
14,094,497
1307.3047
e92771cf6244a4b5965f3cae60d16131774b794c
LINEAR CODES OVER Z 4 + uZ 4 : MACWILLIAMS IDENTITIES, PROJECTIONS, AND FORMALLY SELF-DUAL CODES 11 Jul 2013 Bahattin Yildiz Suat Karadeniz LINEAR CODES OVER Z 4 + uZ 4 : MACWILLIAMS IDENTITIES, PROJECTIONS, AND FORMALLY SELF-DUAL CODES 11 Jul 20131 Linear codes are considered over the ring Z 4 + uZ 4 , a non-chain extension of Z 4 . Lee weights, Gray maps for these codes are defined and MacWilliams identities for the complete, symmetrized and Lee weight enumerators are proved. Two projections from Z 4 + uZ 4 to the rings Z 4 and F 2 + uF 2 are considered and self-dual codes over Z 4 +uZ 4 are studied in connection with these projections. Finally three constructions are given for formally self-dual codes over Z 4 + uZ 4 and their Z 4 -images together with some good examples of formally self-dual Z 4 -codes obtained through these constructions. Introduction Codes over rings have been a common research topic in coding theory. Especially after the appearance of [5], a lot of research went towards studying codes over Z 4 . The rich algebraic structure of rings has allowed researchers get good results in coding theory. The rings studied have varied over the recent years, but codes over Z 4 remain a special topic of interest in coding theory because of their relation to lattices, designs, cryptography and their many applications. There is a vast literature on codes over Z 4 ; for some of the works done in this direction we refer to [3], [4], [7], [10], [11], etc. Recently, several families of rings have been introduced in coding theory, rings that are not finite chain but are Frobenius. These rings have a rich algebraic structure and they lead to binary codes with large automorphism groups and in some cases new binary self-dual codes( [13], [1], [8], [9]). F 2 + uF 2 is a size 4 ring that also has generated a lot of interest among coding theorists starting with [2]. There is an interesting connection between Z 4 and F 2 + uF 2 . Both are commutative rings of size 4, they are both finite-chain rings and they have both been studied quite extensively in relation to coding theory. Some of the main differences between these two rings are that their characteristic is not the same, F 2 is a subring of F 2 + uF 2 but not that of Z 4 and the Gray images of Z 4 -codes are usually not linear while the Gray images of F 2 + uF 2 -codes are linear. Inspired by this similarity(and difference) between the two rings, and our previous works on the non-chain size 16 ring F 2 + uF 2 + vF 2 + uvF 2 , we study codes over the ring Z 4 + uZ 4 in this work. It turns out that this ring is also a non-chain, commutative ring of size 16 but with characteristic 4. The ideal structure turns out to be very similar to that of F 2 + uF 2 + vF 2 + uvF 2 . We define linear Gray maps from Z 4 + uZ 4 to Z 2 4 , extend the Lee weight from Z 4 to Z 4 + uZ 4 and we define and prove the MacWilliams identities for all the weight enumerators involved. We also study self-dual codes over Z 4 + uZ 4 and the images of these codes under several maps. In particular, we introduce two projections, one from Z 4 + uZ 4 to Z 4 and the other from Z 4 + uZ 4 to F 2 + uF 2 . We study certain properties of the projections in terms of the minimum weights and self-duality. The last part of the paper is about three constructions for formally self-dual codes over Z 4 + uZ 4 whose Gray images are also formally self-dual over Z 4 . We tabulate some good formally self-dual codes over Z 4 obtained from formally self-dual codes over Z 4 + uZ 4 through two of the constructions. It has a total of 6 ideals given by (2.1) Linear I 0 = {0} ⊆ I 2u = 2u(Z 4 + uZ 4 ) = {0, 2u} ⊆ I u , I 2 , I 2+u ⊆ I 2,u ⊆ I 1 = Z 4 + uZ 4 where I u = u(Z 4 + uZ 4 ) = {0, u, 2u, 3u}, I 2 = 2(Z 4 + uZ 4 ) = {0, 2, 2u, 2 + 2u}, I 2+u = (2 + u)(Z 4 + uZ 4 ) = {0, 2 + u, 2u, 2 + 3u} I 2,u = {0, 2, u, 2u, 3u, 2 + u, 2 + 2u, 2 + 3u}. It is clear that Z 4 +uZ 4 is a local ring with I 2,u as its maximal ideal. The residue field is given by (Z 4 + uZ 4 )/I 2,u ≃ F 2 . Since Ann(I 2,u ) = {0, 2u}, and this has dimension 1 over the residue field, we have from [12] that Theorem 2.1. Z 4 + uZ 4 is a local Frobenius ring. The ideal 2, u is not principal and the ideals 2 and u are not related via inclusion, which means that Z 4 + uZ 4 is not a finite chain ring or a principal ideal ring. We divide the units of Z 4 + uZ 4 into two subsets U 1 and U 2 calling them units of first type and second type, respectively, as follows: (2.2) U 1 = {1, 3, 1 + 2u, 3 + 2u} and (2.3) U 2 = {1 + u, 3 + u, 1 + 3u, 3 + 3u}. The reason that we distinguish between the units is the following observation that can easily be verified: (2.4) ∀a ∈ Z 4 + uZ 4 , a 2 =    0 if a is a non-unit 1 if a ∈ U 1 1 + 2u if a ∈ U 2 . 2.2. Linear Codes over Z 4 + uZ 4 , the Lee weight and the Gray map. Definition 2.2. A linear code C of length n over the ring Z 4 + uZ 4 is a Z 4 + uZ 4submodule of (Z 4 + uZ 4 ) n . Since Z 4 + uZ 4 is not a finite chain ring, we cannot define a standard generating matrix for linear codes over Z 4 + uZ 4 . In the case of F 2 + uF 2 , the Lee weight was defined in [2] as w(0) = 0, w(1) = w(1 + u) = 1, w(u) = 2, and accordingly a Gray map from (F 2 + uF 2 ) n to F 2n 2 was defined by sending a + ub to (b, a + b) with a, b ∈ F n 2 . We will adopt a similar technique here. We will generalize the Gray map and define the weight in such a way that will give us a distance preserving isometry. Define φ : (Z 4 + uZ 4 ) n → Z 2n 4 by (2.5) φ(a + ub) = (b, a + b), a, b ∈ Z n 4 . We now define the Lee weight w L on Z 4 + uZ 4 by letting w L (a + ub) = w L ((b, a + b)), where w L ((b, a + b)) describes the usual Lee weight on Z 2 4 . The Lee distance is defined accordingly. Note that with this definition of the Lee weight and the Gray map we have the following main theorem: Theorem 2.3. φ : (Z 4 + uZ 4 ) n → Z 2n 4 is a distance preserving linear isometry. Thus, if C is a linear code over Z 4 + uZ 4 of length n, then φ(C) is a linear code over Z 4 of length 2n and the two codes have the same Lee weight enumerators. Throughout the paper we will use the notation w L to denote the Lee weight on Z 4 + uZ 4 , as well as Z 4 and F 2 + uF 2 . The Dual, The Complete Weight Enumerator and MacWilliams Identities 3.1. The Dual of linear codes over Z 4 + uZ 4 . First take the Euclidean inner product on (Z 4 + uZ 4 ) n by taking (3.1) (x 1 , x 2 , . . . , x n ), (y 1 , y 2 , . . . , y n ) = x 1 y 1 + x 2 y 2 + · · · + x n y n where the operations are performed in the ring Z 4 + uZ 4 . We are now ready to define the dual of a linear code C over Z 4 + uZ 4 : Definition 3.1. Let C be a linear code over Z 4 + uZ 4 of length n, then we define the dual of C as C ⊥ := {y ∈ (Z 4 + uZ 4 ) n | y, x = 0, ∀x ∈ C}. Note that from the definition of the inner product, it is obvious that C ⊥ is also a linear code over Z 4 + uZ 4 of length n. Since Z 4 + uZ 4 is a frobenius ring we also have |C| · |C ⊥ | = 16 n . We next study the MacWilliams identities for codes over Z 4 + uZ 4 : The Complete Weight Enumerator and MacWilliams Identities. Let Z 4 + uZ 4 = {g 1 , g 2 , . . . , g 16 } be given as Z 4 +uZ 4 = {0, u, 2u, 3u, 1, 1+u, 1+2u, 1+3u, 2, 2+u, 2+2u, 2+3u, 3, 3+u, 3+2u, 3+3u}. Definition 3.2. The complete weight enumerator of a linear code C over Z 4 + uZ 4 is defined as cwe C (X 1 , X 2 , . . . , X 16 ) = c∈C (X ng 1 (c) 1 X ng 2 (c) 2 . . . X ng 16 (c) 16 ) where n gi (c) is the number of appearances of g i in the vector c. Remark 3.3. Note that cwe C (X 1 , X 2 , . . . , X 16 ) is a homogeneous polynomial in 16 variables with the total degree of each monomial being n, the length of the code. Since 0 ∈ C, we see that the term X n 1 always appears in cwe C (X 1 , X 2 , . . . , X 16 ). We also observe that (3.2) cwe C (1, 1, . . . , 1) = |C|, and (3.3) cwe C (a, 0, . . . , 0) = a n . The complete weight enumerator gives us a lot of information about the code. Now, since Z 4 + uZ 4 is a Frobenius ring, the MacWilliams identities for the complete weight enumerator hold. To find the exact identities we define the following character on Z 4 + uZ 4 : Definition 3.4. Define χ : Z 4 + uZ 4 → C × by χ(a + bu) = i a+b . It is easy to verify that φ is a non-trivial character when restricted to each non-zero ideal, hence it is a generating character for Z 4 + uZ 4 . Then we make up the 16 × 16 matrix T , by letting T (i, j) = χ(g i g j ). The matrix T is given as follows: T =                             1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i i i i −1 −1 −1 −1 −i −i −i −i 1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 −i −i −i −i −1 −1 −1 −1 i i i i 1 i −1 −i i −1 −i 1 −1 −i 1 i −i 1 i −1 1 i −1 −i −1 −i 1 i 1 i −1 −i −1 −i 1 i 1 i −1 −i −i 1 i −1 −1 −i 1 i i −1 i 1 1 i −1 −i 1 i −1 −i 1 i −1 −i 1 i −1 i 1 −1 1 −1 −1 1 −1 1 1 −1 1 −1 −1 1 −1 1 1 −1 1 −1 −i i −i i −1 1 −1 1 i −i i −i 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 i −i i −i −1 1 −1 1 −i i −i i 1 −i −1 i −i −1 i 1 −1 i 1 −i i 1 −i −1 1 −i −1 i 1 −i −1 i 1 −i −1 i 1 −i −1 i 1 −i −1 i i 1 i −1 −1 i 1 −i −i −1 i 1 1 −i −1 i −1 i 1 −i 1 −i −1 i −1 i 1 −i                             . The following theorem then follows from [12] quite easily: Theorem 3.5. Let C be a linear code over Z 4 + uZ 4 of length n and suppose C ⊥ is its dual. Then we have cwe C ⊥ (X 1 , X 2 , . . . , X 16 ) = 1 |C| cwe C (T · (X 1 , X 2 , . . . , X 16 ) t ), where () t denotes the transpose. 3.3. The Symmetrized Weight enumerator and The Lee Weight enumerator. Since in Z 4 , w L (1) = w L (3) = 1, the symmetrized weight enumerator for codes over Z 4 was defined as swe C (X, Y, Z) = cwe C (X, Y, Z, Y ). Adopting the same idea, we will define the symmetrized weight enumerator of codes over Z 4 + uZ 4 . To do this we need the following table which gives us the elements of Z 4 + uZ 4 , their Lee weights and the corresponding variables: a Lee Weight of a The corresponding variable 0 0 X 1 u 2 X 2 2u 4 X 3 3u 2 X 4 1 1 X 5 1 + u 3 X 6 1 + 2u 3 X 7 1 + 3u 1 X 8 2 2 X 9 2 + u 2 X 10 2 + 2u 2 X 11 2 + 3u 2 X 12 3 1 X 13 3 + u 1 X 14 3 + 2u 3 X 15 3 + 3u 3 X 16 So, looking at the elements that have the same weights we can define the symmetrized weight enumerator as follows: Definition 3.6. Let C be a linear code over Z 4 + uZ 4 of length n. Then define the symmetrized weight enumerator of C as Here X represents the elements that have weight 0 (the 0 element); Y represents the elements with weight 4 (the element 2u); Z represents the elements of weight 3 (the elements 1 + u, 1 + 2u, 3 + 2u and 3 + 3u); W represents the elements of weight 1 (the elements 1, 1 + 3u, 3 and 3 + u) and finally S represents the elements of weight 2 (the elements 2, u, 3u, 2 + u, 2 + 2u and 2 + 3u). Now, combining Theorem 3.5 and the definition of the symmetrized weight enumerator, we obtain the following theorem: Theorem 3.7. Let C be a linear code over Z 4 + uZ 4 of length n and let C ⊥ be its dual. Then we have swe C ⊥ (X, Y, Z, W, S) = 1 |C| swe C (6S+4W +X+Y +4Z, 6S−4W +X+Y −4Z, −2W +X−Y +2Z, 2W +X−Y −2Z, −2S+X+Y ). We next define the Lee weight enumerator of a code over Z 4 + uZ 4 : Definition 3.8. Let C be a linear code over Z 4 . Then the Lee weight enumerator of C is given by (3.5) Lee C (W, X) = c∈C W 4n−wL(c) X wL(c) . Considering the weights that the variables X, Y, Z, W, S of the symmetrized weight enumerator represent, we easily get the following theorem: Theorem 3.9. Let C be a linear code over Z 4 + uZ 4 of length n. Then Lee C (W, X) = swe C (W 4 , X 4 , W X 3 , W 3 X, W 2 X 2 ). Now combining Theorem 3.7 and Theorem 3.9 we obtain the following theorem: Theorem 3.10. Let C be a linear code over Z 4 + uZ 4 of length n and C ⊥ be its dual. With Lee C (W, X) denoting its Lee weight enumerator as was given in (3.5), we have Lee C ⊥ (W, X) = 1 |C| Lee C (W + X, W − X). Proof. Looking at Theorem 3.7 and Theorem 3.9, the proof is complete after observing the following identities: 6W 2 X 2 + 4W 3 X + W 4 + X 4 + 4W X 3 = (W + X) 4 , 6W 2 X 2 − 4W 3 X + W 4 + X 4 − 4W X 3 = (W − X) 4 , −2W 3 X + W 4 − X 4 + 2W X 3 = (W + X)(W − X) 3 , 2W 3 X + W 4 − X 4 − 2W X 3 = (W + X) 3 (W − X), and finally −2W 2 X 2 + W 4 + X 4 = (W + X) 2 (W − X) 2 . 4. Self-dual Codes over Z 4 + uZ 4 , Projections, lifts and the Z 4 -images We start by recalling that a linear code C over Z 4 + uZ 4 is called self-orthogonal if C ⊆ C ⊥ and it will be called self-dual if C = C ⊥ . Since the code of length 1 generated by u is a self-dual code over Z 4 + uZ 4 , by taking the direct sums, we see that Theorem 4.1. Self-dual codes over Z 4 + uZ 4 of any length exist. We next observe: Theorem 4.2. (i) If C is self-orthogonal, then for every codeword c ∈ C, n Ui (c) must be even. Here, n Ui (c) denotes the number of units of the ith type(in U i ) that appear in c (ii) If C is self-dual of length n, then the all 2u-vector of length n must be in C. Proof. (i) If C is self-orthogonal, then c, c = 0 for all c ∈ C. But by 2.4 we see that c, c = n U1 (c) + n U2 (c) + n U2 (c) · 2u = 0 in Z 4 + uZ 4 implies that n U2 (c) must be even and since n U1 (c) + n U2 (c) must also be even, we see that n U1 (c) is also even. (ii) If C is self-dual, then again c, c = 0 for all c ∈ C and so by the above equation, the number of units in c, which we denote by n U (c) must be even. From the ring structure we see that unit · (2u) = 2u and (non − unit) · (2u) = 0, thus if we denote the all 2u-vector of length n by 2u, then c, 2u = n U (c) · (2u) = 0 in Z 4 + uZ 4 since n U (c) is even. This shows that 2u ∈ C ⊥ = C since C is selfdual. Define two maps from (Z 4 + uZ 4 ) n to Z n 4 as follows: (4.1) µ(a + ub) = a and (4.2) ν(a + ub) = b. Note that µ is a projection of Z 4 + uZ 4 to Z 4 . We can define another projection by defining α : Z 4 + uZ 4 → F 2 + uF 2 by reducing elements of Z 4 + uZ 4 modulo 2. The map α can be extended linearly like µ. Any linear code over Z 4 + uZ 4 has two projections defined in this way. Since these maps are linear, we see that The following theorem describes the images of self-dual codes over Z 4 + uZ 4 : Theorem 4.4. Let C be a self-dual code over Z 4 + uZ 4 of length n. Then a) φ(C) is a formally self-dual code over Z 4 of length 2n. b) µ(C) is a self-orthogonal code over Z 4 of length n and α(C) is self orthogonal over F 2 + uF 2 . c) If ν(C) is self-orthogonal, then φ(C) is a self-dual code of length 2n. Proof. a) We know that φ(C) is a linear code over Z 4 of length 2n. Since C is self-dual, we know by the MacWilliams identity that was proved in section 3, that the weight enumerator of C is invariant under the MacWilliams transform. The result now follows because φ is weight-preserving. b) Suppose a 1 and a 2 are in µ(C). This means that there exist b 1 , b 2 in Z n 4 such that a 1 + ub 1 , a 2 + ub 2 ∈ C. However since C is self-dual we must have a 1 + ub 1 , a 2 + ub 2 = 0 which means a 1 · a 2 + u(a 1 · b 2 + a 2 · b 1 ) = 0 from which a 1 · a 2 = 0 follows. Here, · stands for the euclidean dot product in Z 4 . A similar proof can be done for α as well. c) Now since C is self-dual, for all a 1 + ub 1 , a 2 + ub 2 ∈ C, we have a 1 + ub 1 , a 2 + ub 2 = a 1 · a 2 + u(a 1 · b 2 + a 2 · b 1 ) = 0, from which it follows that (4.3) a 1 · a 2 = a 1 · b 2 + a 2 · b 1 = 0. But by the hypothesis, we also have b 1 · b 2 = 0 for any such codewords. Combining these all, we get φ(a 1 + ub 1 ) · φ(a 2 + ub 2 ) = (b 1 , a 1 + b 1 ) · (b 2 , a 2 + b 2 ) = 2b 1 · b 2 + a 1 · a 2 + a 1 · b 2 + a 2 · b 1 = 0. This proves that φ(C) is self-orthogonal. But since φ is an isometry, both C and φ(C) have the same size, so |C| = |φ(C)| = 16 n/2 = 4 n , which proves that φ(C) is self-dual. Corollary 4.5. If C is a self-dual code over Z 4 + uZ 4 , generated by a matrix of the form [I n |A], then µ(C) and α(C) are self-dual over Z 4 and F 2 + uF 2 respectively. If µ(C) = D and α(C) = E, we say that C is a lift of D and E. One way of obtaining good codes over Z 4 + uZ 4 is to take the good ones over Z 4 and F 2 + uF 2 and take their lift over Z 4 + uZ 4 . The following theorem gives us a bound on how good the lift can be: Theorem 4.6. Let D be a non-zero linear code over Z 4 and E be a non-zero linear code over F 2 + uF 2 such that µ(C) = D and α(C) = E. Let d, d ′ , d ′′ denote the minimum Lee weights of C, D and E respectively. Then d ≤ 2d ′ and d ≤ 2d ′′ . Proof. Let x ∈ D be such that w L (x) = d ′ . Since µ(C) = D, ∃y ∈ Z n 4 such that x + uy ∈ D. Since C is linear, we have u(x + uy) = ux ∈ C. But by definition of the Lee weight on Z 4 + uZ 4 , we have w L (ux) = w L (x, x) = 2d ′ . Thus the inequality d ≤ 2d ′ is proved. The second inequality is proved in exactly the same way. As the theorem suggests, to construct good codes over Z 4 + uZ 4 by lifting from Z 4 and F 2 + uF 2 , we need to take codes over the projection rings that have high minimum distances. We illustrate this in the following example: Example Let D be the Z 4 -code generated by G ′ = [I 8 |A ′ ] and E be the F 2 +uF 2linear code generated by G ′′ = [I 8 |A ′′ ] where A ′ =                        , A ′′ =             u u 1 u 0 1 1 u u u u 1 u 0 1 1 1 u u u 1 u 0 1 1 1 u u u 1 u 0 0 1 1 u u u 1 u u 0 1 1 u u u 1 1 u 0 1 1 u u u u 1 u 0 1 1 u u             . D and E are both codes of length 16, size 4 8 and minimum Lee distance 8. We consider a common lift of D and E over Z 4 + uZ 4 to obtain C that is generated by G = [I 8 |A], where A =             u 2 + u 3 + 2u u 0 1 3 2 + u 2 + u u 2 + u 3 + 2u u 0 1 3 3 2 + u u 2 + u 3 + 2u u 0 1 1 3 2 + u u 2 + u 3 + 2u u 0 0 1 3 2 + u u 2 + u 3 + 2u u u 0 1 3 2 + u u 2 + u 3 + 2u 3 + 2u u 0 1 3 2 + u u 2 + u 2 + u 3 + 2u u 0 1 3 2 + u u             . C is a linear code over Z 4 + uZ 4 of length 16, size (16) 8 = 4 16 and minimum Lee distance 12. Taking the Gray image, we get φ(C) to be a formally self-dual Z 4 -code of length 32 and minimum Lee distance 12. 5. Three constructions for formally self-dual codes over Z 4 + uZ 4 Because of Theorem 3.10 we can easily show that the Gray image of formally self-dual codes over Z 4 + uZ 4 are formally self-dual over Z 4 as well. Some of the construction methods described in [6] for binary codes can be extended to Z 4 + uZ 4 as well. Let v be the i-th row of G and w be the j-th row of G ′ . Then v, w k = −A ji + A ij = 0 since A T = A. Therefore C ′ = C ⊥ and C is equivalent to C ′ = C ⊥ . Since w L (−a) = w L (a) for all a ∈ Z 4 + uZ 4 , this is a weight preserving equivalence. Theorem 5.2. Let M be a circulant matrix over Z 4 + uZ 4 of order n. Then the matrix [I n | M ] generates a formally self-dual code over Z 4 + uZ 4 . This is called the double circulant construction. Proof. Let C be the code generated by [I n | M ] = G and let C ′ be the code generated by [−M T | I n ] = G ′ . It can similarly be shown that C ′ = C ⊥ . Observe that C ′ is equivalent over Z 4 + uZ 4 to the code C ′′ generated by [M T |I n ] = G ′′ . So to show the equivalence of C and C ′ , we will show the equivalence of C and C ′′ . There is a permutation σ of rows such that after applying it to G ′′ , the first column of σ(M T ) is the same as the first column of M . Namely, (2) , · · · , M 1σ(n) ). Since the matrix M is circulant, every column of M is then equal to a column of σ(M T ). Then apply the necessary column permutation τ so that τ (σ(M T )) = M . We apply another column permutation ρ so that the matrix found by σ(I n ) is returned to identity. Note that τ does not affect this part. Thus we obtain G from G ′′ by the consecutive applications of the permutations σ, τ and ρ, which means that C is equivalent to C ′ = C ⊥ , and again as in the first theorem this is a weight preserving equivalence. Now, applying Theorem 2.3, we have the following corollary: Corollary 5.3. Let C be a linear code over Z 4 + uZ 4 generated by a matrix of the form [I n |A], where A is an n × n matrix. If A is symmetric or circulant, then C is formally self-dual and hence φ(C) is a formally self-dual code over Z 4 of length 4n. (M 11 , M 21 , · · · , M n1 ) = (M T σ(1)1 , M T σ(2)1 , · · · , M T σ(n)1 ) = (M 1σ(1) , M 1σ Theorem 5.4. Let M be a circulant matrix over Z 4 + uZ 4 of order n − 1. Then the matrix G =         α β β ... β γ I n γ M . . γ         , where α, β, γ ∈ Z 4 + uZ 4 such that γ = ±β, generates a formally self-dual code of length 2n over Z 4 + uZ 4 whose Gray image is a formally self-dual code over Z 4 of length 4n. This is called the bordered double circulant construction. Proof. Let G ′ =         −α −γ −γ ... −γ −β −β −M T I n . . −β         . It is easy to see that G ′ generates C ⊥ . By the same method as was done in the previous theorem the parts of G and G ′ except the β and γ can be made equuivalent. Multiplying all the columns except the I n by −1 we see that C ⊥ is equivalent to a code C * with generator matrix of the form G * =         α γ γ ... γ β I n β M . . β         . If γ = β, it is easy to see that C and C * will be the same codes. If γ = −β, then multiply all but the first row of G * by −1. The resulting matrix still generates C * . But since w L (a) = w L (−a) for all a ∈ Z 4 + uZ 4 , we see that C and C * will have the same weight enumerator. Hence in both cases we see that C and C ⊥ have the same weight enumerators. We finish with some examples of formally self-dual Z 4 -codes thus obtained: Formally Self-dual codes from double circulant constructions We present in the following, some good formally self-dual codes over Z 4 obtained from formally self-dual codes constructed over Z 4 + uZ 4 by the purely double circulant matrices. So the code C is generated over Z 4 + uZ 4 by a matrix of the form [I n |M ] where M is a circulant matrix, hence in the tables we will only give the first row of M . The length indicates the length of the code over Z 4 + uZ 4 , hence the length of the Z 4 -image is doubled. We will also indicate the minimum Lee distance of the codes: (0,0,1 + 2u,1 + 2u,1,1,3u,1 + u) 12 18 (0,0,1,1,1 + 2u,3 + 3u,2 + 2u,1 + u,2) 12 20 (0,0,1,3,1,3 + 2u,u,3 + 2u,u,2 + u) 14 22 (0,0,1,1,1,1,2,1,2 + 2u,1 + 3u,3 + 2u) 14 24 (0,0,1,1,1,1,0,1,0,2,2u,2 + 3u) 14 26 (0,0,1,1,1,1,0,3,1 + u,2u,3u,1 + 2u,3 + 2u) 15 Formally Self-dual codes from bordered double circulant constructions We present in the following, some good formally self-dual codes over Z 4 obtained from formally self-dual codes constructed over Z 4 +uZ 4 by bordered double circulant matrices as given in Theorem 5.4. So the code C is generated over Z 4 + uZ 4 by a matrix of the form given in Theorem 5.4.Hence in the tables we will give α, β, γ values as well as the first row of M . Again the length of the Z 4 -images is double the length indicated. d denotes the minimum Lee distance: (1 + 2u,1,2,1 + 3u,3) (u, 1 + 2u, 1 + 2u) 10 14 (0,0,u,u,2,3 + 2u) (3 + u, 1 + 2u, 1 + 2u) 10 16 (1,1,0,1,3 + u,3u,1 + 2u) (3 + 2u, 1, 1) 11 18 (0,0,0,0,2 + 2u,3u,1,3 + 2u) (3 + u, 3 + 2u, 3 + 2u) 12 20 (0,0,0,0,u,1 + 3u,1 + u,u,2 + 2u) (1 + u, 3 + 2u, 3 + 2u) 12 22 (0,0,0,0,2u,1 + u,3 + u,1 + 3u,2 + 2u,2 + 3u) (1 + u, 3 + 2u, 3 + 2u) 14 24 (0,0,0,0,0,1,u,2u,2 + 2u,2 + 3u,3) (1, 1 + 2u, 1 + 2u) 14 Remark 5.5. The first construction, namely the construction by symmetric matrices did not lead to good numerical results. That is why we did not tabulate codes from the first construction. Remark 5.6. In both the tables given above, an exhaustive search was made through all possible codes of the given form up to length 14, but for higher lengths the search was not exhaustive. For these higher lengths, an exhaustive search can be made using more powerful computational resources with a possibility of better minimum distances. ( 3 . 4 ) 34swe C (X, Y, Z, W, S) = cwe C (X, S, Y, S, W, Z, Z, W, S, S, S, S, W, W, Z, Z). Theorem 4. 3 . 3If C is a linear code over Z 4 + uZ 4 of length n, then µ(C), ν(C) are both linear codes over Z 4 of length n, while α(C) is a linear code over F 2 + uF 2 of length n. Theorem 5 . 1 . 51Let A be an n × n matrix over Z 4 + uZ 4 such that A T = A. Then the code generated by the matrix [I n | A] is a formally self-dual code of length 2n. Proof. Consider the matrix [−A T | I n ] = [−A | I n ] = G ′ and let G = [I n | A]. Both G and G ′ generate codes of size 16 n . Moreover the codes generated are equivalent over Z 4 + uZ 4 . So if C = G and C ′ = G ′ , all we need to do to finish the proof is to show that C ′ = C ⊥ . Codes over Z 4 + uZ 4 2.1. The Ring Z 4 + uZ 4 . Z 4 + uZ 4 is constructed as a commutative, characteristic 4 ring with u 2 = 0 and it is clear that Z 4 + uZ 4 ∼ = Z 4 [x]/(x 2 ).Its units are given by {1, 1 + u, 1 + 2u, 1 + 3u, 3, 3 + u, 3 + 2u, 3 + 3u}, while the non-units are {0, 2, u, 2u, 3u, 2 + u, 2 + 2u, 2 + 3u}. Table 1 : 1The Lee Weights of the elements of Z 4 + uZ 4 . Table 1 . 1Good f.s.d Z 4 -codes obtained from double circulant matrices over Z 4 + uZ 4Length First Row of M d 4 (2,1 + 2u) 4 6 (2,1,3u) 6 8 (3 + 3u,3u,2u,2 + 3u) 8 10 (1,0,2,3u,2 + u) 8 12 (0,2,3,2u,3,u) 10 14 (3 + 3u,3 + 3u,1 + 2u,1,2 + 2u,3,3) 11 16 Table 2 . 2Good f.s.d Z 4 -codes obtained from bordered double circulant matrices over Z 4 + uZ 4Length First Row of M (α, β, γ) d 4 (0) (0, 1 + 2u, 1 + 2u) 4 6 (2u,1) (3 + 3u, 1 + 3u, 1 + 3u) 6 8 (3 + 3u,3 + 2u,u) (2, 3 + 2u, 3 + 2u) 8 10 (0,0,1 + 2u,1) (3, 1 + 2u, 1 + 2u) 8 12 Codes over R k , Gray Maps and their Binary Images. S T Dougherty, B Yildiz, S Karadeniz, Finite Fields Appl. 17S.T.Dougherty, B.Yildiz and S.Karadeniz, Codes over R k , Gray Maps and their Binary Im- ages, Finite Fields Appl., 17, 205-219 (2011). Type II codes over F 2 + uF 2. S T Dougherty, P Gaborit, M Harada, P Solé, IEEE Trans. Inform. Theory. 45S.T. Dougherty, P. Gaborit, M. Harada and P. Solé, Type II codes over F 2 + uF 2 , IEEE Trans. Inform. Theory, 45, 32-45 (1999). On the Optimal Z 4 -codes of TypeII and length 16. I M Duursma, M Greferath, S E Schmidt, J. Combin. Theory, Series A. 92I.M. Duursma, M. Greferath and S. E. Schmidt, On the Optimal Z 4 -codes of TypeII and length 16, J. Combin. Theory, Series A, 92, 77-82 (2000). T A Gulliver, M Harada, Optimal Double Circulant Z 4 -codes. 2227T.A. Gulliver and M. Harada, Optimal Double Circulant Z 4 -codes, LNCS:AAAAECC, 2227, 122-128 (2001). The Z 4 -linearity of Kerdock, Preparata, Goethals and related codes. A R Hammons, V Kumar, A R Calderbank, N J A Sloane, P Solé, IEEE Trans. Inform. Theory. 40A.R. Hammons, V. Kumar, A.R. Calderbank, N.J.A. Sloane, and P. Solé, The Z 4 -linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory, 40, 301-319 (1994). W C Huffman, V Pless, Fundamentals of Error Correcting Codes. Cambridge: University PressW.C. Huffman and V. Pless, Fundamentals of Error Correcting Codes. Cambridge: University Press, 2003. Decompositions and extremal Type II codes over Z 4. W C Huffman, IEEE Trans. Inform. Theory. 44W.C. Huffman, Decompositions and extremal Type II codes over Z 4 , IEEE Trans. Inform. Theory, 44, 800-809 (1998). New extremal binary self-dual codes of length 68 from R2-lifts of binary self-dual codes. S Karadeniz, B Yildiz, Adv. Math. Commun. 7S. Karadeniz and B. Yildiz, New extremal binary self-dual codes of length 68 from R2-lifts of binary self-dual codes, Adv. Math. Commun., 7, 219-229 (2013). Double-Circulant and Double-Bordered-Circulant constructions for self-dual codes over R 2. S Karadeniz, B Yildiz, Adv. Math. Commun. 6S.Karadeniz and B.Yildiz, Double-Circulant and Double-Bordered-Circulant constructions for self-dual codes over R 2 , Adv. Math. Commun., 6, 193-202 (2012). Z X Wan, Series on Applied Mathematics:Quaternary Codes. World ScientificZ.X. Wan, Series on Applied Mathematics:Quaternary Codes, World Scientific, 1997. Negacyclic and Cyclic Codes over Z 4. J Wolfmann, IEEE Trans. Inform. Theory. 45J.Wolfmann, Negacyclic and Cyclic Codes over Z 4 , IEEE Trans. Inform. Theory, 45, 2527- 2532 (1999). Duality for modules over finite rings and applications to coding theory. J Wood, Amer. J. Math. 121J. Wood, Duality for modules over finite rings and applications to coding theory. Amer. J. Math., 121, 555-575 (1999). Linear codes over F 2 + uF 2 + vF 2 + uvF 2. B Yildiz, S Karadeniz, Des. Codes Crypt. 54B. Yildiz and S. Karadeniz, Linear codes over F 2 + uF 2 + vF 2 + uvF 2 , Des. Codes Crypt., 54, 61-81 (2010).
[]
[ "ANTI-INVARIANT HOLOMORPHIC STATISTICAL SUBMERSIONS", "ANTI-INVARIANT HOLOMORPHIC STATISTICAL SUBMERSIONS" ]
[ "Sema Kazan ", "Kazuhiko Takano " ]
[]
[]
Our purpose in this article is to study anti-invariant statistical submersions from holomorphic statistical manifolds. Firstly we introduce holomorphic statistical submersions satisfying the certain condition, after we give anti-invariant statistical submersions satisfying the certain condition. And we supported our results with examples.Date: 2022. 2020 Mathematics Subject Classification. 53B05, 53B12, 53C25.
10.1007/s00025-023-01904-8
[ "https://export.arxiv.org/pdf/2209.03627v1.pdf" ]
252,118,517
2209.03627
40ff579dfb18bba99af6f77acef24fd109cee059
ANTI-INVARIANT HOLOMORPHIC STATISTICAL SUBMERSIONS 8 Sep 2022 Sema Kazan Kazuhiko Takano ANTI-INVARIANT HOLOMORPHIC STATISTICAL SUBMERSIONS 8 Sep 2022 Our purpose in this article is to study anti-invariant statistical submersions from holomorphic statistical manifolds. Firstly we introduce holomorphic statistical submersions satisfying the certain condition, after we give anti-invariant statistical submersions satisfying the certain condition. And we supported our results with examples.Date: 2022. 2020 Mathematics Subject Classification. 53B05, 53B12, 53C25. Introduction In 1945, the theory of statistical manifolds has started with a paper of C.R. Rao [12]. It is known that the theory of statistical manifolds is called as information geometry. The information geometry, which is typically deals with the study of various geometric structures on a statistical manifold, has begun as a study of the geometric structures possessed by a statistical model of probability distributions. Nowadays, the information geometry has an important application area, such as, information theory, stochastic processes, dynamical systems and times series, statistical physics, quantum systems and the mathematical theory of neural networks [32]. Also, some applications of statistical manifolds in information geometry have been handled in many studies. In [35], the authors have presented an analytical computation of the asymptotic temporal behavior of the information geometric complexity of finite dimensional Gaussian statistical manifolds in the presence of microcorrelations (correlations between microvariables) and in [20], the author has presented an extension of the ergodic, mixing and Bernoulli levels of the ergodic hierarchy for statistical models on curved manifolds, making use of elements of the information geometry. The notion of dual connection (or conjugate connection) in affine geometry, has been first introduced into statistics by S. Amari [34] in 1985. A statistical model equipped with a Riemannian metric together with a pair of dual affine connections is called a statistical manifold. For more information about statistical manifolds and information geometry, we refer to [3], [16], [19], [30] [33], [37] and etc. Considering these notions, the differential geometry of statistical manifolds are being studying by geometers by adding different geometric structures to these manifolds. For instance, in [1] quaternionic Kähler-like statistical manifold have been studied and in [18], the authors have introduced the notion of Sasakian statistical structure and obtained the condition for a real hypersurface in a holomorphic statistical manifold to admit such a structure. In [2], the author has studied conformally-projectively flat trans-Sasakian statistical manifolds. Also, the authors have examined Sasakian statistical manifolds with semi-symmetric metric connection in [36]. Nowadays, some authors has studied statistical submersions. The notion of statistical submersion between statistical manifolds has introduced in 2001 by N. Abe and K. Hasegawa [31], the authors generalizing some basic results of B. O'Neill ( [5], [7]) concerning Riemannian submersions and geodesics. Later, K.Takano has introduced statistical manifolds with almost complex structures and its submersions [22] in 2004. Also, in [24], Takano has given examples of the statistical submersion and in [23] has studied statistical submersions of statistical manifolds with almost contact structures. Quaternionic Kähler-like statistical submersions has been given in [1]. In [14], G.E. Vilcu has studied para-Kähler-like statistical submersions. For other works see [4], [15]. Later such submersions have been considered between manifolds with differentiable structures. B. Watson defined almost Hermitian submersions between almost Hermitian manifolds and he showed that the base manifold and each fibre have the same kind of structure as the total space, in most cases [11]. And, many authors have studied on submersions, see [8], [9], [10], [13], [21] [25], [26], [27], [28]. In Sect.2, we introduce a brief introduction about statistical manifolds and we give the definition and example of the holomorphic statistical manifolds. In Sect.3, we investigate holomorphic statistical submersions satisfying the certain condition. We give an example of holomorphic statistical submersion. In Sect.4, we define the anti-invariant statistical submersion from holomorphic statistical manifolds and we study anti-invariant statistical submersions satisfying the certain conditions. We give an example and some results. Holomorphic Statistical Manifolds An m-dimensional semi-Riemannian manifold is a smooth manifold M m furnished with a metric g, where g is a symmetric nondegenerate tensor field on M of constant index. The common value ν of index g on M is called the index of M (0 ≤ ν ≤ m) and we denote a semi-Riemannian manifold by M m ν . If ν = 0, then M is a Riemannian manifold. The pair (∇, g) is called a statistical structure on M, if ∇ is torsion-free and for vector fields E, F, G on M (∇ E g)(F, G) = (∇ F g)(E, G) (2.1) holds. (2.1) is generally called Codazzi equation. The triple (M, ∇, g) is called a statistical manifold. For the statistical manifold (M, ∇, g), we define another affine connection ∇ * by Eg(F, G) = g(∇ E F, G) + g(F, ∇ * E G). (2.2) The affine connection ∇ * is called conjugate or dual of ∇ with respect to g. The affine connection ∇ * is torsion-free and satisfies (∇ * ) * = ∇. It is easy to see that∇ = 1 2 (∇ + ∇ * ) is a metric connection. The pair (∇, g) is a statistical structure on M if and only if so is (∇ * , g). Clearly, the triple (M, ∇ * , g) is statistical manifold. We denote by R and R * the curvature tensors on M with respect to the affine connection ∇ and its conjugate ∇ * , respectively. Then, we find g(R(E, F )G, H) = −g(G, R * (E, F )H), (2.3) where R(E, F )G = [∇ E , ∇ F ]G − ∇ [E,F ] G. We put S E F = ∇ E F − ∇ * E F. (2.4) Then S E F = S F E and g(S E F, G) = g(F, S E G) hold. An almost complex structure on M is a tensor field J of type (1, 1) such that J 2 = −I, where I stands for the identity transformation. An almost complex manifold is such a manifold with a fixed almost complex structure. An almost complex manifold is necessarily orientable and must have an even dimension. If J preserves the metric g, that is, g(JE, JF ) = g(E, F ), (2.5) then (M, g, J) is an almost Hermitian manifold. Moreover, if J is parallel with respect to the Levi-Civita connection∇, that is, (∇ E J)F = 0, (2.6) then (M, g, J) is called a Kählerian manifold [38]. Let (M, g, J) be a Kählerian manifold and ∇ an affine connection of M. We put ω(E, F ) = g(E, JF ) and (∇ E ω)(F, G) = Eω(F, G) − ω(∇ E F, G) − ω(F, ∇ E G). If (∇, g) is a statistical structure and ω is a ∇-parallel 2-form on M, then (M, ∇, g, J) is called a holomorphic statistical manifold [29]. It is known that the following result [17]: Lemma A. The following hold for a holomorphic statistical manifold (M, ∇, g, J): ∇ E (JF ) = J∇ * E F, (2.7) R(E, F )JG = JR * (E, F )G. (2.8) From (2.7), we find S E (JF ) = −J(S E F ). Example 1. Let R 4 2 be a smooth manifold with local coordinate system (x 1 , x 2 , x 3 , x 4 ), which admits the following almost complex structure J : J =     0 0 1 0 0 0 0 1 −1 0 0 0 0 −1 0 0     . The triple (R 4 2 , g, J) is an almost Hermitian manifold with g =     −1 0 0 0 0 e −x 2 0 0 0 0 −1 0 0 0 0 e −x 2     . We set ∇ ∂ 1 ∂ 1 = −∇ ∂ 3 ∂ 3 = −∂ 2 , ∇ ∂ 1 ∂ 2 = ∇ ∂ 2 ∂ 1 = −∇ ∂ 3 ∂ 4 = −∇ ∂ 4 ∂ 3 = e −x 2 ∂ 1 + e x 1 ∂ 4 , ∇ ∂ 1 ∂ 3 = ∇ ∂ 3 ∂ 1 = ∂ 4 , ∇ ∂ 1 ∂ 4 = ∇ ∂ 4 ∂ 1 = ∇ ∂ 2 ∂ 3 = ∇ ∂ 3 ∂ 2 = e x 1 ∂ 2 − e −x 2 ∂ 3 , ∇ ∂ 2 ∂ 2 = −∇ ∂ 4 ∂ 4 = −e x 1 −x 2 ∂ 3 , ∇ ∂ 2 ∂ 4 = ∇ ∂ 4 ∂ 2 = −e x 1 −x 2 ∂ 1 − ∂ 4, where ∂ i = ∂/∂x i (i = 1, 2, 3, 4). Then (R 4 2 , ∇, g, J) is a holomorphic statistical manifold. Holomorphic Statistical Submersions Let M and B be semi-Riemannian manifolds. A surjective mapping π : M → B is called a semi-Riemannian submersion if π has maximal rank and π * preserves lenghts of horizontal vectors. Let π : M → B be a semi-Riemannian submersion. We put dim M = m and dim B = n. For each point x ∈ B, semi-Riemannian submanifold π −1 (x) with the induced metric g is called a fiber and denoted by M x or M simply. We notice that the dimension of each fiber is always m − n (= s). A vector field on M is vertical if it is always tangent to fibers, horizontal if always orthogonal to fibers. We denote the vertical and horizontal subspace in the tangent space T p M of the total space M by V p (M) and We call a vector field X on M projectable if there exists a vector field X * on B such that π * (X p ) = X * π(p) for each p ∈ M, and say that X and X * are π-related. Also, a vector field X on M is called basic if it is projectable and horizontal. Then, we have ( [5], [6]) Lemma B. If X and Y are basic vector fields on M which are π-related to X * and Y * on B, then i) g(X, Y ) = g(X * , Y * )•π, where g is the metric on M and g the metric on B, ii) H[X, Y ] is basic and is π-related to [X * , Y * ]. Let (M, ∇, g) be a statistical manifold and π : M → B be a semi-Riemannian submersion. We denote the affine connections of M by ∇ and ∇ * . Notice that ∇ U V and ∇ * U V are welldefined vertical vector fields on M for vertical vector fields U and V on M, more precisely ∇ U V = V∇ U V and ∇ * U V = V∇ * U V . Moreover, ∇ and ∇ * are torsion-free and conjugate to each other with respect to g. Let (M, ∇, g) and π : M → B be a statistical manifold and a semi-Riemannian submersion, respectively. We call that π : (M, ∇, g) → (B, ∇, g) is a statistical submersion if π : M → B satisfies π * (∇ X Y ) p = ( ∇ X * Y * ) π(p) for basic vector fileds X, Y and p ∈ M. The letters U, V, W will always denote vertical vector fields, and X, Y, Z horizontal vector fields. The tensor fields T and A of type (1,2) defined by T E F = H∇ VE VF + V∇ VE HF, A E F = H∇ HE VF + V∇ HE HF for vector fields E and F on M . Changing ∇ to ∇ * in the above equations, we set T * and A * , respectively. Then we find (T * ) * = T and (A * ) * = A. For vertical vector fields, T and T * have the symmetry property. For X, Y ∈ H(M) and U, V ∈ V(M), we obtain g(T U V, X) = −g(V, T * U X), g(A X Y, U ) = −g(Y, A * X U ).i) HS V X = A X V − A * X V , ii) VS X V = T V X − T * V X, iii) (M, ∇, g) is a statistical manifold for each x ∈B, iv) (B, ∇, g) is a statistical manifold. For the statistical submersion π : (M, ∇, g) → (B, ∇, g), we have the following Lemmas ( [22]) Lemma D. If X and Y are horizontal vector fields, then A X Y = −A * Y X. Lemma E. For X, Y ∈ H(M) and U, V ∈ V(M) we have ∇ U V = T U V + ∇ U V, ∇ * U V = T * U V + ∇ * U V, ∇ U X = H∇ U X + T U X, ∇ * U X = H∇ * U X + T * U X, ∇ X U = A X U + V∇ X U, ∇ * X U = A * X U + V∇ * X U, ∇ X Y = H∇ X Y + A X Y, ∇ * X Y = H∇ * X Y + A * X Y. Furthermore, if X is basic, then H∇ U X = A X U and H∇ * U X = A * X U . We define the covariant derivatives ∇T and ∇A by (∇ E T ) F G = ∇ E (T F G) − T ∇ E F G − T F (∇ E G), (∇ E A) F G = ∇ E (A F G) − A ∇ E F G − A F (∇ E G) for E, F, G ∈ T M. We change ∇ to ∇ * , then the covariant derivatives ∇ * T, ∇ * A are defined simiraly. We consider the curvature tensor on the statistical submersion. Let R (resp. R * ) be the curvature tensor with respect to the induced affine connection ∇ (resp. ∇ * ) of each fiber. Also, let R(X, Y )Z (resp. R * (X, Y )Z) be horizontal vector field such that π * ( R(X, Y )Z) = R(π * X, π * Y )π * Z (resp. π * ( R * (X, Y )Z) = R * (π * X, π * Y )π * Z) at each p ∈ M, where R (resp. R * ) is the curvature tensor on B of the affine connection ∇ (resp. ∇ * ). Then we have ( [22]) Theorem F. If π: (M, ∇, g) → (B, ∇, g) is a statistical submersion, then we get for X, Y, Z, Z ′ ∈ H(M) and U, V, W, W ′ ∈ V(M) g(R(U, V )W, W ′ ) = g(R(U, V )W, W ′ ) + g(T U W, T * V W ′ ) − g(T V W, T * U W ′ ), g(R(U, V )W, X) = g((∇ U T ) V W, X) − g((∇ V T ) U W, X), g(R(U, V )X, W ) = g((∇ U T ) V X, W ) − g((∇ V T ) U X, W ), g(R(U, V )X, Y ) = g((∇ U A) X V, Y ) − g((∇ V A) X U, Y ) + g(T U X, T * V Y ) − g(T V X, T * U Y ) − g(A X U, A * Y V ) + g(A X V, A * Y U ), g(R(X, U )V, W ) = g([V∇ X , ∇ U ]V, W ) − g(∇ [X,U ] V, W ) − g(T U V, A * X W ) + g(T * U W, A X V ), g(R(X, U )V, Y ) = g((∇ X T ) U V, Y ) − g((∇ U A) X V, Y ) + g(A X U, A * Y V ) − g(T U X, T * V Y ), g(R(X, U )Y, V ) = g((∇ X T ) U Y, V ) − g((∇ U A) X Y, V ) + g(T U X, T V Y ) − g(A X U, A Y V ), g(R(X, U )Y, Z) = g((∇ X A) Y U, Z) − g(T U X, A * Y Z) − g(T U Y, A * X Z) + g(A X Y, T * U Z), g(R(X, Y )U, V ) = g([V∇ X , V∇ Y ]U, V ) − g(∇ [X,Y ] U, V ) + g(A X U, A * Y V ) − g(A Y U, A * X V ), g(R(X, Y )U, Z) = g((∇ X A) Y U, Z) − g((∇ Y A) X U, Z) + g(T * U Z, θ X Y ), g(R(X, Y )Z, U ) = g((∇ X A) Y Z, U ) − g((∇ Y A) X Z, U ) − g(T U Z, θ X Y ), g(R(X, Y )Z, Z ′ ) = g( R(X, Y )Z, Z ′ ) − g(A Y Z, A * X Z ′ ) + g(A X Z, A * Y Z ′ ) + g(θ X Y, A * Z Z ′ ), where we put θ X = A X + A * X . Remark G. We find V[X, Y ] =θ X Y. Let (M, ∇, g, J) be a holomorphic statistical manifold and (B, ∇, g) be a statistical manifold. The statistical submersion π : (M, ∇, g, J) → (B, ∇, g) is called a holomorphic statistical submersion. For X ∈ H(M) and U ∈ V(M) we put JX = P X + F X, JU = tU + f U,(3.2) where P X, tU ∈ H(M) and F X, f U ∈ V(M). From J 2 = −I, we get P 2 = −I − tF, F P + f F = 0, P t + tf = 0, f 2 = −I − F t. Because of g(JE, G) + g(E, JG) = 0 for E, G ∈ T M, we find g(P Y, Z) + g(Y, P Z) = 0, (3.3) g(F X, U ) + g(X, tU ) = 0, (3.4) g(f V, W ) + g(V, f W ) = 0. (3.5) Moreover, we obtain g((H∇ X P )Y, Z) + g(Y, (H∇ * X P )Z) = 0, g((H∇ U P )Y, Z) + g(Y, (H∇ * U P )Z) = 0, g((V∇ X f )V, W ) + g(V, (V∇ * X f )W ) = 0, g((∇ U f )V, W ) + g(V, (∇ * U f )W ) = 0. Hence we have Lemma 3.1. If π : (M, ∇, g, J) → (B, ∇, g) is a holomorphic statistical submersion, then we have i) H∇ X P = 0 (resp. H∇ U P = 0) is equivalent to H∇ * X P = 0 (resp. H∇ * U P = 0). ii) V∇ X f = 0 (resp. ∇ U f = 0) is equivalent to V∇ * X f = 0 (resp. ∇ * U f = 0). Using (2.7), we can get Lemma 3.2. Let π : (M, ∇, g, J) → (B, ∇, g) be a holomorphic statistical submersion. Then we have H∇ U (tV ) + T U (f V ) = P (T * U V ) + t(∇ * U V ), (3.6) T U (tV ) + ∇ U (f V ) = F (T * U V ) + f (∇ * U V ), (3.7) H∇ U (P X) + T U (F X) = P (H∇ * U X) + t(T * U X), (3.8) T U (P X) + ∇ U (F X) = F (H∇ * U X) + f (T * U X), (3.9) H∇ X (tU ) + A X (f U ) = P (A * X U ) + t(V∇ * X U ), (3.10) A X (tU ) + V∇ X (f U ) = F (A * X U ) + f (V∇ * X U ), (3.11) H∇ X (P Y ) + A X (F Y ) = P (H∇ * X Y ) + t(A * X Y ), (3.12) A X (P Y ) + V∇ X (F Y ) = F (H∇ * X Y ) + f (A * X Y ). (3.13) Furthermore, if X is basic, then H∇ * U X = A * X U .T * U V = −P {H∇ U (tV ) + T U (f V )} − t{T U (tV ) + ∇ U (f V )}, (3.14) ∇ * U V = −F {H∇ U (tV ) + T U (f V )} − f {T U (tV ) + ∇ U (f V )}, (3.15) H∇ * U X = −P {H∇ U (P X) + T U (F X)} − t{T U (P X) + ∇ U (F X)}, (3.16) T * U X = −F {H∇ U (P X) + T U (F X)} − f {T U (P X) + ∇ U (F X)}, (3.17) A * X U = −P {H∇ X (tU ) + A X (f U )} − t{A X (tU ) + V∇ X (f U )}, (3.18) V∇ * X U = −F {H∇ X (tU ) + A X (f U )} − f {A X (tU ) + V∇ X (f U )}, (3.19) H∇ * X Y = −P {H∇ X (P Y ) + A X (F Y )} − t{A X (P Y ) + V∇ X (F Y )}, (3.20) A * X Y = −F {H∇ X (P Y ) + A X (F Y )} − f {A X (P Y ) + V∇ X (F Y )}. (3.21) We put (∇ U f )V = ∇ U (f V ) − f (∇ U V ), (H∇ U P )X = H∇ U (P X) − P (H∇ U X), (V∇ X f )U = V∇ X (f U ) − f (V∇ X U ), (H∇ X P )Y = H∇ X (P Y ) − P (H∇ X Y ).(∇ U f )V = −f (V(S U V )) + F (T * U V ) − T U (tV ), (H∇ U P )X = −P (H(S U X)) + t(T * U X) − T U (F X), (V∇ X f )U = −f (V(S X U )) + F (A * X U ) − A X (tU ), (H∇ X P )Y = −P (H(S X Y )) + t(A * X Y ) − A X (F Y ).T U (tV ) + f (∇ U V ) = F (T * U V ) + f (∇ * U V ) if ∇ U f = 0, P (H∇ U X) + T U (F X) = P (H∇ * U X) + t(T * U X) if H∇ U P = 0, A X (tU ) + f (V∇ X U ) = F (A * X U ) + f (V∇ * X U ) if V∇ X f = 0, P (H∇ X Y ) + A X (F Y ) = P (H∇ * X Y ) + t(A * X Y ) if H∇ X P = 0. Now, we can give an example of the holomorphic statistical submersion: 1, 2). Considering the holomorphic statistical manifold (R 4 2 , ∇, g, J) given in Example1, we define a holomorphic statistical submersion π : (R 4 2 , ∇, g, J) → (R 2 1 , ∇, g) by Example 2. Let (R 2 1 , g) be a semi-Riemannian manifold with local coordinate system (x 1 , x 2 ), where g = −1 0 0 e −x 2 . If we put ∇ ∂ 1 * ∂ 1 * = −∂ 2 * , ∇ ∂ 1 * ∂ 2 * = ∇ ∂ 2 * ∂ 1 * = e −x 2 ∂ 1 * , ∇ ∂ 2 * ∂ 2 * = 0, then (R 2 1 , ∇, g) is a statistical manifold, where ∂ i * = ∂/∂x i (i =π(x 1 , x 2 , x 3 , x 4 ) = (x 1 , x 2 ). Moreover, for ∂ 1 , ∂ 2 ∈ Γ(H) and ∂ 3 , ∂ 4 ∈ Γ(V), we get T ∂ 3 ∂ 3 = ∂ 2 , ∇ ∂ 3 ∂ 3 = 0, T ∂ 3 ∂ 4 = T ∂ 4 ∂ 3 = −e −x 2 ∂ 1 , ∇ ∂ 3 ∂ 4 = ∇ ∂ 4 ∂ 3 = −e x 1 ∂ 4 , T ∂ 4 ∂ 4 = 0, ∇ ∂ 4 ∂ 4 = e x 1 −x 2 ∂ 3 , H∇ ∂ 3 ∂ 1 = 0, T ∂ 3 ∂ 1 = ∂ 4 , H∇ ∂ 3 ∂ 2 = H∇ ∂ 4 ∂ 1 = e x 1 ∂ 2 , T ∂ 3 ∂ 2 = T ∂ 4 ∂ 1 = −e −x 2 ∂ 3 , H∇ ∂ 4 ∂ 2 = −e x 1 −x 2 ∂ 1 , T ∂ 4 ∂ 2 = −∂ 4 , A ∂ 1 ∂ 3 = 0, V∇ ∂ 1 ∂ 3 = ∂ 4 , A ∂ 1 ∂ 4 = A ∂ 2 ∂ 3 = e x 1 ∂ 2 , V∇ ∂ 1 ∂ 4 = V∇ ∂ 2 ∂ 3 = −e −x 2 ∂ 3 , A ∂ 2 ∂ 4 = −e x 1 −x 2 ∂ 1 , V∇ ∂ 2 ∂ 4 = −∂ 4 , H∇ ∂ 1 ∂ 1 = −∂ 2 , A ∂ 1 ∂ 1 = 0, H∇ ∂ 1 ∂ 2 = H∇ ∂ 2 ∂ 1 = e −x 2 ∂ 1 , A ∂ 1 ∂ 2 = A ∂ 2 ∂ 1 = e x 1 ∂ 4 , H∇ ∂ 2 ∂ 2 = 0, A ∂ 2 ∂ 2 = −e x 1 −x 2 ∂ 3 . Let π : (M, ∇, g, J) → (B, ∇, g) be a holomorphic statistical submersion. We consider the curvature with respect to the affine connection ∇ of the total space satisfies R(E, F )G = c 4 {g(F, G)E − g(E, G)F + g(JF, G)JE − g(JE, G)JF + 2g(E, JF )JG} (3.22) for E, F, G ∈ T M, where c is a constant. From (2.8), we find g(R(U, V )W, W ′ ) + g(T U W, T * V W ′ ) − g(T V W, T * U W ′ ) (3.23) = c 4 {g(V, W )g(U, W ′ ) − g(U, W )g(V, W ′ ) + g(f V, W )g(f U, W ′ ) − g(f U, W )g(f V, W ′ ) + 2g(U, f V )g(f W, W ′ )}, g((∇ U T ) V W, X) − g((∇ V T ) U W, X) (3.24) = c 4 {g(f V, W )g(tU, X) − g(f U, W )g(tV, X) + 2g(U, f V )g(tW, X)}, g((∇ U T ) V X, W ) − g((∇ V T ) U X, W ) (3.25) = c 4 {g(tV, X)g(f U, W ) − g(tU, X)g(f V, W ) + 2g(U, f V )g(F X, W )}, g((∇ U A) X V, Y ) − g((∇ V A) X U, Y ) + g(T U X, T * V Y ) − g(T V X, T * U Y ) (3.26) − g(A X U, A * Y V ) + g(A X V, A * Y U ) = c 4 {g(tV, X)g(tU, Y ) − g(tU, X)g(tV, Y ) + 2g(U, f V )g(P X, Y )}, g([V∇ X , ∇ U ]V, W ) − g(∇ [X,U ] V, W ) − g(T U V, A * X W ) + g(A X V, T * U W ) = c 4 {g(f U, V )g(F X, W ) − g(F X, V )g(f U, W ) + 2g(X, tU )g(f V, W )}, g((∇ X T ) U V, Y ) − g((∇ U A) X V, Y ) + g(A X U, A * Y V ) − g(T U X, T * V Y ) (3.27) = c 4 {g(U, V )g(X, Y ) + g(f U, V )g(P X, Y ) − g(F X, V )g(tU, Y ) + 2g(X, tU )g(tV, Y )}, g((∇ X T ) U Y, V ) − g((∇ U A) X Y, V ) − g(A X U, A Y V ) + g(T U X, T V Y ) (3.28) = c 4 {−g(X, Y )g(U, V ) + g(tU, Y )g(F X, V ) − g(P X, Y )g(f U, V ) + 2g(X, tU )g(F Y, V )}, g((∇ X A) Y U, Z) − g(T U X, A * Y Z) − g(T U Y, A * X Z) + g(A X Y, T * U Z) (3.29) = c 4 {g(tU, Y )g(P X, Z) − g(P X, Y )g(tU, Z) + 2g(X, tU )g(P Y, Z)}, g([V∇ X , V∇ Y ]U, V ) − g(∇ [X,Y ] U, V ) + g(A X U, A * Y V ) − g(A Y U, A * X V ) (3.30) = c 4 {g(F Y, U )g(F X, V ) − g(F X, U )g(F Y, V ) + 2g(X, P Y )g(f U, V )}, g((∇ X A) Y U, Z) − g((∇ Y A) X U, Z) + g(T * U Z, θ X Y ) (3.31) = c 4 {g(F Y, U )g(P X, Z) − g(F X, U )g(P Y, Z) + 2g(X, P Y )g(tU, Z)}, g((∇ X A) Y Z, U ) − g((∇ Y A) X Z, U ) − g(T U Z, θ X Y ) (3.32) = c 4 {g(P Y, Z)g(F X, U ) − g(P X, Z)g(F Y, U ) + 2g(X, P Y )g(F Z, U )}, g( R(X, Y )Z, Z ′ ) − g(A Y Z, A * X Z ′ ) + g(A X Z, A * Y Z ′ ) + g(θ X Y, A * Z Z ′ ) (3.33) = c 4 {g(Y, Z)g(X, Z ′ ) − g(X, Z)g(Y, Z ′ ) + g(P Y, Z)g(P X, Z ′ ) − g(P X, Z)g(P Y, Z ′ ) + 2g(X, P Y )g(P Z, Z ′ )} for X, Y, Z, Z ′ ∈ H(M ) and U, V, W, W ′ ∈ V(M ), where θ X Y = A X Y + A * X Y . Let π : (M, ∇, g, J) → (B, ∇, g) be a holomorphic statistical submersion with isometric fiber, that is, T = 0. We get from (3.24) c{g(f V, W )g(tU, X) − g(f U, W )g(tV, X) + 2g(U, f V )g(tW, X)} = 0. Thus we find c = 0 or g(f V, W )g(tU, X) − g(f U, W )g(tV, X) + 2g(U, f V )g(tW, X) = 0. (3.34) Because of (3.34), we get g(F X, f V ) = 0 which yields that f F = 0 and tf = 0. Changing W (resp. V ) to f W (resp. f V ), equation (3.34) are g(f 2 V, W )g(tU, X) − g(f 2 U, W )g(tV, X) = 0, g(f 2 V, W )g(tU, X) + 2g(U, f 2 V )g(tW, X) = 0, respectively. Thus we obtain g(f 2 U, W )g(tV, X) = 0 which means that t = 0 or f 2 = 0. For each p ∈ M, we denote by {U 1 , ..., U s } local orthonomal bases of V p (M), where s =dimM. . Thus, we find P 2 = −I − tF, F P = 0, P t = 0 and F t = −I. We assume H∇ U P = 0. Then we get from Corollary 3.4 When f 2 = 0, we get f 2 = ε α g(f U α , f U α ) = − ε α g(f 2 U α , U α ) = 0, that is, f = 0. Hence we haveP (H(S U X)) − t(T * U X) + T U (F X) = 0. (4.1) If we operate F to (4.1), we obtain T * U X = −F (T U (F X)).T U X = −F (T * U (F X)), T * U X = −F (T U (F X)), T U V = −t(T * U (tV )), T * U V = −t(T U (tV )).t(T U X) = T * U (F X), t(T * U X) = T U (F X), P (T U V ) = 0, P (T * U V ) = 0, F (T U V ) = T * U (tV ) = T * V (tU ), F (T * U V ) = T U (tV ) = T V (tU ), T U (P X) = 0, T * U (P X) = 0, P (H∇ U X) = P (H∇ * U X). Using (2.2) and (2.6), we get g(∇ U V, X) = g(J(∇ U V ), JX) = g(∇ * U (JV ), JX) = U g(V, X) − g(tV, ∇ U (P X)) − g(tV, ∇ U (F X)) = g(V, F ((H∇ U P )X)) + g(V, F (T U (F X))). Then we find F ((H∇ U P )X) + T * U X + F (T U (F X)) = 0. Thus we find Lemma 4.3. Let π : (M, ∇, g, J) → (B, ∇, g) be an anti-invariant holomorphic statistical submersion. If V is a totally geodesic foliation on M, then H∇ X P = 0 holds. Next, if H∇ X P = 0, then we get from Corollary 3.4 P (H(S X Y )) − t(A * X Y ) + A X (F Y ) = 0. (4.3) Operating F to (4.3) A * X Y = F (A X (F Y )). (4.4) Hence we have from (A * ) * = A and (3.1) Lemma 4.4. π : (M, ∇, g, J) → (B, ∇, g) be an anti-invariant holomorphic statistical submersion. If H∇ X P = 0, then we get A X Y = −F (A * X (F Y )), A * X Y = −F (A X (F Y )), A X U = −t(A * X (tU )), A * X U = −t(A X (tU ) ). Corollary 4.5. Let π : (M, ∇, g, J) → (B, ∇, g) be an anti-invariant holomorphic statistical submersion. If H∇ X P = 0, then we find F (A X U ) = A * X (tU ), F (A * X U ) = A X (tU ), A X (P Y ) = 0, A * X (P Y ) = 0, P (A X U ) = 0, P (A * X U ) = 0, t(A X Y ) = A * X (F Y ), t(A * X Y ) = A X (F Y ), A P Y U = 0, A * P Y U = 0, P (H∇ X Y ) = P (H∇ * X Y ) . Owing to (3.15), (3.19) and (∇ * ) * = ∇, we get Lemma 4.6. Let π : (M, ∇, g, J) → (B, ∇, g) be an anti-invariant holomorphic statistical submersion. We have ∇ U V = −F (H∇ * U (tV )), ∇ * U V = −F (H∇ U (tV )) , moreover, if H∇ U P = 0, we get ∇ U (F X) = F (H∇ * U X) and ∇ * U (F X) = F (H∇ U X). Furthermore, if X is basic, then ∇ U (F X) = F (A * X U ) and ∇ * U (F X) = F (A X U ). Lemma 4.7. Let π : (M, ∇, g, J) → (B, ∇, g) be an anti-invariant holomorphic statistical submersion. We have V∇ X U = −F (H∇ * X (tU )), V∇ * X U = −F (H∇ X (tU )), moreover, if H∇ X P = 0, we get V∇ X (F Y ) = F (H∇ * X Y ) and V∇ * X (F Y ) = F (H∇ X Y ). The mean curvature vector of the affine connection ∇ is given by N = ε α T Uα U α . If π is an anti-invariant holomorphic statistical submersion with conformal fiber, that is, T U V = k g(U, V ), then we find k = N s , namely, T U V = 1 s g(U, V )N (4.5) which yields from (3.1) that T * U X = − 1 s g(N, X)U. (4.6) Changing X to tV in (4.6), we get from Corollary 4.2 T U V = 1 s g(N, tV )tU. (4.7) Because of (4.5) and (4.7), we find g(U, V )N = g(N, tV )tU which yields from P N = 0 that sN = ε α g(N, tU α )tU α = −tF N = N, that is, (s − 1)N = 0. Thus we have Proposition 4.8. Let π : (M, ∇, g, J) → (B, ∇, g) be an anti-invariant holomorphic statistical submersion with conformal fiber. If H∇ U P = 0, then we get i) the dimension of each fiber is one, or ii) π is an anti-invariant holomorphic statistical submersion with isometric fiber. Let π : (M, ∇, g, J) → (B, ∇, g) be an anti-invariant holomorphic statistical submersion which the curvature tensor with respect to the affine connection ∇ of the total space satisfies (3.22) and H∇ X P = 0. Changing Z to P Z in (3.29), we find c{g(tU, Y )g(P X, P Z) + 2g(tU, X)g(P Y, P Z)} = 0 which means that c = 0 or P = 0. Hence we have Theorem 4.9. Let π : (M, ∇, g, J) → (B, ∇, g) be an anti-invariant holomorphic statistical submersion. If the total space satisfies the condition (3.22) and H∇ X P = 0, then i) the total space is flat, or ii) P = 0. Next, we discuss a holomorphic statistical submersion such that P = 0. Then we find tF = −I, f F = 0, tf = 0 and f 2 = −I − F t. From Lemma 3.2, we can get Lemma 4.10. If π : (M, ∇, g, J) → (B, ∇, g) is a holomorphic statistical submersion satisfying P = 0, then we have Lemma 4.13. If π : (M, ∇, g, J) → (B, ∇, g) is a holomorphic statistical submersion satisfying P = 0 and V∇ X f = 0, then we have A X U = −t(A * X (tU )) = t(A tU X), A * X U = −t(A X (tU )) = t(A * tU X), A X (f U ) = 0, A * X (f U ) = 0, A X Y = −F (A * X (F Y )), A * X Y = −F (A X (F Y )), f (A X Y ) = 0, f (A * X Y ) = 0. Let π : (M, ∇, g, J) → (B, ∇, g) be a holomorphic statistical submersion satisfying (3.22). We assume ∇ U f = 0. Changing W to f W in (3.25), we obtain from Lemma 4.12 c{g(tV, X)g(f U, f W ) − g(tU, X)g(f V, f W )} = 0 which means that c = 0 or g(tV, X)g(f U, f W )−g(tU, X)g(f V, f W ) = 0. Thus we get || f || 2 g(F X, U ) = 0. From F = 0, we get f = 0. Hence we have Theorem 4.14. Let π : (M, ∇, g, J) → (B, ∇, g) be a holomorphic statistical submersion satisfying P = 0. If the total space satisfies the condition (3.22) and ∇ U f = 0, then i) the total space is flat, or ii) f = 0. Because of Theorems 4.9 and 4.14, we have Finally, we give an example of anti-invariant holomorphic statistical submersion. Example 3. Let π : (R 4 2 , ∇, g, J) → (R 2 1 , ∇, g) be a holomorphic statistical submersion given in Example2. Then π is an anti-invariant. H p (M) for each point p ∈ M, and the vertical and horizontal distributions in the tangent bundle T M of M by V(M) and H(M), respectively. Then T M is the direct sum of V(M) and H(M). The projection mappings are denoted V : T M → V(M) and H : T M → H(M) respectively. T (resp. A) vanishes identically if and only if T * (resp. A * ) vanishes identically. Since A is related to the integrability of H(M), if it is identically zero, then H(M) is integrable with respect to ∇. Moreover, if A and T vanish identically, then the total space is a locally product space of the base space and the fiber. It is known that ([31]) Theorem C. Let π : M → B be a semi-Riemannian submersion. Then (M, ∇, g) is a statistical manifold if and only if the following conditions hold: Corollary 3. 3 . 3Let π : (M, ∇, g, J) → (B, ∇, g) a holomorphic statistical submersion. Then we get From (3. 7 ) 7,(3.8),(3.11) and (3.12), we obtain Corollary 3. 4 . 4Let π : (M, ∇, g, J) → (B, ∇, g) a holomorphic statistical submersion. Then we get Corollary 3. 5 . 5Let π : (M, ∇, g, J) → (B, ∇, g) a holomorphic statistical submersion. Then we get Theorem 3. 6 . 6Let π : (M, ∇, g, J) → (B, ∇, g) be a holomorphic statistical submersion with isometric fiber. If the total space satisfies the condition (3.22), then i) the total space is flat, or ii) each fiber is an invariant submanifold of M satisfying the condition (3.22), or iii) each fiber is an anti-invariant submanifold of M which is of constant curvature c 4 . 4. Anti-Invariant Holomorphic Statistical Submersions The holomorphic statistical submersion π : (M, ∇, g, J) → (B, ∇, g) is called an antiinvariant holomorphic statistical submersion if M is an anti-invariant submanifold of M, namely, J(V(M)) ⊂ H(M) have from (T * ) * = T and (3.1) Lemma 4.1. Let π : (M, ∇, g, J) → (B, ∇, g) be an anti-invariant holomorphic statistical submersion. If H∇ U P = 0, then we get Corollary 4. 2 . 2Let π : (M, ∇, g, J) → (B, ∇, g) be an anti-invariant holomorphic statistical submersion. If H∇ U P = 0, then we find Theorem 4 . 15 . 415Let π : (M, ∇, g, J) → (B, ∇, g) be a holomorphic statistical submersion which the total space satisfies the condition (3.22) with non-zero constant c. If H∇ X P = 0 and ∇ U f = 0, then P = 0 is equivalent to f = 0. (4.8)(3.20)and (∇ * ) * = ∇, we find Lemma 4.11. 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[ "Thin 4 He films on alkali substrates: where do 3 He atoms bind?", "Thin 4 He films on alkali substrates: where do 3 He atoms bind?" ]
[ "Massimo Boninsegni " ]
[]
[]
The possible occurrence of bound states of 3 He atoms in the vicinity of a weakly attractive substrate coated with a thin superfluid 4 He film is investigated by first principle computer simulations. No evidence is seen of such bound states, even in the case of the weakest substrate, i.e., Cs; a single 3 He atom always binds to the free 4 He surface, regardless of the thickness of the 4 He film. A comparison of 4 He density profiles computed in this work with those yielded by the Density Functional approach that led to the prediction of 3 He bound states near the substrate, shows that the latter may not have afforded a sufficiently accurate structural description of the adsorbed 4 He film.The existence of a bound state of a 3 He atom at a free liquid 4 He surface was first proposed by Andreev [1], as an essentially ad hoc hypothesis to account for the observed behaviour of the surface tension of isotopic helium mixtures, significantly different from that of bulk 4 He [2]. The variational theory of Lekner [3] provided a qualitative and semi-quantitative microscopic explanation for the occurrence of such bond state, tying it to the monotonic decrease of the local 4 He density on approaching the surface.Subsequently, Pavloff and Treiner [4,5]proposed that a similar physical mechanism could also underlie a 3 He bound state at the interface of superfluid 4 He and a weakly attractive (e.g., alkali) substrate, on which no solid 4 He layer forms. Using a ground state density functional (DFT) approach, they computed the binding energy of a 3 He bound state at the interface between
10.1007/s10909-022-02914-6
[ "https://export.arxiv.org/pdf/2209.03412v1.pdf" ]
252,118,556
2209.03412
8e463c9be4871bf155193f2693280617a8032ae7
Thin 4 He films on alkali substrates: where do 3 He atoms bind? Massimo Boninsegni Thin 4 He films on alkali substrates: where do 3 He atoms bind? Received: date / Accepted: dateNoname manuscript No. (will be inserted by the editor)Superfluidity · Quantum Monte Carlo simulations · Helium mixtures · 3 He The possible occurrence of bound states of 3 He atoms in the vicinity of a weakly attractive substrate coated with a thin superfluid 4 He film is investigated by first principle computer simulations. No evidence is seen of such bound states, even in the case of the weakest substrate, i.e., Cs; a single 3 He atom always binds to the free 4 He surface, regardless of the thickness of the 4 He film. A comparison of 4 He density profiles computed in this work with those yielded by the Density Functional approach that led to the prediction of 3 He bound states near the substrate, shows that the latter may not have afforded a sufficiently accurate structural description of the adsorbed 4 He film.The existence of a bound state of a 3 He atom at a free liquid 4 He surface was first proposed by Andreev [1], as an essentially ad hoc hypothesis to account for the observed behaviour of the surface tension of isotopic helium mixtures, significantly different from that of bulk 4 He [2]. The variational theory of Lekner [3] provided a qualitative and semi-quantitative microscopic explanation for the occurrence of such bond state, tying it to the monotonic decrease of the local 4 He density on approaching the surface.Subsequently, Pavloff and Treiner [4,5]proposed that a similar physical mechanism could also underlie a 3 He bound state at the interface of superfluid 4 He and a weakly attractive (e.g., alkali) substrate, on which no solid 4 He layer forms. Using a ground state density functional (DFT) approach, they computed the binding energy of a 3 He bound state at the interface between superfluid 4 He and different alkali substrates, obtaining a value close to 4 K for all of them [5]. In general, the existence of this kind of a bound state of a 3 He atom, localized either at a free liquid 4 He surface, or at the interface between crystalline and superfluid phases, is an issue of interest not only for its relevance to different aspects of the phenomenology of superfluid or solid 4 He, but also as a pathway to the stabilization of a quasi-2D 3 He gas with novel, fascinating superfluid properties [6,7,8]. There is some experimental evidence of 3 He substrate bound states, but it seems fair to define it indirect, and not entirely conclusive [9,10,11,12,13,14,15]. Only recently, over three decades after the original study of Ref. [5], has an independent microscopic study been carried out [16], attempting to assess quantitatively the predictions, and possibly overcoming (some of) the limitations, of the calculation of Ref. [5], which is based on a heuristic density functional. The study of Ref. [16] made use of first principle, Quantum Monte Carlo (QMC) computer simulations and utilized the same mathematical model of Ref. [5], focusing however on the limit of a macroscopically thick 4 He film on a 4 He substrate. While that calculation confirmed qualitatively some of the DFT predictions, significant quantitative differences were reported. In particular, the 4 He density profile n(z) in the direction perpendicular to the substrate computed in Ref. [16] displays considerably more marked oscillations in the vicinity of the substrate, compared to that yielded by DFT. This can significantly affect the occurrence of a bound state of a 3 He atom near the substrate, as one can semi-quantitatively understand based on Lekner's approach, in which n(z) is related to the effective potential "seen" by a 3 He atom, then used in a single-particle Schrödinger equation [3,5]. In this paper, we extend the study of Ref. [16] and carry out a more extensive comparison of the results with those of Ref. [5]. Specifically, we consider weakly attractive (alkali) substrates, this time for thin (i.e., 10 layers) 4 He films. The main goal of this study is to investigate the binding of a single 3 He atom to a thin 4 He film wetting the substrate, and specifically whether the bound state(s) are primarily residing on the free 4 He surface, or in some cases closer to the substrate (obviously for films of sufficient thickness for which such a distinction is meaningful). Here too, we make use of essentially the same theoretical model adopted in Ref. [5] (see below for details). Our results yield no evidence of 3 He "substrate" states for thin 4 He films adsorbed on the alkali substrates considered here, namely Cs and Li, which are, respectively, the most weakly and strongly attractive; we have also carried out a few calculations for a Na substrate as well, in order to compare our results directly to those of Ref. [5]. In particular, we find that in the low temperature limit, for the substrates considered here (and, by extension, all alkali substrates), and all film thicknesses, the probability density of position of the 3 He atom is strongly peaked at the free 4 He surface. We find once again major quantitative differences between the computed 4 He density profiles and those reported in Ref. [5], suggesting that the scenario of substrate bound states may well be a spurious results, a consequence of a crude approxima-tions built into the original DFT approach. Subsequent versions thereof (see, for instance, Ref. [17]) might afford better quantitative agreement with first principle calculations [18], and it may be interesting to utilize them to reassess the prediction of the existence of substrate bound states of 3 He. This paper is organized as follows: in section 2 we describe the model of interest and briefly review the computational technique utilized; in section 3 we present our results, outlining our conclusions in section 4. Model and Methodology The model system simulated here comprises N He atoms, one of them being of the light isotope 3 He and all others of 4 He; all the atoms are assumed to be point-like. The system is enclosed in a parallelepipedal cell of sizes L × L × L z . The substrate is modeled as a flat surface (i.e., corrugation is neglected), located at z = 0; periodic boundary conditions are used in all three directions, but the length L z of the cell along z is taken long enough to make boundary conditions irrelevant in that direction. The nominal helium coverage is θ = N/L 2 . The interaction between two He atoms is described by the accepted Aziz pair potential [19], while that of the He atoms with the substrate is accounted for by means of a potential that only depends on the distance of the atom from the substrate. In most of our calculations, we made use of the atomsubstrate potential of Chizmeshya et al. (CCZ) [20], but we also performed a few calculations with the simpler, widely adopted "3-9" potential, in order to compare our results directly to those of Ref. [5], in which such a potential was used. The main difference between the two potentials is that, for a given alkali substrate, the CCZ model interaction has a significantly deeper attractive well, yielding values of wetting temperature and contact angle in closer agreement with experiment. The most important approximation contained in this model, one that is invariably made when investigating helium adsorption on weak substrates, is the neglect of substrate corrugation, which seems justified on account of the relatively large distance ( 4Å) at which the atoms on the first adsorbed layer sit, even for the most attractive alkali substrate, namely Li [18]. The QMC methodology adopted here is the canonical [21,22] continuousspace Worm Algorithm [23,24], a finite temperature (T ) quantum Monte Carlo (QMC) technique. Although we are clearly interested in low temperature (T → 0) physics, finite temperature methods have been repeatedly demonstrated to have several advantages over ground state ones, for investigating Bose systems (for an extensive discussion of this subject, see for instance Ref. [25]); in particular, they are unaffected by the bias of existing ground state methods (e.g., Diffusion Monte Carlo), arising from the use of a trial wave function, as well as from the control of a population of walkers [26,27], often leading to incorrect predictions (see, for instance, Refs. [28,29,30]). The results presented in this work were obtained by carrying out simulations at temperature T = 0.25 K, which we empirically found to be low enough that the physical estimates obtained at this temperature can be regarded as ground state estimates. Other details of the simulations carried out in this work are standard, and therefore the reader is referred to the original references. We used the fourthorder approximation for the short-time imaginary-time propagator (see, for instance, Ref. [31]), and observed convergence of all the physical estimates for a value of time step τ = 3.125 × 10 −3 K −1 . The quantities of interest in this work are mainly structural, namely the integrated 4 He density profile n(z) computed along the direction perpendicular to the substrate, as well as the probability density of position of the lone 3 He atom, used to assess the nature of the bound state. Most of the results shown here were obtained by simulating systems comprising up to 242 atoms (this for the largest coverage considered here, namely θ = 0.6Å −2 ). Experience accumulated in decades of computer simulations of thin 4 He films adsorbed on a variety of substrates, gives us reasonable confidence that the system size utilized here affords quantitatively reliable energetic and structural estimates. Results We now proceed with the illustration of our results, for convenience organized by substrate. Sodium. Fig. 1 shows the 4 He density profile n(z) computed by QMC for a helium film of coverage θ = 0.4Å −2 . For this particular case, we compare the results obtained with two different model potentials describing the interaction of helium atoms with the substrate, namely the CCZ and the 3-9, on which the calculation of Ref. [5] is based 1 . While there are noticeable differences between the 4 He density profiles computed with the two different potentials, chiefly higher peaks yielded by the CCZ (which is more attractive), such differences are negligible compared to the much wider discrepancy existing between the QMC and the DFT result (dashed line in Fig. 1, the data were read off Fig. 2 of Ref. [5]). In particular, besides the fact that the density peaks yielded by the DFT calculation are considerably less pronounced than those computed by QMC (the height of the main peak is lower by as much as ∼ 30 %), DFT predicts altogether a 4 He film whose average density is very low (shown for comparison in Fig. 1 is the equilibrium density of superfluid 4 He at T = 0, dotted line), extending some ∼ 5Å further away from the substrate, compared the QMC-computed n(z), with either choice of helium-substrate potential. Also shown in Fig. 1 is the probability density of position |Ψ(z)| 2 of the single 3 He atom, which is strongly peaked in correspondence of the free 4 He surface. What about substrate states? For the case of sodium, within DFT the 3 He substrate state was found to be rather sharply localized in the proximity of the substrate, at a slightly lower distances than the first adsorbed 4 He layer, and to have a binding energy comparable to that of the bound state located at the 4 He surface (Fig. 2 of Ref. [5]) (roughly close to 5 K). A substantial underestimation by the DFT approach of the local 4 He density, which the results shown in Fig. 1 clearly suggest, may drastically alter quantitatively the physical scenario of Ref. [5], mainly by increasing significantly the kinetic energy of localization associated to the putative substrate bound state. In any case, no evidence of any substrate bound state was found in our QMC study. Lithium. This is the most strongly attractive of the alkali substrates, one on which 4 He has been predicted to form a quasi-two-dimensional superfluid monolayer at low temperature [18]. Fig. 2 shows 4 He density profiles n(z) computed on a Li substrate, at T = 0.25 K, for four different coverages, from the T = 0 equilibrium density θ = 0.052Å −2 [32] to 0.600Å −2 , together with the probability density of position of a substitutional 3 He atom. At the equilibrium density, 4 He forms a (superfluid) quasi-2D monolayer on a Li substrate, and, as shown in Fig. 2, the single 3 He atom lies very nearly on the same plane of the 4 He atoms. As the coverage is increased, no solid layer forms, due to the weakness of the helium-substrate potential. Crystallization is preempted by promotion of atoms to the second layer, with the concurrent displacement of the 3 He atom to the surface of the film, where it remains as successive layers form, i.e., the 3 He atom only binds to the film near the substrate in the presence of a single 4 He layer. There is no evidence of a bound state of 3 He localized near the substrate for any film thickness exceeding one layer. The 4 He density profiles for the two largest coverages shown in Fig. 2 suggest that, in the limit of a thick film, n(z) fairly quickly stabilizes to a value close to the superfluid 4 He T = 0 equilibrium bulk value, displaying three welldefined oscillations in a substrate region of width ∼ 10Å. If we tentatively assume that the result shown in Fig. 2 for θ = 0.6Å −2 provides a quantitatively reliable representation of n(z) in the substrate region for the case of a thick film, then we may compare this result to that shown in Fig. 3 of Ref. [5]. In this case too, just like for the Na substrate, the original DFT study yielded considerably less marked density oscillations in the substrate region; for example, the height of the first (second) peak is underestimated by ∼ 10 (20)%. Cesium. On the weakest substrate, namely Cs, 4 He is not expected to form a thin film at low T , due to the shallowness of the attractive well of the heliumsubstrate potential [33]. Indeed, experimental evidence shows that θ ∼ 0.9Å −2 (roughly eleven layers) is the minimum coverage for which a 4 He film will form on a Cs substrate [34]; for lower coverages, the film breaks down into droplets, characterized by a well-defined contact angle with the surface [33,35,36,37]. For a thick 4 He film on Cs, substrate states for 3 He atoms were predicted to exist, in Ref. [5]. It was proposed in Ref. [38] that a small concentration of 3 He alters the above state of affairs, stabilizing thin films of helium on Cs at low T . This prediction was verified experimentally for helium mixtures with a 3 He concentration of a few percent [11,13]. Based on this prediction and its observation, we have carried out simulations of thin 4 He films on a Cs substrate, assuming a low 3 He concentration, sufficient to stabilize the film 2 . Fig. 3 shows the results of our computer simulations for a Cs substrate. 2 It should be noted that the methodology utilized in this work allows one to observe, in a computer simulation, a thin film of pure 4 He on a Cs substrate "bead up" and form a single droplet, at the temperature considered in this study. However, this requires that systems of sufficiently large size be studied. For N 200 and/or at low coverage, periodic boundary conditions stabilize a uniform thin film. There is little qualitative difference with respect to the results of Fig. 2, i.e., for a Li substrate. If anything, the tendency for the 3 He atom to be pushed to the surface appears accentuated, compared to what observed on a Li substrate. And in this case too, the comparison of the result for n(z) at the highest coverage considered, namely θ = 0.6Å −2 , with those of Ref. [5] for a thick film (Fig. 3 therein), points to a significant underestimation (as much as 30%) by the DFT calculation of the amplitude of the local density oscillations in the vicinity of the substrate. It is worth pointing out the the results for the four coverages shown in Fig. 2 and Fig. 3 are only representative; results for many other intermediate coverages are available, and none of them yield any evidence of a competition between the surface and the substrate, in terms of where the 3 He impurity may reside. Discussion and Conclusions First principle numerical studies of adsorption of 4 He films on alkali substrates at low temperature yield no evidence of the substrate bound states of 3 He atoms predicted almost thirty decades ago, by a DFT calculation. These calculations are based on the same microscopic Hamiltonian of which the original DFT calculations made use. Comparison of the 4 He density profiles in the direction perpendicular to the substrate shows that the DFT study quantitatively underestimated the local 4 He density oscillations in the vicinity of the substrate. As shown by Lekner's theory, the prediction of a possible 3 He bound state near the substrate hinges on a rather delicate energy balance, which heavily relies on an accurate representation of the local 4 He density near the substrate. To our knowledge no subsequent study of this problem has been carried out, based on DFT, following that of Ref. [5]; it seems conceivable that the original DFT prediction may have to be revised, possibly with the aid of an improved density functional. Indeed, much more satisfactory agreement between QMC and DFt was eported in subsequent works [18]. At the same time, it must also be noted that some experimental evidence that can be interpreted as supporting the existence of 3 He substrate states has been reported [11,13]. While that evidence may not be entirely conclusive, nonetheless it seems fair to conclude that the issue of the existence of substrate states is to some extent still open. Fig. 1 4 1He density profile n(z) in the direction perpendicular to a Na substrate, computed in this work, for a helium film of coverage θ = 0.4Å −2 at T = 0.25 K. Results are shown for two different choices of the potential describing the interaction of the helium atoms with the substrate, namely the CCZ and the 3-9 (see text). Dashed line shows the DFT result from Ref.[5]. Also shown (labeled 3 He), is the probability density of position |Ψ(z)| 2 of the single 3 He atom (right axis, arbitrary units). The dotted horizontal line shows the 4 He T = 0 equilibrium density n = 0.021837Å −3 . Fig. 2 4 2He density profile n(z) in the direction perpendicular to a Li substrate, computed in this work, for helium films of coverage θ from 0.052 to 0.600Å −2 at T = 0.25 K. Also shown (dashed lines), is the probability density of position |Ψ(z)| 2 of the single 3 He atom (right axis, arbitrary units). Horizontal lines show the 4 He T = 0 equilibrium density. Fig. 3 4 3He density profile n(z) in the direction perpendicular to a Cs substrate, computed in this work, for helium films of coverage θ from 0.058 to 0.600Å −2 at T = 0.25 K. Also shown (dashed lines), is the probability density of position |Ψ(z)| 2 of the single 3 He atom (right axis, arbitrary units). Horizontal line shows the 4 He T = 0 equilibrium density. The parameters for this particular potential are taken fromTable 1of Ref.[5]. They are different from those reported inTable 1of Ref.[20]. AcknowledgmentsThis work was supported by the Natural Sciences and Engineering Research Council of Canada. Computing support of ComputeCanada is acknowledged.Conflict of interestThe author declares no conflict of interest. Surface tension of weak helium isotope solutions. A F Andreev, Sov. Phys. 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F Mezzacapo, M Boninsegni, 10.1103/PhysRevLett.97.045301Phys. Rev. Lett. 97445301F. Mezzacapo, M. Boninsegni, Superfluidity and quantum melting of p-H 2 clusters, Phys. Rev. Lett. 97(4), 045301 (2006). DOI 10.1103/PhysRevLett.97.045301 Structure, superfluidity, and quantum melting of hydrogen clusters. F Mezzacapo, M Boninsegni, DOI10.1103/PhysRevA.75.033201Phys. Rev. A. 75333201F. Mezzacapo, M. Boninsegni, Structure, superfluidity, and quantum melting of hydro- gen clusters, Phys. Rev. A 75(3), 033201 (2007). DOI 10.1103/PhysRevA.75.033201 Worm Algorithm for Continuous-Space Path Integral Monte Carlo Simulations. M Boninsegni, N Prokof&apos;ev, B Svistunov, 10.1103/ PhysRevLett.96.070601Phys. Rev. Lett. 96770601M. Boninsegni, N. Prokof'ev, B. Svistunov, Worm Algorithm for Continuous-Space Path Integral Monte Carlo Simulations, Phys. Rev. Lett. 96(7), 070601 (2006). DOI 10.1103/ PhysRevLett.96.070601 Worm algorithm and diagrammatic Monte Carlo: A new approach to continuous-space path integral Monte Carlo simulations. M Boninsegni, N V Prokof&apos;ev, B V Svistunov, 10.1103/PhysRevE.74.036701Phys. Rev. E. 74336701M. Boninsegni, N.V. Prokof'ev, B.V. Svistunov, Worm algorithm and diagrammatic Monte Carlo: A new approach to continuous-space path integral Monte Carlo simula- tions, Phys. Rev. E 74(3), 036701 (2006). DOI 10.1103/PhysRevE.74.036701 Path integrals in the theory of condensed helium. D M Ceperley, 10.1103/RevModPhys.67.279Rev. Mod. Phys. 672279D.M. Ceperley, Path integrals in the theory of condensed helium, Rev. Mod. Phys. 67(2), 279 (1995). DOI 10.1103/RevModPhys.67.279 Population size bias in diffusion Monte Carlo. M Boninsegni, S Moroni, 10.1103/PhysRevE.86.056712Phys. Rev. E. 86556712M. Boninsegni, S. Moroni, Population size bias in diffusion Monte Carlo, Phys. Rev. E 86(5), 056712 (2012). DOI 10.1103/PhysRevE.86.056712 Phase Separation in Mixtures of Hard Core Bosons. M Boninsegni, 10.1103/PhysRevLett.87.087201Phys. Rev. Lett. 87887201M. Boninsegni, Phase Separation in Mixtures of Hard Core Bosons, Phys. Rev. Lett. 87(8), 087201 (2001). DOI 10.1103/PhysRevLett.87.087201 Ground state phase diagram of parahydrogen in one dimension. M Boninsegni, 10.1103/PhysRevLett.111.235303Phys. Rev. Lett. 11123235303M. Boninsegni, Ground state phase diagram of parahydrogen in one dimension, Phys. Rev. Lett. 111(23), 235303 (2013). DOI 10.1103/PhysRevLett.111.235303 Classical and quantum filaments in the ground state of trapped dipolar bose gases. F Cinti, M Boninsegni, 10.1103/PhysRevA.96.013627Phys. Rev. A. 96713627F. Cinti, M. Boninsegni, Classical and quantum filaments in the ground state of trapped dipolar bose gases, Phys. Rev. A 96(7), 013627 (2017). DOI 10.1103/PhysRevA.96. 013627 Absence of superfluidity in 2d dipolar bose striped crystals. F Cinti, M Boninsegni, 10.1007/s10909-019-02209-3J. Low Temp. Phys. 1965-6413F. Cinti, M. Boninsegni, Absence of superfluidity in 2d dipolar bose striped crystals, J. Low Temp. Phys. 196(5-6), 413 (2019). DOI 10.1007/s10909-019-02209-3 Permutation sampling in path integral Monte Carlo. M Boninsegni, 10.1007/s10909-005-7513-0J. Low Temp. Phys. 141127M. Boninsegni, Permutation sampling in path integral Monte Carlo, J. Low Temp. Phys. 141(1-2), 27 (2005). DOI 10.1007/s10909-005-7513-0 Structure and energetics of helium films on alkali substrates. M Boninsegni, L Szybisz, 10.1103/PhysRevB.70.024512Phys. Rev. B. 70224512M. Boninsegni, L. Szybisz, Structure and energetics of helium films on alkali substrates, Phys. Rev. B 70(2), 024512 (2004). DOI 10.1103/PhysRevB.70.024512 Helium prewetting and nonwetting on weak-binding substrates. E Cheng, M W Cole, W F Saam, J Treiner, DOI10.1103/physrevlett.67.1007Phys. Rev. Lett. 6781007E. Cheng, M.W. Cole, W.F. Saam, J. Treiner, Helium prewetting and nonwetting on weak-binding substrates, Phys. Rev. Lett. 67(8), 1007 (1991). DOI 10.1103/physrevlett. 67.1007 Anomalous wetting of helium on cesium. K S Ketola, S Wang, R B Hallock, DOI10.1103/physrevlett.68.201Phys. Rev. Lett. 682201K.S. Ketola, S. Wang, R.B. Hallock, Anomalous wetting of helium on cesium, Phys. Rev. Lett. 68(2), 201 (1992). DOI 10.1103/physrevlett.68.201 Contact angle of liquid 4 He on a cs surface. J Klier, P Stefanyi, A F G Wyatt, DOI10.1103/physrevlett.75.3709Phys. Rev. Lett. 75203709J. Klier, P. Stefanyi, A.F.G. Wyatt, Contact angle of liquid 4 He on a cs surface, Phys. Rev. Lett. 75(20), 3709 (1995). DOI 10.1103/physrevlett.75.3709 Optical measurement of contact angle of liquid helium on cesium. E Rolley, C Guthmann, DOI10.1007/bf02396813J. Low. Temp. Phys. 1081-21E. Rolley, C. Guthmann, Optical measurement of contact angle of liquid helium on cesium, J. Low. Temp. Phys. 108(1-2), 1 (1997). 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[]
[ "Early Release Science of the exoplanet WASP-39b with JWST NIRISS", "Early Release Science of the exoplanet WASP-39b with JWST NIRISS" ]
[ "Adina D Feinstein ", "✉ ", "Michael Radica ", "Luis Welbanks ", "Catriona Anne Murray ", "Kazumasa Ohno ", "Louis-Philippe Coulombe ", "Néstor Espinoza ", "Jacob L Bean ", "Johanna K Teske ", "Björn Benneke ", "Michael R Line ", "Zafar Rustamkulov ", "Arianna Saba ", "Angelos Tsiaras ", "Joanna K Barstow ", "Jonathan J Fortney ", "Peter Gao ", "Heather A Knutson ", "Ryan J Macdonald ", "Thomas Mikal-Evans ", "Benjamin V Rackham ", "Jake Taylor ", "Vivien Parmentier ", "Natalie M Batalha ", "Zachory K Berta-Thompson ", "Aarynn L Carter ", "Quentin Changeat ", "Leonardo A Dos Santos ", "Neale P Gibson ", "Jayesh M Goyal ", "Laura Kreidberg ", "Mercedes López-Morales ", "Joshua D Lothringer ", "Yamila Miguel ", "Karan Molaverdikhani ", "Sarah E Moran ", "Giuseppe Morello ", "Sagnick Mukherjee ", "David K Sing ", "Kevin B Stevenson ", "Hannah R Wakeford ", "Eva-Maria Ahrer ", "Munazza K Alam ", "Lili Alderson ", "Natalie H Allen ", "Natasha E Batalha ", "Taylor J Bell ", "Jasmina Blecic ", "Jonathan Brande ", "Claudio Caceres ", "S L Casewell ", "Katy L Chubb ", "Ian J M Crossfield ", "Nicolas Crouzet ", "Patricio E Cubillos ", "Leen Decin ", "Jean-Michel Désert ", "Joseph Harrington ", "Kevin Heng ", "Thomas Henning ", "Nicolas Iro ", "Eliza ", "M.-R Kempton ", "Sarah Kendrew ", "James Kirk ", "Jessica Krick ", "Pierre-Olivier Lagage ", "Monika Lendl ", "Luigi Mancini ", "Megan Mansfield ", "E M May ", "N J Mayne ", "Nikolay K Nikolov ", "Enric Palle ", "Dominique J M Petit Dit De La Roche ", "Caroline Piaulet ", "Diana Powell ", "Seth Redfield ", "Laura K Rogers ", "Michael T Roman ", "Pierre-Alexis Roy ", "Matthew C Nixon ", "Everett Schlawin ", "Xianyu Tan ", "P Tremblin ", "Jake D Turner ", "Olivia Venot ", "William C Waalkes ", "Peter J Wheatley ", "Xi Zhang " ]
[]
[ "670 | Nature |" ]
The Saturn-mass exoplanet WASP-39b has been the subject of extensive efforts to determine its atmospheric properties using transmission spectroscopy 1-4 . However, these efforts have been hampered by modelling degeneracies between composition and cloud properties that are caused by limited data quality 5-9 . Here we present the transmission spectrum of WASP-39b obtained using the Single-Object Slitless Spectroscopy (SOSS) mode of the Near Infrared Imager and Slitless Spectrograph (NIRISS) instrument on the JWST. This spectrum spans 0.6-2.8 μm in wavelength and shows several water-absorption bands, the potassium resonance doublet and signatures of clouds. The precision and broad wavelength coverage of NIRISS/SOSS allows us to break model degeneracies between cloud properties and the atmospheric composition of WASP-39b, favouring a heavy-element enhancement ('metallicity') of about 10-30 times the solar value, a sub-solar carbon-to-oxygen (C/O) ratio and a solar-to-super-solar potassium-to-oxygen (K/O) ratio. The observations are also best explained by wavelength-dependent, non-grey clouds with inhomogeneous coverageof the planet's terminator.We observed a transit of WASP-39 b using the NIRISS 10 on the JWST as part of the Transiting Exoplanet Community Early Release Science Program 11,12 . Our observations spanned 8.2 h starting on 26 July 2022 20:45 UTC, covering the 2.8-h transit as well as 3.0 h before and 2.4 h after the transit to establish a flux baseline. The data were taken in the SOSS mode, which simultaneously covers the wavelength range from 0.6 to 2.8 μm across two spectral orders on the same detector. Order 1 contains the spectral range between 0.6 and 2.8 μm at an average resolving power of R ≣ λ/Δλ = 700, whereas order 2 delivers the spectral range of 0.6-1.4 μm at an average resolving power of R = 1,400. In the SOSS mode, the spectra are spread across more than 20 pixels in the cross-dispersion direction by means of a cylindrical defocusing lens (see Extended DataFig. 1), thus allowing longer integration times and reducing the impact of pixel-level differences in the detector response. However, this defocus results in the physical overlap of both orders on the detector. The time-series observation was composed of 537 integrations of 49.4 s (nine groups per integration), corresponding to a duty cycle of 89%.We extracted the stellar spectra from the time-series observations using six different pipelines to test the impact of differences in spectral order tracing, 1/f noise correction, background removal and spectrum extraction methodology (see Methods and Extended DataFigs. 2 and 3). We created spectrophotometric light curves for each pipeline(Fig. 1)and summed the data to create white-light curves per spectral order
10.1038/s41586-022-05674-1
[ "https://export.arxiv.org/pdf/2211.10493v1.pdf" ]
253,734,322
2211.10493
813fdd3ca30e683ebeaa8d99a2d9ab536b405f4e
Early Release Science of the exoplanet WASP-39b with JWST NIRISS 23 February 2023 Adina D Feinstein ✉ Michael Radica Luis Welbanks Catriona Anne Murray Kazumasa Ohno Louis-Philippe Coulombe Néstor Espinoza Jacob L Bean Johanna K Teske Björn Benneke Michael R Line Zafar Rustamkulov Arianna Saba Angelos Tsiaras Joanna K Barstow Jonathan J Fortney Peter Gao Heather A Knutson Ryan J Macdonald Thomas Mikal-Evans Benjamin V Rackham Jake Taylor Vivien Parmentier Natalie M Batalha Zachory K Berta-Thompson Aarynn L Carter Quentin Changeat Leonardo A Dos Santos Neale P Gibson Jayesh M Goyal Laura Kreidberg Mercedes López-Morales Joshua D Lothringer Yamila Miguel Karan Molaverdikhani Sarah E Moran Giuseppe Morello Sagnick Mukherjee David K Sing Kevin B Stevenson Hannah R Wakeford Eva-Maria Ahrer Munazza K Alam Lili Alderson Natalie H Allen Natasha E Batalha Taylor J Bell Jasmina Blecic Jonathan Brande Claudio Caceres S L Casewell Katy L Chubb Ian J M Crossfield Nicolas Crouzet Patricio E Cubillos Leen Decin Jean-Michel Désert Joseph Harrington Kevin Heng Thomas Henning Nicolas Iro Eliza M.-R Kempton Sarah Kendrew James Kirk Jessica Krick Pierre-Olivier Lagage Monika Lendl Luigi Mancini Megan Mansfield E M May N J Mayne Nikolay K Nikolov Enric Palle Dominique J M Petit Dit De La Roche Caroline Piaulet Diana Powell Seth Redfield Laura K Rogers Michael T Roman Pierre-Alexis Roy Matthew C Nixon Everett Schlawin Xianyu Tan P Tremblin Jake D Turner Olivia Venot William C Waalkes Peter J Wheatley Xi Zhang Early Release Science of the exoplanet WASP-39b with JWST NIRISS 670 | Nature | 61423 February 202310.1038/s41586-022-05674-1Received: 2 November 2022 Accepted: 20 December 2022 Published online: 9 January 2023 Open access Check for updatesA list of affiliations appears at the end of the paper. The Saturn-mass exoplanet WASP-39b has been the subject of extensive efforts to determine its atmospheric properties using transmission spectroscopy 1-4 . However, these efforts have been hampered by modelling degeneracies between composition and cloud properties that are caused by limited data quality 5-9 . Here we present the transmission spectrum of WASP-39b obtained using the Single-Object Slitless Spectroscopy (SOSS) mode of the Near Infrared Imager and Slitless Spectrograph (NIRISS) instrument on the JWST. This spectrum spans 0.6-2.8 μm in wavelength and shows several water-absorption bands, the potassium resonance doublet and signatures of clouds. The precision and broad wavelength coverage of NIRISS/SOSS allows us to break model degeneracies between cloud properties and the atmospheric composition of WASP-39b, favouring a heavy-element enhancement ('metallicity') of about 10-30 times the solar value, a sub-solar carbon-to-oxygen (C/O) ratio and a solar-to-super-solar potassium-to-oxygen (K/O) ratio. The observations are also best explained by wavelength-dependent, non-grey clouds with inhomogeneous coverageof the planet's terminator.We observed a transit of WASP-39 b using the NIRISS 10 on the JWST as part of the Transiting Exoplanet Community Early Release Science Program 11,12 . Our observations spanned 8.2 h starting on 26 July 2022 20:45 UTC, covering the 2.8-h transit as well as 3.0 h before and 2.4 h after the transit to establish a flux baseline. The data were taken in the SOSS mode, which simultaneously covers the wavelength range from 0.6 to 2.8 μm across two spectral orders on the same detector. Order 1 contains the spectral range between 0.6 and 2.8 μm at an average resolving power of R ≣ λ/Δλ = 700, whereas order 2 delivers the spectral range of 0.6-1.4 μm at an average resolving power of R = 1,400. In the SOSS mode, the spectra are spread across more than 20 pixels in the cross-dispersion direction by means of a cylindrical defocusing lens (see Extended DataFig. 1), thus allowing longer integration times and reducing the impact of pixel-level differences in the detector response. However, this defocus results in the physical overlap of both orders on the detector. The time-series observation was composed of 537 integrations of 49.4 s (nine groups per integration), corresponding to a duty cycle of 89%.We extracted the stellar spectra from the time-series observations using six different pipelines to test the impact of differences in spectral order tracing, 1/f noise correction, background removal and spectrum extraction methodology (see Methods and Extended DataFigs. 2 and 3). We created spectrophotometric light curves for each pipeline(Fig. 1)and summed the data to create white-light curves per spectral order The Saturn-mass exoplanet WASP-39b has been the subject of extensive efforts to determine its atmospheric properties using transmission spectroscopy [1][2][3][4] . However, these efforts have been hampered by modelling degeneracies between composition and cloud properties that are caused by limited data quality [5][6][7][8][9] . Here we present the transmission spectrum of WASP-39b obtained using the Single-Object Slitless Spectroscopy (SOSS) mode of the Near Infrared Imager and Slitless Spectrograph (NIRISS) instrument on the JWST. This spectrum spans 0.6-2.8 μm in wavelength and shows several water-absorption bands, the potassium resonance doublet and signatures of clouds. The precision and broad wavelength coverage of NIRISS/SOSS allows us to break model degeneracies between cloud properties and the atmospheric composition of WASP-39b, favouring a heavy-element enhancement ('metallicity') of about 10-30 times the solar value, a sub-solar carbon-to-oxygen (C/O) ratio and a solar-to-super-solar potassium-to-oxygen (K/O) ratio. The observations are also best explained by wavelength-dependent, non-grey clouds with inhomogeneous coverageof the planet's terminator. We observed a transit of WASP-39 b using the NIRISS 10 on the JWST as part of the Transiting Exoplanet Community Early Release Science Program 11,12 . Our observations spanned 8.2 h starting on 26 July 2022 20:45 UTC, covering the 2.8-h transit as well as 3.0 h before and 2.4 h after the transit to establish a flux baseline. The data were taken in the SOSS mode, which simultaneously covers the wavelength range from 0.6 to 2.8 μm across two spectral orders on the same detector. Order 1 contains the spectral range between 0.6 and 2.8 μm at an average resolving power of R ≣ λ/Δλ = 700, whereas order 2 delivers the spectral range of 0.6-1.4 μm at an average resolving power of R = 1,400. In the SOSS mode, the spectra are spread across more than 20 pixels in the cross-dispersion direction by means of a cylindrical defocusing lens (see Extended Data Fig. 1), thus allowing longer integration times and reducing the impact of pixel-level differences in the detector response. However, this defocus results in the physical overlap of both orders on the detector. The time-series observation was composed of 537 integrations of 49.4 s (nine groups per integration), corresponding to a duty cycle of 89%. We extracted the stellar spectra from the time-series observations using six different pipelines to test the impact of differences in spectral order tracing, 1/f noise correction, background removal and spectrum extraction methodology (see Methods and Extended Data Figs. 2 and 3). We created spectrophotometric light curves for each pipeline (Fig. 1) and summed the data to create white-light curves per spectral order (Extended Data Fig. 4). The spectrophotometric and white-light curves are largely free of instrumental systematics except for a constant-rate linear trend in time and an exponential ramp effect within the first 15 min of the time series. The fitted transit depths were binned into 80 spectral wavelength changes in order 1 and 20 in order 2 to create transmission spectra at R ≈ 300. We present the spectra from the nirHiss, supreme-SPOON and transitspectroscopy reduction pipelines in Fig. 2. We find consistent results between the pipelines, with the derived spectra also being in agreement with previous Hubble Space Telescope (HST) observations (see also Extended Data Fig. 5). We investigated the atmospheric properties of WASP-39b by comparing our measured transmission spectrum from the nirHiss pipeline to grids of one-dimensional, radiative-convective-thermochemical equilibrium models. These models explore the impact of atmospheric Fig. 1 | Selection of systematics-corrected spectrophotometric light curves and residuals for the transit of WASP-39b observed with NIRISS/SOSS for orders 1 and 2. An exoplanet transit model (solid line) was fitted to each light curve with chromatic_fitting using a quadratic limb-darkening law. The limbdarkening coefficients, planet-to-star radius ratio (R p /R * ) and out-of-transit flux were varied in each wavelength channel, whereas all other parameters were fixed. The residuals to the best-fit models are shown for each light curve. The wavelength range for each channel is denoted in panel a, whereas parts-per-million (ppm) scatter in the residuals is denoted in panel b. We calculate the ppm as the standard deviation of the out-of-transit residuals. We quote the ratio of the predicted photon noise for each bin in brackets. The reductions are from the nirHiss and chromatic_fitting routines described in Methods. We define our errors as the 1σ uncertainties extracted from the stellar spectra. (https://github.com/afeinstein20/wasp39b_niriss_paper/blob/ main/scripts/figure1.py). Article metallicity (M/H), carbon-to-oxygen ratio (C/O), potassium-to-oxygen ratio (K/O), heat redistribution (f ) and cloud coverage on the transmission spectrum of the planet. We explored several cloud models ranging from parametric treatments 13,14 to a droplet sedimentation model 15 that calculates the vertical distributions of cloud mass mixing ratio and mean particle size from the balance between gravitational sedimentation and eddy diffusion of cloud particles. Using a Bayesian inference framework (see Methods), we compared these grids of models to the observations and inferred the ranges of M/H, C/O ratio, K/O ratio and f that best explain the data while marginalizing over different cloud treatments. WASP-39, the host star, has a metallicity equal to that of the Sun within measurement precision [16][17][18][19] , so we reference the atmospheric abundances of the planet to the solar pattern of elemental abundances 20 . We compared the grid spectra computed by various models (PICASO, ATMO, PHOENIX and ScCHIMERA) with an observational spectrum obtained from each data-reduction pipeline and obtained broadly consistent results on the inferred atmospheric properties. We report the results from the comparison between the nirHiss spectrum and ScCHIMERA grid that allows the most comprehensive treatments of cloud properties. Our best-fitting model to the NIRISS/SOSS transmission spectrum of WASP-39 b is presented in Fig. 3. The spectral maxima at 0.9, 1.15, 1.4 and 1.8 μm owing to water absorption result in a >30σ detection of the molecule (see Methods). Similarly, the potassium doublet at 0.768 μm is detected in the data at 6.8σ. Signatures of CO and/or CO 2 are identified because of their contribution to the spectrum past 2.3 μm. We find a 3.6σ significance model preference for CO and no significant preference for CO 2 (see Methods). From the chemical equilibrium models considered, we find that the observations are best explained by a sub-solar C/O ratio (see Fig. 4a). Across the different spectroscopic resolutions and atmospheric models, the best-fit C/O ratio is 0.2, which is the lowest ratio explored in the grid of models. We rule out super-solar C/O ratio because of the lack of CH 4 features at about 1.7 μm and about 2.3 μm, at which they would be expected for C/O ratio ≳ 0.7. Overall, solar-to-super-solar C/O ratios fail to explain the transmission spectrum at the shortest (≲1 μm) and longest (≳2 μm) wavelengths. Our best-fit C/O ratio is broadly consistent with the observations of WASP-39b with NIRCam (2.4-4.0 μm; ref. 21 ), NIRSpec G395H (3-5 μm; ref. 22 ) and NIRSpec PRISM (0.5-5.0 μm; ref. 23 We find that the observations are best explained by an atmospheric metallicity of 10-30 times solar. Metallicity inferences over the wavelength range of these observations are largely driven by the size and shape of the water-vapour features, with some minor contributions because of CO and/or CO 2 at longer wavelengths (>2 μm; see Figs. 3 and 4b). The preferred range of metallicities provides the best fit to the shape and size of the muted water-vapour features shortward of 2 μm in combination with the larger water and CO/CO 2 feature longward of 2 μm, regardless of the assumed cloud treatment in our models. Owing to the simultaneous detection of potassium and water vapour, we are able to place constraints on the K/O ratio, which is a refractory-to-volatile elemental ratio, being a solar-to-super-solar value. Because the refractory elements are condensed into solids in most parts of protoplanetary disks, the disk gas accretion tends to cause a sub-stellar refractory elemental abundance 24 . By contrast, solid accretion, such as planetesimal accretion, acts to increase the refractory elemental abundance and refractory-to-volatile elemental ratio 25 , although the latter depends on the composition of the accreted solids 26 . We anticipate that the K/O ratio diagnoses to what degree the solid accretion enriched the atmosphere during the formation stage. All of our fitted models find that the WASP-39b observations are well described by solar-to-super-solar K/O ratios, which is in agreement with previous inferences for this planet obtained through observations with limited spectral coverage 27 . We do not expect the K feature to be affected by stellar chromospheric magnetic activity given the effective temperature of the star of approximately 5,300 K (ref. 28 ) and the general quietness of WASP-39 (see ref. 21 ). It is also in line with larger population studies of hot giant planets that broadly found solar-to-super-solar refractory abundances and solar-to-sub-solar H 2 O abundances 27,29 . The shape and strength of the potassium doublet are best explained by K/O ratios of 0.1-0.5, equivalent to 1-3 times solar (see Extended Data Fig. 8), whereas the suggested K/O ratio might be a lower limit owing to the photoionization of K at upper atmospheres 30 . The NIRISS/SOSS observations enable the detection of clouds in the atmosphere of WASP-39b. Clear atmosphere models cannot explain the amplitudes of all of the water-vapour features simultaneously, which strongly indicates the presence of clouds (see Methods and Extended Data Fig. 6). The atmospheric models explored here indicate the presence of non-grey and non-homogeneous clouds, with model preferences of 8σ and greater for models with both non-grey and non-homogeneous clouds over models with grey homogeneous clouds only. This model preference is driven by the decrease in transit depth between 2.0 and 2.3 μm (see Extended Data Fig. 7a), which cannot be explained by grey clouds uniformly distributed along the terminator (see Extended Data Fig. 7a). Moreover, in the various cloud treatments tested here (grey, grey + power law and flux-balanced clouds; see Methods), both parametric and droplet sedimentation models indicate a preference for inhomogeneous cloud coverage of roughly 50-70% around the planetary day-night terminator because it better explains the decrease in transit depth between 2.0 and 2.3 μm. Atmospheric circulation and cloud microphysical models have predicted that the cloud structure varies substantially along the terminators of hot Jupiters [31][32][33] . In particular, different compositions of clouds have different condensation temperatures and thus probably have different cloud coverage at the terminator 31 . Further studies combining temperature difference of east-west terminators to microphysical cloud models may be able to use the measured cloud coverage to determine the cloud composition of WASP-39b. Previous indications of non-grey or non-homogeneous clouds [34][35][36][37][38][39] have relied on a single or small number of spectroscopic points, making our inference here for WASP-39b of non-grey cloud with inhomogeneous terminator coverage in the transmission spectrum of an exoplanetary atmosphere the most confident so far. These constraints on the physical properties of clouds, alongside several spectral features across a broad wavelength coverage, are key to breaking well-known degeneracies between the metallicity and cloud cover in atmospheric models 8,14,40 and deriving constraints on the bulk atmospheric properties. The high precision of NIRISS/SOSS in combination with broadest wavelength coverage <2.8 μm for any JWST instrument, minimal systematics and no issue with saturation allows us to obtain more precise and robust constraints on atmospheric composition and tracers of planet formation than most previous transmission spectroscopy observations. The super-solar metallicity of WASP-39 b and the solar-to-super-solar K/O ratio are in agreement with previous studies of mass-metallicity trends in transiting exoplanets 3,7,27,41 . If confirmed with further detailed modelling, a super-solar K/O ratio in the atmosphere of WASP-39 b would probably indicate enrichment resulting from the accretion of planetesimals [25][26][27] , although the measurements of potassium and oxygen abundances for the host star are also needed to establish this result. Similarly, the suggestion of sub-solar C/O ratio and super-solar metallicity may be compatible with a planetesimal accretion scenario, for example, refs. [42][43][44] . The combination of a super-solar metallicity, super-solar K/O ratio and sub-solar C/O ratio may suggest that the planet formed beyond the H 2 O snow line followed by inward migration, for which theory predicts efficient accretion of planetesimals at approximately 2-10 AU, for example, refs. 25,45 . At those orbital distances, the planetesimals probably contain K rock (for example, alkali feldspar KAlSi 3 O 8 (refs. 46,47 )) and H 2 O ice but almost no CO ice, for example, refs. 48 4 , the latter having spectroscopic signatures incompatible with the observations. To mute these incompatible CH 4 features at high C/O ratios, the model requires a higher degree of cloudiness that also mutes any remaining H 2 O features in the spectrum. b, The same as for panel a but instead we vary the metallicity parameter. The metallicity constraint is driven by the λ > 2 μm data; the high-metallicity models (M/H > 2) expect larger transit depths than that seen in the data. The same reference model is plotted as a thick black line in both panels. We define our errors as the 1σ uncertainties extracted from the 16th and 84th percentiles of the transit depths fit from each pipeline. (https://github.com/afeinstein20/ wasp39b_niriss_paper/blob/main/scripts/figure4.py). Article pathways of this planet requires statistical constraints on the complete chemical inventory of the planet and the relative abundances of the carbon-bearing, oxygen-bearing and alkali-bearing species. Such efforts will be possible when applying retrieval techniques to the complete transmission spectrum of WASP-39b from 0.5 to 5.5 μm that is being produced by the Transiting Exoplanet Community Early Release Science Program. Our results validate the JWST's NIRISS/SOSS as an instrument mode fully capable of producing excellent exoplanet atmosphere measurements. Online content Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41586-022-05674-1. Methods Given the newness of the data, we applied six independent datareduction and light-curve-fitting routines to the data: nirHiss, supreme-SPOON, transitspectroscopy, NAMELESS, iraclis and FIREFLy. Each pipeline extracts the stellar spectra from orders 1 and 2 (λ = 0.6-2.8 μm) with the exception of FIREFLy, which only extracts data from order 1. There is an extra order 3 that has a spectral range of λ = 0.6-0.95 μm (ref. 50 ). However, the signal of order 3 is generally weak and, because it provides no new wavelength information beyond what is covered in orders 1 and 2, is not used by any of the presented pipelines. Below, we first describe the important reduction steps taken by each, followed by their light-curve-fitting methodologies. We note here that, in each pipeline, the position of the SOSS trace was found to match almost perfectly with that measured during commissioning (Fig. 1). Furthermore, each pipeline trimmed the first 10-15 integrations to remove the effects of the exponential ramp in the fitting routines. We present a summary of all pipelines in Extended Data Table 1. The nirHiss pipeline nirHiss is a Python open-source package that uses the stage 2 outputs from the Eureka! pipeline and performs further background and cosmic-ray removal, as well as extraction of the stellar spectra. For this analysis, we took the uncalibrated images and ran our own stages 1 and 2 calibration using Eureka! 51 , an open-source package that performs spectral extraction and light-curve fitting for several JWST instruments. We use the default steps presented in Eureka!, which includes detector-level corrections, production of count-rate images, application of physical corrections and calibrations to individual exposures. Next, nirHiss removes background noise sources in a multistep process. The zodiacal background is first removed by applying the background model provided on the STScI JDox User Documentation website (https://jwst-docs.stsci.edu/). The background is scaled to a small region of each science integration in which there was no contamination from any of the orders; in this case, x ∈ [190,250], y ∈ [200,500]. The average scaling-calculated here to be 0.881-is applied to all science integrations. Second, a model of zeroth-order contaminants is built using the F277W integrations. The F277W integrations were taken after the transit of WASP-39b with the GR700XD/CLEAR pupil element and the F277W filter (throughput centred at λ = 2.776 μm with a bandwidth of λ = 0.715 μm). These observations consist of ten integrations with an exposure time of 49.4 s. Observations with the F277W filter contain only the spectral trace of order 1 in the region in which x ≤ 460 pixels, thus allowing for the detection and modelling of zeroth-order contaminants across most of the detector. A median F277W frame is created to identify and mask any bad-data-quality pixels. To ensure that no further noise is added from the median F277W frame, we create a 2D background model map using photutils.Back-ground2D. To identify regions of the background, we masked the upper-left corner, in which the trace is located, and any regions >1.5σ, which includes the zeroth-order sources. For photutils.Background2D, we used a filter size of (3, 2) pixels and a box size of (2, 2) pixels. Once the background is removed from the median F277W frame, we apply a Gaussian filter with a width of 2 to smooth out any further small-scale background noise. To apply the median F277W frame to the stage 2 science integrations, we scaled it to two isolated zeroth-order sources in the science integrations at x 1 ∈ [900, 1,100], y 1 ∈ [150, 250] and x 2 ∈ [1,800, 2,000], y 2 ∈ [150, 250]. We applied the average scaling to all integrations. We found the average F277W background scaling to be 2.81. We apply the scaled background frame to each time-series observation integration. Once the zeroth-order contaminants are removed, we trace the location of orders 1 and 2. The spatial profile for NIRISS/SOSS along the column is double-peaked, with a slight dip in the middle. We developed a routine to identify the trace locations using a three-step approach to identifying each order. For each column in the first order trace, we identify the locations of the two peaks, or 'ears', and assume that the middle of the trace is the median row pixel between the two ears. We repeat this process for the third and second orders in that sequence, masking orders once they have been traced. We chose to identify the third order before the second order because it is better spatially resolved and does not overlap with any other orders. The routine creates one main set of traces from a median frame of all observations, which is used to extract the stellar spectra. As an extra output, we track the changes in the (x, y) pixel positions of each order on the detector across all integrations. After the traces are identified, we continue our reduction to remove any extra noise and cosmic rays/bad pixels. We perform further 1/f noise correction following the routine presented in transitspectroscopy (described below). Finally, nirHiss identifies and interpolates over cosmic rays. To identify cosmic rays, we used the L.A.Cosmic technique wrapped into ccdproc 52,53 , which identifies pixels based on a variation of the Laplacian edge detection. We identify cosmic rays as pixels with σ > 4 using this method. We interpolate over any further bad pixels by taking the median value of the two surrounding pixels along the column. We extract the spectra using a box-extraction routine and ignore any contaminants from overlapping orders or from any potential background orders. We use a box diameter of 24 pixels for both orders 1 and 2. The supreme-SPOON pipeline In parallel, we reduce the WASP-39 b time-series observation with the independent supreme-SPOON (supreme-Steps to Process sOss ObservatioNs) pipeline, which processes SOSS time-series observations from the raw, uncalibrated detector images to extracted 1D light curves. An outline of the specific steps is presented below. For detector-level processing, supreme-SPOON closely follows stage 1 of the jwst pipeline. All default steps, up to and including the reference pixel correction, are run using their default settings. The reference pixel step is known to provide an inadequate correction of 1/f noise for SOSS observations; however, we include it to remove group-to-group variations in the bias level, as well as even-odd row variations. At this stage, we remove the zodiacal background from each group. This is accomplished by first calculating a group-wise median frame and scaling the model background provided in the STScI JDox to the flux level of each group in this median, yielding eight background models, one for each group. The region chosen to calculate the scaling was x ∈ [300, 500], y ∈ [210,250], in which there is minimal contamination from any of the SOSS orders. The nth background model is then subtracted from the corresponding group of each integration. We then proceed to a more in-depth treatment of 1/f noise. Unlike the other pipelines used in this work, supreme-SPOON treats 1/f noise at the group level instead of at the integration level. 1/f noise is a time-varying noise source introduced by the voltage amplifiers during the readout of the detector and therefore the 1/f pattern will vary from group to group, even within a given integration. To perform the 1/f correction, first a median out-of-transit frame is calculated for each group. This group-wise median is then scaled to the flux level of each frame in a given group by means of the transit curve and subtracted, showing the characteristic 1/f striping in the residuals. A column-wise median of this residual map is then subtracted from the original frame. The trace residuals as well as any bad pixels are masked in the median calculation. From this point, we once again proceed with the standard stage 1 steps of the jwst pipeline, with the exception of the dark current step, to obtain the supreme-SPOON stage 1 outputs. The dark current subtraction step is skipped as it was found to reintroduce 1/f noise into the data. The dark current level is also extremely small (several tens of electrons s −1 compared with many thousands for the target signal) and can thus be safely ignored. supreme-SPOON only applies the assign_wcs, srctype and flat_field steps of the stage 2 jwst pipeline to the stage 1 products. The background subtraction was already performed as part of stage 1 calibrations. Furthermore, the flux calibration steps (pathloss, which accounts for light incident on the telescope primary mirror that falls outside the SUBSTRIP256 subarray, and photom, which performs the actual photometric flux calibration) are skipped, both because an absolute flux calibration is unnecessary for relative spectrophotometric measurements and a wavelength-dependent flux calibration is nonsensical for SOSS, in which contributions from several wavelengths from all orders affect a single pixel. At this point, supreme-SPOON identifies any remaining hot pixels through median filtering of a median stack of all frames and interpolates them by means of the median of a surrounding box. These products are the supreme-SPOON stage 2 results. Stage 3 of the supreme-SPOON pipeline is the 1D extraction. This can be performed through two different methods: the first is a simple box aperture extraction on each order, ignoring the order contamination. The second uses ATOCA (Algorithm to Treat Order ContAmination) 50 to explicitly model the order contamination. Briefly, ATOCA constructs a linear model for each pixel on the detector, including contributions from the first and second diffraction orders, allowing for the decontamination of the SOSS detector-that is, ATOCA constructs models of both the first and second orders individually, thereby allowing a box extraction to be performed on each free from the effects of order contamination. Although the effects of this order contamination for differential measurements (such as exoplanet atmosphere observations) are predicted to be small (about 1% of the amplitude of the expected spectral features) 50,54 , in the quest to obtain the most accurate possible transmission spectra, this contamination effect is important to take into account. ATOCA is at present built into the Extract1dStep of the jwst pipeline, although it is not the default option and must be toggled to on by means of the 'soss_atoca' parameter. To improve the performance of ATOCA, we do not use the default specprofile reference file included in the jwst pipeline but instead construct estimates of the underlying spatial profiles of the first and second orders, on which ATOCA relies, using the APPLESOSS (A Producer of ProfiLEs for SOSS) algorithm 54 . We determine the centroid positions for each order on a median stack using the 'edgetrigger' algorithm 54 , and these positions are found to match to within a pixel with the default centroids contained in the jwst_niriss_spectrace_0023.fits reference file; the spectrace file is available on the JWST Calibration Reference Data System (CRDS). The SOSS trace position is furthermore highly stable over the course of this time-series observation, with root mean square (RMS) variations in x and y positions of approximately 5 mpix and RMS rotation of about 0.3″. We therefore fix the 'soss_transform' parameter to [0, 0, 0] and perform the extraction with a box size of 25 pixels. Any remaining >5σ outliers in the resulting spectra are then identified and clipped. At present, supreme-SPOON does not explicitly treat contamination from zeroth orders of background stars that intersect the trace. The transitspectroscopy pipeline This third pipeline analysis combines the jwst pipeline stage 1 'rateints. fits' files with transitspectroscopy 55 . transitspectroscopy completes stellar spectral extraction as well as transit fitting. The trace positions for NIRISS orders 1 and 2 were determined using transitspectroscopy.trace_spectrum. This routine cross-correlates an input function with each column in the detector to find the centre of the different traces by means of the maximum of the resulting cross-correlation function. To follow the shape of the NIRISS order profiles, an input function consisting of a double Gaussian was used with parameters that were trained on the NIRISS/SOSS observations of HAT-P-14 b ( JWST Program ID 1541; principal investigator Espinoza): μ 1 = −7.5; σ 1 = 3.0; μ 2 = 7.5; σ 2 = 3.0. The trace for order 2 was not fit for pixels ≤1,040, as the throughput is not high enough for the method to robustly fit the trace without incorporating nearby contaminants. After identifying the trace positions with this method for both orders 1 and 2, both traces were smoothed using a series of spline functions. We find that the best-fit parameters for order 1, which were trained on the HAT-P-14 b observations, are: x knots,1 = [ [6,1,, [1,200,1, The zodiacal background was removed by scaling the model background provided on the STScI JDox User Documentation. This model was compared with a small region of the median science integrations in which there was little to no contamination from the orders (x ∈ [500, 800], y ∈ [210,250]). The ratio of all the pixels in this region versus the pixels in the background model was computed, ordered and the median ratio of all the second quartile pixels was used as the scaling factor between the background model and the data, which was found to be 0.909. All the integrations had this scaled background subtracted. Each integration is corrected for 1/f noise with the following procedure. First, all the out-of-transit, background-corrected integrations are median combined and scaled by the relative flux decrease produced by the transit event at each integration (that is, 1.0 for out-of-transit integrations or about 0.976 for mid-transit). These scaled median frames are then subtracted from each individual integration, which then leaves in the frame only detector-level effects, such as 1/f noise. We then go column by column and take the median of all pixels in these residual frames within a distance of 20-35 pixels from the centre of the trace, and use this as an estimate of the contribution from 1/f noise to that given column. This value is then removed from each pixel within 20 pixels from the trace on that column. No correction for order 1 contamination on order 2 was made, as the contribution is negligibly small in this case 50 -similarly for order 1 contamination in order 2 in our extraction. To extract the resulting background-corrected and 1/f-corrected spectrum, we used the transitspectroscopy.spectroscopy.getSim-pleSpectrum routine with a 30-pixel total aperture for both orders. To handle obvious outliers in the resulting spectrum because of, for example, uncorrected cosmic rays and/or deviating pixels, we median-normalized the spectra for each integration and combined them to form a 'master' 1D spectrum for both orders 1 and 2. The median was taken at each wavelength, as well as the error on that median, and this was then used to search for 5σ outliers on each individual integration at each wavelength. If outliers were found, they were replaced by the rescaled version of this median master spectrum. The NAMELESS pipeline Starting from the jwst pipeline stage 1 products, we use the NAMELESS (Niriss dAta reduction MEthod for exopLanEt SpectroScopy) pipeline to go through the jwst pipeline stage 2 with the addition of custom correction routines. First, we go through the assign_wcs, srctype and flat_field steps of the jwst pipeline stage 2, opting for a custom background-subtraction routine and skipping the pathloss and photom steps as absolute flux calibration is not needed. After flat-field correction, we scale the model background provided on the STScI JDox User Documentation to a region of the median frame in which the contribution from the tail of the three orders is lowest (x ∈ [200, 250], y ∈ [400, 600]). From the distribution of the scaling values of all pixels within the defined region, we take the 16th percentile as our scaling value and subtract the scaled background frame from all integrations. We subsequently correct for 1/f noise by performing a columnby-column subtraction for each median-frame-subtracted integrations. The median frame is computed from the out-of-eclipse integrations (integration # ∈ [200, 400]) and scaled to each individual integration by dividing the sum of the pixels in the first order by that of the median frame. We then subtract the scaled median frame from all integrations, perform the column-by-column subtraction on the residual frames and add back the scaled median frame to the corrected residual frames to obtain the 1/f-corrected integrations. We detect outliers frame by frame using the product of the second derivatives in the column and row directions. This method works particularly well for isolated outliers, as this leads to a strong inflexion that corresponds to a large second derivative. Because the spectral orders also lead to larger second-derivative values, we divide the frames into windows of 4 × 4 pixels, compute the local second-derivative median and standard deviation and flag any pixel that is more than four standard deviations away from the median. Furthermore, we also flag pixels with null or negative flux. All identified outliers are set equal to the median value of the window in which it was identified. Finally, we proceed with spectral extraction of the corrected frames by first tracing the sections of the spectral orders that we wish to extract. We trace orders 1 and 2 from x 1 ∈ [4, 2,043] and x 2 ∈ [4, 1,830], respectively. The centre of the traces is found for each individual column by performing a convolution of the profile with a Gaussian filter, in which we use the maximum of the convolved profile as the centre of the trace. For the tracing of the second order, we keep the centre of the trace fixed below x = 900, as the flux from the first order can bias the tracing method. Furthermore, we smooth the positions of the trace centroids using a spline function with 11 and 7 knots for the first and second orders, respectively. We perform spectral extraction of the first and second orders at all integrations using the transitspectroscopy. spectroscopy.getSimpleSpectrum routine with an aperture width of 30 pixels. The iraclis pipeline We used the jwst pipeline stage 1 rateints.fits files with modified routines from iraclis 5,56 , which was initially designed for the HST. The modified routines will be part of the version 2 of iraclis, which will become publicly available in the near future. The routines applied to the rateints. fits files were flat fielding, bad-pixels and cosmic-rays correction, sky background subtraction, 1/f noise correction, X-drift and Y-drift detection, light-curve extraction, light-curve modelling and planetary spectrum decontamination. We started our analysis by dividing the images by the appropriate flat-field frame ( jwst_niriss_flat_0275.fits), as provided by the JWST CRDS. The next step was the bad-pixels and cosmic-rays correction. For bad pixels, we used those with a positive DQ flag in the rateints.fits files, excluding the warm pixels, as their large number did not allow for a reliable correction. We also identified extra outliers (cosmic rays or other artefacts) by calculating two flags for each pixel: the difference from the average of the ten horizontally neighbouring pixels (x-flag) and the difference from the average of the ten vertically neighbouring pixels (y-flag). If a pixel's x-flag was 5σ larger than the other pixels in the column and its y-flag 5σ was larger than the other pixels in the row, it was identified as a cosmic ray (see also ref. 56 ). Both bad pixels and outliers were replaced with the value of a 2D interpolation function, created from the rest of the pixels, similarly to analyses with the HST 56 . We then subtracted a column-based sky background frame and a column-based 1/f noise frame from each image. For each image, we first used a trace filter (value >0.001 in the jwst_niriss_spectrace_0023.fits, provided by the JWST CRDS) and a column-based 1 × median absolute deviation filter to find the illuminated pixels. Then, we calculated the column-based median of the image-using only the unilluminated pixels-and subtracted it from the image. Finally, we calculated the column-based median of the IMFD (Image-MedianFrame Difference)using only the unilluminated pixels-and subtracted it from the image. This process is not efficient in subtracting 100% of the background contamination, which was removed during the last analysis step (spectrum decontamination). X-pixel and Y-pixel trace drifts were detected relative to the first image by comparing the sums along the columns and the rows, respectively, similarly to the HST 56 . The drifts are on the order of pixels without any evident trend in motion. Because this is below the subpixel size used in the iraclis extraction, we find that there is no marked impact of not correcting these drifts. For each spectroscopic image, we initially divided each pixel into a 100 × 100 grid of subpixels and, for each subpixel, we calculated the distance from the trace (CD) and the wavelength (λ), creating the CD map and the λ map , respectively. λ was assigned to each subpixel directly from the wavelength solution (interpolated wavelength solution from the jwst_niriss_wavemap_0013.fits file, provided by the JWST CRDS, shifted by the detected X and Y drifts). CD was calculated as the distance between the centre of the subpixel and the point of the trace with the same distance (interpolated trace from the jwst_niriss_spectrace_0023.fits file, provided by the JWST CRDS, shifted by the detected X and Y drifts). Our high-resolution bins had a λ width of 10 Å, ranging between 0.62 and 0.85 μm for order 2 and between 0.85 and 2.8 μm for order 1, and a CD width of 1.5 pixels, ranging from −25 to 25 pixels. Finally, to construct the light curve of each bin, we applied the following smoothed aperture mask on each spectroscopic image and summed the values of all the subpixels. We chose a smoothed aperture, similarly to the HST to reduce the effects of jitter noise: in which CD 1 , CD 2 and σ CD are the bin boundaries and the smoothing factor along the cross-dispersion axis, respectively, and λ 1 , λ 2 and σ λ are the bin boundaries and the smoothing factor along the dispersion axis, respectively. For the smoothing factors, we used the values of σ CD = 0.015 pixels and Å -that is, about 10% of the bin size. We chose these values for the smoothing factors because lower values would effectively create a sharp-edge aperture, whereas larger values would force the bins to overlap substantially. FIREFLy Although FIREFLy (Fast InfraRed Exoplanet Fitting for Light curves) 57 was written and optimized for reducing NIRSpec-PRISM and G395H time-series observations, it worked well on the NIRISS/SOSS dataset, in which it selected and processed the spectrophotometry from order 1 only with minimal tuning or intervention. FIREFLy is not written in such a way to extract data from order 2 (λ < 0.9 μm). In our reduction, we perform standard calibrations on the raw data using the jwst pipeline for stages 1 and 2 reduction. On the jwst stage 2 outputs, we perform bad-pixel and cosmic-ray cleaning on each integration. We perform 1/f destriping and background subtraction using a pixel mask generated from the temporally medianed image that selects regions of the data image below a specified count threshold. We extract the spectrophotometry using an optimized aperture extraction width that is constant in wavelength. The aperture width is selected such that the scatter of the resulting out-of-transit white-light photometry is minimized. Light-curve fitting and transmission spectra We used a suite of light-curve-fitting routines to fit the extracted light curves. Each routine fits for orbital parameters from the broadband white-light curves for each order (see Extended Data Fig. 8). We fixed the orbital period to the best-fitting value from P = 4.05528 days (ref. 1 ) for all pipeline fits. For the spectroscopic light curves, most routines (nirHiss/chromatic_fitting, supreme-SPOON/juliet, transitspectroscopy/juliet and NAMELESS/ExoTEP) fixed the orbital parameters (that is, the mid-transit time, t 0 , semi-major axis to stellar radius ratio a/R * , impact parameter b, eccentricity e) to the same values to ensure consistency. These parameters were fixed to their best-fitting values from the transitspectroscopy/juliet white-light-curve fit, except for t 0 which was fixed to the value obtained from the white-light curve in each case. This left the planet-to-star radius ratio R p /R * , the limb-darkening coefficients and parameters for any further systematics models to vary. These four routines also fit spectroscopic light curves at the native instrument resolution. However, two routines, iraclis and FIREFLy, instead fixed the orbital parameters in their spectroscopic fits to values obtained through their white-light curve fits. iraclis also fits directly for the orbital inclination, i, as opposed to b and a/R * like the other routines. iraclis fits for their spectrophotometric light curves at the pixel resolution, whereas FIREFLy binned the spectroscopic light curves first and fits for the transit parameters. We present all of the best-fit white-light-curve parameters for order 1 in Extended Data Table 2. Furthermore, for the spectroscopic light-curve fits, we only considered the region of order 2 with wavelength <0.85 μm, as the 0.85-1.0-μm range is covered at higher signal-to-noise ratio (SNR) by order 1. All errors on each parameter are representative of 1σ (lower 16th and upper 84th percentiles) of the fit. chromatic_fitting chromatic_fitting is an open-source Python tool for modelling multiwavelength photometric light curves. This tool is built on the framework of chromatic, a package for importing, visualizing and comparing spectroscopic datasets from a variety of sources, including Eureka! and the jwst pipeline. In this paper, we applied chromatic_fitting to the nirHiss reduction. chromatic_fitting uses the PyMC3 (NUTS) sampler 58 to fit the exoplanet transit model to the light curves. First, we fit the white-light curves for order 1. The white-light curve was generated using an inverse variance-weighted average of the unbinned data. We fixed the orbital period to 4.05528 days (ref. 1 ) and assumed a circular orbit. We fit for the mid-transit epoch t 0 , the stellar mass M * and radius R * , the impact parameter b, the planet-to-star radius ratio R p /R * , quadratic limb-darkening coefficients (u 1 , u 2 ) and out-of-transit flux F 0 . For the fitted parameters t 0 , M * , R * , R p /R * and F 0 , we assumed normal priors N(2459787.56, 0.02 2 ), N(0.934, 0.056 2 ), N(0.932, 0.014 2 ), N(0.146, 0.05 2 ) and N(1.0, 0.01 2 ), respectively. For b, we used a uniform prior between 0 and 1.146, in which b ≤ 1 + R p /R * . For the limb-darkening coefficients, we calculated the theoretical values from 3D models in ExoTIC-LD 59-61 (based on the stellar parameters T eff = 5,512 K, logg = 4.47 dex and Fe/H = 0.0 dex (ref. 17 )) and assumed normal priors around these values with σ = 0.05. We also included a second-order polynomial in time to describe the systematics with a fixed constant term of 0.0 and normal priors on the first-order and second-order coefficients c 1 and c 2 of N(0.0, (1e −4 ) 2 ). Using the NUTS implementation of PyMC3, we ran 4,000 tuning steps and 4,000 draws, with four walkers, for the white-light curve and the mean parameter values are shown in Extended Data Table 2. We checked for convergence using the rank-normalized R-hat diagnostic 62,63 . For each spectroscopic light curve, we fixed the period P, transit epoch t 0 , eccentricity e, semi-major axis in stellar radii a/R * and impact parameter b to the white-light parameter values from the transitspectroscopy/juliet routine (Extended Data Table 2). We then fit for the planet-to-star radius ratio R p /R * , quadratic limb-darkening coefficients (u 1 , u 2 ) and out-of-transit flux F 0 -for all of these parameters, we assumed the same normal priors as for the white-light curve. We also included a second-order polynomial in time with the same priors as the white-light-curve fit. For each wavelength, we ran 2,000 tuning steps and 2,000 draws, also with four walkers. The final transmission spectrum was taken as the mean value drawn from the posterior distribution for the planet-to-star radius ratio with 1σ uncertainties extracted from the 16th to 84th highest density interval region. The SNR in the spectrophotometric light curves from nirHiss for order 1 at 1.34 μm is 165 and that for order 2 at 0.71 μm is 103. We define the SNR as n σ σ × / bins intransit outoftransit , in which σ is the standard deviation. juliet We applied the juliet package 64 for light-curve fitting to the products of several reduction pipelines described above. Here we give a general overview of the methods and include exact details for each fit when appropriate. For the supreme-SPOON reduced stellar spectra, we fit the white-light curve for the mid-transit time, t 0 , the impact parameter, b, the scaled orbital semi-major axis, a/R * and the scaled planetary radius, R p /R * , assuming a circular orbit. We also fit two parameters of a quadratic limb-darkening model following the parameterization of ref. 65 , as well as an additive scalar jitter and the two parameters of a linear trend with time. We therefore fit seven parameters to the white-light curve for each order, using wide, flat priors for each case. We then proceeded to fit the light curves from each individual wavelength bin at the native detector resolution-that is, two pixels per bin to roughly account for the extent of the point spread function in the spectral direction. This results in 1,020 bins for order 1 and 520 bins for order 2, as we only consider wavelengths <0.85 μm. For the spectroscopic fits, we fixed the central transit time to the white-light-curve value, and the other orbital parameters were as described for chromatic_fitting. For the linear trend with time, we used the white-light posterior for each of the two parameters as prior distributions for all wavelength bins, whereas for the limb-darkening parameters, we adopted a Gaussian prior centred around the predictions of the ExoTiC-LD package 61,66 with a width of 0.1. As the SOSS throughput files included with ExoTiC-LD did not cover the full wavelength range of both orders, we instead used the throughputs determined during commissioning and included in the spectrace reference file of the jwst pipeline. We truncated the Gaussian prior at 0 and 1, to prevent the limb-darkening parameters from straying into unphysical regions of the parameter space. We then used flat, uninformative priors for the remaining two parameters, the scaled planetary radius and the scalar jitter. The supreme-SPOON white-light-curve fits have χ = 1.15 ν 2 for order 1 and χ = 1.11 ν 2 for order 2. For the transitspectroscopy reduced stellar spectra, we first fit the white-light curves of orders 1 and 2 separately. For these, as suggested above, the period was fixed but the other parameters were allowed to vary. In particular, we set a normal prior on the time-of-transit centre of N(2459787.5, 0.2 2 ) days, in which the first value denotes the mean and the second the variance of the prior. A normal prior was also set on a/R * ~ N(11.37,0.5 2 ), in which '~' denotes 'distributed as', and a truncated normal between 0 and 1 was set for the impact parameter b ~ TN(0.447, 0.1 2 ), in which the means were set following the work of ref. 67 but the variances are large to account for the variation of these parameters in the literature between different authors. We set a uniform prior for the planet-to-star radius ratio between 0 and 0.2 and fixed eccentricity to 0. As well as those, we fit for a mean out-of-transit offset with a normal prior of N(0, 0.1 2 ) and a jitter term added in quadrature to the error bars with a log-uniform prior between 10 and 1,000 ppm. To account for systematic trends in the data, we use a Gaussian process by means of celerite 68 with a simple Matérn 3/2 kernel to parameterize those trends. We set log-uniform priors for both the amplitude of this Gaussian process from 10 −5 to 1,000 ppm and for the timescale from 10 −3 days to 0.5 days. We use the framework of ref. 65 to parameterize limb darkening through a square-root law, which, following ref. 69 , is one of the laws that should give the best results at this level of precision. For the wavelength-dependent light curves, we used a similar setup with two main differences. The first is that we fix the time-of-transit centre, a/R * and b to their white-light values. The second is that we set truncated normal priors on the transformed limb-darkening coefficients (q 1 , q 2 ) between 0 and 1, with standard deviations of 0.1 and means obtained by the following method. First, we obtain the nonlinear limb-darkening coefficients using an ATLAS stellar model with the closest properties to those of WASP-39 using the limb-darkening software 70 . Then, the square-root law limb-darkening coefficients are obtained following the algorithm of ref. 71 , which are transformed to the (q 1 , q 2 ) parameterization using the equations in ref. 65 . These are then set as the mean for each wavelength-dependent light curve. We note that we fit the light curves at the pixel level, which means fitting one light curve per detector column. We fit them in parallel using the transitspectroscopy.transitfitting.fit_lightcurves routine. ExoTEP For the NAMELESS reduction, we perform light-curve fitting on the extracted spectrophotometric observations using the ExoTEP framework 72 . We first fit the white-light curves of both orders 1 and 2 separately. We fit for the mid-transit time t 0 , the planet-to-star radius ratio R p /R * and quadratic limb-darkening coefficients (u 1 , u 2 ) 73,74 , while fixing the impact parameter b and semi-major axis a/R * to the values of the best order 1 white-light-curve fit from the transitspectroscopy/juliet analysis. We also fit for the scatter σ, as well as a linear systematics model with an offset c and slope v. Uniform priors are considered for all parameters. Furthermore, we only discard the first 10 min of observations (ten integrations) to remove the exponential ramp. For all light curves, we compute the rolling median for a window size of 11 integrations and bring any data point that is more than four standard deviations away from it to the median value. We fit the light curves using the Markov chain Monte Carlo (MCMC) ensemble sampler emcee 75 for 1,000 steps using four walkers per free parameter. The first 600 steps, 60% of the total amount, are discarded as burn-in. We then fit the spectroscopic light curves, keeping t 0 fixed to its white-light value, at a resolution of three pixels per bin for order 1 (680 bins) and one pixel per bin for order 2 from about 0.6-0.9 μm (675 bins). We used 1,000 steps for the spectroscopic fits, once again discarding the first 600 as burn-in. iraclis We analysed all the light curves using the open-source Python package PyLightcurve 76 . For every light curve, PyLightcurve: (1) calculates the limb-darkening coefficients using the ExoTETHyS package 77,78 , the wavelength range of the bin, the response curves for each of the NIRISS orders ( jwst_niriss_spectrace_0023.fits file, provided by the JWST CRDS) and the stellar parameters (T eff = 5,540 K, logg = 4.42 cm s −2 , Fe/H = 0.14 dex (ref. 79 )); (2) finds the maximum-likelihood model for the data (an exposure-integrated transit model together with a quadratic trend model using the Nelder-Mead minimization algorithm included in the SciPy package 80 ; (3) removes outliers that deviate from the maximum-likelihood model by more than three times the standard deviation of the normalized residuals; (4) scales the uncertainties by the RMS of the normalized residuals, to take into account any extra scatter; (5) and, finally, performs an MCMC optimization process using the emcee package 75 . We initially modelled the first-order white-light curve (sum of all bins above 0.85 μm with out-of-transit fluxes above 20 data numbers per second (DN s −1 ) and fit for the white R p /R * , the orbital parameters, a/R * and i, and the transit mid-time. We then modelled the spectral light curves, fitting only for the R p /R * , fixing the orbital parameters, a/R * and i, and the transit mid-time to the above white results. In both cases, the models also included a quadratic detrending function that was multiplied by the transit model. After modelling, we applied a spectral decontamination step, taking advantage of the varying total flux across the spectral traces. Owing to the contamination, we have (R p /R * ) 2 × (TF − x)/TF, in which TF is the out-of-transit flux (star and contamination) and x is the flux of the contaminating source. Hence, for each wavelength, we fitted for x and applied the correc- . This procedure is effective in removing uniform contamination. The uniform contamination fixes issues of sky background overcorrection or undercorrection. It also corrects for order overlap. After the decontamination described above, there was still a contaminating source affecting the spectrum around 0.72 μm, which was not uniform because of the point spread function. To separate this source, we applied independent component analysis on the stellar spectra extracted from various distances from the trace. We used two components to describe the contaminating source and one to describe the stellar spectrum. Finally, we estimated the (R p /R * ) 2 for each wavelength bin using the weighted average of all the bins that had the same wavelength range. We only took into account the bins that had out-of-transit fluxes above 20 DN s −1 . This choice effectively applied an optimal aperture size for each wavelength bin. FIREFLy To extract the transmission spectrum, we bin the cleaned spectrophotometric light curves by wavelength first to create 120 variable-width spectral channels with roughly equal counts in each. We fit for the transit depth of the planet in each channel using a joint light curve and systematics model. The systematics model accounts for spectral shifts in the X and Y directions 57 . We use the orbital parameters recovered from an MCMC fit to the white-light curve and fix them at each wavelength channel for our fit. We fit for the two quadratic limb-darkening terms a and b at each wavelength channel. We find that the best-fit limb-darkening coefficients are uniquely determined and deviate by a constant offset relative to model coefficients. Our fits are performed iteratively using the Python package lmfit. The light curves show a typical photometric scatter of 0.3% per integration and the typical transit-depth uncertainties vary between 150 and 300 ppm, which is in line with near-photon-limited precision. More details of the FIREFLy fitting routine can be found in ref. 57 and in ref. 23 . Atmospheric models To interpret the measured transmission spectrum, we performed an extensive comparison with grids of synthetic transmission spectra. We tested several independent atmospheric models to avoid any model-dependent interpretation of the data. Unless otherwise noted, all of our grids have assumed radiative-convective-thermochemical equilibrium to estimate atmospheric compositions. The exploration of atmospheric models with fewer assumptions (for example, without the assumption of chemical equilibrium with metallicity and C/O ratio as free parameters) and those considering other effects of disequilibrium chemistry is left for future work. We derive basic interpretations for the observed spectrum based on four independent model grids, ATMO, PHOENIX, PICASO and ScCHI-MERA. Each grid contains precomputed transmission spectra at various atmospheric properties, such as M/H, C/O ratio and cloud properties, using from grey to Mie-scattering cloud opacity (see next subsection for details). The ScCHIMERA grid considers further model advancements: (1) various cloud treatments, including grey cloud, grey + power-law cloud opacity and physically motivated (that is, droplet sedimentation) cloud model, (2) the impact of inhomogeneous cloud coverage along the planetary terminator and (3) K/O ratio as a grid dimension. ScCHIMERA provides the best fit to the observations compared with the other three grids and informs the results presented in the main text. Grid search with precomputed forward models Here we introduce the independent grids of precomputed transmission spectra, their model description and the main results from these grid fits. We first present the three grids that assume horizontally homogeneous clouds. ATMO The atmospheric pressure-temperature (PT) profile is computed using the 1D radiative-convective equilibrium model ATMO [81][82][83] . The model includes the molecular/atomic opacity of CH 4 , CO, CO 2 , C 2 H 2 , Cs, FeH, HCN, H 2 O, H 2 S, K, Li, Na, NH 3 , PH 3 , Rb, SO 2 , TiO and VO, for which the adopted line list is summarized in ref. 84 . The line lists of several key species are: H 2 O (ref. 85 ), CH 4 (ref. 86 ), CO 2 (ref. 87 ), CO from HITEMP2010 (ref. 88 ) and K from VALD3 (ref. 89 ratio, which is fixed then scaled to solar metallicity. The cloudy models include small particle opacity as the Rayleigh scattering gas opacity enhanced by a factor of either 0 or 10, whereas large particle opacity is equated to the H 2 Rayleigh scattering opacity at 0.35 μm enhanced by a factor of 0.5, 1.0, 5.0, 10.0, 30.0 and 50.0. In total, the ATMO grid consists of 484 cloud-free and 6,292 cloudy atmosphere models. We only consider horizontally homogeneous clouds in the ATMO grid fits. PHOENIX The atmospheric PT profile is computed using the 1D radiative-convective equilibrium model PHOENIX [90][91][92] 93 ) and atomic lines from the database of Kurucz and Bell 94 . For cloudy models, the small non-grey cloud particle opacity is treated as a sum of Rayleigh scattering opacity of all gas species enhanced by a factor of either 0 (clear atmosphere) or 10; large grey particle opacity is treated as grey cloud deck pressure levels of 0.3, 3.0 and 10.0 mbar. In total, the PHOENIX grid consists of 95 cloud-free and 380 cloudy atmosphere models. We only consider horizontally homogeneous clouds in the PHOENIX grid fits. PICASO 3.0 Similarly to the grids of models presented above, we precomputed atmospheric PT profiles using the 1D radiative-convective equilibrium model PICASO 3.0 (refs. 95 99 ), CH 4 (ref. 100 ), CO 2 (ref. 101 ), CO (ref. 102 ) and K from VALD3 (ref. 89 ). For cloudy models, we post-processed the computed PT profiles using the droplet sedimentation model Virga 15,103 , which determines the vertical distributions of cloud-mass mixing ratio and mean particle size from the balance between downward mass flux of gravitational sedimentation and upward mass flux of eddy diffusion. We vary vertically constant eddy diffusion coefficients of K zz = 10 5 , 10 7 , 10 9 and 10 11 and vertically constant sedimentation parameters of f sed = 0.6, 1.0, 3.0, 6.0 and 10.0. The f sed value is defined as the ratio of the mass-averaged sedimentation velocity of cloud particles to the mean upward velocity of the atmosphere, with a smaller f sed yielding more vertically extended clouds 99 ; see, for example, refs. 103,104 . We have assumed horizontally homogeneous clouds and accounted for the formation of MgSiO 3 , MnS and Na 2 S clouds. Then, we post-processed the atmospheric properties to compute synthetic transmission spectra. We note that the optical properties of the flux-balanced cloud are computed by the Mie theory 105 under the assumption of a log-normal particle-size distribution with a mean particle size translated from f sed (ref. 15 ). In total, the PICASO grid consists of 192 cloud-free and 3,840 cloudy atmosphere models. We compare the NIRISS/SOSS spectrum (binned to R = 300) to each of these model grids and summarize the best fits in the top panel of Extended Data Fig. 6. For each cloudy and clear model we tested, we compute χ 2 /N obs = 2. 98-8.55 between the data and the models, with specific values per model indicated in the legend of Extended Data Fig. 6. All of our forward model grids consistently indicate super-solar metallicity (M/H = 1-2) and sub-solar C/O ratio. Each best-fit spectrum shows different structures at >2 μm, as the spectra at these wavelengths are more sensitive to the treatment of cloud properties (see next subsection for details). The best-fit spectra from PICASO, ATMO and PHOENIX indicate atmospheric metallicities of M/H = 1.7, 1.0 and 2.0, respectively. These models also consistently indicate that the C/O ratio is between 0.229 and 0.389, corresponding to the lowest C/O ratio grid point in each grid (see the main text for why models prefer lower C/O ratios). Thus, the super-solar metallicity and sub-solar C/O ratio of WASP-39b are consistent across the different model interpretations of the NIRISS/ SOSS transmission spectrum. We also find that clear atmospheric models fail to fit the observed spectrum even at very high metallicity (M/H = 2.0), as shown in the bottom panel of Extended Data Fig. 6. The clear models fail to match the amplitudes of H 2 O absorption features at λ = 0.90, 1.15, 1.40 and 1.80 μm simultaneously. The clear ATMO models fit the data better than the clear PICASO and PHOENIX models because the ATMO grid allows lower heat redistribution factors (that is, cooler atmosphere). The clear models also overestimate the transit depth at λ ≈ 2 μm because of a strong CO 2 absorption resulting from the inferred high metallicity (M/H = 2.0). The inability of clear atmosphere models to fit the overall NIRISS spectrum strongly indicates the presence of clouds in the atmosphere and emphasizes the ability of the NIRISS wavelength coverage to break the cloud property-metallicity degeneracy. The best-fit cloud properties are f sed = 1 and K zz = 10 9 cm 2 s −1 for Virga clouds in PICASO, a grey cloud opacity of five times the H 2 Rayleigh scattering opacity at 0.35 μm for ATMO and a grey cloud top pressure of 3 × 10 −4 bar for PHOENIX. Grid search with ScCHIMERA The NIRISS transmission spectrum offers key insights into the atmospheric properties of WASP-39b over a broad wavelength range. The simultaneous detection of H 2 O and K, alongside possible indications of carbon-bearing species, allows us to explore equilibrium models for which the K/O ratio is an extra dimension besides the commonly used C/O ratio and metallicity parameters. Furthermore, as explained in the previous subsection (see also Fig. 3 demonstrating how clouds contribute to the NIRISS spectrum), the broad wavelength coverage of these NIRISS observations makes it possible to explore more complex cloud models beyond traditional grey and homogeneous cloud models. To explore these considerations, we implement the ScCHIMERA grid as explained below. ScCHIMERA Previous implementations of this framework include refs. [106][107][108] , in which the methods are described in detail. Implementations of this procedure to the JWST data include ref. 109 . For a given set of planetary parameters, our methods precompute the temperature-pressure structure of the planetary atmosphere and the thermochemical equilibrium gas-mixing-ratio profiles. The computations are performed on a grid of atmospheric metallicity (M/H, for example, log 10 enrichment relative to solar 20 ) spaced at 0.125 dex values between 0 and 2.25 (for example, 1-177 times solar) and C/O ratios at values of 0.20, 0.35, 0.45, 0.55, 0.65, 0.70 and 0.80. Unlike previous implementations of this framework, and to better understand the NIRISS/SOSS observations presented, we include a dimension to our grid exploring the K/O ratio (that is, log 10 enrichment relative to solar 20 ) with spacing of 0.5 dex between −1 and 0 and 0.1 dex between 0 and 1, overall spanning a range from −1 to 1 or 0.1-10 times solar. In these calculations, the atmospheric metallicity scales the sum of K, C and O. This sum determines the final elemental abundances after scaling metallicity, C/O ratio and K/O ratio. That is, the total oxygen elemental abundance is O′ = M/H K/O + C/O + 1 , the total carbon elemental abundance is C′ = O′ × C/O and the total potassium elemental abundance is K′ = O′ × K/O. Furthermore, we explore the energy redistribution (f ) between the day and night sides of the planet 110 , with values of 0.657, 0.721, 0.791, 0.865, 1.000, 1.030, 1.120, 1.217 and 1.319 in our grid, for which f = 1.0 and 2.0 correspond to full day-to-night heat redistribution and dayside-only redistribution, respectively. The transmission spectrum of the planet is computed with CHIMERA 104,111-115 using the converged atmospheric structures. We compare the observations to these models in a Bayesian inference framework using the nested sampling algorithm MultiNest 108 through its Python implementation PyMultiNest 109 and obtain an optimal set of M/H, C/O ratio, K/O ratio and f through nearest-neighbour search in the grid. When computing the transmission spectrum for a given set of (M/H, C/O ratio, K/O ratio, f), we also adjust the 1-bar planetary radius controlling the absolute transit depth (an arbitrary pressure with no direct impact on the inferred properties; see, for example, ref. 8 99,117 ), CO 2 (ref. 117 ), CO (ref. 88 ), CH 4 (ref. 88 ), H 2 S (ref. 118 ), HCN (ref. 119 ), Na (refs. 120,121 ) and K (refs. 120,122 ), which were computed following the methods described in refs. 123,124 . The cloud models considered are: (1) a basic cloud model with a grey, uniformly vertically distributed cloud opacity (κ cloud ); (2) a grey + power-law cloud model that accounts for non-grey opacity of small-size particles as a vertically uniform power-law opacity (that is, a parameter for the scattering slope and a Rayleigh enhancement factor; for example, refs. 13,36,39,125 ) in addition to grey cloud component, which is expressed by a grey cloud deck of infinite opacity at a given atmospheric pressure; and (3) a droplet sedimentation model 15 (assuming enstatite grains) in which parameters capture the eddy diffusion coefficient and the ratio of sedimentation velocity to characteristic vertical mixing velocity (see also the description of PICASO above). For cloud treatments 2 and 3, we also consider the possibility of inhomogeneities around the planetary limb by considering a linear combination of clear and cloudy models 14 , which is key for breaking degeneracies between metallicity and cloud properties 8,40 . We assume the same PT profile for both cloudy and clear limbs in the inhomogeneous cloud models and leave investigation on the possibility of different PT profiles in those regions to future studies. Identification of absorbers and model selection We perform our Bayesian inference using all model combinations with the ScCHIMERA grid on four different data resolutions for the nirHiss transmission spectrum: R = 100, R = 300, native instrument resolution (R order 1 = 910; R order 2 = 830) and pixel-level resolution (R order 1 = 1,820; R order 2 = 1,660). Resolutions are given at the reference wavelengths of λ = 1.791 μm for order 1 and 0.744 μm for order 2. We test the robustness of our inferences against different binning and convolution strategies and find the results, that is, the bulk atmospheric properties M/H, C/O ratio and K/O ratio, to be consistent regardless of the resolution of the data. We find a fiducial combination of parameters that can best explain the spectrum (that we call the reference model) with full redistribution (f = 1, matching predictions that planets in this temperature regime are unlikely to possess strong day-to-night temperature contrast [126][127][128], M/H = 1.375 (that is, about 20 times solar), C/O ratio of 0.2 and K/O ratio of 0.1. With these atmospheric properties, the data are best explained by the droplet sedimentation model (ScCHIMERA cloud model 3) and inhomogeneous cover. However, the grey + power-law model (ScCHIMERA cloud model 2) with inhomogeneous cover provides a comparable fit to the data. We compare sets of models by computing their Bayes factor and converting to a 'sigma' detection significance using the prescription in refs. 39,40 . Using R = 300 data, the homogeneous droplet sedimentation model (model 3) is preferred over homogeneous grey cloud (model 1) at ≳8σ, which strongly indicates the non-grey nature of cloud opacity. Meanwhile, the inhomogeneous droplet sedimentation model is preferred over the homogeneous droplet sedimentation model cloud at 5σ. This is evidence that, for the same model 3, inhomogeneous cloud coverage is preferred. The inhomogeneous droplet sedimentation model is preferred over all the other tested models across all aforementioned resolutions tested. We explore the contribution of different chemical species to our reference model by performing the Bayesian inference using the inhomogeneous cloud model 3 and artificially disabling the contribution of a selected chemical species, one at a time. By redoing the Bayesian inference, we are able to compare the Bayesian evidence by computing the Bayes factor and converting to a 'sigma' detection significance described above. We detected H 2 O at >30σ, K at 6.8σ and CO at 3.6σ, but no notable detections of Na, CH 4 , CO 2 , HCN and H 2 S. The best-fit metallicity across all models is about 10-30 times solar, the best fit K/O ratio 1-2 times solar and C/O ratio 0.2. Taking the average and standard deviation of the best-fit results for all 20 runs (that is, five models on four data resolutions), we find an average M/H = 19 times solar with a standard deviation of 5 times solar and an average K/O ratio 1.5 times solar with a standard deviation of 0.26 times solar. Wavelength sensitivity to inferences We investigate the dependence of the inferred atmospheric properties on the spectral range of the observations by performing the same Bayesian inferences described above on the spectrum blueward of 2 μm (see Extended Data Fig. 7b). This exercise is repeated on all 20 model-data combinations from ScCHIMERA. With the exception of the solar-to-super-solar K/O ratio, inferences about the atmospheric metallicity, C/O ratio and clouds are primarily driven by the shallower transit depth seen in the λ range 2.1-2.3 μm. This wavelength region is that in which the traces of orders 1 and 2 overlap on the detector. To assess the robustness of our results, we explore different data treatments that could affect the final spectrum. First, we find that there are no zeroth-order background contaminants that could be diluting the transit depth in this region. Second, we extract the transmission spectra and fit for dilution between the orders (supreme-SPOON data reduction) and without accounting for the overlap (supreme-SPOON, nirHiss and transitspectroscopy). The evidence for minimal dilution stems from reducing the data through both methods with the same pipeline (supreme-SPOON), which uses the same steps for the entire reduction process along the way, with the exception of fitting and not fitting for dilution. Both techniques yield similar results in the λ range 2.1-2.3 μm. We note that the contamination from order 2 into order 1 was previously shown to be between 8 and 12 ppm (ref. 50 ) and is therefore negligible. We find that, without the data redward of 2 μm, the M/H value is more scattered across models and resolutions with an average metallicity of 61 times solar for the 20 runs and a standard deviation of 28 times solar. On the other hand, the inference on the C/O ratio remains consistently 0.2 across all models and resolutions. Similarly, the K/O ratio remains solar-to-super-solar, with an average of 1.89 times solar and a standard deviation of 0.29 times solar. These results confirm the necessity for the broad wavelength coverage of NIRISS to constrain the atmospheric metallicity of a planet 8,14,40 . Without the transit depth decrease at 2.1 μm, our models do not exhibit a preference for cloud models 2 and 3 over cloud model 1, nor do they prefer the presence of inhomogeneities in the cloud cover. Without these constraints on the cloud properties, a wide range of metallicities can provide an equally good fit to the observations blueward of 2 μm when combined with different cloud properties, preventing reliable constraints on the metallicity. The exploration of these models is summarized in Extended Data Fig. 7. The top panel shows the different cloud treatments and their goodness of fit to the data. Overall, models with inhomogeneous cloud cover best explain the data, with the flux-balanced cloud of model 3 giving the lowest χ 2 . The bottom panel contrasts the reference model against the results from all cloud models when using data blueward of 2 μm only. Without the information contained in the dip in transit depth at 2.1 μm, all cloud treatments provide an equally good fit and overestimate the transit depth between 2.0 and 2.3 μm. K/O ratio inferences We explore the possibility of constraining the K/O ratio using NIRISS/ SOSS. As explained above, across different models and data resolutions, our results indicate that the observations of WASP-39b are best explained by a solar-to-super-solar K/O ratio. To further explore this, we repeat our Bayesian inference for all 20 model-data configurations (five models each at four resolutions) using the observations blueward of 0.8 μm. From high-resolution to low-resolution observations and for all cloud model configurations, we find that all 20 runs prefer models with solar or super-solar K/O ratios for WASP-39b ranging from 1 to 10 times solar. The average across the 20 runs is 2.12 times solar and a standard deviation of 2.33 times solar, with the relatively larger standard deviation resulting from two inferences of highly super-solar K/O ratios of 7 times solar or greater for observations at pixel-level resolution. Using the reference model atmospheric properties (such as M/H = 1.37, C/O ratio 0.2, full redistribution f = 1), we search for the best-fit K/O ratio while simultaneously adjusting the 1-bar radius and the parameters for the inhomogeneous cloud model 3, when only using the observations blueward of 0.8 μm. The best-fit K/O ratio of 0.4 is consistent with the inferences using all the data and the data blueward of 2.0 μm only. This model is shown in Extended Data Fig. 8 in green. For the best-fit cloud parameters and 1-bar radius, we compute a series of K/O ratios spanning sub-solar and super-solar values. We compute the fit of each model to the data using χ 2 statistics. We then convert the resulting χ 2 value to a P-value. These P-values allow us to estimate the agreement between each model and the data. Our results find that sub-solar K/O ratios are disfavoured to 2σ, whereas super-solar values ≳0.7 are disfavoured to 5σ. Data availability The raw data from this study are publicly available at the Space Telescope Science Institute's Mikulski Archive for Space Telescopes (https:// archive.stsci.edu/). The data used to create all of the figures in this manuscript are freely available on Zenodo and GitHub (Zenodo link: https://github.com/afeinstein20/wasp39b_niriss_paper). All further data are available on request. Code availability The following are open-source pipelines written in Python available either through the Python Package Index (PyPI) or GitHub that were used throughout this work: Eureka! (https://github.com/ kevin218/Eureka); nirHiss (https://github.com/afeinstein20/nirhiss); supreme-SPOON (https://github.com/radicamc/supreme-spoon); transitspectroscopy (https://github.com/nespinoza/transitspectroscopy/ tree/dev); iraclis (https://github.com/ucl-exoplanets/Iraclis); juliet (https://github.com/nespinoza/juliet); chromatic (https://github.com/ zkbt/chromatic); chromatic_fitting (https://github.com/catrionamurray/chromatic_fitting); ExoTiC-LD 13,57 (https://github.com/Exo-TiC/ ExoTiC-LD); ExoTETHyS 125 (https://github.com/ucl-exoplanets/Exo-TETHyS); PICASO 92,93 (https://github.com/natashabatalha/picaso); Virga 98,99 (https://github.com/natashabatalha/virga); CHIMERA (https://github.com/mrline/CHIMERA); PyMultiNest (https://github. com/JohannesBuchner/PyMultiNest); MultiNest (https://github.com/ JohannesBuchner/MultiNest). ). We considered atmospheric metallicities M/H = −1.0, +0.0, +1.0, +1.7, +2.0 and +2.3, C/O ratios 0.35, 0.55, 0.7, 0.75, 1.0 and 1.5, planetary intrinsic temperatures T int = 100, 200, 300 and 400 K and day-night energy redistribution factors of 0.25, 0.50, 0.75 and 1.00, for which full heat redistribution corresponds to 0.5. The model varies the O/H ratio to achieve each C/O ratio with a fixed C/H . We considered atmospheric metallicities of M/H = −1.0, +0.0, +1.0 and +2.0, C/O ratio ranging from 0.3 to 1.0 divided into 136 grid points, planetary intrinsic temperatures T int = 200 and 400 K and day-night energy redistribution factors of 0.172, 0.25 and 0.351, in which full heat redistribution corresponds to 0.25. The model varies the C/H ratio to achieve each C/O ratio with a fixed O/H ratio, which is also scaled to solar metallicity. -98 ) for atmospheric metallicities M/H = −1.0, −0.5, +0.0, +0.5, +1.0, +1.5, +1.7 and +2.0, atmospheric bulk C/O ratios 0.229, 0.458, 0.687 and 1.100, planetary intrinsic temperatures T int = 100, 200 and 300 K and heat redistribution factors of 0.5 and 0.4, in which full heat redistribution corresponds to 0.5. The model fixes the sum of C and O abundances (for example, the (C+O)/H ratio) to that scaled by the metallicity and solar C+O abundance. The model includes 29 chemical species: CH 4 , CO, CO 2 , K, Li, LiCl, LiH, MgH, NH 3 , N 2 , Na, PH 3 , Rb, SiO, TiO and VO. The line lists of several key species are: H 2 O (ref. ). We define our errors as the 1σ uncertainties extracted from the 16th and 84th percentiles of the transit depths fit from each pipeline. The JWST and HST data agree across the three broad H 2 O features that they have in common. We find evidence of a K absorption feature at 0.76 μm in the new JWST data, which was proposed in the previous HST data 18 . (https://github. com/afeinstein20/wasp39b_niriss_paper/blob/main/scripts/figure2.py).Fig. 3 | Interpretation of the constituents of the NIRISS WASP-39b transmission spectrum. a,b, Panel a shows the comparison of the transmission spectrum of WASP-39b from the nirHiss reduction (grey points) with respect to the best-fit reference model (black line). This model assumes an atmospheric metallicity of M/H = 1.38 (23 times the solar value), C/O ratio of 0.2 (0.55 times the solar value 20 ), K/O ratio of 0.1 (1.26 times the solar value), full day-night heat redistribution ( f = 1) and flux-balanced clouds with inhomogeneous terminator coverage. Each coloured line removes a key constituent found in our best-fit reference model to demonstrate how the spectrum would change were these features not included. The removal of clouds and H 2 O absorption from the reference model result in large-scale changes to the shape and depth of the transmission spectrum. Other sources of opacity with an impact on the spectrum are K, CO and CO 2 . Residuals between the data and the reference model are plotted in panel b. c,d, These two panels show the molecular absorption crosssections for a selection of gases observable within the NIRISS bandpass. Panel c highlights gases inferred by our analysis of the spectrum of WASP-39b. Panel d highlights some gases that were not identified in these data but may be present in future observations of other exoplanets. We define our errors as the 1σ uncertainties extracted from the 16th and 84th percentiles of the transit depths fit from each pipeline. (https://github.com/afeinstein20/wasp39b_niriss_paper/ blob/main/scripts/figure3.py).0.60 0.88 1.16 1.44 1.72 2.00 2.30 2.80 Wavelength (μm) 2.05 2.10 2.15 2.20 2.25 Transit depth (%) K H 2 O H 2 O H 2 O H 2 O H 2 O Order 2 O rder 1 HST (Wakeford et al. 2018) nirHiss supreme-SPOON transitspectroscopy Fig. 2 | NIRISS transmission spectra for WASP-39b obtained by three data- reduction pipelines. We find broad agreement in the overall structure of the transmission spectra between several reduction pipelines, a sample of which are presented here (see Extended Data Fig. 5 for all reductions). The JWST data are shown in the coloured points, whereas previous HST observations of WASP-39b (ref. 18 ) are shown in white. We note that we only consider wavelengths <0.85 μm for order 2, as order 1 has much higher fidelity in the overlapping 0.85-1.0-μm range. 2.00 2.05 2.10 2.15 2.20 2.25 Transit depth (%) a Clear atmospheric contribution No CO No CO 2 No H 2 O No K Reference −50 0 50 Residuals (ppm) b −28 −25 −22 log 10 (m 2 ) c H 2 O CO CO 2 Na K 0.60 1.07 1.53 2.00 2.80 Wavelength (μm) −28 −25 −22 log 10 (m 2 ) d CH 4 NH 3 SO 2 PH 3 H 2 S ,49 , which explains the sub-solar C/O ratio and super-solar K/O ratio, along with a super-solar metallicity if a sufficient amount of planetesimals was accreted. However, fully understanding the possible formation2.05 2.10 2.15 2.20 2.25 Transit depth (%) a Carbon-to-oxygen ratio (C/O) 0.20 (ref.) 0.55 0.70 0.80 0.60 0.88 1.16 1.44 1.72 2.00 2.30 2.80 Wavelength (μm) 2.05 2.10 2.15 2.20 2.25 Transit depth (%) b Metallicity (M/H) 1.38 (ref.) 0 1.00 2.00 2.25 Fig. 4 | Impact of the C/O ratio and metallicity on the JWST-NIRISS spectrum of WASP-39b. a, Variation of the C/O ratio in the best-fit reference model while keeping the metallicity, redistribution and K/O ratio parameters from the reference model the same and fitting for the cloud parameters and scaled planetary radius to best explain the observations. Under these equilibrium conditions, increasing the C/O ratio results in less H 2 O and more CH ], [1,500, 1,700-5], [1,700, 2,041]]; n knots,1 = [4, 2, 3, 4] and for order 2: x knots,2 = [[601, 850-5], [850, 1,100-5], [1,100, 1,749]]; n knots,2 = [2, 2, 5]. TiO, VO and atoms up to U. The line list of H 2 O is from BT2 (ref. 85 ), other molecular lines from HITRAN 2008 (ref.The model includes various chemical species: CH, CH 4 , CN, CO, CO 2 , COF, C 2 , C 2 H 2 , C 2 H 4 , C 2 H 6 , CaH, CrH, FeH, HCN, HCl, HF, HI, HDO, HO 2 , H 2, H 2 S, H 2 O, H 2 O 2, H 3 + , MgH, NH, NH 3 , NO, N 2 , N 2 O, OH, O 2 , O 3 , PH 3 , SF 6 , SiH, SiO, SiO 2 , TiH, ) and model different cloud treatments. The opacity sources considered are H 2 -H 2 and H 2 -He CIA 116 , H 2 O (refs. Competing interestsThe authors declare no competing interests.Additional informationCorrespondence and requests for materials should be addressed to Adina D. Feinstein. Peer review information Nature thanks the anonymous reviewers for their contribution to the peer review of this work. Reprints and permissions information is available at http://www.nature.com/reprints.Fig. 1| Comparison of the (x, y) positions of NIRISS orders 1 and 2 across the detector as modelled from different reduction pipelines. a,c, Each pipeline traces the curvature of orders 1 and 2 using different methods. We show the best-fit trace for order 1 in panel a and order 2 in panel b. We highlight zoomed-in regions to further examine differences. We note that iraclis uses the JWST-provided spectral trace. There is generally good agreement between the traces across the entire detector (<1 pixel deviations), with the strongest deviations towards the ends of each trace (for example, x pixel position <500 for orders 1 and 2). This demonstrates the reliability of order spectral traces across all pipelines. b,d, We provide an example spatial profile along the column for order 1 (b) and order 2 (d) at x = 1,250. (https://github.com/afeinstein20/ wasp39b_niriss_paper/blob/main/scripts/edfigure1.py).Extended DataArticleExtended DataFig. 2| Comparison of averaged background frames computed for each reduction pipeline. a, A median out-of-transit integration from the stage 2 output files of the jwst pipeline in data numbers per second (DN s −1 ). b1,c1,d1,e1, Estimated median background frames for four example pipelines: nirHiss (b), supreme-SPOON (c), transitspectroscopy (d) and iraclis (e). All reduction pipelines use the predefined zodiacal background model provided on the STScI JDox User Documentation website. nirHiss estimates the zeroth-order contaminants by taking a smoothed median from the F277W filter integrations. We note that the background frame from supreme-SPOON (c1) for group eight is shown here and was scaled by a factor of about 0.02 to lie on the same scale as the background from the other three pipelines. iraclis subtracts a median per column to remove further 1/f noise. b2,c2,d2,e2, Background-subtracted median integrations for each pipeline, plotted in DN s −1 but scaled from −5 to 5 to highlight subtle changes in the background. For these integrations, we define out of transit as integrations 1-200 and 400-518. (https://github.com/afeinstein20/wasp39b_niriss_paper/blob/main/scripts/ edfigure2.py).Fig. 3| Example extracted stellar spectra from different reduction pipelines. The insets highlight the peak of the spectra. The supreme-SPOON spectra are scaled by a factor of 72 to compare the overall shape of the spectra, rather than the extracted flux counts. a, The extracted stellar spectra from order 1. All pipelines are in relatively good agreement, whereas the shape of the iraclis data changes slightly at λ < 1.1 μm. This is probably owing to different traces that were used in the spectral extraction routine. b, The extracted stellar spectra from order 2. Differences at λ = 0.725 and 0.860 μm are because of differences in removing zeroth-order contaminants in the background. The iraclis pipeline does not extract data past λ > 0.85 μm, which is where the order overlap region begins. Across all pipelines, the shape of the spectra, as well as overall absorption features, cosmic-ray-removal techniques and noise levels are consistent. (https://github.com/afeinstein20/wasp39b_niriss_paper/blob/ main/scripts/edfigure3.py).Extended DataArticleExtended Data Fig. 5 | Transmission spectra for WASP-39b for all reduction techniques.Our best-fit reference model to the nirHiss spectrum (red) is plotted as a solid black line in all panels and the spectra are separated into three panels for ease of reading. Wavelengths that overlap with zeroth-order contaminants are masked. (https://github.com/afeinstein20/wasp39b_niriss_ paper/blob/main/scripts/edfigure5.py).ArticleExtended DataFig. 6| A summary of precomputed forward model fits to the NIRISS/SOSS spectrum. a, Each coloured line shows the best-fit spectrum from the PICASO, ATMO and PHOENIX cloudy grid. The χ 2 values per number of data points (N obs = 327) are χ 2 /N obs = 2.98, 3.24 and 3.51 for the PICASO, ATMO and PHOENIX grids, respectively. All grid models consistently indicate a supersolar metallicity of M/H = 1-2 and a sub-solar C/O ratio. b, The same as the top panel but for the best-fit clear atmosphere models. The clear models yield noticeably worse fits to the data, χ 2 /N obs = 7.02, 4.11 and 8.55 for the PICASO, ATMO and PHOENIX grids, respectively, which strongly indicates the presence of clouds in the atmosphere. (https://github.com/afeinstein20/wasp39b_ niriss_paper/blob/main/scripts/edfigure6.py).Fig. 7| A demonstration of how the redder wavelength coverage of NIRISS/SOSS drives the inference on cloud structure for WASP-39b. We fit the NIRISS/SOSS spectrum (grey) using a suite of cloud models to derive the best-fit C/O ratio and metallicity. Here we demonstrate how the best-fit model for each cloud treatment changes as a function of what wavelength region we fit. a, The best-fit models when using the entire wavelength coverage of NIRISS/SOSS. b, The best-fit models when using λ < 2 μm, which excludes the overlapping region between orders on the detector. The reference spectrum (black) on both panels corresponds to the best-fit inhomogeneous droplet sedimentation model for the entire wavelength coverage. The fitted data are presented as dark-grey points. The quoted numbers in brackets in the legend are the respective χ 2 /N for each fit for the top (left value) and bottom (right value). The difference between cloud models is within the noise of the NIRISS/ SOSS data when fitting to λ < 2 μm. It is clear that fitting the entire NIRISS/SOSS wavelength coverage results in a lower χ 2 /N and better fit. (https://github.com/ afeinstein20/wasp39b_niriss_paper/blob/main/scripts/edfigure7.py).Extended DataArticleExtended Data Fig. 8 | Evidence for super-solar K/O ratio in WASP-39b.We fit for the K/O ratio while keeping the rest of the model parameters (for example, C/O ratio, metallicity and redistribution) the same as our reference model and fitting for the cloud parameters and scaled planetary radius. Here we present the different K/O ratio models (solid lines) we fit against the transmission spectrum at R = 300 (black and white points). We represent the respective fit of each model in the orange shading. (https://github.com/ afeinstein20/wasp39b_niriss_paper/blob/main/scripts/edfigure8.py).Table 1| An outline of reduction and fitting pipelines used to produce transmission spectra for WASP-39b with NIRISS/SOSS The size of the box aperture is listed in parentheses when appropriate. 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[ "Lambda hyperon production and polarization in collisions of p (3.5 GeV) + Nb HADES collaboration", "Lambda hyperon production and polarization in collisions of p (3.5 GeV) + Nb HADES collaboration" ]
[ "G Agakishiev \nJoint Institute of Nuclear Research\n141980DubnaRussia\n", "O Arnold \nExcellence Cluster 'Origin and Structure of the Universe'\n85748GarchingGermany\n", "A Balanda \nSmoluchowski Institute of Physics\nJagiellonian University of Cracow\n30-059KrakówPoland\n", "D Belver \nLabCAF F. Física\nUniv. de Santiago de Compostela\n15706Santiago de CompostelaSpain\n", "A V Belyaev \nJoint Institute of Nuclear Research\n141980DubnaRussia\n", "J C Berger-Chen \nExcellence Cluster 'Origin and Structure of the Universe'\n85748GarchingGermany\n", "A Blanco \nLIP-Laboratório de Instrumentação\n\n", "M Böhmer \nPhysik Department E12\nTechnische Universität München\n85748GarchingGermany\n", "J L Boyard \nInstitut de Physique Nucléaire (UMR 8608)\nCNRS\nIN2P3\nUniversité Paris Sud\nF-91406Orsay CedexFrance\n", "P Cabanelas \nLabCAF F. Física\nUniv. de Santiago de Compostela\n15706Santiago de CompostelaSpain\n\nalso at Nuclear Physics Center\nExtreMe Matter Institute EMMI\nDipartimento di Fisica\nUniversity of Lisbon\n1649-013, 64291Lisboa, DarmstadtPortugal, Germany\n\nUniversità di Milano\n20133MilanoItaly\n\nTechnische Universität Dresden\n01062DresdenGermany\n\nFrederick University\n1036NikosiaCyprus\n", "S Chernenko \nJoint Institute of Nuclear Research\n141980DubnaRussia\n", "A Dybczak \nSmoluchowski Institute of Physics\nJagiellonian University of Cracow\n30-059KrakówPoland\n", "E Epple \nExcellence Cluster 'Origin and Structure of the Universe'\n85748GarchingGermany\n", "L Fabbietti \nExcellence Cluster 'Origin and Structure of the Universe'\n85748GarchingGermany\n", "O V Fateev \nJoint Institute of Nuclear Research\n141980DubnaRussia\n", "P Finocchiaro \nInstituto Nazionale di Fisica Nucleare -Laboratori Nazionali del Sud\n95125CataniaItaly\n", "P Fonte \nLIP-Laboratório de Instrumentação\n\n", "J Friese \nPhysik Department E12\nTechnische Universität München\n85748GarchingGermany\n", "I Fröhlich \nInstitut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany\n", "T Galatyuk \nTechnische Universität Darmstadt\n64289DarmstadtGermany\n", "J A Garzón \nLabCAF F. Física\nUniv. de Santiago de Compostela\n15706Santiago de CompostelaSpain\n", "R Gernhäuser \nPhysik Department E12\nTechnische Universität München\n85748GarchingGermany\n", "K Göbel \nInstitut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany\n", "M Golubeva \nInstitute for Nuclear Research\nRussian Academy of Science\n117312MoscowRussia\n", "D González-Díaz \nTechnische Universität Darmstadt\n64289DarmstadtGermany\n", "F Guber \nInstitute for Nuclear Research\nRussian Academy of Science\n117312MoscowRussia\n", "M Gumberidze \nTechnische Universität Darmstadt\n64289DarmstadtGermany\n\nInstitut de Physique Nucléaire (UMR 8608)\nCNRS\nIN2P3\nUniversité Paris Sud\nF-91406Orsay CedexFrance\n", "T Heinz \nGSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany\n", "T Hennino \nInstitut de Physique Nucléaire (UMR 8608)\nCNRS\nIN2P3\nUniversité Paris Sud\nF-91406Orsay CedexFrance\n", "R Holzmann \nGSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany\n", "A Ierusalimov \nJoint Institute of Nuclear Research\n141980DubnaRussia\n", "I Iori \nIstituto Nazionale di Fisica Nucleare\nSezione di Milano\n20133MilanoItaly\n", "A Ivashkin \nInstitute for Nuclear Research\nRussian Academy of Science\n117312MoscowRussia\n", "M Jurkovic \nPhysik Department E12\nTechnische Universität München\n85748GarchingGermany\n", "B Kämpfer \nFísica Experimental de Partículas\n3004-516CoimbraPortugal\n\nInstitut für Strahlenphysik\nHelmholtz-Zentrum Dresden-Rossendorf01314DresdenGermany\n", "T Karavicheva \nInstitute for Nuclear Research\nRussian Academy of Science\n117312MoscowRussia\n", "I Koenig \nGSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany\n", "W Koenig ", "B W Kolb \nGSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany\n", "G Kornakov \nGSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany\n\nLabCAF F. Física\nUniv. de Santiago de Compostela\n15706Santiago de CompostelaSpain\n", "R Kotte \nInstitut für Strahlenphysik\nHelmholtz-Zentrum Dresden-Rossendorf01314DresdenGermany\n", "A Krása \nNuclear Physics Institute\nAcademy of Sciences of Czech Republic\n25068RezCzech Republic\n", "F Krizek \nNuclear Physics Institute\nAcademy of Sciences of Czech Republic\n25068RezCzech Republic\n", "R Krücken \nPhysik Department E12\nTechnische Universität München\n85748GarchingGermany\n", "H Kuc \nSmoluchowski Institute of Physics\nJagiellonian University of Cracow\n30-059KrakówPoland\n\nInstitut de Physique Nucléaire (UMR 8608)\nCNRS\nIN2P3\nUniversité Paris Sud\nF-91406Orsay CedexFrance\n", "W Kühn \nII.Physikalisches Institut\nJustus Liebig Universität Giessen\n35392GiessenGermany\n", "A Kugler \nNuclear Physics Institute\nAcademy of Sciences of Czech Republic\n25068RezCzech Republic\n", "A Kurepin \nInstitute for Nuclear Research\nRussian Academy of Science\n117312MoscowRussia\n", "V Ladygin \nJoint Institute of Nuclear Research\n141980DubnaRussia\n", "R Lalik \nExcellence Cluster 'Origin and Structure of the Universe'\n85748GarchingGermany\n", "S Lang \nGSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany\n", "K Lapidus \nExcellence Cluster 'Origin and Structure of the Universe'\n85748GarchingGermany\n", "A Lebedev \nInstitute of Theoretical and Experimental Physics\n117218MoscowRussia\n", "T Liu \nInstitut de Physique Nucléaire (UMR 8608)\nCNRS\nIN2P3\nUniversité Paris Sud\nF-91406Orsay CedexFrance\n", "L Lopes \nLIP-Laboratório de Instrumentação\n\n", "M Lorenz \nInstitut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany\n", "L Maier \nPhysik Department E12\nTechnische Universität München\n85748GarchingGermany\n", "A Mangiarotti \nLIP-Laboratório de Instrumentação\n\n", "J Markert \nInstitut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany\n", "V Metag \nII.Physikalisches Institut\nJustus Liebig Universität Giessen\n35392GiessenGermany\n", "B Michalska \nSmoluchowski Institute of Physics\nJagiellonian University of Cracow\n30-059KrakówPoland\n", "J Michel \nInstitut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany\n", "C Müntz \nJoint Institute of Nuclear Research\n141980DubnaRussia\n", "L Naumann \nInstitut für Strahlenphysik\nHelmholtz-Zentrum Dresden-Rossendorf01314DresdenGermany\n", "Y C Pachmayer \nInstitut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany\n", "M Palka \nSmoluchowski Institute of Physics\nJagiellonian University of Cracow\n30-059KrakówPoland\n", "Y Parpottas \nDepartment of Physics\nUniversity of Cyprus\n1678NicosiaCyprus\n", "V Pechenov \nGSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany\n", "O Pechenova \nInstitut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany\n", "J Pietraszko \nGSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany\n", "W Przygoda \nSmoluchowski Institute of Physics\nJagiellonian University of Cracow\n30-059KrakówPoland\n", "B Ramstein \nInstitut de Physique Nucléaire (UMR 8608)\nCNRS\nIN2P3\nUniversité Paris Sud\nF-91406Orsay CedexFrance\n", "A Reshetin \nInstitute for Nuclear Research\nRussian Academy of Science\n117312MoscowRussia\n", "A Rustamov \nInstitut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany\n", "A Sadovsky \nInstitute for Nuclear Research\nRussian Academy of Science\n117312MoscowRussia\n", "P Salabura \nSmoluchowski Institute of Physics\nJagiellonian University of Cracow\n30-059KrakówPoland\n", "A Schmah \nExcellence Cluster 'Origin and Structure of the Universe'\n85748GarchingGermany\n\nnow at Lawrence Berkeley National Laboratory\nBerkeleyUSA\n", "E Schwab \nGSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany\n", "J Siebenson \nExcellence Cluster 'Origin and Structure of the Universe'\n85748GarchingGermany\n", "Yu G Sobolev \nNuclear Physics Institute\nAcademy of Sciences of Czech Republic\n25068RezCzech Republic\n", "S Spataro \nII.Physikalisches Institut\nJustus Liebig Universität Giessen\n35392GiessenGermany\n\nnow at Dipartimento di Fisica Generale and INFN\nUniversità di Torino\n10125TorinoItaly\n", "B Spruck \nII.Physikalisches Institut\nJustus Liebig Universität Giessen\n35392GiessenGermany\n", "H Ströbele \nInstitut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany\n", "J Stroth \nGSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany\n\nInstitut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany\n", "C Sturm \nGSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany\n", "A Tarantola \nInstitut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany\n", "K Teilab \nInstitut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany\n", "P Tlusty \nNuclear Physics Institute\nAcademy of Sciences of Czech Republic\n25068RezCzech Republic\n", "M Traxler \nGSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany\n", "R Trebacz \nSmoluchowski Institute of Physics\nJagiellonian University of Cracow\n30-059KrakówPoland\n", "H Tsertos \nDepartment of Physics\nUniversity of Cyprus\n1678NicosiaCyprus\n", "T Vasiliev \nJoint Institute of Nuclear Research\n141980DubnaRussia\n", "V Wagner \nNuclear Physics Institute\nAcademy of Sciences of Czech Republic\n25068RezCzech Republic\n", "M Weber \nPhysik Department E12\nTechnische Universität München\n85748GarchingGermany\n", "C Wendisch [email protected] \nFísica Experimental de Partículas\n3004-516CoimbraPortugal\n\nInstitut für Strahlenphysik\nHelmholtz-Zentrum Dresden-Rossendorf01314DresdenGermany\n", "J Wüstenfeld \nInstitut für Strahlenphysik\nHelmholtz-Zentrum Dresden-Rossendorf01314DresdenGermany\n", "S Yurevich \nGSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany\n", "Y V Zanevsky \nJoint Institute of Nuclear Research\n141980DubnaRussia\n" ]
[ "Joint Institute of Nuclear Research\n141980DubnaRussia", "Excellence Cluster 'Origin and Structure of the Universe'\n85748GarchingGermany", "Smoluchowski Institute of Physics\nJagiellonian University of Cracow\n30-059KrakówPoland", "LabCAF F. Física\nUniv. de Santiago de Compostela\n15706Santiago de CompostelaSpain", "Joint Institute of Nuclear Research\n141980DubnaRussia", "Excellence Cluster 'Origin and Structure of the Universe'\n85748GarchingGermany", "LIP-Laboratório de Instrumentação\n", "Physik Department E12\nTechnische Universität München\n85748GarchingGermany", "Institut de Physique Nucléaire (UMR 8608)\nCNRS\nIN2P3\nUniversité Paris Sud\nF-91406Orsay CedexFrance", "LabCAF F. Física\nUniv. de Santiago de Compostela\n15706Santiago de CompostelaSpain", "also at Nuclear Physics Center\nExtreMe Matter Institute EMMI\nDipartimento di Fisica\nUniversity of Lisbon\n1649-013, 64291Lisboa, DarmstadtPortugal, Germany", "Università di Milano\n20133MilanoItaly", "Technische Universität Dresden\n01062DresdenGermany", "Frederick University\n1036NikosiaCyprus", "Joint Institute of Nuclear Research\n141980DubnaRussia", "Smoluchowski Institute of Physics\nJagiellonian University of Cracow\n30-059KrakówPoland", "Excellence Cluster 'Origin and Structure of the Universe'\n85748GarchingGermany", "Excellence Cluster 'Origin and Structure of the Universe'\n85748GarchingGermany", "Joint Institute of Nuclear Research\n141980DubnaRussia", "Instituto Nazionale di Fisica Nucleare -Laboratori Nazionali del Sud\n95125CataniaItaly", "LIP-Laboratório de Instrumentação\n", "Physik Department E12\nTechnische Universität München\n85748GarchingGermany", "Institut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany", "Technische Universität Darmstadt\n64289DarmstadtGermany", "LabCAF F. Física\nUniv. de Santiago de Compostela\n15706Santiago de CompostelaSpain", "Physik Department E12\nTechnische Universität München\n85748GarchingGermany", "Institut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany", "Institute for Nuclear Research\nRussian Academy of Science\n117312MoscowRussia", "Technische Universität Darmstadt\n64289DarmstadtGermany", "Institute for Nuclear Research\nRussian Academy of Science\n117312MoscowRussia", "Technische Universität Darmstadt\n64289DarmstadtGermany", "Institut de Physique Nucléaire (UMR 8608)\nCNRS\nIN2P3\nUniversité Paris Sud\nF-91406Orsay CedexFrance", "GSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany", "Institut de Physique Nucléaire (UMR 8608)\nCNRS\nIN2P3\nUniversité Paris Sud\nF-91406Orsay CedexFrance", "GSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany", "Joint Institute of Nuclear Research\n141980DubnaRussia", "Istituto Nazionale di Fisica Nucleare\nSezione di Milano\n20133MilanoItaly", "Institute for Nuclear Research\nRussian Academy of Science\n117312MoscowRussia", "Physik Department E12\nTechnische Universität München\n85748GarchingGermany", "Física Experimental de Partículas\n3004-516CoimbraPortugal", "Institut für Strahlenphysik\nHelmholtz-Zentrum Dresden-Rossendorf01314DresdenGermany", "Institute for Nuclear Research\nRussian Academy of Science\n117312MoscowRussia", "GSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany", "GSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany", "GSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany", "LabCAF F. Física\nUniv. de Santiago de Compostela\n15706Santiago de CompostelaSpain", "Institut für Strahlenphysik\nHelmholtz-Zentrum Dresden-Rossendorf01314DresdenGermany", "Nuclear Physics Institute\nAcademy of Sciences of Czech Republic\n25068RezCzech Republic", "Nuclear Physics Institute\nAcademy of Sciences of Czech Republic\n25068RezCzech Republic", "Physik Department E12\nTechnische Universität München\n85748GarchingGermany", "Smoluchowski Institute of Physics\nJagiellonian University of Cracow\n30-059KrakówPoland", "Institut de Physique Nucléaire (UMR 8608)\nCNRS\nIN2P3\nUniversité Paris Sud\nF-91406Orsay CedexFrance", "II.Physikalisches Institut\nJustus Liebig Universität Giessen\n35392GiessenGermany", "Nuclear Physics Institute\nAcademy of Sciences of Czech Republic\n25068RezCzech Republic", "Institute for Nuclear Research\nRussian Academy of Science\n117312MoscowRussia", "Joint Institute of Nuclear Research\n141980DubnaRussia", "Excellence Cluster 'Origin and Structure of the Universe'\n85748GarchingGermany", "GSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany", "Excellence Cluster 'Origin and Structure of the Universe'\n85748GarchingGermany", "Institute of Theoretical and Experimental Physics\n117218MoscowRussia", "Institut de Physique Nucléaire (UMR 8608)\nCNRS\nIN2P3\nUniversité Paris Sud\nF-91406Orsay CedexFrance", "LIP-Laboratório de Instrumentação\n", "Institut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany", "Physik Department E12\nTechnische Universität München\n85748GarchingGermany", "LIP-Laboratório de Instrumentação\n", "Institut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany", "II.Physikalisches Institut\nJustus Liebig Universität Giessen\n35392GiessenGermany", "Smoluchowski Institute of Physics\nJagiellonian University of Cracow\n30-059KrakówPoland", "Institut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany", "Joint Institute of Nuclear Research\n141980DubnaRussia", "Institut für Strahlenphysik\nHelmholtz-Zentrum Dresden-Rossendorf01314DresdenGermany", "Institut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany", "Smoluchowski Institute of Physics\nJagiellonian University of Cracow\n30-059KrakówPoland", "Department of Physics\nUniversity of Cyprus\n1678NicosiaCyprus", "GSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany", "Institut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany", "GSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany", "Smoluchowski Institute of Physics\nJagiellonian University of Cracow\n30-059KrakówPoland", "Institut de Physique Nucléaire (UMR 8608)\nCNRS\nIN2P3\nUniversité Paris Sud\nF-91406Orsay CedexFrance", "Institute for Nuclear Research\nRussian Academy of Science\n117312MoscowRussia", "Institut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany", "Institute for Nuclear Research\nRussian Academy of Science\n117312MoscowRussia", "Smoluchowski Institute of Physics\nJagiellonian University of Cracow\n30-059KrakówPoland", "Excellence Cluster 'Origin and Structure of the Universe'\n85748GarchingGermany", "now at Lawrence Berkeley National Laboratory\nBerkeleyUSA", "GSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany", "Excellence Cluster 'Origin and Structure of the Universe'\n85748GarchingGermany", "Nuclear Physics Institute\nAcademy of Sciences of Czech Republic\n25068RezCzech Republic", "II.Physikalisches Institut\nJustus Liebig Universität Giessen\n35392GiessenGermany", "now at Dipartimento di Fisica Generale and INFN\nUniversità di Torino\n10125TorinoItaly", "II.Physikalisches Institut\nJustus Liebig Universität Giessen\n35392GiessenGermany", "Institut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany", "GSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany", "Institut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany", "GSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany", "Institut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany", "Institut für Kernphysik\nJohann Wolfgang Goethe-Universität\n60438FrankfurtGermany", "Nuclear Physics Institute\nAcademy of Sciences of Czech Republic\n25068RezCzech Republic", "GSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany", "Smoluchowski Institute of Physics\nJagiellonian University of Cracow\n30-059KrakówPoland", "Department of Physics\nUniversity of Cyprus\n1678NicosiaCyprus", "Joint Institute of Nuclear Research\n141980DubnaRussia", "Nuclear Physics Institute\nAcademy of Sciences of Czech Republic\n25068RezCzech Republic", "Physik Department E12\nTechnische Universität München\n85748GarchingGermany", "Física Experimental de Partículas\n3004-516CoimbraPortugal", "Institut für Strahlenphysik\nHelmholtz-Zentrum Dresden-Rossendorf01314DresdenGermany", "Institut für Strahlenphysik\nHelmholtz-Zentrum Dresden-Rossendorf01314DresdenGermany", "GSI Helmholtzzentrum für Schwerionenforschung GmbH\n64291DarmstadtGermany", "Joint Institute of Nuclear Research\n141980DubnaRussia" ]
[]
Results on Λ hyperon production are reported for collisions of p (3.5 GeV) + Nb, studied with the High Acceptance Di-Electron Spectrometer (HADES) at SIS18 at GSI Helmholtzzentrum for Heavy-Ion Research, Darmstadt. The transverse mass distributions in rapidity bins are well described by Boltzmann shapes with a maximum inverse slope parameter of about 90 MeV at a rapidity of y = 1.0, i.e. slightly below the center-of-mass rapidity for nucleonnucleon collisions, ycm = 1.12. The rapidity density decreases monotonically with increasing rapidity within a rapidity window ranging from 0.3 to 1.3. The Λ phase-space distribution is compared with results of other experiments and with predictions of two transport approaches which are available publicly. None of the present versions of the employed models is able to fully reproduce the experimental distributions, i.e. in absolute yield and in shape. Presumably, this finding results from an insufficient modelling in the transport models of the elementary processes being relevant for Λ production, rescattering and absorption. The present high-statistics data allow for a genuine two-dimensional investigation as a function of phase space of the self-analyzing Λ polarization in the weak decay Λ → pπ − . Finite negative values of the polarization in the order of 5 − 20 % are observed over the entire phase space studied. The absolute value of the polarization increases almost linearly with increasing transverse momentum for pt > 300 MeV/c and increases with decreasing rapidity for y < 0.8. PACS. 25.75.Dw, 25.75.Gz
10.1140/epja/i2014-14081-2
[ "https://arxiv.org/pdf/1404.3014v2.pdf" ]
53,554,206
1404.3014
9e04afdb6a133f5dff781cb45c79d204e4b0756e
Lambda hyperon production and polarization in collisions of p (3.5 GeV) + Nb HADES collaboration 14 Apr 2014 G Agakishiev Joint Institute of Nuclear Research 141980DubnaRussia O Arnold Excellence Cluster 'Origin and Structure of the Universe' 85748GarchingGermany A Balanda Smoluchowski Institute of Physics Jagiellonian University of Cracow 30-059KrakówPoland D Belver LabCAF F. Física Univ. de Santiago de Compostela 15706Santiago de CompostelaSpain A V Belyaev Joint Institute of Nuclear Research 141980DubnaRussia J C Berger-Chen Excellence Cluster 'Origin and Structure of the Universe' 85748GarchingGermany A Blanco LIP-Laboratório de Instrumentação M Böhmer Physik Department E12 Technische Universität München 85748GarchingGermany J L Boyard Institut de Physique Nucléaire (UMR 8608) CNRS IN2P3 Université Paris Sud F-91406Orsay CedexFrance P Cabanelas LabCAF F. Física Univ. de Santiago de Compostela 15706Santiago de CompostelaSpain also at Nuclear Physics Center ExtreMe Matter Institute EMMI Dipartimento di Fisica University of Lisbon 1649-013, 64291Lisboa, DarmstadtPortugal, Germany Università di Milano 20133MilanoItaly Technische Universität Dresden 01062DresdenGermany Frederick University 1036NikosiaCyprus S Chernenko Joint Institute of Nuclear Research 141980DubnaRussia A Dybczak Smoluchowski Institute of Physics Jagiellonian University of Cracow 30-059KrakówPoland E Epple Excellence Cluster 'Origin and Structure of the Universe' 85748GarchingGermany L Fabbietti Excellence Cluster 'Origin and Structure of the Universe' 85748GarchingGermany O V Fateev Joint Institute of Nuclear Research 141980DubnaRussia P Finocchiaro Instituto Nazionale di Fisica Nucleare -Laboratori Nazionali del Sud 95125CataniaItaly P Fonte LIP-Laboratório de Instrumentação J Friese Physik Department E12 Technische Universität München 85748GarchingGermany I Fröhlich Institut für Kernphysik Johann Wolfgang Goethe-Universität 60438FrankfurtGermany T Galatyuk Technische Universität Darmstadt 64289DarmstadtGermany J A Garzón LabCAF F. Física Univ. de Santiago de Compostela 15706Santiago de CompostelaSpain R Gernhäuser Physik Department E12 Technische Universität München 85748GarchingGermany K Göbel Institut für Kernphysik Johann Wolfgang Goethe-Universität 60438FrankfurtGermany M Golubeva Institute for Nuclear Research Russian Academy of Science 117312MoscowRussia D González-Díaz Technische Universität Darmstadt 64289DarmstadtGermany F Guber Institute for Nuclear Research Russian Academy of Science 117312MoscowRussia M Gumberidze Technische Universität Darmstadt 64289DarmstadtGermany Institut de Physique Nucléaire (UMR 8608) CNRS IN2P3 Université Paris Sud F-91406Orsay CedexFrance T Heinz GSI Helmholtzzentrum für Schwerionenforschung GmbH 64291DarmstadtGermany T Hennino Institut de Physique Nucléaire (UMR 8608) CNRS IN2P3 Université Paris Sud F-91406Orsay CedexFrance R Holzmann GSI Helmholtzzentrum für Schwerionenforschung GmbH 64291DarmstadtGermany A Ierusalimov Joint Institute of Nuclear Research 141980DubnaRussia I Iori Istituto Nazionale di Fisica Nucleare Sezione di Milano 20133MilanoItaly A Ivashkin Institute for Nuclear Research Russian Academy of Science 117312MoscowRussia M Jurkovic Physik Department E12 Technische Universität München 85748GarchingGermany B Kämpfer Física Experimental de Partículas 3004-516CoimbraPortugal Institut für Strahlenphysik Helmholtz-Zentrum Dresden-Rossendorf01314DresdenGermany T Karavicheva Institute for Nuclear Research Russian Academy of Science 117312MoscowRussia I Koenig GSI Helmholtzzentrum für Schwerionenforschung GmbH 64291DarmstadtGermany W Koenig B W Kolb GSI Helmholtzzentrum für Schwerionenforschung GmbH 64291DarmstadtGermany G Kornakov GSI Helmholtzzentrum für Schwerionenforschung GmbH 64291DarmstadtGermany LabCAF F. Física Univ. de Santiago de Compostela 15706Santiago de CompostelaSpain R Kotte Institut für Strahlenphysik Helmholtz-Zentrum Dresden-Rossendorf01314DresdenGermany A Krása Nuclear Physics Institute Academy of Sciences of Czech Republic 25068RezCzech Republic F Krizek Nuclear Physics Institute Academy of Sciences of Czech Republic 25068RezCzech Republic R Krücken Physik Department E12 Technische Universität München 85748GarchingGermany H Kuc Smoluchowski Institute of Physics Jagiellonian University of Cracow 30-059KrakówPoland Institut de Physique Nucléaire (UMR 8608) CNRS IN2P3 Université Paris Sud F-91406Orsay CedexFrance W Kühn II.Physikalisches Institut Justus Liebig Universität Giessen 35392GiessenGermany A Kugler Nuclear Physics Institute Academy of Sciences of Czech Republic 25068RezCzech Republic A Kurepin Institute for Nuclear Research Russian Academy of Science 117312MoscowRussia V Ladygin Joint Institute of Nuclear Research 141980DubnaRussia R Lalik Excellence Cluster 'Origin and Structure of the Universe' 85748GarchingGermany S Lang GSI Helmholtzzentrum für Schwerionenforschung GmbH 64291DarmstadtGermany K Lapidus Excellence Cluster 'Origin and Structure of the Universe' 85748GarchingGermany A Lebedev Institute of Theoretical and Experimental Physics 117218MoscowRussia T Liu Institut de Physique Nucléaire (UMR 8608) CNRS IN2P3 Université Paris Sud F-91406Orsay CedexFrance L Lopes LIP-Laboratório de Instrumentação M Lorenz Institut für Kernphysik Johann Wolfgang Goethe-Universität 60438FrankfurtGermany L Maier Physik Department E12 Technische Universität München 85748GarchingGermany A Mangiarotti LIP-Laboratório de Instrumentação J Markert Institut für Kernphysik Johann Wolfgang Goethe-Universität 60438FrankfurtGermany V Metag II.Physikalisches Institut Justus Liebig Universität Giessen 35392GiessenGermany B Michalska Smoluchowski Institute of Physics Jagiellonian University of Cracow 30-059KrakówPoland J Michel Institut für Kernphysik Johann Wolfgang Goethe-Universität 60438FrankfurtGermany C Müntz Joint Institute of Nuclear Research 141980DubnaRussia L Naumann Institut für Strahlenphysik Helmholtz-Zentrum Dresden-Rossendorf01314DresdenGermany Y C Pachmayer Institut für Kernphysik Johann Wolfgang Goethe-Universität 60438FrankfurtGermany M Palka Smoluchowski Institute of Physics Jagiellonian University of Cracow 30-059KrakówPoland Y Parpottas Department of Physics University of Cyprus 1678NicosiaCyprus V Pechenov GSI Helmholtzzentrum für Schwerionenforschung GmbH 64291DarmstadtGermany O Pechenova Institut für Kernphysik Johann Wolfgang Goethe-Universität 60438FrankfurtGermany J Pietraszko GSI Helmholtzzentrum für Schwerionenforschung GmbH 64291DarmstadtGermany W Przygoda Smoluchowski Institute of Physics Jagiellonian University of Cracow 30-059KrakówPoland B Ramstein Institut de Physique Nucléaire (UMR 8608) CNRS IN2P3 Université Paris Sud F-91406Orsay CedexFrance A Reshetin Institute for Nuclear Research Russian Academy of Science 117312MoscowRussia A Rustamov Institut für Kernphysik Johann Wolfgang Goethe-Universität 60438FrankfurtGermany A Sadovsky Institute for Nuclear Research Russian Academy of Science 117312MoscowRussia P Salabura Smoluchowski Institute of Physics Jagiellonian University of Cracow 30-059KrakówPoland A Schmah Excellence Cluster 'Origin and Structure of the Universe' 85748GarchingGermany now at Lawrence Berkeley National Laboratory BerkeleyUSA E Schwab GSI Helmholtzzentrum für Schwerionenforschung GmbH 64291DarmstadtGermany J Siebenson Excellence Cluster 'Origin and Structure of the Universe' 85748GarchingGermany Yu G Sobolev Nuclear Physics Institute Academy of Sciences of Czech Republic 25068RezCzech Republic S Spataro II.Physikalisches Institut Justus Liebig Universität Giessen 35392GiessenGermany now at Dipartimento di Fisica Generale and INFN Università di Torino 10125TorinoItaly B Spruck II.Physikalisches Institut Justus Liebig Universität Giessen 35392GiessenGermany H Ströbele Institut für Kernphysik Johann Wolfgang Goethe-Universität 60438FrankfurtGermany J Stroth GSI Helmholtzzentrum für Schwerionenforschung GmbH 64291DarmstadtGermany Institut für Kernphysik Johann Wolfgang Goethe-Universität 60438FrankfurtGermany C Sturm GSI Helmholtzzentrum für Schwerionenforschung GmbH 64291DarmstadtGermany A Tarantola Institut für Kernphysik Johann Wolfgang Goethe-Universität 60438FrankfurtGermany K Teilab Institut für Kernphysik Johann Wolfgang Goethe-Universität 60438FrankfurtGermany P Tlusty Nuclear Physics Institute Academy of Sciences of Czech Republic 25068RezCzech Republic M Traxler GSI Helmholtzzentrum für Schwerionenforschung GmbH 64291DarmstadtGermany R Trebacz Smoluchowski Institute of Physics Jagiellonian University of Cracow 30-059KrakówPoland H Tsertos Department of Physics University of Cyprus 1678NicosiaCyprus T Vasiliev Joint Institute of Nuclear Research 141980DubnaRussia V Wagner Nuclear Physics Institute Academy of Sciences of Czech Republic 25068RezCzech Republic M Weber Physik Department E12 Technische Universität München 85748GarchingGermany C Wendisch [email protected] Física Experimental de Partículas 3004-516CoimbraPortugal Institut für Strahlenphysik Helmholtz-Zentrum Dresden-Rossendorf01314DresdenGermany J Wüstenfeld Institut für Strahlenphysik Helmholtz-Zentrum Dresden-Rossendorf01314DresdenGermany S Yurevich GSI Helmholtzzentrum für Schwerionenforschung GmbH 64291DarmstadtGermany Y V Zanevsky Joint Institute of Nuclear Research 141980DubnaRussia Lambda hyperon production and polarization in collisions of p (3.5 GeV) + Nb HADES collaboration 14 Apr 2014Received: April 15, 2014EPJ manuscript No. (will be inserted by the editor) 2 The HADES collaboration (G. Agakishiev et al.): Lambda hyperon production and polarization in collisions of p (3.5 GeV) + Nb Results on Λ hyperon production are reported for collisions of p (3.5 GeV) + Nb, studied with the High Acceptance Di-Electron Spectrometer (HADES) at SIS18 at GSI Helmholtzzentrum for Heavy-Ion Research, Darmstadt. The transverse mass distributions in rapidity bins are well described by Boltzmann shapes with a maximum inverse slope parameter of about 90 MeV at a rapidity of y = 1.0, i.e. slightly below the center-of-mass rapidity for nucleonnucleon collisions, ycm = 1.12. The rapidity density decreases monotonically with increasing rapidity within a rapidity window ranging from 0.3 to 1.3. The Λ phase-space distribution is compared with results of other experiments and with predictions of two transport approaches which are available publicly. None of the present versions of the employed models is able to fully reproduce the experimental distributions, i.e. in absolute yield and in shape. Presumably, this finding results from an insufficient modelling in the transport models of the elementary processes being relevant for Λ production, rescattering and absorption. The present high-statistics data allow for a genuine two-dimensional investigation as a function of phase space of the self-analyzing Λ polarization in the weak decay Λ → pπ − . Finite negative values of the polarization in the order of 5 − 20 % are observed over the entire phase space studied. The absolute value of the polarization increases almost linearly with increasing transverse momentum for pt > 300 MeV/c and increases with decreasing rapidity for y < 0.8. PACS. 25.75.Dw, 25.75.Gz Introduction During the past two decades strangeness carrying particles which are produced in relativistic heavy-ion reactions at energies around the corresponding production thresholds in nucleon-nucleon (NN) collisions attracted strong attention from both experimental and theoretical sides [1,2,3,4,5] since their production and propagation are expected to provide insight into the nature of hot and dense nuclear matter. For instance, at energies available at SIS18/GSI Darmstadt densities up to three times that of ground-state nuclear matter and temperatures up to ∼ 100 MeV can be achieved. In the pioneering work of the KaoS collaboration at SIS18 on sub-threshold kaon production [6] and based on comparisons to transport approaches [7,8] a soft equation of state (EoS) was extracted as predicted already long before [9,10]. A strong focus was put on the study of in-medium properties of kaons and antikaons. Especially, their collective flow and spectral shapes at low transverse momenta are expected to be altered within the nuclear environment. The corresponding nuclear mean-field potential acting on the kaons and antikaons are predicted to be weakly positive (∼ 20 to 40 MeV) and strongly negative (∼ −50 to −80 MeV), respectively [11,12]. Kaon and antikaon phase-space distributions [1,13,14,15,16,17,18] as well as azimuthal emission (flow) patterns of kaons [1,19,20,21] support these predictions while the antikaon flow data are not conclusive yet. The tight interplay of experimental and and theoretical efforts revealed the importance of strangeness-exchange reactions for antikaon production (Y N →KN N ) and absorption (KN → Y π). Also Λ hyperons, being co-produced with kaons, are predicted to feel a weakly attractive potential (∼ −30 MeV) [22,23,24]. However, experimental data on the flow properties [25] and on spectral shapes [15,26,27] of hyperons produced in heavy-ion collisions near or below threshold are scarce. Evidently, not only nucleus-nucleus but also nucleon-nucleus collisions, though proceeding at densities not exceeding nuclear-matter ground-state density, may contribute to the discussion on medium modifications of meson and baryon properties. Here, valuable information has been collected by the KaoS collaboration, too [28]. The availability of such information is of great importance, since the reliable characterization of strangeness production and propagation in pA collisions is an indispensable prerequisite for the understanding of heavy-ion collisions. No experimental information is available on hyperon production in case of near-threshold nucleon-nucleus collisions which is assumed to serve as a link between elementary NN and heavy-ion collisions. Such data would allow for an improvement of the predictive power on the strangeness sector of state-of-the-art transport approaches which partially lack reliable input information on elementary processes. Parity conservation in strong interaction requires that the spin of produced Λ hyperons is aligned perpendicularly to the production plane. In the parity-nonconserving weak process, Λ → pπ − , the Λ polarization causes a significant up-down asymmetry of the decay proton w.r.t. that plane. The Λ polarization, surprisingly appearing in inclusive reactions with unpolarized beams and targets, was first observed in 1976 at Fermilab in collisions of p (300 GeV) + Be [29]. The observation of the negative polarization and the strong (almost lin-ear) transverse-momentum dependence of its magnitude was rapidly confirmed in p (24 GeV/c) + Pt collisions at CERN-PS [30], in p (400 GeV) + Be again at the Fermilab neutral hyperon beam (Λ andΛ) [31], in p (28.5 GeV/c) + Ir at AGS [32], and in p (12 GeV) + W at KEK [33]. Λ polarization measurements in pp interactions at the CERN intersecting storage rings [34] showed no obvious dependence on √ s which was varied from 31 to 62 GeV (which would correspond to 510 and 2050 GeV beam protons on a stationary proton target). However, the polarization was found to increase strongly with Feynman-x (x F ) and with transverse momentum approaching -40 % when both quantities are maximum. Experiments with polarized beams followed soon, e.g. p + Be collisions at 13.3 and 18.5 GeV/c at the AGS [35]. It is worth mentioning that all of these early experiments consisted of magnetic spectrometers set to fixed scattering angles. But more importantly, they suffered from a rather small solid-angle coverage resulting in a high degree of correlation between the transverse (p t ) and longitudinal (x F or rapidity) coordinates (e.g. in pN reactions at 450 GeV at CERN-SPS [36]). Any larger phase-space coverage implied a time-consuming setting of the corresponding spectrometer to different scattering angles with proper relative normalization as tried at Fermilab in p (400 GeV) collisions on Be, Cu, and Pb targets with the main focus on the behaviour of the Λ andΛ polarization at large transverse momentum [37]. The first exclusive measurements of the Λ polarization were performed with the E766 experiment at the AGS in pp → K + Λ(π + π − ) n (n=1-4) at 27.5 GeV/c [38,39]. Λ polarization in pp collisions is antisymmetric in x F by virtue of rotational invariance. Since the detector used had uniform acceptance in x F , the same polarization magnitude was measured for positive and negative x F . The behaviour of the polarization in x F and p t was found the same for all the studied reactions. First polarization studies of Λ hyperons produced in heavy-ion collisions (10.7A GeV Au + Au) were reported by the E896 collaboration at BNL-AGS [40]. Their results revealed a dependence of the polarization on the transverse momentum and x F being consistent with previous measurements in pp and pA collisions. The polarization of Λ hyperons produced inclusively by a Σ − beam of 340 GeV/c at CERN-SPS [41], however, exhibited the striking feature of the mostly positive sign of the polarization which is opposite to what has been observed in Λ production by protons or neutrons. The HERMES collaboration at DESY determined Λ and Λ polarizations in quasireal photoproduction at the 27.6 GeV positron beam of the HERA collider on an internal gas target [42]. Surprisingly, the Λ polarization, averaged over the acceptance of the spectrometer, appeared positive, in contrast to almost all other experiments, while theΛ polarization appeared compatible with zero. A linear rise of the Λ polarization magnitude with increasing transverse momentum was found similar to that observed in earlier experiments. At lower beam energies only few hyperon polarization data exist. The DISTO collaboration at SATURNE investigated strangeness production in elementary pp collisions with polarized proton beam at 3.67 GeV/c [43]. First polarization transfer measurements for exclusive hyperon production reactions were reported in ref. [44]. At COSY, the TOF collaboration studied Λ polarization in the elementary pp → pKΛ reaction with polarized beams at 2.75 and 2.95 GeV/c [45,46,47]. At higher energy and similar to DISTO [43] and E766 at AGS [38], the experiments revealed a negative (positive) polarization in the beam (target) fragmentation region, while the polarization seemed to vanish at the lower energy. Besides the polarization of the Λ hyperon, the Λ analyzing power and the spin-transfer coefficient, also referred to as Λ depolarization, could be determined. The lowest beam energy so far for which a Λ polarization value could be determined amounts to 1.8 GeV. For central nucleus-nucleus collisions of Ar+KCl at this beam kinetic energy per incident nucleon, the streamer-chamber group at the BEVALAC reported an average Λ polarization value of about −0.10 ± 0.05 [48]. For recent reviews including parameterizations of various polarization dependences on kinematic quantities we refer the reader to refs. [49,50,51]. While the Λ polarization seems to be established as an experimental fact, its origin remains a mystery. Various models are proposed, assigned either to the quark-exchange [52,53,54,55,56,57] or to the meson-exchange picture [58,59,60,61,62]. But, there is still no theoretical description which is able to explain the experimental observations consistently [49]. Summarizing the experimental situation on Λ production and polarization, we conclude that the high-acceptance spectrometer HADES [63] would be an appropriate experimental device allowing for I) the investigation of the Λ phase-space distribution in proton-nucleus collisions and II) the study of a genuine two-dimensional (transverse, longitudinal) dependence of the Λ polarization over a large phase-space region, feasible with one and the same apparatus setting. The present paper is organized as follows. In Sect. 2 we give an overview of the HADES experiment on p + Nb collisions at 3.5 GeV beam kinetic energy. We proceed with the data analysis in Sect. 3. In Sect. 3.1 we present the method to identify the Λ hyperons from their weak decay into proton-π − pairs, while in Sect. 3.2 we describe the analysis chain for the extraction of experimental Λ phase-space distributions and the corresponding predictions by transport models. In Sect. 3.3 the results on the Λ polarization and its phase-space dependence will be presented. Finally, in Sect. 4 we summarize our results. The experiment The experiment was performed with the High Acceptance Di-Electron Spectrometer (HADES) at the Schwerionensynchrotron SIS18 at GSI, Darmstadt. HADES, although primarily optimized to measure di-electrons [64], offers excellent hadron identification capabilities [18,27,65,66,67] allowing for a profound correlation analysis. A detailed description of the spectrometer is presented in ref. [63]. The present results are based on a dataset which was previously analyzed with respect to e + e − [68] as well as to pion and η production [69,70] in collisions of p + Nb at 3.5 GeV; the production of K 0 mesons, focussing on the K 0 phase-space distribution and its alteration due to the influence of a kaon-nucleon potential at nuclearmatter ground-state density, will be reported elsewhere. In the following we summarize the main features of the apparatus. HADES consists of a 6-coil toroidal magnet centered on the beam axis and six identical detection sections located between the coils and covering polar angles from 18 to 85 degrees. The six sectors consist of hadron blind Ring-Imaging Cherenkov (RICH) detectors (not used for the present investigation), four planes of Multi-wire Drift Chambers (MDCs) for track reconstruction, and two time-of-flight walls, TOFino (polar angles 18 • < θ < 44 • ) and TOF (44 • < θ < 85 • ), supplemented at forward polar angles with Pre-Shower chambers. The TOF and TOFino+Pre-Shower detectors were combined into a Multiplicity and Electron Trigger Array (META). A reconstructed track in the spectrometer is composed of straight inner and outer track segments in the MDCs. The pointing vector of the outer track segment is used for matching with a META hit. Possible trajectories through pairs of inner and outer track segments are combined to track candidates. A Runge-Kutta algorithm allows to calculate the momentum of each track candidate making use of the track deflection in the magnetic field between the inner and outer segments. The quality of the METAhit matching and the Runge-Kutta fitting (characterized by χ 2 values) is used to create an ordered list of track candidates. The track candidate with the lowest product of both χ 2 values is selected as the true track. Its segments and associated track candidates are then deleted from the candidate list. This procedure is repeated until no track candidates are left in the list. Particle identification of protons and π − mesons is based on the correlation of their momenta and energy loss in the MDCs. Two-dimensional cuts in the corresponding correlation plots are used to select the different particle species. For more details, e.g. the quality of kaon identification, see refs. [65,66]. Finally, the momentum calculated from the track curvature is corrected for the energy loss of the charged particles in the target, beam pipe and detector materials. In the present experiment, a proton beam of about 2 × 10 6 particles per second with kinetic energy of 3.5 GeV (corresponding to an excess energy w.r.t. the threshold for Λ pro- duction in NN collisions of √ s N N − √ s N N,Λ = 0.63 GeV) was incident on a 12-fold segmented target of natural niobium ( 93 Nb). The choice of the target was the result of simulations aimed at optimizing the di-electron experiment [68], i.e. compromising on the ratio of the vector-meson production and the combinatorial background due to γ conversion. However, the usage of this medium-size target is of advantage also for the present Λ investigations, since the p-π − combinatorial background increases stronger with target mass than the Λ yield which in turn is tightly connected to the production of the associated kaons [28]. The data readout was started by different trigger decisions [69]. For the present analysis, we employ only the data of the first-level trigger (LVL1, downscaled by a factor of three), requiring a charged-particle multiplicity ≥ 3 in the TOF/TOFino detectors. We processed about N ev = 3.2 × 10 9 of such LVL1 events. The total reaction cross section of σ pN b = (848 ± 127) mb is provided by measuring charged pions and by interpolating known pion production cross sections [68,70]. Analysis Λ identification It is important to mention that Σ 0 hyperons decay almost exclusively into Λ's via the decay Σ 0 → Λγ (branching ratio BR = 100 %, lifetime cτ = 2.22 × 10 −11 m [71]), with the photon not being detected in the present experiment. Hence, throughout the paper, any "Λ yield" has to be understood as that of Λ+Σ 0 . Correspondingly, in case of transport model simulations (cf. Sect. 3.2.3), where the individual particle species are known, the yields of Λ and Σ 0 hyperons are summed up. Furthermore, we note that the actual Λ polarization may be considerably larger than the measured one presented in Sect. 3.3, since Λ hyperons from Σ 0 decay are expected to carry, on average, -1/3 the polarization of the Σ 0 [30]. The latter one is predicted [31,52] and measured [72] to be opposite to that of the Λ. In the present analysis, we identify the Λ hyperons through their weak decay Λ → pπ − (BR = 63.9 %, cτ = 7.89 cm [71]), with the charged daughter particles detected in HADES [27,67]. The long lifetime of the Λ causes a sizeable fraction of these particles to decay away from the primary vertex. The precision of the track reconstruction with HADES is sufficient to resolve these secondary vertices [65,67]. For the selection of Λ's, topological cuts are used. As a compromise between a high Λ yield and a reasonable signal-to-background ratio (> 0.1 all over the investigated phase space), we choose i) a minimum value of the Λ decay vertex distance to the primary vertex, d V > 43 mm, ii) minimum values of the proton and π − shortest track distances to the primary vertex, d p > 4 mm, d π − > 10 mm, and iii) an upper threshold of the proton-π − minimum track distance, d t < 10 mm. Here, the off-vertex cut i) is the main condition responsible for the extraction of a Λ signal with good (∼ 1) signal-tobackground ratio. Figure 1 shows the invariant-mass distribution of all proton-π − pairs which pass the cuts listed above. To extract the Λ yield, the invariant-mass spectrum is fitted, typically in the range from 1090 to 1200 MeV, with a combination of different functions describing both the signal and the combinatorial background of uncorrelated proton-π − pairs. The signal peak is parametrized by two Gaussians (with identical mean values but different widths to account for a certain broadening of the peak at its base), while the background is approximated by a Tsallis (q-exponential) function, f T (x) ∝ (1 − (1 − q)x) 1/(1−q) . Here, x is a linear function of the invariant mass, m pπ − , and q is a shape parameter which may account for the phase-space limitation at large invariant masses, due to energy-momentum conservation. (Approximating instead the combinatorial background with the event-mixing technique, a quite similar quality of reproduction of the invariant-mass distribution over almost the entire mass ranges below and above the peak (|m pπ − − m Λ | > 4σ Λ ) is possible, with a slight exception of some underestimate of the data near (m pπ − < 1085 MeV/c 2 ) to the kinematical limit, i.e. the sum of the proton and π − masses.) From the Gaussian fit to the peak, the pole mass is determined to be 1115.6 ± 0.1 MeV/c 2 , in very good agreement with the value of 1115.683 ± 0.006 MeV/c 2 listed by the Particle Data Group [71]. The peak width, which is a pure apparatus effect, is taken as the weighted average of the sigma widths of both Gaussians. It amounts to 3.1 MeV/c 2 , being only slighty larger than the corresponding widths measured in previous HADES analyses of the system Ar (1.76A GeV) + KCl [27,67] where, on average, lower momenta and hence tracks with higher curvature are involved. In total, for the cuts listed above and after background subtraction, about 1.1 × 10 6 Λ hyperons were reconstructed within a ±2σ window around the peak mass, with a mean signal-to-background ratio of 1.1. Fig. 1. Proton-π − invariant mass distribution (symbols) with the Λsignal peak in the reaction p (3.5 GeV) + Nb. The full curve shows the result of a common fit with two Gaussians for the peak and a Tsallis function for the background (see text). The dashed curve represents the background. The vertical dotted lines bound the 2σ window for signal counting. Λ inclusive phase-space distribution Experimental data The bottom right panel of Fig. 2 shows the two-dimensional raw data yield (i.e. Λ peak yields extracted from the proton-π − invariant-mass distributions after subtraction of the combinatorial background) as a function of transverse momentum and rapidity. Corrections for detector acceptance and reconstruction efficiency were performed with Monte-Carlo simulations involving, as appropriate event generator, the UrQMD transport approach [73,74] and the GEANT [75] package accounting for the proper particle decays and the finite detector acceptance, granularity, resolution, etc. The minor missing experimental yield outside the 2σ window around the Λ-peak mass (cf. Fig. 1) is well considered by applying the same cut to the simulation which exhibits a similar broadened peak base proved to be due to small-angle scattering in air and in the detector materials of the Λ decay products. The bottom left panel of Fig. 2 displays the GEANT output, while the middle left panel shows the UrQMD output. The ratio of both distributions of simulated Λ data delivers the corresponding reconstruction efficiency matrix (top left panel). Note that also the bias on the Λ yield due to the effect of the LVL1 trigger is estimated with the help of UrQMD+GEANT simulations. An enhancement of the LVL1 triggered yield over the minimumbias value of 1.53 ± 0.02 has been found which is corrected for when determining the final (minium-bias) Λ yield as a function of phase-space population. Within the given marginal uncertainty, the correction factor itself exhibits no dependence on phase space. Finally, after dividing -for each phase-space cell -the experimental raw data by the corresponding efficiency, the distribution corrected for acceptance, reconstruction efficiency, LVL1 trigger bias, and non-target interaction ( 17 % events w/o tracks), is derived and displayed in the middle right panel of Fig. 2. Note that this phase-space distribution is the prerequisite for the two-dimensional investigation of the Λ polarization presented in Sect. 3.3. To extrapolate into the unmeasured region at low transverse momentum, we follow the widely applicable and commonly used recipe of approximating the p t spectra by Maxwell-Boltzmann distributions. To do so, we start with the triple differential yield d 3 N/d 3 p ∝ exp (−E/T ). Here, E = ((pc) 2 + (m 0 c 2 ) 2 ) 1/2 = m t c 2 cosh y is the total energy, y = tanh −1 (p c/E) is the lab. rapidity, m t = ((p t /c) 2 + m 2 0 ) 1/2 is the transverse mass, p t (p ) is the transverse (longitudinal in beam direction) momentum, and m 0 is the rest mass of the particle of interest. Finally, c gives the speed of light in vacuum. Transformation from spherical to cylindrical coordinates (being more appropriate for an almost rotational symmetric apparatus) and integration over the azimuthal angle delivers the two-dimensional yield d 2 N m 2 t dm t dy = C B (y) exp(− m t c 2 T B (y) ).(1) Fits with a simple exponential to these transverse-mass spectra in slices of rapidity deliver constants, C B (y), and inverse (Boltzmann) slope parameters, T B (y). Figure 3 shows the Λ m t distributions for the indicated rapidity regions. The lines are Boltzmann fits to the data according to Eq. (1). For the determination of the dN/dy distribution, the experimental data, where available, are integrated, and the yield in the unmeasured region is obtained from the fit. Before we present the rapidity dependences of the inverse slope parameter, T B (y), and the rapidity density distribution, dN/dy, of Λ hyperons produced in p+Nb collisions, we show the result of a self-consistency check, i.e. using UrQMD simulations for the acceptance and reconstruction efficiency corrections and analyzing another simulation similarly to the experimental data. For this purpose, we used the fireball option of the event generator Pluto [76]. Based on the expected experimental phase-space distribution, we generated Λ hyperon events populating the phase space according to an isotropic thermal source with a temperature of T = 50 MeV centered at an average rapidity of y = 0.56. The Pluto events were tracked through GEANT to account for the detector response and the geometrical decay topology. The GEANT output was embedded into experimental events to account for a proper track environment and, finally, passed through the entire analysis chain with the correction matrix taken from UrQMD simulation. Figure 4 shows the result for the rapidity distribution. Though the shapes of both simulations differ significantly, the initial shape of the phase-space distribution generated by Pluto is well recovered. Hence, we are convinced that our reconstruction method is based on solid grounds. Furthermore, we note Fig. 4. Rapidity density distribution of Pluto simulated Λ hyperons (arbitrarily scaled) before (dashed curve) and after (symbols) the full analysis chain as applied for experimental data. The full curve represents the corresponding spectral shape of the UrQMD simulations used for acceptance and reconstruction efficiency correction. The arrow indicates the center-of-mass rapidity in nucleon-nucleon collisions at 3.5 GeV. that the experimental distributions given below represent the average of two analyses from two independent groups [77,78]. The given systematic errors comprise the slight differences of the corresponding analysis results and those from a variation of the cut values given in Sect. 3.1 within reasonable limits (±20%). The statistical errors are often smaller than the symbols displayed in the figures. Figure 5 presents the dependence of the Λ inverse (Boltzmann) slopes on rapidity, T B (y) (with different transport model predictions overlaid, for discussion see Sect. 3.2.3). The error bars represent the uncertainties arising from the variations of the topological cuts to select the Λ hyperons and of the borders of the Boltzmann fits applied to the m t (p t ) dis-tributions displayed in Fig. 3 (7). We find slope parameters, i.e. apparent transverse temperatures, from 55 to 92 MeV with the maximum at a rapidity of y max = 1.0, that is below the center-of-mass rapidity of the nucleon-nucleon reference system, y cm = 1.12. Note that in symmetric heavy-ion collisions, e.g. of Ar + KCl at 1.76A GeV (y cm = 0.86, HADES [27]) and Ni + Ni at 1.93A GeV (y cm = 0.89, FOPI [15]) at SIS18 or of Au+Au at 10.7A GeV at AGS (y cm = 1.60, E896 [79]), the rapidity dependence of the Λ inverse slope parameter was found to follow well the thermal model prediction, i.e. a decline of the slope with increasing distance from mid-rapidity as T B = T / cosh(y − y cm ). Corresponding mid-rapidity values T = (95.5 ± 2), (106 ± 5), and (237 ± 5) MeV have been reported in refs. [27], [15], and [79], respectively. Note that, in a recent analysis of η meson production in our reaction, p (3.5 GeV) + Nb, we found a similar symmetric rapidity dependence of the inverse slope, which could be parametrized with the above 1/cosh(y) dependence, however with the maximum located well below y cm , i.e. T B,η (y) = 84 MeV / cosh(y − 0.96) [69]. In contrast, the present rapidity dependence of the Boltzmann slope parameter for Λ hyperons does not follow the isotropic thermal model prediction. Rather, it falls faster than ∝ 1/ cosh(y − y max ), cf. dotted curve in Fig. 5. Integrating the two-dimensional phase-space distribution over transverse momentum, the Λ rapidity-density distribution, dN/dy, is derived. It is displayed in Fig. 6. Except for the first data point, the statistical errors are smaller than the symbols. The given error bars represent the systematic errors. The grayshaded band displays the uncertainty of the absolute normalization of about 12 %. In contrast to symmetric heavy-ion collisions at comparable beam energies, where the rapidity distributions of secondary particles are symmetric around maxima at the center-of-mass rapidity of the nucleon-nucleon system [15,26,27,79,80,81], the present Λ yield at rapidities of y > 0.3 decreases monotonically with increasing rapidity. Unfortunately, the interesting region around target rapidity is not covered by the detector, due to its limiting upper polar angle, θ < 85 degrees. For comparison, we note that our pion and η meson analyses in the same collision system, p (3.5 GeV) + Nb, showed Gaussian shaped rapidity distribution centered, however, at rapidities of about 0.95 [69], i.e. well below y cm . Even slower emission sources (β 0.5−0.6) are estimated by KaoS for K + and K − production in p+Au collisions at 3.5 GeV [28]. It is worth to be noted that this observation is confirmed by our preliminary data on K 0 meson production in p+Nb exhibiting a rapidity density distribution centered around y 0.6 [82]. Finally, we try to estimate the total production probability of Λ + Σ 0 hyperons. Since the shape of the experimental distribution at the acceptance limits is not precisely known, especially at large polar angles, i.e. around target rapidity, y ∼ 0, this attempt is possible only in a model-dependent way. Since the dN/dy distribution is declining almost linearly, we performed a straight-line fit to the data points and extrapolated the distribution for y > 1.3 linearly. Similarly, the dN/dy distribution at low rapidities, y < 0.1, is assumed to increase linearly from zero at y = −0.3 to the first data point. The resulting total yield amounts to 0.017 ± 0.003, where the error comprises both, experimental and systematic errors. With this extrapolation, about 20 % of the yield is outside of the experimentally accessible rapidity range. For comparison, our preliminary K 0 yield in p+Nb, as estimated from the integral of a Gaussian function fitted to the K 0 S rapidity distribution [82], amounts to 0.011 ± 0.002. With the yield ratios of (Λ + Σ + + Σ 0 + Σ − )/(Λ + Σ 0 ) = 1.44, (K + + K 0 )/K 0 = 2.24, and K − /K 0 = 0.013 taken from UrQMD, we find strangeness balance, i.e. the equality of the total number of s ands quarks, being nicely fulfilled on average. Comparison with other data Looking for other experimental results on Λ production in pA collisions, no data could be found at beam energies below 9 GeV. The results next to ours are derived with the JINR Dubna 2 m propane bubble chamber for collisions of p+C at 10 GeV/c [83,84], i.e. already at an excess energy w.r.t. the Λ threshold in NN collisions of √ s N N − √ s N N,Λ = 2 GeV. The authors report a Λ production probability of (0.053 ± 0.005). The transverse momentum spectrum was found slightly harder than ours, as expected from the higher beam energy allowing for more energetic particles and hence larger transverse momenta. The rapidity distribution appeared asymmetric with the upper tail reaching to rapidities of 2.6. The maximum and mean values are located at about 0.8 and 1.0, respectively, i.e. both are well below the corresponding c.m. rapidity for NN collisions (y cm = 1.53), qualitatively similar to our observation (Fig. 6). Similar results have already been reported earlier in central collisions of carbon and oxygen at 4.5 GeV/c beam momentum per incident nucleon on different target nuclei, as measured with the 2 m streamer chamber SKM-200 at the Synchrophasotron in Dubna [85,86]. Systematically increasing the target mass, the authors found a Λ rapidity distribution which steadily became asymmetric and shifted towards target rapidity. To be able to compare our Λ production probability to the strange particle yields of another proton-nucleus experiment performed at the same beam energy but with different target nuclei, i.e. KaoS data on K + and K − production in p+C and p+Au collisions [28], we normalize the yields to the number of participants, A part . Using a nuclear overlap (Glauber) model [87] (with a Woods-Saxon density profile, an inelastic nucleonnucleon cross section of σ inel N N = 30 mb, and an impact parameter range of b = 0 − 10 fm), we get A part = 2.5 (3.3) for p+Nb (p+Au). Hence, we derive for our system, p+Nb, a normalized Λ + Σ 0 yield being about 1.9 times larger than the K + yield per number of participants for p+Au collisions at the same beam energy [28]. Transport model predictions Two transport approaches are compared with the experimental data, i.e. the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) model [73,74] and the Giessen Boltzmann Uehling Uhlenbeck (GiBUU) model [2,88]. For UrQMD 1 , we used code release version 3.3p1, while for GiBUU 2 , we worked with release 1.6.6179 in real-particle mode (w/o mean-field baryon-baryon potential). In GiBUU, a threshold energy of √ s = (3.4 ± 0.1) GeV (default) steers, for baryon-baryon collisions, the smooth transition from the resonance model to the PYTHIA implementation (version 6.4.26). The rapidity dependence of the inverse slope dependence presented in Fig. 5 is neither reproduced by the UrQMD nor by the GiBUU model. Both approaches significantly overestimate this transverse shape parameter of the phase-space distribution for most of the accessible rapidities. In contrast, the rapidity density distribution, dN/dy, presented in Fig. 6 is fairly well described by UrQMD, not only in shape but also on an absolute scale, over a large part of the experimentally accessible rapidity range, while GiBUU system-atically underestimates (overestimates) the yield below (above) rapidities of y 0.8 and does not reproduce the shape. The strong yield around target rapidity visible in both models, however, can not be compared with experimental data due to acceptance limitations. Inspecting the transverse-momentum distributions within rapidity slices, d 2 N/dy dp t , as displayed on a linear scale in Fig. 7, one realizes that the highest predictive power of the transport models would result from the low transverse momenta, where, however, the detector acceptance often prevents reliable experimental data. For rapidities y 0.3, both models exhibit p t spectra with shapes similar to those of the singleslope Maxwell-Boltzmann distributions fitted to the experimental data (dotted curves). In the target-rapidity range, however, the models show deviations from the one-slope shape, i.e. a superposition of differently hard spectra. The two main sources of these spectra are the contribution from first chance collisions yielding a rapidity spectrum centered around y cm and the contribution from subsequent hyperon-nucleon (YN) collisions slowing down the Λ's down to target rapidity, y = 0. This finding could be well established by inspecting the subprocesses governing the Λ rapidity distributions generated with the models (cf. Fig. 6). Thus, increasing in GiBUU the YN cross sections by a factor of two the Λ yield could be partially redistributed to lower rapidities leading to a stronger enhancement around target rapidity. However, neither the shape nor the absolute yield of the experimental rapidity distribution could be reproduced. We tested the influence of the energy threshold at which GiBUU switches from the resonance model to PYTHIA in baryon-baryon collisions. Decreasing this threshold, hence increasing the operating range of PYTHIA, the inverse slope parameter T B (y) derived from Boltzmann fits to the model distributions decreased (i.e. the spectra got softer) and the rapidity density dN/dy increased, for all rapidities. E.g., for a threshold of 2.6 GeV (default value in release 1.5), T B (y) from the simulation is throughout smaller than the experimental data with a maximum of 75 MeV at y 0.7, while the yield dN/dy increases roughly by an overall factor of 1.5. Apparently, in PYTHIA some elementary cross sections being relevant for Λ production are larger than the corresponding ones implemented into the resonance model. For a threshold of 3.3 GeV, the average Λ yield within the detector acceptance would match the experimental one. The shape of the experimental dN/dy distribution, however, could not be reproduced, i.e. the model distribution falls slower with rapidity than the experimental one (cf. Fig. 6). Finally, on an absolute scale, the UrQMD approach does a better job than the GiBUU model, since it largely reproduces the experimental transverse-momentum and rapiditydensity distributions. The ongoing analysis of inclusive K 0 and Λ production in pp collisions at the same kinetic beam energy of 3.5 GeV will provide cross section measurements of quite a number of elementary reaction channels (especially pp → ∆ ++ K 0 Λ/Σ 0 ) and perhaps even of new channels (e.g. channels involving Σ(1385) or Λ(1405) [89,90]) which may help to improve the resonance-model part of GiBUU. Λ hyperons might be polarized perpendicularly to the production plane, i.e. a plane defined by the momentum vector of the beam and the momentum vector of the hyperon. The normal vector of that plane is defined as Λ Polarization n = p beam × p Λ |p beam × p Λ | .(2) The relevant angle to be studied is the angle ζ between n and the momentum vector p * p of the decay proton in the Λ rest frame (marked by an asterisk), i.e. cos ζ = p * p · n |p * p · n| .(3) The corresponding experimental angular distribution is displayed in the upper panel of Fig. 8. It is derived similarly to the filling of the raw phase-space distribution of the Λ, i.e. performing, for each angular bin d cos ζ, a combinatorialbackground subtraction on the proton-π − invariant-mass distribution with subsequent Λ-peak integration. The corresponding (polarization-free) UrQMD-simulated angular distribution is given in the middle panel of Fig. 8. The yield suppression around cos ζ ∼ 0 is the result of the detector acceptance. Both, the experimental and simulated, angular distributions exhibit a high degree of mirror symmetry w.r.t. the Λ production plane. Since the Λ statistics is copious, we extract the Λ polarization P by fitting a straight line dN d cos ζ = C (1 + αP cos ζ)(4) to the distribution of the corrected (experimental/simulation) angular distribution shown in the lower panel of Fig. 8. Note that in Eq. (4), the asymmetry parameter, α = 0.642 ± 0.013, of the parity-violating weak decay of the Λ hyperon is a measure of the interference between s and p waves of the final state [71]. The Λ polarization, averaged over the available phase space, is determined with different methods, thus allowing for a consistence check of the various analyses. Method (1) implies first the averaging over a set of Λ phase-space distributions resulting from a reasonable, i.e. ±20%, variation of the topological cuts to select the Λ candidates (cf. Sect. 3.1) and then one overall fit to the mean angular distribution. It delivers P = −0.115 ± 0.005 ± 0.021. Method (2) involves first individual angular fits (each for a certain geometrical cut setting) yielding a variety of polarization numbers and then an average (i.e. weighted with the proper yield) over this set of fit results. Here, we get P = −0.123 ± 0.005 ± 0.012, in agreement with the value derived with method (1). In both cases, statistical and systematic errors are given, where the latter ones include the variations of the topological cuts and of the fit ranges applied to the p-π − invariant-mass distributions. Finally, method (3) makes use of the mirror symmetry mentioned above. Though it is possibly not fully met due to slight acceptance differences at the detector edges of experiment and simulation, it can be used to determine the polarization by simply integrating the corrected angular distribution over the positive (Up) and negative (Down) cosine of the angle ζ, i.e. P = 2 α Up − Down Up + Down . (Another, equivalent, approach to the polarization is based on the relation αP = cos ζ / cos 2 ζ , where the angle brackets imply the average over the entire angular range and all events.) The corresponding overall polarization derived with Eq. (5) amounts to −0.104 ± 0.008 ± 0.027, in agreement with the values derived with the linear fit. The mean value following from the various methods amounts to P = −0.119±0.005±0.016 with the given statistical and systematic errors. For the differential Λ polarization investigated in the following, the linear fit with Eq. (4) is used, and the systematic errors are accounted for by method (1). Now, we study the phase-space dependence of the Λ polarization. Figure 9 shows the Λ polarization as a function of rapidity and transverse momentum, P(y, p t ). Note that, for this (quasi triple-differential) figure, sixteen hundred invariantmass distributions of proton-π − pairs have been analyzed (i.e., 8 topological cut sets, 8 cos ζ bins, 5 p t bins, 5 y bins). We exclusively observe negative central polarizations values over the entire phase space displayed. Even when incorporating the given errors, this finding keeps almost unchanged. Though the statistics is copious, the absolute value of the polarization is somewhat fluctuating, i.e. it shows differences between the central values determined in neighbouring phase-space cells. Typically, these deviations of the polarization values from the average trends in slices of rapidity or transverse momentum (determined, e.g., by linear regression), are well within the errors. A clear tendency from the lower right to the upper left corner is found: The polarization is strongest at low rapidities and large transverse momenta. Its absolute value increases with transverse momentum for the two lowest rapidity bins while it is almost independent of p t within the remaining y region; and it decreases with rapidity for the three upper transversemomentum bins while it hardly changes with rapidity for the two low-p t bins. To make this finding more obvious, we studied the onedimensional dependences. For that reason, we included also the regions outside the sharp upper and lower rapidity and transverse-momentum limits of Fig. 9 which were applied for statistical reasons. Integrating over all experimentally populated rapidities, Fig. 10 shows the dependence of the Λ polarization on the Λ transverse momentum. Its absolute value clearly increases with p t . The dotted line is a linear fit to the data with the simplest formula ensuring a vanishing polarization at zero transverse momentum, i.e. P(p t ) = D p t . The corresponding slope amounts to D = (−0.18 ± 0.02) (GeV/c) −1 . Looking for phase-space dependences of earlier measurements one realizes that, because of the fixed Λ production angle in most of the high-energy experiments, a definite Feynman-x value corresponds to a certain transverse momentum. Hence, there the p t dependence of the Λ polarization reflects both p tand x F -dependences. Nevertheless, our slope D is well in between the p t slopes of about -0.05 to -0.30 (GeV/c) −1 found for pN-reactions at energies from 400 to 800 GeV and presented in the compilation of ref. [36]. Integrating instead over all experimentally accessible transverse momenta, Fig. 11 gives the rapidity dependence of the Λ polarization, P(y). Its absolute value is smallest at an intermediate rapidity, y 0.82 ± 0.09, as derived from a parabolic fit (dotted curve) to the experimental data, and increases weakly both with decreasing and increasing rapidity, whereby the behaviour at the upper rapidities is compatible with a constant polarization. It is worth to recall that Λ polarization in pp collisions is odd in x F (or (y-y cm )) with negative (positive) sign in the beam (target) fragmentation region [38,43,47]. In the present experiment, however, the Λ hyperons carry exclusively negative polarizations while they populate mostly the target hemisphere (Fig. 6). Hence, the number of Λ's arising from primary (first chance) pN collisions is negligible, and the polarization we see at low rapidities is supposed to come from Λ's which have scattered, which are produced by slowed-down beam protons, or which are produced by "unwounded" protons on a cluster of many nucleons from the target. Summary Summarizing, we presented high-statistics data on production and polarization of Λ hyperons produced in collisions of p (3.5 GeV) + Nb. The data were taken with HADES at SIS18/GSI. The Λ phase-space distribution was compared with corresponding results of other experiments and with transport model predictions. The rapidity density was found to decrease monotonically within a rapidity window ranging from 0.3 to 1.3, i.e. from values above the target rapidity to values beyond the center-of-mass rapidity for nucleon-nucleon collisions. None of the employed models was able to reproduce fully the experimental distributions, i.e. in absolute yield and in shape. We link this observation to an insufficient modelling in the transport approaches of the elementary processes being relevant for Λ production, rescattering and absorption. For the first time, the data allow for a genuine two-dimensional 11. Λ polarization as a function of the Λ rapidity, P(y). The vertical bars associated with symbols represent the combined statistical and systematic errors. The dotted curve is a fit to the data with a parabola (cf. text). The arrow indicates the center-of-mass rapidity in nucleon-nucleon collisions at 3.5 GeV. investigation of the Λ polarization as a function of the large phase space covered by HADES. We found finite negative values of the polarization in the order of 5 − 20% over the entire phase space with its magnitude increasing almost linearly with increasing transverse momentum for p t > 300 MeV/c, −P(p t ) = (0.18 ± 0.02) (GeV/c) −1 p t , and increasing with decreasing rapidity for y < 0.8. The average polarization amounts to P = −0.119 ± 0.005 (stat) ± 0.016 (syst). Assuming that the Λ polarization origins from interactions of the primary beam protons with target nucleons, the surprise is the survival of the Λ spin orientation in subsequent rescattering processes until kinetic freeze-out. Even if the Λ hyperons are not produced in primary (first chance) collisions, they carry the information on the beam projectile. Fig. 2 . 2Bottom right: Raw Λ yield d 2 N/dpt dy. Bottom left: GEANT output of UrQMD simulations. Middle left: GEANT input (UrQMD output). Top left: Reconstruction efficiency matrix as derived from the ratio of the distributions below. Middle right: Experimental Λ yield after correction with the efficiency matrix. Fig. 3 . 3Experimental Λ yield distribution m −2 t d 2 N/dmt dy (symbols with error bars) in slices of rapidity as given in the legend. The lines represent fits with an exponential distribution according to Eq. (1). Full lines cover the corresponding fit regions. Extended dashed lines display the part used for extrapolation. Fig. 5 . 5Rapidity dependence of the experimental Λ inverse (Boltzmann) slope parameter, TB(y), (symbols) as derived from the exponential fits inFig. 3. The error bars represent the systematic errors. The full and dashed curves display similar dependences resulting from corresponding fits to the spectra of the transport approaches UrQMD and GiBUU, respectively. The dotted curve represents the function TB(y) = 92 MeV / cosh(y − 1.0). The arrow indicates the center-ofmass rapidity in nucleon-nucleon collisions at 3.5 GeV. Fig. 6 . 6Experimental rapidity-density distribution, dN/dy, of Λ hyperons (symbols). The error bars show the systematic errors. The grayshaded band represents the uncertainty due of the absolute normalization. The model curves and the arrow have the same meaning as inFig. 5. Fig. 7 . 7Experimental Λ transverse momentum distribution (symbols) for successive windows in rapidity (given on top). The model curves have the same meaning as inFig. 5. The dotted curves represent Maxwell-Boltzmann distributions fitted to the data. Fig. 8 . 8Top: Raw angular distribution of the proton in the Λ rest frame relative to the Λ production plane normal. Middle: The same for polarization-free simulation data normalized to the corresponding input distribution. Bottom: Corrected and arbitrarily normalized angular distribution. The displayed error bars comprise both, statistical and systematic, errors. The dashed line is a fit according to Eq. (4) used to extract the polarization. Fig. 9 . 9Λ polarization as a function of rapidity and transverse momentum, P(y, pt). The errors attached to the polarization values comprise both, statistical and systematic, errors. The axis on the right displays the linear color coding applied to the polarization values only, not involving the corresponding errors. Fig. 10 . 10Λ polarization as a function of the Λ transverse momentum, P(pt). The vertical bars associated with symbols represent the combined statistical and systematic errors. The dotted curve is a fit to the data with a straight line (cf. text). Fig. Fig. 11. Λ polarization as a function of the Λ rapidity, P(y). 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[]
[ "Improving Item Cold-start Recommendation via Model-agnostic Conditional Variational Autoencoder ACM Reference Format", "Improving Item Cold-start Recommendation via Model-agnostic Conditional Variational Autoencoder ACM Reference Format" ]
[ "Xu Zhao [email protected] ", "Yi Ren ", "Ying Du [email protected] ", "Shenzheng Zhang ", "Nian Wang ", "Xu Zhao ", "Yi Ren ", "Ying Du ", "Shenzheng Zhang ", "Nian Wang ", "\nTencent News\nBeijingChina\n", "\nTencent News\nBeijingChina\n", "\nTencent News\nBeijingChina\n", "\nTencent News\nBeijingChina\n", "\nTencent News\nBeijingChina\n" ]
[ "Tencent News\nBeijingChina", "Tencent News\nBeijingChina", "Tencent News\nBeijingChina", "Tencent News\nBeijingChina", "Tencent News\nBeijingChina" ]
[ "Proceedings of the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR '22)" ]
Embedding & MLP has become a paradigm for modern large-scale recommendation system. However, this paradigm suffers from the cold-start problem which will seriously compromise the ecological health of recommendation systems. This paper attempts to tackle the item cold-start problem by generating enhanced warmed-up ID embeddings for cold items with historical data and limited interaction records. From the aspect of industrial practice, we mainly focus on the following three points of item cold-start: 1) How to conduct cold-start without additional data requirements and make strategy easy to be deployed in online recommendation scenarios.2) How to leverage both historical records and constantly emerging interaction data of new items. 3) How to model the relationship between item ID and side information stably from interaction data. To address these problems, we propose a model-agnostic Conditional Variational Autoencoder based Recommendation(CVAR) framework with some advantages including compatibility on various backbones, no extra requirements for data, utilization of both historical data and recent emerging interactions. CVAR uses latent variables to learn a distribution over item side information and generates desirable item ID embeddings using a conditional decoder. The proposed method is evaluated by extensive offline experiments on public datasets and online A/B tests on Tencent News recommendation platform, which further illustrate the advantages and robustness of CVAR.CCS CONCEPTS• Information systems → Recommender systems.
10.1145/3477495.3531902
[ "https://arxiv.org/pdf/2205.13795v1.pdf" ]
249,151,934
2205.13795
1f5aa593cb7c374c5e91b32bc97e2badee3653f7
Improving Item Cold-start Recommendation via Model-agnostic Conditional Variational Autoencoder ACM Reference Format ACMCopyright ACMJuly 11-15, 2022. July 11-15, 2022 Xu Zhao [email protected] Yi Ren Ying Du [email protected] Shenzheng Zhang Nian Wang Xu Zhao Yi Ren Ying Du Shenzheng Zhang Nian Wang Tencent News BeijingChina Tencent News BeijingChina Tencent News BeijingChina Tencent News BeijingChina Tencent News BeijingChina Improving Item Cold-start Recommendation via Model-agnostic Conditional Variational Autoencoder ACM Reference Format Proceedings of the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR '22) the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR '22)Madrid, Spain; Madrid, Spain; New York, NY, USA, 6 pagesACMJuly 11-15, 2022. July 11-15, 202210.1145/3477495.3531902ACM ISBN 978-1-4503-8732-3/22/07. . . $15.00. 2022. Im-proving Item Cold-start Recommendation via Model-agnostic Conditional Variational Autoencoder. InCold-Start RecommendationConditional Variational AutoencoderItem ID embedding Embedding & MLP has become a paradigm for modern large-scale recommendation system. However, this paradigm suffers from the cold-start problem which will seriously compromise the ecological health of recommendation systems. This paper attempts to tackle the item cold-start problem by generating enhanced warmed-up ID embeddings for cold items with historical data and limited interaction records. From the aspect of industrial practice, we mainly focus on the following three points of item cold-start: 1) How to conduct cold-start without additional data requirements and make strategy easy to be deployed in online recommendation scenarios.2) How to leverage both historical records and constantly emerging interaction data of new items. 3) How to model the relationship between item ID and side information stably from interaction data. To address these problems, we propose a model-agnostic Conditional Variational Autoencoder based Recommendation(CVAR) framework with some advantages including compatibility on various backbones, no extra requirements for data, utilization of both historical data and recent emerging interactions. CVAR uses latent variables to learn a distribution over item side information and generates desirable item ID embeddings using a conditional decoder. The proposed method is evaluated by extensive offline experiments on public datasets and online A/B tests on Tencent News recommendation platform, which further illustrate the advantages and robustness of CVAR.CCS CONCEPTS• Information systems → Recommender systems. INTRODUCTION In the era of mobile internet, various online applications continuously emerge and explosively grow, in which recommendation systems play a key role in connecting users and content. Due to the excellent scalability and convenience of handling massive features, Embedding & MLP [4,12,13,33] has become a paradigm for modern large-scale recommendation systems [5,6,23,54,55]. However, this paradigm is data demanding and suffers from coldstart problem [9,17,34]. Concretely, for a large amount of new emerging items with limited interactions, their embeddings are insufficiently trained which leads to poor recommendation performance. The cold-start problem has become a crucial obstacle for online recommendation. Under the influence of Pareto [31] effect in our industrial system, A small portion of well-trained items tend to obtain more accurate recommendations and more impressions, which will further compromise the distributing efficiency of system. Some approaches have been proposed to address the item coldstart challenge. CLCRec [46] proposes to address cold-start problem by maximizing two kinds of mutual information using contrastive learning technology. Heater [58] uses the sum squared error (SSE) loss to model the collaborative embedding. Notice that CLCRec and Heater are only designed for the CF-based backbone models. There are also some model-agnostic methods that could be widely equipped to various backbones. DropoutNet [39] applies dropout technology to relieve the model's dependency on item ID. Meta Embedding [26] focuses on learning how to learn the ID embedding for new items with meta learning technology. MWUF [57] proposes meta Scaling and meta Shifting Networks to warm up cold ID embeddings. Although these methods are model-agnostic, they all have extra and strict requirements on data. For instance, DropoutNet and MWUF require the interacted user set of items for cold-start, while Meta Embedding requires building two mini-batches containing the same items for training. In the online scenario of industrial recommendation, these extra requirements on data stream make the deployment process rather difficult. Generally, there are two ways to solve the cold-start problem: the first is to mine the distribution patterns hidden in historical data [26,43,49,53,57], such as learning the transformation relationship between side information and item ID [26,46,57,58]. The second is to improve the learning efficiency with limited samples of cold items, such as methods based on meta learning [19,22,25,48]. However, previous works rarely consider both directions at the same time. In other words, methods in the first category only focus on the initialization of embedding, while the methods in the second category usually ignore patterns hidden in historical data. Considering the issues mentioned above in previous research along with our industrial practice, we summarize three key points for the design of cold-start method: 1) How to conduct cold-start without additional data requirements and make strategy easy to be deployed in online recommendation scenario. 2) How the cold-start method can leverage both historical records and constantly emerging interaction data of new items. 3) How to model the relationship between item ID and side information stably from interaction data and minimize the discrepancy between cold item ID embedding and fully trained embedding space. To achieve these desiderata, we propose a Conditional Variational Autoencoder based model-agnostic Recommendation (CVAR) framework. As an independent framework, CVAR could be equipped on various backbone models and be trained in an end-to-end way, using the same samples as backbone. Thus CVAR makes no redundant requirements for training data. In addition to giving desirable initialization in the cold-start phase, CVAR will leverage the continuously updating item ID embeddings from the backbone to generate enhanced warmed-up embeddings with superior quality. Therefore CVAR can fulfill the second concern mentioned above. As for the third problem mentioned before, CVAR aligns the representation of item ID and side information in the latent space of Conditional Variational AutoEncoder(CVAE) [36,40,50], which is shown in Figure 1. Specifically, previous works [26,57,58] usually train a learnable mapping from item side information to item ID. However, item ID contains not only content information, but also lots of interaction information which makes it difficult to learn a precise mapping directly. Inspired by Denoise Autoencoder [38], we first conduct dimension reduction of item ID embedding with Encoder-Decoder paradigm and get the denoised representation of item ID in a latent space. Then the side information is transformed to the latent space and aligned with the denoised representation of item ID. Corresponding with the design in CVAE, latent representation is defined as normal distribution which could maintain some Exploit-Exposure ability for CVAR. The main contributions of this work are summarized into four folds: (1) We propose a model-agnostic CVAR to warm up cold item ID embeddings. CVAR has no extra data requirements which makes it easy to be deployed in online scenario. (2) CVAR not only learns the pattern in historical data but also leverages the continuously updating item ID embedding from the backbone to generate enhanced warmed-up embeddings with superior quality. (3) We propose to model the relationship between item id and side information in the latent space and generate desirable ID embeddings using a conditional decoder. (4) Extensive offline and online experiments are conducted to demonstrate the effectiveness and compatibility of CVAR. PROPOSED METHOD In this section, we propose CVAR, a model-agnostic framework to warm up ID embeddings for new items. CVAR is designed based on the Click-Through-Rate(CTR) prediction task [10,32], predicting the click/watch/purchase behavior in recommendation scenario, which is usually formulated as a supervised binary classification task. Each sample in CTR task consists of multiple input features x and the binary label . Generally, the input features x could be splitted into three parts, i.e. x = ( , X). • , item ID, a unique number or string to identify each item in recommendation system. • X = { 1 , ..., |X | }, features which are used for CTR predict , may be categorical or continuous, such as the user attributes and contextual information. Moreover, We take part of features from X as item side information I ⊂ X, which will be consumed in item cold-start procedure. Standard feature preprocessing [10,32] has been applied in this work. Continuous features are normalized to range between 0 and 1. Following the embedding technology [18,24], categorical features are transformed to dense vectors, called embeddings. Normalized values of continuous features and dense embeddings of categorical features are concatenated together to constitute the final representation of input features. We denote the representations of , I, X as ∈ R , I ∈ R ×|I | and X ∈ R ×|X | respectively. Note that I and X are the concatenation of multiple feature embeddings. The CTR target is to approximate the probabilityˆ= ( = 1|x) by a discriminative function (·): = ( , X ; )(1) where denotes the parameters of the backbone model (·). Then the Binary Cross Entropy [7] is used to format the loss function: L ( , ) = − logˆ− (1 − ) log(1 −ˆ)(2) where denotes the parameters of the embedding layer, including , I and X . As the parameters is trained by historical data, the item ID embeddings of recent emerging items are rarely updated and stay around the initial point, leading to a low testing accuracy. It is known as the item cold-start problem. As the relationship between cold-start and warm-up in recommendation system is confusing, we will give a brief discussion here. Generally speaking, cold-start strategies are applied to items which first appear in the system, while warm-up strategies are applied to items whose exposure numbers are lower than a threshold. Note that this threshold is inconsistent across different systems. In other words, warm-up is a subsequent procedure of cold-start. In this work, We conduct cold-start and warm-up in a unified framework CVAR. The structure of CVAR is shown in Figure 1. The basic design of CVAR is generating a better embedding for item ID and replace the original unsufficiently trained embedding. The Item ID in Figure 1 denotes the original item ID embedding trained by backbone model, while the Warm Item ID denotes the enhanced warmedup embeddingˆgenerated by CVAR. Instead of directly learning a transformation from I to , CVAR aligns the representations transformed from and I in the autoencoder's latent space [8,44,47]. Following the design of CVAE [36], the autoencoder applied to is formulated as: , = ( ; ); ∈ R , ∈ R (3) ∼ N ( , Σ); Σ ∈ R × , (Σ) = (4) = ( , ; );ˆ∈ R(5) where and correspond to the Regular Encoder and Decoder in Figure 1, and denote the parameters of and , denotes the dimension of latent space. Notice that as shown in Equation (3) the latent representation is defined as a multivariate normal distribution N ( , Σ) with mean and diagonal covariance matrix Σ whose trace is . In Equation (4), latent representation is sampled from N ( , Σ) using the reparameterization trick [28]. Decoder takes along with conditional frequency as input and reconstructs item ID embedding asˆ. Since the item frequency information has a direct impact on the distribution of ID embedding and CVAR is operated in the entire warm-up phase, unlike traditional CVAE design [36,40,50], we only filter from the full side information as the condition of decoder to emphasize its impact. Reconstruction Loss between andˆis formulated by Euclidean distance: L ( , ) = ∥ −ˆ∥ 2 2(6) Notice that rarely updated of almost cold items in Equation (6) may mislead the training procedure. However, with as condition to , the generated embedding will be restricted to a reasonable space according to which could relieve this misleading effect. Thus we do not deal with this case separately. Besides, in inference stage, we set as a huge value which will help to produce warm ID embeddings. Through the autoencoder structure described above, we obtain the information-compressed latent distribution N ( , Σ) of . Aligning embeddings or distributions in autoencoder's latent space is a generally used technology which has been proven effective in various fields [8]. Thus we consider mapping the side information embedding I into the same latent space and aligning with N ( , Σ). Specifically, Prior Encoder ′ maps I to N ( ′ , Σ ′ ) and a wasserstein loss [27,35] function is applied to aligning distribution N ( , Σ) with N ( ′ , Σ ′ ): ′ , ′ = ′ ( I ; ′ ); ′ ∈ R , ′ ∈ R (7) ′ ∼ N ′ ( ′ , Σ ′ ); Σ ′ ∈ R × , (Σ ′ ) = ′ (8) ′ = ( ′ , ; );ˆ′ ∈ R (9) L ( , ′ ) = 2 (N ( , Σ), N ′ ( ′ , Σ ′ ))(10) where ′ denotes the parameters of ′ , 2 (·, ·) denotes the Wasserstein Distance. Instead of KL Divergence [2] or other distribution measurement, we choose Wasserstein Distance considering its symmetry and stability which is widely used in various scenarios [20,45,51,52]. We will give an additional discussion about the design of . Since that it's not feasible to extract frequency information from side information by ′ in item cold start scenario, the latent space ideally should contain no frequency information. However, the frequency information is already utilized in the origin item ID embedding and automatically compressed to the latent space under the Encoder-Decoder framework. Thus we set as the a independent condition of decoder to reduce the proportion of frequency information in latent space. In addition to L and L , we replace in Equation (1) with enhanced warmed-upˆ′ as item ID input to backbone and get the CTR loss L by forward computation: = (ˆ′, X ; ) (11) L ( , ) = − logˆ− (1 − ) log(1 −ˆ) (12) To avoid disturbing the recommendation of hot items, CVAR is taken as an independent module with backbone. During the training of CVAR, optimization of L in Equation (12) is only applied to and , while the parameters of (·) are fixed. We finally get the loss function to train CVAR: L ( , , ′ ) = L + L + L(13) where and are hyperparameters to fuse the losses. Moreover, training of CVAR is along with the backbone's training. As shown in Equation (13), CVAR will be trained by optimizing , , ′ to minimize L , using the same samples as training backbone, without any additional requirements on data. For a coming batch of samples, it's first fed to backbone to update the original item ID embedding , then used to train CVAR and update , , ′ . Updated is also consumed in the training of CVAR at each step. Thus we claim that CVAR not only learns the pattern in historical data but also uses the information of update at each step to relieve the cold-start issue. In inference phase of recent emerging items, their item ID embeddings are not well tained. Therefore we obtain ′ by sampling from N ′ which is generated by item side information and get enhanced warmed-up ID itemˆ′ by passing ′ to as Equation (7) and (8) shown. Then we could replace with ′ for testing of recent emerging items. As marked in Figure 1, Equation (4) and (5) are operated in training of CVAR, while Equation (8) and (9) play a by splitting the datasets following [26] and [57]. We divide items into two groups, old and new based on their frequency, where items with more than labeled instances are old and others are new. We use of 200 and 2000 for Movielens-1M and Taobao Ad data. Note that the ratio of new items to old items is approximately 8:2, which is similar to the definition of long-tail items [3]. Besides, new item instances sorted by timestamp are divided into four groups denoted as warm-a, -b, -c, and test set following [26] and [57]. Backbones and Baselines. Because CVAR is model-agnostic, it can be applied to various existing models in the Embedding & MLP paradigm. Thus we conduct experiments upon the following representative backbones: FM [30], DeepFM [12], Wide&Deep [4], DCN [42], IPNN [29], OPNN [29]. Meanwhile we choose some State-Of-The-Art(SOTA) methods for the item cold-start problem as baselines: DropoutNet [39], Meta embedding(Meta-E) [26], MWUF [57]. We reproduce each baseline based on open source code or their publications if the code is unavailable. We open all of the related source code on Github 4 . Implementation Details. For a fair comparison, we use the same setting for all methods. The MLPs in backbones and cold-start modules use the same structure with two dense layers (hidden units 16). The embedding size of each feature is fixed to 16. Learning rate and mini-batch size are set to 0.001 and 2048 respectively. At the inference stage, of new emerging items is set to the largest item frequency in the corresponding dataset to generate warmedup embeddings. Training is done with Adam [15] optimizer over shuffled samples. In experiments, we firstly use old item instances to pretrain the backbone model as well as the cold-start module and evaluate on the test set(Initialization phase). Then we in turn feed warm-a, -b, -c data to train the backbone or CVAR and evaluate models on the test set step by step. We take the AUC score [21] and the F1 score [14] as evaluation metrics. Experimental Results. We compare the cold-start effectiveness of CVAR with backbone and three SOTA cold-start baselines including DropoutNet, Meta-E, MWUF. Meanwhile, we choose two most famous CTR prediction methods as backbones, DeepFM and Wide&Deep. To evaluate the quality of initial embedding generated by CVAR and further prove the effectiveness of CVAR in leveraging the changing id embedding for better warm-up, we conduct a version of contrast experiment for CVAR denoted as CVAR (Init Only), where CVAR only plays a role in initialization phase and is disabled in following three warm-up phases. We conduct experiments on two datasets and evaluate the mean results over three runs. The main experimental results are shown in Table 1. Notice that CVAR (Init Only) outperforms other baselines in most cases, which indicates a high quality of initial embeddings generated by CVAR. Moreover, superior performance of CVAR (Init Only) proves that distribution alignment in latent space is better than directly mapping side information to item id which is adopted in Meta-E and MWUF. Except for a better initialization, CVAR can significantly improve the prediction performance in warmup stages which is proven by the outstanding results of CVAR in Table 1. This phenomenon demonstrates that CVAR can indeed produce high quality warmed-up embedding based on the evolving item ID embedding learned by backbone in warm-up stage. Method compatibility. Since CVAR is model-agnostic, we conduct experiments in more scenarios to verify its compatibility. Results on six popular backbones and two datasets in Figure 2 demonstrate the compatibility and robustness of CVAR. Comparison of different frequency condition As mentioned before, is set as the condition of decoder in CVAR and has a direct impact on the distribution of ID embedding. Considering the final is uncertain for new emerging items at inference stage, we compare the performance of CVAR with different . Because is normalized before using, we conduct five experiments with equal 0.01, 0.1, 0.25, 0.5 and 1. Experimental results are shown in Table 2. It's shown that performance of CVAR increases gradually with the increase of , which explains why we set as a huge value at inference stage for cold start. Online A/B tests To verify the effectiveness of CVAR, we further conduct online A/B tests [16] for 7 days on Tencent News recommendation platform 5 . As usually adopted in industry recommendation, the whole coarseto-fine recommendation progress can be divided into four stages : candidate generation, coarse-grained ranking, fine-grained ranking, and re-ranking. In our industrial scenario, CVAR is applied to the ranking stage whose backbone is MMOE [23,37]. Consistent with the four phases in offline experiments, we group online items into four groups: cold, warm-a, -b, -c, with increasing exposed frequency. In our system, there are two mediums of news: Article and Video. We focus on the following metrics [11] (from high importance to low): Exposure Rate, Watch Time, Page(Video) Views. Exposure Table 3: Online A/B results of item groups with increasing warm-up level: cold, warm-a, warm-b and warm-c. Red results mean they are statistically significant (whose p-value in hypothesis testing [1] is stable less than 0.05.) Metrics Cold Warm-a Warm-b Warm-c Total We show the online results on various item groups in Table 3. It's apparent that metrics of cold items are significantly improved, which proves the effectiveness of CVAR. As expected, the gain of CVAR gradually diminishes as the exposed frequency increases. Moreover, exposed items' Gini coefficient [56] on various item categories reduces from 0.7413 to 0.7369 after applying CVAR, which indicates CVAR could alleviate the Matthew Effect [41] to some extent. CONCLUSION In this paper, we proposed the model-agnostic CVAR for item coldstart which uses latent variables to learn a distribution over side information and generates desirable ID embeddings using a conditional decoder. For better cold-start performance, CVAR not only learns the pattern in historical data but also leverages the continuously updating item ID embedding from the backbone to generate enhanced warmed-up embeddings with superior quality. From the aspect of industrial practice, we claim that additional strict data requirements of cold-start methods will make the deployment process rather difficult in online scenario. Thus CVAR is designed to be trained with the same raw samples as training main prediction model. Note that the proposed CVAR is a general framework that can be applied to various backbones. Finally, extensive offline experiments on public datasets and online A/B tests show the effectiveness and compatibility of CVAR. Figure 1 : 1The proposed model-agnostic CVAR Framework, where Backbone Model can be any CTR prediction model with embedding layer. Figure 2 : 2AUC curves through warming-up on two datasets, over six popular backbone models, three runs for each. Table 1 : 1Model Comparison of cold-start effectiveness on two datasets(Movielens 1M and Taobao Ad), under two backbones(DeepFM and Wide&Deep), three runs for each. The best improvements are highlighted in bold.role for recent emerging items in testing phase. For a single item, besides just searching for a better initialization of ID embedding, this replacement operation will continue until is fully trained. Dataset Split. To demonstrate the recommendation performance in both cold-start and warm-up phases, we conduct the experimentsMethods Cold phase Warm-a phase Warm-b phase Warm-c phase AUC F1 AUC F1 AUC F1 AUC F1 Dataset: Movielens 1M & Backbone: DeepFM DeepFM 0.7267 0.6231 0.7424 0.6383 0.7574 0.6503 0.7694 0.6608 DropoutNet 0.7387 0.6339 0.7491 0.6441 0.7587 0.6531 0.7673 0.6599 Meta-E 0.7327 0.6344 0.7441 0.6432 0.7544 0.6519 0.7633 0.6592 MWUF 0.7316 0.6289 0.7462 0.6413 0.7589 0.6521 0.7701 0.6616 CVAR (Init Only) 0.7401 0.6353 0.7518 0.6454 0.7624 0.6547 0.7717 0.6622 CVAR 0.7419 0.6356 0.7927 0.6789 0.8021 0.6856 0.8041 0.6878 Dataset: Movielens 1M & Backbone: Wide&Deep Wide&Deep 0.7071 0.5972 0.7232 0.6164 0.7354 0.6273 0.7461 0.6372 DropoutNet 0.7125 0.6038 0.7228 0.6159 0.7313 0.6244 0.7390 0.6314 Meta-E 0.6727 0.5287 0.7201 0.6120 0.7345 0.6280 0.7450 0.6374 MWUF 0.7063 0.5966 0.7230 0.6157 0.7355 0.6275 0.7459 0.6366 CVAR (Init Only) 0.7020 0.5795 0.7255 0.6160 0.7375 0.6293 0.7473 0.6380 CVAR 0.6937 0.5643 0.7627 0.6525 0.7756 0.6639 0.7840 0.6712 Dataset: Taobao Ad & Backbone: DeepFM DeepFM 0.5983 0.1350 0.6097 0.1378 0.6207 0.1401 0.6311 0.1438 DropoutNet 0.5989 0.1352 0.6098 0.1374 0.6203 0.1396 0.6302 0.1435 Meta-E 0.5982 0.1346 0.6093 0.1377 0.6195 0.1400 0.6294 0.1428 MWUF 0.5986 0.1348 0.6082 0.1374 0.6184 0.1399 0.6279 0.1429 CVAR (Init Only) 0.5987 0.1350 0.6098 0.1376 0.6204 0.1398 0.6306 0.1432 CVAR 0.5978 0.1347 0.6198 0.1408 0.6308 0.1477 0.6380 0.1503 Dataset: Taobao Ad & Backbone: Wide&Deep Wide&Deep 0.6081 0.1360 0.6129 0.1427 0.6207 0.1455 0.6287 0.1484 DropoutNet 0.6095 0.1359 0.6184 0.1427 0.6246 0.1454 0.6312 0.1474 Meta-E 0.6082 0.1378 0.6122 0.1443 0.6190 0.1477 0.6259 0.1506 MWUF 0.6089 0.1382 0.6125 0.1423 0.6210 0.1457 0.6285 0.1483 CVAR (Init Only) 0.6027 0.1359 0.6065 0.1429 0.6163 0.1471 0.6232 0.1496 CVAR 0.6051 0.1368 0.6220 0.1457 0.6290 0.1495 0.6336 0.1511 3 EXPERIMENTS 3.1 Offline Experiments 3.1.1 Experiment Setup. In this section, we will introduce the of- fline experiment setup. Public Datasets. For offline experiments, We evaluate CVAR on the following two public datasets MovieLens-1M 1 and Taobao Ad 2 . • MovieLens-1M: One of the most well-known recommenda- tion benchmark dataset. The data consists of 1 million movie ranking instances over thousands of movies and users. Fea- tures of a movie include its title, year of release, and genres which are seen as item side information. Titles and genres are lists of tokens. Each user has features including the user's ID, age, gender and occupation. • Taobao Display Ad Click: It randomly samples 1140000 users from 26 million ad display / click records on Taobao 3 web- site to construct the dataset. We take each Ad as a item for CTR prediction, with 4 categorical attributes as side in- formation, including category ID, campaign ID, brand ID, advertiser ID. Each user has features including Micro group ID, cms_group_id, gender, age, consumption grade, shopping depth, occupation and city level. Table 2 : 2CVAR performance(AUC) with different on Movielens1M and Wide&Deep, three runs for each. The best improvements are highlighted in bold.Cold Warm-a Warm-b Warm-c 0.01 0.6936 0.7627 0.7756 0.7839 0.1 0.6939 0.7629 0.7756 0.7837 0.25 0.6946 0.7627 0.7754 0.7842 0.5 0.6956 0.7638 0.7754 0.7844 1.0 0.6973 0.7649 0.7757 0.7845 39% +0.38% Article Watch Time +2.39% +4.51% +2.08% +0.16% +0.13% Video Watch Time +2.60% +1.78% +0.72% +0.68% +0.66% Rate measures the distribution percentage of the item group. Watch Time and Page(Video) Views reflect how users are attracted by recommended content.Exposure Rate +1.48% -0.21% -0.04% +0.17% - Watch Time +2.49% +2.90% +1.40% +0.Total Page Views +4.46% +2.87% +1.42% +0.62% +1.09% Article Page Views +3.58% +3.37% +1.74% +0.31% +0.82% Video Views +5.84% +2.25% +1.05% +1.01% +1.35% http://www.grouplens.org/datasets/movielens 2 https://tianchi.aliyun.com/dataset/dataDetail?dataId=56 3 https://www.taobao.com https://github.com/BestActionNow/CVAR https://news.qq.com/ P value and the theory of hypothesis testing: an explanation for new researchers. David Jean Biau, Brigitte M Jolles, Raphaël Porcher, Clinical Orthopaedics and Related Research®. 468David Jean Biau, Brigitte M Jolles, and Raphaël Porcher. 2010. P value and the theory of hypothesis testing: an explanation for new researchers. Clinical Orthopaedics and Related Research® 468, 3 (2010), 885-892. Estimation of KL divergence: Optimal minimax rate. 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[ "https://github.com/BestActionNow/CVAR" ]
[ "Graded Algebraic Theories", "Graded Algebraic Theories" ]
[ "Satoshi Kura [email protected] \nNational Institute of Informatics\nTokyoJapan\n\nThe Graduate University for Advanced Studies (SOKENDAI)\nKanagawaJapan\n" ]
[ "National Institute of Informatics\nTokyoJapan", "The Graduate University for Advanced Studies (SOKENDAI)\nKanagawaJapan" ]
[]
We provide graded extensions of algebraic theories and Lawvere theories that correspond to graded monads. We prove that graded algebraic theories, graded Lawvere theories, and finitary graded monads are equivalent via equivalence of categories, which extends the equivalence for monads. We also give sums and tensor products of graded algebraic theories to combine computational effects as an example of importing techniques based on algebraic theories to graded monads.
10.1007/978-3-030-45231-5_21
null
211,133,140
2002.06784
a407a7b77ee9dc08623d218baf5994f18a1fa8a9
Graded Algebraic Theories Satoshi Kura [email protected] National Institute of Informatics TokyoJapan The Graduate University for Advanced Studies (SOKENDAI) KanagawaJapan Graded Algebraic Theories 10.1007/978-3-030-45231-5 We provide graded extensions of algebraic theories and Lawvere theories that correspond to graded monads. We prove that graded algebraic theories, graded Lawvere theories, and finitary graded monads are equivalent via equivalence of categories, which extends the equivalence for monads. We also give sums and tensor products of graded algebraic theories to combine computational effects as an example of importing techniques based on algebraic theories to graded monads. Introduction In the field of denotational semantics of programming languages, monads have been used to express computational effects since Moggi's seminal work [18]. They have many applications from both theoretical and practical points of view. Monads correspond to algebraic theories [5]. This correspondence gives natural presentations of many kinds of computational effects by operations and equations [21], which is the basis of algebraic effect [20]. The algebraic perspective of monads also provides ways of combining [9], reasoning about [22], and handling computational effects [23]. Graded monads [27] are a refinement of monads and defined as a monadlike structure indexed by a monoidal category (or a preordered monoid). The unit and multiplication of graded monads are required to respect the monoidal structure. This structure enables graded monads to express some kind of "abstraction" of effectful computations. For example, graded monads are used to give denotational semantics of effect systems [12], which are type systems designed to estimate scopes of computational effects caused by programs. f ∈ Σ n,m t i ∈ T Σ m (X) for each i ∈ {1, . . . , n} f (t 1 , . . . , t n ) ∈ T Σ m⊗m (X) This paper provides a graded extension of algebraic theories that corresponds to monads graded by small strict monoidal categories. This generalizes Ngraded theories in [17]. The main ideas of this extension are the following. First, we assign to each operation a grade, i.e., an object in a monoidal category that represents effects. Second, our extension provides a mechanism (Fig 1) to keep track of effects in the same way as graded monads. That is, if an operation f with grade m is applied to terms with grade m , then the grade of the whole term is the product m ⊗ m . For example, graded algebraic theories enable us to estimate (an overapproximation of) the set of memory locations computations may access. The sideeffects theory [21] is given by operations lookup l and update l,v for each location l ∈ L and value v ∈ V together with several equations, and each term represents a computation with side-effects. Since lookup l and update l,v only read from or write to the location l, we assign {l} ∈ 2 L as the grade of the operations in the graded version of the side-effects theory where 2 L is the join-semilattice of subsets of locations L. The grade of a term is (an overapproximation of) the set of memory locations the computations may access thanks to the rule in Fig 1. We also provide graded Lawvere theories that correspond to graded algebraic theories. The intuition of a Lawvere theory is a category whose arrows are terms of an algebraic theory. We use this intuition to define graded Lawvere theories. In graded algebraic theories, each term has a grade, and substitution of terms must respect the monoidal structure of grades. To characterize this structure of "graded" terms, we consider Lawvere theories enriched in a presheaf category. Like algebraic theories brought many concepts and techniques to the semantics of computational effects, we expect that the proposed graded algebraic theories will do the same for effect systems. We look into one example out of such possibilities: combining graded algebraic theories. The main contributions of this paper are summarized as follows. -We generalize (N-)graded algebraic theories of [17] to M-graded algebraic theories and also provide M-graded Lawvere theories where M is a small strict monoidal category. We show that there exist translations between these notions and finitary graded monads, which yield equivalences of categories. -We extend sums and tensor products of algebraic theories [9] to graded algebraic theories. We define sums in the category of M-graded algebraic theories, and tensor products as an M × M -graded algebraic theory made from an M-graded and an M -graded algebraic theory. We also show a few properties and examples of these constructions. Preliminaries Enriched Category Theory We review enriched category theory and introduce notations. See [13] for details. Let V 0 = (V 0 , ⊗, I) be a (not necessarily symmetric) monoidal category. V 0 is right closed if (−) ⊗ X : V 0 → V 0 has a right adjoint [X, −] for each X ∈ obV 0 . Similarly, V 0 is left closed if X ⊗ (−) has a right adjoint X, − for each X ∈ obV 0 . V 0 is biclosed if V 0 is left and right closed. Let V 0 t denote the monoidal category (V 0 , ⊗ t , I) where ⊗ t is defined by X ⊗ t Y := Y ⊗ X. Note that V 0 t is right closed if and only if V 0 is left closed. We define V 0 -category, V 0 -functor and V 0 -natural transformation as in [13]. If V 0 is right closed, then V 0 itself enriches to a V 0 -category V with homobject given by V(X, Y ) := [X, Y ]. We use the subscript (−) 0 to distinguish the enriched category V from its underlying category V 0 . Assume that V 0 is biclosed and let A be a V 0 -category. The opposite cat- egory A op is the V 0 t -category defined by A op (X, Y ) = A(Y, X). For any X ∈ obA, A(X, −) : A → V 0 is a V 0 -functor where A(X, −) Y,Z : A(Y, Z) → [A(X, Y ), A(X, Z)] is defined by transposing the composition law • of A. A V 0 t -functor A(−, X) is defined by A op (X, −) : A op → V 0 t . Let A be a V 0 -category. For each X ∈ V 0 and C ∈ A, a tensor X ⊗ C is an object in A together with a counit morphism ν : X → A(C, X ⊗ C) such that a V 0 -natural transformation A(X ⊗ C, −) → X, A(C, −) obtained by transposing (•) • (A(X ⊗ C, B) ⊗ ν) is isomorphic where • is the composition in the V 0 -category A. A cotensor X C is a tensor in A op . For example, if V 0 = Set, then tensors X ⊗ C are copowers X · C, and cotensors X C are powers C X . A V 0 -functor F : A → B is said to preserve a tensor X ⊗ C if F C,X⊗C • ν : X → B(F C, F (X ⊗ C)) is again a counit morphism. F preserves cotensors if F op preserves tensors. Let Φ be a collection of objects in V 0 . A V 0 -functor F : A → B is said to preserve Φ-(co)tensors if F preserves (co)tensors of the form X ⊗ C (X C) for each X ∈ Φ and C ∈ obA. Graded Monads We review the notion of graded monad in [12,7], and then define the category GMnd M of finitary M-graded monads. Throughout this section, we fix a small strict monoidal category M = (M, ⊗, I). Definition 1 (graded monads). An M-graded monad on C is a lax monoidal functor M → [C, C] where [C, C] is a monoidal category with composition as multiplication. That is, an M-graded monad is a tuple ( * , η, µ) of a functor * : M × C → C and natural transformations η X : X → I * X and µ m1,m2,X : m 1 * (m 2 * X) → (m 1 ⊗ m 2 ) * X such that the following diagrams commute. m * X I * (m * X) m * (I * X) m * X η m * η µ µ m 1 * (m 2 * (m 3 * X)) m 1 * ((m 2 ⊗m 3 ) * X) (m 1 ⊗m 2 ) * (m 3 * X) (m 1 ⊗m 2 ⊗m 3 ) * X m1 * µ µ µ µ A morphism of M-graded monad is a monoidal natural transformation α : ( * , η, µ) → ( * , η , µ ), i.e. a natural transformation α : * → * that is compatible with η and µ. An intuition of graded monads is a refinement of monads: m * X is a computation whose scope of effect is indicated by m and whose result is in X. The monoidal category M defines the granularity of the refinement, and a 1-graded monad is just an ordinary monad. Note that we do not assume that M is symmetric because some of graded monads in [12] require M to be nonsymmetric. We also deal with such a nonsymmetric case in Example 25. A finitary functor is a functor that preserves filtered colimits. In this paper, we focus on finitary graded monads on Set. A morphism in GMnd M is determined by the restriction to ℵ 0 ⊆ Set where ℵ 0 is the full subcategory of Set on natural numbers. Lemma 3. Let T = ( * , η, µ) and T = ( * , η , µ ) be finitary M-graded monads. There exists one-to-one correspondence between the following. 1. Morphisms α : T → T . 2. Natural transformations β : * • (M × i) → * • (M × i) (where i : ℵ 0 → Set is the inclusion functor) such that the following diagrams commute for each n, n ∈ ℵ 0 , m 1 , m 2 ∈ M and f : n → m 2 * n . n I * n I * n ηn η n β m 1 * n m 1 * n m 1 * (m 2 * n ) m 1 * (m 2 * n ) m 1 * (m 2 * n ) (m 1 ⊗ m 2 ) * n (m 1 ⊗ m 2 ) * n β m1 * f m1 * f m1 * β µ µ β Proof. By the equivalence [Set, Set] f [ℵ 0 , Set] induced by restriction and the left Kan extension along the inclusion i : ℵ 0 → Set. Day Convolution We describe a monoidal biclosed structure on the (covariant) presheaf category Note that since we do not assume M to be symmetric, neither is [M, Set] 0 . Note also that the twisting and the above construction commute: there is an isomorphism [M, Set] 0 t ∼ = [M t , Set] 0 of monoidal categories. Categories Enriched in a Presheaf Category We rephrase the definitions of [M, Set] 0 -enriched category, functor and natural transformation in elementary terms. An [M, Set] 0 -category is, so to say, an "Mgraded" category: each morphism has a grade m ∈ obM and the grade of the composite of two morphisms with grades m and m is the product m ⊗ m of the grades of each morphism. Likewise, [M, Set] 0 -functors and [M, Set] 0 -natural transformations can be also understood as an "M-graded" version of ordinary functors and natural transformations. Specifically, the following lemma holds [2]. Lemma 6. There is a one-to-one correspondence between (1) an [M, Set] 0category C and (2) the following data satisfying the following conditions. -A class of objects obC. -For each X, Y ∈ obC, a hom objects C(X, Y ) ∈ [M, Set] 0 . -For each X ∈ obC, an element 1 X ∈ C(X, X)I. -For each X, Y, Z ∈ obC, a family of morphisms • m1,m2 : These data must satisfy the identity law C(Y, Z)m 1 × C(X, Y )m 2 → C(X, Z)(m 1 ⊗ m 2 ) m1,1 Y • f = f = f • 1 X for each f ∈ C(X, Y )m and the associativity (h • g) • f = h • (g • f ) for each f ∈ C(X, Y )m 1 , g ∈ C(Y, Z)m 2 and h ∈ C(Z, W )m 3 . Proof. The identity 1 X : y(I) → C(X, X) in C corresponds to 1 X ∈ C(X, X)I by the Yoneda lemma, and the composition • : C(Y, Z)⊗ C(X, Y ) → C(X, Z) in Cf ∈ C T (Y, Z)m and g ∈ C T (X, Y )m is µ • (m * g) • f . The definition of C T is similar to the definition of the Kleisli categories for ordinary monads. Actually, C T can be constructed via the Kleisli category C T for the graded monad T presented in [7] (although C T itself is not enriched). This can be observed by C T ((I, X), (m, Y )) ∼ = C T (X, Y )m. Graded Algebraic Theories We explain a framework of universal algebra for graded monads, which is a natural extension of [27,17]. The key idea of this framework is that each term is associated with not only an arity but also a "grade", which is represented by an object in a monoidal category M. We also add coercion construct for terms that changes the grade of terms along a morphism of the monoidal category M. Then, a mapping that takes m ∈ M and a set of variables X and returns the set of terms with grade m (modulo the equational axioms) yields a graded monad. We fix a small strict monoidal category M = (M, ⊗, I) throughout this section. We sometimes identify n ∈ N with {1, . . . , n}, or {x 1 , . . . , x n } if it is used as a set of variables. Equational Logic A signature is a family of sets of symbols Σ = (Σ n,m ) n∈N,m∈M . An element f ∈ Σ n,m is called an operation with arity n and grade m. We define a sufficient structure to interpret operations in a category C as follows. Definition 8. M-model condition is defined by the following conditions on a tuple (C, ( , η , µ )). -C is a category with finite power. -( , η , µ ) is a strong M t -action (i.e. an M t -graded monad whose unit and multiplication are invertible). A model A = (A, | · | A ) of Σ in a category C satisfying M-model condition consists of an object A ∈ C and an interpretation |f | A : A n → m A for each f ∈ Σ n,m . A homomorphism α : A → B between two models A, B is a morphism α : A → B in C such that (m α) • |f | A = |f | B • α n for each f ∈ Σ n,m . Definition 10. Let X be a set of variables. The set of (M-graded) Σ-terms T Σ m (X) for each m ∈ M is defined inductively as follows. x ∈ X x ∈ T Σ I (X) t ∈ T Σ m (X) w : m → m cw(t) ∈ T Σ m (X) f ∈ Σn,m ∀i ∈ {1, . . . , n}, ti ∈ T Σ m (X) f (t1, . . . , tn) ∈ T Σ m⊗m (X) That is, we build Σ-terms from variables by applying operations in Σ and coercions c w while keeping track of the grade of terms. When applying operations, we sometimes write f (λi ∈ n.t i ) or f (λi.t i ) instead of f (t 1 , . . . , t n ). Definition 11. Let A be a model of a signature Σ. For each m ∈ M and s ∈ T Σ m (n), the interpretation |s| A : A n → m A is defined as follows. -For any variable x i , |x i | A = η • π i where π i : A n → A is the i-th projection. -For each w : m → m and s ∈ T Σ m ({x 1 , . . . , x n }), |c w (s)| A = (w A) • |s| A . -If f ∈ Σ k,m and t i ∈ T Σ m ({x 1 , . . . , x n }) for each i ∈ {1, . . . , k}, then |f (t 1 , . . . , t k )| A is defined by the following composite. A n (m A) k m A k m (m A) (m ⊗m ) A |t1|,...,|t k | ∼ = m |f | µ When we interpret a term t ∈ T Σ m (X), we need to pick a finite set n such that fv(t) ⊆ n ⊆ X where fv(t) is the set of free variables in t, but the choice of the finite set does not matter when we consider only equality of interpretations by the following fact. If σ : n → n is a renaming of variables and σ : T Σ m (n) → T Σ m (n ) is a mapping induced by the renaming σ, then for each t ∈ T Σ m (n), |σ(t)| A = |t| A • A σ , which implies that equality of the interpretations of two terms s, t is preserved by renaming: |s| = |t| implies |σ(s)| = |σ(s)|. An equational axiom is a family of sets E = (E m ) m∈M where E m is a set of pairs of terms in T Σ m (X). We sometimes identify E with its union m∈M E m . A presentation of an M-graded algebraic theory (or an M-graded algebraic theory) is a pair T = (Σ, E) of a signature and an equational axiom . A model A of (Σ, E) is a model of Σ that satisfies |s| A = |t| A for each (s = t) ∈ E. Let Mod T (C) denote the category of models of T in C and homomorphisms between them. To obtain a graded monad on Set from T , we need a strict left action of M on Mod T ([M, Set] 0 ) and an adjunction between Mod T ([M, Set] 0 ) and Set. The former is defined by the following, while the latter is described in §3.2. Proof. [M, Set] 0 has M t × M-action defined by (m 1 , m 2 ) * F = F (m 1 ⊗ (−) ⊗ m 2 ). Thus, M-action for Mod T ([M, Set] 0 ) is obtained by Lemma 12. Substitution s[t 1 /x 1 , . . . , t k /x k ] for M-graded Σ-terms can be defined as usual, but we have to take care of grades: given s ∈ T Σ m (k) and t 1 , . . . , t k ∈ T Σ m (n), the substitution s[t 1 /x 1 , . . . , t k /x k ] is defined as a term in T Σ m⊗m (n). We obtain an equational logic for graded theories by adding some additional rules to the usual equational logic. Definition 14. The entailment relation T s = t (where s, t ∈ T m (X)) for an M-graded theory T is defined by adding the following rules to the standard rules i.e. reflexivity, symmetry, transitivity, congruence, substitution and axiom in E (see e.g. [26] for the standard rules of equational logic). s, t ∈ T Σ m (X) T s = t w : m → m T cw(s) = cw(t) t ∈ T Σ m (X) T c1 m (t) = t t ∈ T Σ m (X) w : m → m w : m → m T c w (cw(t)) = c w •w (t) f ∈ Σn,m ti ∈ T Σ m (X) for each i ∈ {1, . . . , n} w : m → m T f (cw(t1), . . . , cw(tn)) = cm⊗w(f (t1, . . . , tn)) Definition 15. Given a model A of T , we denote A s = t if s, t ∈ T Σ m (n) (for some n) and |s| A = |t| A . If C is a category satisfying M-model condition, we denote T , C s = t if A s = t for any model A of T in C. It is easy to verify that the equational logic in Definition 14 is sound. Theorem 1 (soundness). T s = t implies T , C s = t. Free Models We describe a construction of a free model F T X ∈ Mod T ([M, Set] 0 ) of a graded theory T generated by a set X, which induces an adjunction between Mod T ([M, Set] 0 ) and Set. This adjunction, together with the M-action of Corollary 13, gives a graded monad as described in [7]. Definition 16 (free model F T X). Let T = (Σ, E) be an M-graded theory. We define a functor F T X : M → Set by F T Xm := T Σ m (X)/∼ m for each m ∈ M and any X ∈ Set where s ∼ m t is the equivalence relation defined by T s = t and F T Xw([t] m ) := [c w (t)] m for any w : m → m where [t] m is the equivalence class of t ∈ T Σ m (X). For each f ∈ Σ n,m , let |f | F T X : (F T X) n → m F T X be a mapping defined by |f | F T X m ([t 1 ] m , . . . , [t n ] m ) = [f (t 1 , . . . , t n )] m ⊗m for each m ∈ M. We define a model of T by F T X = (F T X, | · | F T X ). The model F T X, together with the mapping η X : X → F T XI defined by x → [x] I , has the following universal property as a free model generated by X. v : F T X → A satisfying v I • η X = v. Corollary 18. Let U : Mod T ([M, Set] 0 ) → Set be the forgetful functor defined by the evaluation at I, that is, U A = A I and U α = α I . The free model functor F T : Set → Mod T ([M, Set] 0 ) is a left adjoint of U . By considering the interpretation in the free model, we obtain the following completeness theorem. Theorem 19 (completeness). T , [M, Set] 0 s = t implies T s = t. Recall that Mod T ([M, Set] 0 ) has a left action (Corollary 13). Therefore the above adjunction induces an M-graded monad as described in [7]. The relationship between Mod T ([M, Set] 0 ) and the Eilenberg-Moore construction is as follows. In [7], the Eilenberg-Moore category C T for any graded monad T on C is introduced together with a left action : M × C T → C T . If C = Set and T is the graded monad obtained from an M-graded theory T , then the Eilenberg-Moore category Set T is essentially the same as Mod T ([M, Set] 0 ). Theorem 20. The comparison functor K : Mod T ([M, Set] 0 ) → Set T (see [7] for the definition) where T is an M-graded theory and T is the graded monad induced from the graded theory T is isomorphic. Moreover, K preserves the Maction: • (M × K) = K • . We define the category GS M of graded algebraic theories as follows. Definition 21. Let T = (Σ, E) and T = (Σ , E ). A morphism α : T → T between graded algebraic theories is a family of mappings α n,m : Σ n,m → F T nm from operations in Σ to Σ -terms such that the equations in E are preserved by α, i.e. for each s, t ∈ T Σ m (X), (s, t) ∈ E implies |s| (F T X,α) = |t| (F T X,α) where (F T X, α) is a model of T induced by α. Definition 22. Given a morphism α : T → T , let F α : F T → F T be a natural transformation defined by F α ([t]) = |t| (F T X,α) for each t ∈ T Σ m (X). Definition 23. We write GS M for the category of graded algebraic theories and morphisms between them. The identity morphisms are defined by 1 T (f ) = [f (x 1 , . . . , x n )] for each f ∈ Σ n,m . The composition of α : T → T and β : T → T is defined by β • α(f ) = F β (α(f )). Examples Example 24 (graded modules). Let M = (N, +, 0) where N is regarded as a discrete category. Given a graded ring A = n∈N A n , let Σ be a set of operations which consists of the binary addition operation + (arity: 2, grade: 0), the unary inverse operation − (arity: 1, grade: 0), the identity element (nullary operation) 0 (arity: 0, grade: 0) and the unary scalar multiplication operation a · (−) (arity: 1, grade: n) for each a ∈ A n . Let E be the equational axiom for modules. A model (F, | · |) of the M-graded theory (Σ, E) in [M, Set] 0 consists of a set F n for each n ∈ N and functions |+| n : (F n ) 2 → F n , |−| n : F n → F n , |0| n ∈ F n and |a · (−)| n : F n → F m+n for each n ∈ N and each a ∈ A m , and these interpretations satisfy E. Therefore models of (Σ, E) in [M, Set] 0 correspond one-to-one with graded modules. m * X = {Er(e) | e ∈ m \ {Ok}} ∪ {Ok(x) | x ∈ X ∧ Ok ∈ m} η X (x) = Ok(x) µ m1,m2,X (Er(e)) = Er(e) µ m1,m2,X (Ok(x)) = x The M-graded theory T ex for the graded exception monad is defined by (Σ ex , ∅) where Σ ex is the set that consists of an operation raise e (arity: 0, grade: {e}) for each e ∈ Ex. The graded monad induced by T ex coincides with the graded exception monad. Indeed, the free model functor F T ex for T ex is given by F T ex Xm = m * X. Here, the operations raise e are interpreted by e ∈ Ex. Intuitively, this can be understood as follows. Since all the operations are of grade I, coercions c w in a term can be moved to the innermost places where variables occur by repeatedly applying c w (f (t 1 , . . . , t n )) = f (c w (t 1 ), . . . , c w (t n )) (see Definition 14). Therefore, we can consider terms of T M as terms of T whose variables are of the form c w (x). An M-graded monad ( * , η, µ) obtained from T M is as follows. |raise e | F T ex X m = Er(e) ∈ F T ex X({e} ⊗ m)m * X = T (M(I, m) × X) η = η T (1 I , −) µ = T (⊗ × X) • µ T • T st Graded Lawvere Theories We present a categorical formulation of graded algebraic theories of §3 in a similar fashion to ordinary Lawvere theories. For ordinary (single-sorted) finitary algebraic theories, a Lawvere theory is defined as a small category L with finite products together with a strict finiteproduct preserving identity-on-objects functor J : ℵ op 0 → L where ℵ 0 is the full subcategory of Set on natural numbers. Intuitively, morphisms in the Lawvere theory L are terms of the corresponding algebraic theory, and objects of L, which are exactly the objects in obℵ 0 , are arities. According to the above intuition, it is expected that a graded Lawvere theory is also defined as a category whose objects are natural numbers and morphisms are graded terms. However, since terms in a graded algebraic theory are stratified by a monoidal category M, mere sets are insufficient to express hom-objects of graded Lawvere theories. Instead, we take hom-objects from the functor category Proof. A cotensor (n · y(I)) (n · y(I)) is a tensor (n · y(I)) ⊗ t (n · y(I)) in [M, Set] t . Since ⊗ t is biclosed, ⊗ t preserves colimits in both arguments. Therefore, (n · y(I)) ⊗ t (n · y(I)) ∼ = (n · n ) · y(I). f → (x → ν(x) • f ). The condition that ν is isomorphic can be rephrased as follows. Proof. The essence of the proof is that the unit morphism ν : n · y(I) → A(n C, C) corresponds to elements π 1 , . . . , π n ∈ A(n C, C)I by [M, Set] 0 (n · y(I), A(n C, C)) ∼ = [M, Set] 0 (y(I), A(n C, C)) n ∼ = A(n C, C)I n . The The latter part of the lemma follows from the former part. If (π 1 , . . . , π n ) ∈ (A(n C, C)I) n satisfies the condition in Lemma 29, we call the element π i ∈ A(n C, C)I the i-th projection of n C. Note that the choice of projections is not necessarily unique. However, when we say that A is an [M, Set] 0 -category with N op M -cotensors, we implicitly assume that there are a chosen cotensor n C and chosen projections (π 1 , . . . , π n ) ∈ (A(n C, C)I) n for each n ∈ obN op M and C ∈ obA. We also assume that 1 X = X without loss of generality. Proof. For any X ∈ C T op and n, the cotensor n X is given by finite power X n , and the i-th projection is given by η • π i ∈ C T op I where π i : X n → X is the i-th projection of the finite power X n . The rest of the proof is routine. Equivalence We have shown three graded notions: graded algebraic theories, graded Lawvere theories and finitary graded monads, which give rise to categories GS M , GLaw M and GMnd M , respectively. This section is about the equivalence of these three notions. We give only a sketch of the proof of the equivalence, and the details are deferred to [14, Appendix A]. Graded Algebraic Theories and Graded Lawvere Theories We prove that the category of graded algebraic theories GS M and the category of graded Lawvere theories GLaw M are equivalent by showing the existence of an adjoint equivalence Th U : GLaw M → GS M . Let M be a small strict monoidal category and T = (Σ, E) be an M-graded algebraic theory. We define ThT (the object part of Th) as an M-graded Lawvere theory whose morphisms are terms of T modulo equational axioms. It is easy to show that ThT has N op M -cotensors (by Lemma 29). Therefore, Th is a mapping from an object in GS M to an object in GLaw M . We define a functor U : GLaw M → GS M by taking all the morphism f ∈ L(n, 1)m in L ∈ GLaw M as operations and all the equations that hold in L as equational axioms. Moreover, the unit and the counit of Th U are isomorphic. Therefore: Theorem 37. Two categories GS M and GLaw M are equivalent. We can also prove the equivalence of the categories of models. Lemma 38. If C is a category satisfying M-model condition, then Mod T (C) is equivalent to Mod(ThT , C T ) where T is the M t -action on C. Graded Lawvere theories and Finitary Graded Monads We prove that the category of graded Lawvere theories GLaw M and the category of finitary graded monads GMnd M are equivalent. Given a graded Lawvere theory, a finitary graded monad is obtained as a coend that represents the set of terms. On the other hand, given a finitary graded monad, a graded Lawvere theory is obtained from taking the full sub-[M, Set] 0 -category on arities ob(N op M ) of the opposite category of the Kleisli(-like) category in Definition 7. These constructions give rise to an equivalence of categories. An M-graded Lawvere theory yields a finitary graded monad by letting m * X be the set of terms of grade m whose variables range over X. Definition 39. Let L be an M-graded Lawvere theory. We define T L = ( * , η, µ) by a (finitary) M-graded monad whose functor part is given as follows. m * X := n∈ℵ0 L(n, 1)m × X n Note that L(−, 1) : ℵ 0 → [M, Set] 0 is a Set-functor here. Given a graded monad, a graded Lawvere theory is obtained as follows. Since L T has N M -cotensors n 1 = n whose projections are given by π i = ( * → η(i)) ∈ Set(1, I * n), L T is a graded Lawvere theory. Given a morphism α : T → T in GMnd M , we define L α : L T → L T by (L α ) n,n ,m = Set(n , α n,m ) : L T (n, n )m → L T (n, n )m. It is easy to prove that L α is a morphism in GLaw M and L (−) : GMnd M → GLaw M is a functor. Theorem 41. Two categories GLaw M and GMnd M are equivalent. Proof. L (−) is an essentially surjective fully faithful functor. Combining Effects Under the correspondence to algebraic theories, combinations of computational effects can be understood as combinations of algebraic theories. In particular, sums and tensor products are well-known constructions [9]. In this section, we show that these constructions can be adapted to graded algebraic theories. By the equivalence GMnd M GLaw M GS M in §5, constructions like sums and tensor products in one of these categories induce those in the other two categories. So, we choose GS M and describe sums as colimits in GS M and tensor products as a mapping GS M1 × GS M2 → GS M1×M2 . Sums We prove that GS M has small colimits. Lemma 42. The category GS M has small coproducts. Proof. Given a family {(Σ (i) , E (i) )} i∈I of objects in GS M , the coproduct is obtained by the disjoint union of operations and equations: i∈I (Σ (i) , E (i) ) = i∈I Σ (i) , i∈I E (i) . Lemma 43. The category GS M has coequalizers. Proof. Let T = (Σ, E) and T = (Σ , E ) be graded algebraic theories and α, β : T → T be a morphism. The coequalizer T of α and β is given by adding the set of equations induced by α and β to T , that is, T := (Σ , E ∪E ) where E = {(s, t) | ∃f ∈ Σ, α(f ) = [s] ∧ β(f ) = [t]}. Since a category has all small colimits if and only if it has all small coproducts and coequalizers, we obtain the following corollary. A free model functor F for T ex + T M is given by F Xm = T (m * ex X). For each n-ary operation f in T , |f | F X m : (T (m * ex X)) n → T (m * ex X) is induced by free models of T , and for each e ∈ Ex, |raise e | F X m : 1 → T ({e} * ex X) is defined by η T {e} * ex X (e) ∈ T ({e} * ex X). It is easy to see that F X defined above is indeed a model of T ex + T M . Therefore, we obtain a graded monad m * X = T (m * ex X). Tensor Products The tensor product of two ordinary algebraic theories (Σ, E) and (Σ , E ) is constructed as (Σ ∪ Σ , E ∪ E ∪ E ⊗ ) where E ⊗ consists of f (λi.g(λj.x ij )) = g(λj.f (λi.x ij ) ) for each f ∈ Σ and g ∈ Σ . However, when we extend tensor products to graded algebraic theories, the grades of the both sides are not necessarily equal. If the grade of f is m and the grade of g is m , then the grades of f (λi.g(λj.x ij )) and g(λj.f (λi.x ij )) are m ⊗ m and m ⊗ m, respectively. Therefore, we have to somehow guarantee that the grade of f ∈ Σ and the grade of g ∈ Σ commute. We solve this problem by taking the product of monoidal categories. That is, we define the tensor product of an M 1 -graded algebraic theory and an M 2 -graded algebraic theory as an M 1 × M 2 -graded algebraic theory. Before defining tensor products, we consider extending an M-graded theory to M -graded theory along a lax monoidal functor G = (G, η G , µ G ) : M → M . Given an M-graded theory T = (Σ, E), we define the M -graded theory G * T = (G * Σ, G * E) by (G * Σ) n,m := {f ∈ Σ n,m | Gm = m } and G * E := {G * (s) = G * (t) | (s = t) ∈ E} where for each term t of T (with grade m), G * (t) is the term of G * T (with grade Gm) defined inductively as follows: if x is a variable, then G * (x) := c η G (x); for each w : m → m and term t, G * (c w (t)) := c Gw (G * (t)); for each f ∈ Σ n,m and terms t 1 , . . . , t n with grade m , G * (f (t 1 , . . . , t n )) := c µ G m,m (f (G * (t 1 ), . . . , G * (t n ))). The tensor product of T 1 ∈ GS M1 and T 2 ∈ GS M2 is defined by first extending T 1 and T 2 to M 1 ×M 2 -graded theories and then adding commutation equations. Definition 46 (tensor product). Let T 1 = (Σ, E) ∈ GS M1 and T 2 = (Σ , E ) ∈ GS M2 . The tensor product T 1 ⊗ T 2 is defined by (K * Σ ∪ K * Σ , K * E ∪ K * E ∪ E T1⊗T2 ) ∈ GS M1×M2 where K : M 1 → M 1 × ME T1⊗T2 := {f (λi.g(λj.x ij )) = g(λj.f (λi.x ij )) | f ∈ (K * Σ) n,m , g ∈ (K * Σ ) n ,m }. That is, if f is an operation in T 1 with grade m 1 ∈ M 1 , then T 1 ⊗ T 2 has the operation f with grade (m 1 , I 2 ) ∈ M 1 × M 2 and similarly for operations in T 2 . The tensor products satisfy the following fundamental property. Example 48. We exemplify the tensor product by showing a graded version of [9,Corollary 6], which claims that the L-fold tensor product of the side-effects theory in [21] with one location is the side-effects theory with L locations. First, we consider the situation where there is only one memory cell whose value ranges over a finite set V . Let 2 the preordered monoid (join-semilattice) ({⊥, }, ≤, ∨, ⊥) where ≤ is the preorder defined by ⊥ ≤ . Intuitively, ⊥ represents pure computations, and represents (possibly) stateful computations. Let T st be a 2-graded theory of two types of operations lookup (arity: V , grade: ) and update v (arity: 1, grade: ) for each v ∈ V and the four equations in [21] for the interaction of lookup and update. Note that we have to insert coercion to arrange the grade of the equation lookup(λv ∈ V.update v (x)) = c ⊥≤ (x). The graded monad ( * , η, µ) induced by T st is as follows. ⊥ * X = X * X = (V × X) V ((⊥ ≤ ) * X)(x) = λv.(v, x) The middle equation can be explained as follows: any term with grade can be presented by a canonical form t f := lookup(λv.update f V (v) (f X (v))) where f = f V , f X : V → V × X is a function, and therefore, the mapping f → t f gives a bijection between (V × X) V and * X = T Σ (X)/∼. The L-fold tensor product of T st , which we denote by T ⊗L st , is a 2 L -graded theory where 2 L = (2 L , ⊆, ∪, ∅) is the join-semilattice of subsets of L. Specifically, T ⊗L st consists of operations lookup l and update l,v with grade {l} for each l ∈ L and v ∈ V with additional three commutation equations in [21]. The induced graded monad is L * ⊗L X = {f : V L → (V L × X) | read(L , f ) ∧ write(L , f )} where L ⊆ L, and read(L , f ) and write(L , f ) assert that f depends only on values at locations in L and does not change values at locations outside L . That is, L * ⊗L X represents computations that touch only memory locations in L . read(L , f ) := ∀σ, σ ∈ V n , (∀l ∈ L , σ(l) = σ (l)) =⇒ f (σ) = f (σ ) write(L , f ) := ∀σ, σ ∈ V n , x ∈ X, (σ , x) = f (σ) =⇒ ∀l / ∈ L , σ(l) = σ (l) Related Work Algebraic theories for graded monads. Graded monads are introduced in [27], and notions of graded theory and graded Eilenberg-Moore algebra appear in [17,4] for coalgebraic treatment of trace semantics. However, these work only deal with N-graded monads where N is regarded as a discrete monoidal category, while we deal with general monoidal categories. The Kleisli construction and the Eilenberg-Moore construction for graded monads are presented in [7] by adapting the 2-categorical argument on resolutions of monads [29]. Algebraic operations for graded monads are introduced in [12] and classified into two types, which are different in how to integrate the grades of subterms. One is operations that take terms with the same grade, and these are what we treated in this paper. The other is operations that take terms with different grades: the grade of f (t 1 , . . . , t n ) is determined by an effect function : M n → M associated to f . Although the latter type of operations is also important to give natural presentations of computational effects, we leave it for future work. Enriched Lawvere theories. There are many variants of Lawvere theories [25,11,24,10,19,15,1,16,28], and most of them share a common pattern: they are defined as an identity-on-objects functor from a certain category (e.g., ℵ op 0 ) which represents arities, and the functor must preserve a certain class of products (or cotensors if enriched). Among the most relevant work to ours are enriched Lawvere theories [24] and discrete Lawvere theories [10]. For a given monoidal category V, a Lawvere V-theory is defined as an identity-on-objects finite cotensor (i.e. V fp -cotensor) preserving V t -functor J : V op fp → L where V fp is the full subcategory of V spanned by finitely presentable objects. If V = [M, Set] 0 t , Lawvere [M, Set] 0 t -theories are analogous to our graded Lawvere theories except that we used N op M instead of ([M, Set] 0 ) fp . Since n · y(I) ∈ N op M is finitely presentable, we can say that the notion of graded Lawvere theory is obtained from enriched Lawvere theories by restricting arities to N op M ⊆ ([M, Set] 0 ) fp . However, the correspondence to finitary graded monads on Set is an interesting point of our graded Lawvere theories compared to Lawvere V-theories, which correspond to finitary V-monads on V. Discrete Lawvere theories restrict arities of Lawvere V-theories to ℵ 0 , that is, a discrete Lawvere V-theory is defined as a (Set-enriched) finite-product preserving functor J : ℵ op 0 → L 0 where L is a V t -category. Actually, discrete Lawvere [M, Set] 0 t -theories are equivalent to graded Lawvere theories because there is a finite-product preserving functor ι : ℵ op 0 → N op M such that the composition with ι gives a bijection between graded Lawvere theories J : N op M → L and discrete Lawvere [M, Set] 0 t -theories J 0 • ι : ℵ op 0 → L 0 . However, we considered not only symmetric monoidal categories but also nonsymmetric ones, which cause a nontrivial problem when we define tensor products of algebraic theories. The problem is that adding commutation equations requires some kind of commutativity of monoidal categories. We solved this problem by considering product monoidal categories and defining the tensor product of an M 1 -graded theory and an M 2 -graded theory as an M 1 × M 2 -graded theory, and the use of two different monoidal categories is new to the best of our knowledge. Conclusions and Future Work To extend the correspondence between algebraic theories, Lawvere theories, and (finitary) monads, we introduced notions of graded algebraic theory and graded Lawvere theory and proved their correspondence with finitary graded monads. We also provided sums and tensor products for graded algebraic theories, which are natural extensions of those for ordinary algebraic theories. Since we do not assume monoidal categories to be symmetric, our tensor products are a bit different from the ordinary ones in that this combines two theories graded by (or enriched in) different monoidal categories. We hope that these results will lead us to apply many kinds of techniques developed for monads to graded monads. As future work, we are interested in "change-of-effects", that is, changing the monoidal category M in M-graded algebraic theory along a (lax) monoidal functor F : M → M . The problem already appeared in §6.2 to define tensor products, but we want to look for more properties of this operation. We are also interested in integrating a more general framework for notions of algebraic theory [6] and obtaining a graded version of the framework. Another direction is exploiting models of graded algebraic theories as modalities in the study of coalgebraic modal logic [17,4] or weakest precondition semantics [8]. Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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 license and indicate if changes were made. The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. Fig. 1 . 1A rule of term formation. [M, Set] 0 where M = (M, ⊗, I) is a small monoidal category [3]. Here, we use the subscript (−) 0 to indicate that [M, Set] 0 is an ordinary (not enriched) category since we also use the enriched version [M, Set] later. The external tensor product F G : M × M → Set is defined by (F G)(m 1 , m 2 ) = F m 1 × Gm 2 for any F, G : M → Set. Definition 4 . 4Let F, G : M → Set be functors. The Day tensor product F⊗ G : M → Set is the left Kan extension Lan ⊗ (F G) of the external tensor product F G : M × M → Set along the tensor product ⊗ : M × M → M. Note that a natural transformation θ : F⊗ G → H is equivalent to a natural transformation θ m1,m2 : F m 1 × Gm 2 → H(m 1 ⊗ m 2 ) by the universal property. The Day convolution induces a monoidal biclosed structure in [M, Set] 0 [3]. Proposition 5. The Day tensor product makes ([M, Set] 0 ,⊗, y(I)) a monoidal biclosed category where y : M op → [M, Set] 0 is the Yoneda embedding y(m) := M(m, −). The left and the right closed structure are given by F, G m = [M, Set] 0 (F, G(m⊗−)) and [F, G] m = [M, Set] 0 (F, G(−⊗m)) for each m ∈ M, respectively. m2∈M which is natural in m 1 and m 2 . The subscripts m 1 and m 2 are often omitted. corresponds to the natural transformation • m1,m2 : C(Y, Z)m 1 ×C(X, Y )m 2 → C(X, Z)(m 1 ⊗ m 2 ) by the universal property of the Day convolution. The rest of the proof is easy. An [M, Set] 0 -functor F : C → D consists of a mapping X → F X and a natural transformation F X,Y : C(X, Y ) → D(F X, F Y ) (for each X, Y ) that preserves identities and compositions of morphisms. An [M, Set] 0 -natural transformation α : F → G is a family of elements α X ∈ D(F X, GX)I X∈ob(C) that satisfies α Y • F f = Gf • α X for each f ∈ C(X, Y )m. Vertical and horizontal compositions of [M, Set] 0 -natural transformations are defined as expected. We introduce a useful construction of [M, Set] 0 t -categories. Given an Mgraded monad (in other words, a lax left M-action) on C, we can define an [M, Set] 0 t -enriched category as follows. Definition 7. Let T = ( * , η, µ) be an M-graded monad on C. An [M, Set] 0 tcategory C T is defined by ob C T := obC and C T (X, Y )m := C(X, m * Y ). The identity morphisms are the unit morphisms η X ∈ C T (X, X)I, and the composite of - For each m ∈ M, m (−) preserves finite powers: m c n ∼ = (m c) n . Example 9. If A is a category with finite powers, then the functor category [M, A] has strong M t -action defined by m F := F (m ⊗ (−)) and satisfies M-model condition. Especially, [M, Set] 0 satisfies M-model condition. Lemma 12 . 12Let C be a category satisfying M 1 ×M 2 -model condition. If T is an M 1 -graded algebraic theory, then C satisfies M 1 -model condition and Mod T (C) satisfies M 2 -model condition. Proof. An M t 1 -action on C is obtained by the composition of M t 1 × M t 2 -action and the strong monoidal functor M t 1 → M t 1 × M t 2 defined by m → (m, I). Finite powers and an M t 2 -action for Mod T (C) are induced by those for C. Corollary 13. Mod T ([M, Set] 0 ) has an M-action, which is given by the precomposition of m ⊗ (−) like the M-action of Example 9. Lemma 17 . 17For any model A in [M, Set] 0 and any mapping v : X → AI, there exists a unique homomorphism Example 25 (graded exception monad[12, Example 3.4]). We give an algebraic presentation of the graded exception monad.Let M and ( * , η, µ) be a preordered monoid and the graded monad defined as follows. Let P + (X) denote the set of nonempty subsets of X. Let Ex be a set of exceptions and M = ((P + (Ex ∪ {Ok}), ⊆), I, ⊗) be a preordered monoid where I = {Ok} and the multiplication ⊗ is defined by m ⊗ m = (m \ {Ok}) ∪ m if Ok ∈ m and m ⊗ m = m otherwise (note that this is not commutative). The graded exception monad ( * , η, µ) is the M-graded monad given as follows. Example 26 ( 26extending an ordinary monad to an M-graded monad). We consider the problem of extending an M -graded theory to an M-graded theory along a lax monoidal functor of type M → M, but here we restrict ourselves to the case of M = 1 and the strict monoidal functor of type 1 → M. Let M = (M, I, ⊗) be an arbitrary small strict monoidal category. Let T = (Σ, E) be a (1-graded) theory and (T, η T , µ T ) be the corresponding ordinary monad. Let T M = (Σ M , E M ) be the M-graded theory obtained when we regard each operation in T as an operation with grade I ∈ M, that is, Σ M n,m := Σ n if m = I and Σ M n,m := ∅ otherwise, and E M := E. The free model functor for T M is F T M X = F T (M(I, −) × X) where F T : Set → Mod T (Set) is the free model functor for T as a 1-graded theory, and the interpretation of an operation f ∈ Σ n in F T M X is defined by the interpretation in the free models of T . |f | F T M X m = |f | F T (M(I,m)×X) : F T (M(I, m) × X) n → F T (M(I, m) × X) Here, ⊗ : M(I, m 1 ) × M(I, m 2 ) → M(I, m 1 ⊗ m 2 ) is induced by ⊗: M × M → M and st X,Y : X × T Y → T (X × Y ) is the strength for T . [M, Set] 0 and define graded Lawvere theories using [M, Set] 0 -categories where [M, Set] 0 is equipped with the Day convolution monoidal structure. Specifically, ℵ 0 (in ordinary Lawvere theories) is replaced with an [M, Set] 0 -category N M , L with an [M, Set] 0 -category, and "finite products" with "N op M -cotensors". So, we first provide an enriched category N M that we use as arities. Since we do not assume that M is symmetric, N M is defined to be an [M, Set] 0 t -category so that the opposite category N op M is an [M, Set] 0 -category. Let [M, Set] t be an [M, Set] 0 t -category induced by the closed structure of [M, Set] 0 t . That is, homobjects of [M, Set] t are given by [M, Set] t (G, H)m = [M, Set] 0 (G, H(− ⊗ m)). Definition 27. An [M, Set] 0 t -category N M is defined by the full sub-[M, Set] 0 tcategory of [M, Set]t whose set of objects is given by obN M = {n · y(I) | n ∈ N} ⊆ ob[M, Set] t where N is the set of natural numbers and n · y(I) is the n-fold coproduct of y(I). We sometimes identify obN M with N via the mapping n → n := n · y(I).Lemma 28. The [M, Set] 0 -category N op M has N op M -cotensors, which are given by n n = n · n for each n and n . N op M -cotensors (i.e. n · y(I) C) behave like an enriched counterpart of finite powers (−) n . We show that N op M -cotensors in a general [M, Set] 0 -category A are characterized by projections satisfying a universal property. Given a unit morphism ν : n → A(n C, C) of the cotensor n C, an [M, Set] 0 -natural transformation ν : A(B, n C) → [n, A(B, C)] is given by Lemma 29 . 29An [M, Set] 0 -category A has N op M -cotensors if and only if for any n ∈ N and C ∈ obA, there exist an object n C ∈ obA and (π 1 , . . . , π n ) ∈ (A(n C, C)I) n such that the following condition holds: for each m, the function f → (π 1 • f, . . . , π n • f ) of type A(B, n C)m → (A(B, C)m) n is bijective.An [M, Set] 0 -functor F : A → B preserves N op M -cotensors if and only if (F n C,C,I • π 1 , . . . , F n C,C,I • π n ) ∈ (B(F (n C), F C)I) n satisfies the same condition for each n and C. [ M , MSet] 0 -natural transformation ν is isomorphic if and only if each component ν m : A(B, n C)m → [n, A(B, C)] m of ν is isomorphic, which is moreover equivalent to the condition that f → (π 1 • f, . . . , π n • f ) : A(B, n C)m → (A(B, C)) n is isomorphic since we have [n, A(B, C)] m ∼ = (A(B, C)m) n . Given n-tuple (f 1 , . . . , f n ) of elements in A(B, C)m, we denote by f 1 , . . . , f n an element in A(B, n C)m obtained by the inverse of f → (π 1 • f, . . . , π n • f ) and call this a tupling. Tuplings and projections for N op M -cotensors behave like those for finite products. The following proposition claims that N op M is a free [M, Set] 0 -category with chosen N op M -cotensors generated by one object. Proposition 30. Let A be an [M, Set] 0 -category with N op M -cotensors and C be an object in A. Then there exists a unique N op M -cotensor preserving [M, Set] 0functor F : N op M → A such that F n = n C and F π i = π i . We define M-graded Lawvere theories in a similar fashion to enriched Lawvere theories. Definition 31. An M-graded Lawvere theory is a tuple (L, J) where L is an [M, Set] 0 -category with N op M -cotensors and J : N op M → L is an identity-onobjects N op M -cotensor preserving [M, Set] 0 -functor. A morphism F : (L, J) → (L , J ) between two graded Lawvere theories is an [M, Set] 0 -functor F : L → L such that F J = J . We denote the category of graded Lawvere theories and morphisms between them by GLaw M . By Proposition 30, the existence of the above J : N op M → L is equivalent to requiring that obL = N and projections in L are chosen in some way. So, we sometimes leave J implicit and just write L ∈ GLaw M for (L, J) ∈ GLaw M . Definition 32. A model of graded Lawvere theory L in an [M, Set] 0 -category A with N op M -cotensor is an N op M -cotensor preserving [M, Set] 0 -functor of type L → A. A morphism α : F → G between two models F, G of graded Lawvere theory L is an [M, Set] 0 -natural transformation. Let Mod(L, A) be the category of models of graded Lawvere theory L in the [M, Set] 0 -category A. In §3, we use a category C satisfying M-model condition to define a model of graded algebraic theory. Actually, M-model condition is sufficient to give an [M, Set] 0 -category with N op M -cotensors. Lemma 33. If C satisfies M-model condition, then the [M, Set] 0 -category C T op defined in Definition 7 has N op M -cotensors. If we apply Lemma 33 to [M, Set] 0 equipped with the M t -action in Example 9 (here denoted by T ), then ([M, Set] 0 ) T op coincides with [M, Set] (i.e. the [M, Set] 0 -category obtained by the closed structure of [M, Set] 0 ). Definition 34 . 34An [M, Set] 0 -category ThT is defined by ob(ThT ) := N and (ThT )(n, n )m := (F T nm) n with composition defined by substitution. Definition 35 . 35A functor U : GLaw M → GS M is defined as follows. -For each L ∈ obGLaw M , U L = (Σ, E) where Σ n,m = L(n, 1)m, E = {(s, t) | |s| L = |t| L } and | · | L : T Σ m (n) → L(n, 1)m is an interpretation of terms defined in the same way as Definition 11. -Given G : L → L , let U G : U L → U L be a functor defined by U G(f ) = [G(f )(x 1 , . . . , x n )] for each f ∈ L(n, 1)m. Then, ThT has the following universal property as a left adjoint of U . Lemma 36. For each T , let η T : T → U ThT be a family of functions η T ,n,m : Σ n,m → F U ThT nm defined by η T ,n,m (f ) = [[f (x 1 , . . . , x n )](x 1 , . . . , x n )]. For any α : T → U L, there exists a unique morphism α : ThT → L such that α = U α • η T . Definition 40 . 40Let T = ( * , η, µ) be an M-graded monad on Set. Let L T be the full sub-[M, Set] 0 -category of ( Set T ) op with ob(L T ) = N. Corollary 44 . 44Three equivalent categories GS M , GMnd M and GLaw M are cocomplete. Example 45. It is known that the sum of an ordinary monad T and the exception monad (−) + Ex (where Ex is a set of exceptions) is given by T ((−) + Ex) [9, Corollary 3]. We show that a similar result holds for the graded exception monad. Let T ex be the theory in Example 25 and M be the preordered monoid used there. We denote ( * ex , η ex , µ ex ) for the graded exception monad. Let T = (Σ, E) be a (1-graded) theory and (T, η T , µ T ) be the corresponding ordinary monad. Let T M = (Σ M , E M ) be the M-graded theory obtained from T as in Example 26. We consider a graded monad obtained as the sum of T ex and T M . 2 and K : M 2 → M 1 × M 2 are lax monoidal functors defined by Km 1 := (m 1 , I 2 ) and K m 2 := (I 1 , m 2 ), and Proposition 47 . 47Let C be a category satisfying M 1 × M 2 -model condition. Let T i be an M i -graded algebraic theory for i = 1, 2. Then we have an isomorphism Mod T1 (Mod T2 (C)) ∼ = Mod T1⊗T2 (C). Proof. Let ((A, | · | ), | · |) ∈ Mod T1 (Mod T2 (C)) be a model. 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[ "HYPERBOLIC 24-CELL 4-MANIFOLDS WITH ONE CUSP", "HYPERBOLIC 24-CELL 4-MANIFOLDS WITH ONE CUSP" ]
[ "John G Ratcliffe ", "Steven T Tschantz " ]
[]
[]
In this paper, we describe all the hyperbolic 24-cell 4-manifolds with exactly one cusp. There are four of these manifolds up to isometry. These manifolds are the first examples of one-cusped hyperbolic 4-manifolds of minimum volume.We next list the sets of indices of the vertices of each of the 24 sides of Q.
10.1080/10586458.2021.1926010
[ "https://arxiv.org/pdf/2004.07284v1.pdf" ]
215,786,263
2004.07284
0e1971e77e2e4bc33eee84f650be541dcb580fa5
HYPERBOLIC 24-CELL 4-MANIFOLDS WITH ONE CUSP 15 Apr 2020 John G Ratcliffe Steven T Tschantz HYPERBOLIC 24-CELL 4-MANIFOLDS WITH ONE CUSP 15 Apr 2020arXiv:2004.07284v1 [math.GT] In this paper, we describe all the hyperbolic 24-cell 4-manifolds with exactly one cusp. There are four of these manifolds up to isometry. These manifolds are the first examples of one-cusped hyperbolic 4-manifolds of minimum volume.We next list the sets of indices of the vertices of each of the 24 sides of Q. Introduction The 24-cell is a regular 4-dimensional polytope in either Euclidean, Spherical or hyperbolic 4-space with exactly 24 sides each of which is a regular octahedron. A hyperbolic 24-cell manifold is a hyperbolic 4-manifold that is obtained from an ideal, regular, hyperbolic 24-cell by gluing each side to another side by an isometry. Examples of hyperbolic 24-cell manifolds are given in our paper [8]. Hyperbolic 24cell manifolds have minimum volume among hyperbolic 4-manifolds. In this paper, all hyperbolic manifolds are assumed to be complete. As a reference for hyperbolic manifolds, see [7]. The main result of this paper is the following classification of all one-cusped hyperbolic 24-cell manifolds: Theorem 1. There are exactly four hyperbolic 24-cell manifolds, with a single cusp, up to isometry. Each of these manifolds is non-orientable. The link of each cusp is affinely equivalent to the second non-orientable closed flat 3-manifold N 3 2 in the Hantzsche-Wendt [3] classification of closed flat 3-manifolds. The volume of a hyperbolic 4-manifold M of finite volume is given by the formula Vol(M ) = 4 3 π 2 χ(M ). Our examples are the first known examples of one-cusped hyperbolic 4-manifolds of Euler characteristic 1, and therefore of minimum volume. The first examples of one-cusped hyperbolic 4-manifold were constructed by Kolpakov and Martelli [4], and in particular, they found an example with χ = 4. Examples of one-cusped hyperbolic 4-manifolds with χ = 2 were constructed by Slavich and Kolpakov [10,5,6]. The existence of our examples answers Question 4.17 of [5] in the affirmative. Our paper is organized as follows: In §2, the flat manifold N 3 2 is described. In §3, the classification of the one-cusped hyperbolic 24-cell manifolds is described. In §4, we discuss how to obtain a presentation for the fundamental group of a hyperbolic 24-cell manifold. In §5, . . . , §8, we give a presentation for the fundamental group of each of the one-cusped hyperbolic 24-cell manifolds. In §9, we determine the volume of the maximum cusp of each of the one-cusped hyperbolic 24-cell manifolds. In §10, we determine the order of the group of isometries of each of the one-cusped hyperbolic 24-cell manifolds. In §11, we describe the orientable double covers of each of the one-cusped hyperbolic 24-cell manifolds. The non-orientable closed flat 3-manifolds are classified up to affine equivalence by their first homology group, with H 1 (N 3 2 ) = Z ⊕ Z. The flat 3-manifold N 3 2 is the Euclidean space-form E 3 /Γ where Γ is the crystallographic group with International Tables Number 9. For the standard affine representation of Γ, see Table 1B of [1]. The flat 3-manifold N 3 2 fibers in various ways (see Table 1 of [9]). In particular N 3 2 is a torus bundle over the circle with monodromy the orientation-reversing isometry of the flat torus E 2 /Z 2 induced by the reflection of E 2 defined by the matrix 0 1 1 0 . Name H 0 H 1 H 2 H 3 H 4 24c1.1 Z Z ⊕ Z 3 ⊕ Z 13 Z 0 0 24c1.2 Z Z ⊕ Z 3 Z ⊕ Z 13 0 0 24c1.3 Z Z 3 2 ⊕ Z 5 Z 2 0 0 24c1.4 Z Z 2 2 ⊕ Z 3 3 Z 3 0 0 Classification of the One-Cusped Hyperbolic 24-Cell Manifolds To prove Theorem 1, we first found all the possible side-pairings of an ideal, regular, hyperbolic 24-cell that yield a hyperbolic 4-manifold by an exhaustive computer search over a search space of order (23!!)48 12 . Our search was successful because the ridge and edge cycle conditions, defining when a side-pairing gives a manifold, each depend on only a few of the side-pairing maps. A backtracking search in the space of partial side-pairings that incrementally checks these conditions can eliminate potential side-pairings early on. The next side to extend a partial sidepairing is chosen strategically to maximize the conditions that can be checked thus limiting the number of search tree branches that need to be examined. We further took advantage of the symmetry of the regular 24-cell. Up to symmetry of the regular 24-cell, we found 13,108 side-pairings that yield a hyperbolic 4-manifold. Only four of these side-pairings yield a manifold with a single cusp. We classified these one-cusped manifolds up to isometry by computing their homology groups using a cellular homology chain complex. The one-cusped hyperbolic 24-cell manifolds are classified by their first homology groups (see Table 1). We now set up notation to describe the side-pairings for the four one-cusped hyperbolic 24-cell manifolds. The hyperboloid model of hyperbolic 4-space is H 4 = {x ∈ R 5 : x 2 1 + · · · + x 2 4 − x 2 5 = −1 and x 5 > 0}. We identify the group of isometries of H 4 with the positive Lorentz group O + (4, 1). We will work with a standard 24-cell Q in H 4 that is centered at the central point (0, 0, 0, 0, 1) of H 4 . The outward normal vectors to the sides of Q, with respect to the Lorentzian inner product x • y = x 1 y 1 + · · · + x 4 y 4 − x 5 y 5 , are taken in the following fixed order: s 1 = (1, 1, 0, 0, 1) s 13 = (1, 0, 0, 1, 1) s 2 = (−1, 1, 0, 0, 1) s 14 = (−1, 0, 0, 1, 1) s 3 = (1, −1, 0, 0, 1) s 15 = (1, 0, 0, −1, 1) s 4 = (−1, −1, 0, 0, 1) s 16 = (−1, 0, 0, −1, 1) s 5 = (1, 0, 1, 0, 1) s 17 = (0, 1, 0, 1, 1) s 6 = (−1, 0, 1, 0, 1) s 18 = (0, −1, 0, 1, 1) s 7 = (1, 0, −1, 0, 1) s 19 = (0, 1, 0, −1, 1) s 8 = (−1, 0, −1, 0, 1) s 20 = (0, −1, 0, −1, 1) s 9 = (0, 1, 1, 0, 1) s 21 = (0, 0, 1, 1, 1) s 10 = (0, −1, 1, 0, 1) s 22 = (0, 0, −1, 1, 1) s 11 = (0, 1, −1, 0, 1) s 23 = (0, 0, 1, −1, 1) s 12 = (0, −1, −1, 0, 1) s 24 = (0, 0, −1, −1, 1) The 24-cell is self-dual, and so Q has 24 ideal vertices, which are taken in the following fixed order: The side-pairing for 24c1.4 is given in terms of indices of vertices as follows: S 1 → S 9 : 13 → 21, Presentations for the corresponding Discrete Groups In this section, we discuss how to obtain a presentation for the fundamental group of a hyperbolic 24-cell manifold from a side-pairing of the 24-cell Q together with an ordering of the sides of Q. A side-pairing map S i → S j determines a side-pairing transformation g i in O + (4, 1) which is the composition r j f i of the symmetry f i of Q that corresponds to the side-pairing map S i → S j followed by the reflection r j of H 4 in the side S j . Note that g j = (g i ) −1 , since (g i ) −1 = (f −1 i r j f i )f −1 i = r i f j , and so we will assume that i < j. By Poincaré's fundamental polyhedron theorem, a side-pairing of Q together with an ordering of the sides of Q determines a set of 12 side-pairing transformations that form a set of generators for a discrete subgroup Γ * of O + (4, 1) whose orbit space H 4 /Γ * is isometric to the hyperbolic 4-manifold M obtained by gluing together the sides of Q by the side-pairing. The fundamental group of M is isomorphic to Γ * . The dihedral angles of the regular polytope Q are all π/2. Therefore Q determines a regular tessellation Q of H 4 with fundamental cell Q. The group of symmetries of the tesselation is a (3, 4, 3, 4) Coxeter simplex reflection group Γ generated by the reflections of H 4 represented by the following matrices in O + (4, 1): 1 2       1 1 1 1 0 1 1 −1 −1 0 1 −1 1 −1 0 1 −1 −1 1 0 0 0 0 0 2       ,       1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 −1 0 0 0 0 0 1       ,            ,       −1 −2 0 0 2 −2 −1 0 0 2 0 0 1 0 0 0 0 0 1 0 −2 −2 0 0 3       . The last matrix represents the reflection of H 4 in Side 1 of Q. The first four matrices generate the group of symmetries of Q, which has order 1152. The group Γ * is a torsion-free subgroup of Γ of index 1152. By Poincaré's fundamental polyhedron theorem, defining relators for the sidepairing transformation generators of Γ * are in one-to-one correspondence with ridge cycles determined by the side-pairing. A ridge is a co-dimension 2 face. The 24-cell Q has 96 ridges, each an ideal triangle, that are partitioned into cycles of order 4, since the dihedral angles of Q are all π/2. Therefore, there are exactly 24 ridge cycles, and 24 corresponding defining relators of length 4 for the group Γ * . Let Γ k be the torsion-free subgroup of Γ of index 1152 determined by the given side-pairing for the manifold 24c1.k for k = 1, . . . , 4. Presentation for the Group Γ 1 The 12 side-pairing transformations g 1 , g 2 , g 3 , g 4 , g 6 , g 7 , g 10 , g 11 , g 13 , g 14 , g 15 , g 16 that generate the group Γ 1 are represented in O + (4, 1) as follows: g 1 = 1 2       −3 −3 1 1 4 −1 1 1 −1 0 −3 −3 −1 −1 4 1 −1 1 −1 0 −4 −4 0 0 6       , g 2 = 1 2       1 1 −1 1 0 3 −3 1 1 4 3 −3 −1 −1 4 1 1 1 −1 0 4 −4 0 0 6       , g 3 = 1 2       1 1 −1 1 0 3 −3 1 1 −4 3 −3 −1 −1 −4 1 1 1 −1 0 −4 4 0 0 6       , g 4 = 1 2       −3 −3 1 1 −4 −1 1 1 −1 0 −3 −3 −1 −1 −4 1 −1 1 −1 0 4 4 0 0 6       , g 6 = 1 2       −1 1 −1 −1 0 −1 −1 −1 1 0 3 −1 −3 −1 4 3 1 −3 1 4 4 0 −4 0 6       , g 7 = 1 2       −1 1 −1 −1 0 −1 −1 −1 1 0 3 −1 −3 −1 −4 3 1 −3 1 −4 −4 0 4 0 6       , g 10 = 1 2       1 1 1 −1 0 −1 −3 3 −1 −4 1 −1 −1 −1 0 −1 3 −3 −1 4 0 4 −4 0 6       , g 11 = 1 2       1 1 1 −1 0 −1 −3 3 −1 4 1 −1 −1 −1 0 −1 3 −3 −1 −4 0 −4 4 0 6       , g 13 = 1 2       −1 −1 −1 1 0 −3 −1 1 −3 4 −1 1 1 1 0 −3 1 −1 −3 4 −4 0 0 −4 6       , g 14 = 1 2       −1 1 1 −1 0 −1 −1 −1 −1 0 −3 −1 1 3 −4 3 −1 1 −3 4 4 0 0 −4 6       , g 15 = 1 2       −1 1 1 −1 0 −1 −1 −1 −1 0 −3 −1 1 3 4 3 −1 1 −3 −4 −4 0 0 4 6       , g 16 = 1 2       −1 −1 −1 1 0 −3 −1 1 −3 −4 −1 1 1 1 0 −3 1 −1 −3 −4 4 0 0 4 6       . Defining relators for the above set of 12 generators for Γ 1 are as follows: g 3 g 16 g 2 4 , g 3 g −2 10 g −1 4 , g 2 g 4 g −1 16 g −1 15 , g 4 g 16 g 11 g 14 , g 7 g −1 15 g −2 16 , g 4 g −1 14 g 13 g −1 7 , g 2 3 g −1 7 g −1 16 , g 1 g −1 7 g 3 g 10 , g 4 g −1 10 g −2 14 , g 3 g −1 15 g −1 6 g −1 10 , g 1 g −1 13 g −1 14 g 3 , g 2 6 g −1 10 g 14 , g 2 g 11 g 4 g −1 6 , g 1 g −1 15 g 16 g −1 6 , g 2 g −1 16 g −1 10 g 6 , g 1 g 13 g 10 g 15 , g 6 g −1 14 g −2 13 , g 1 g 2 11 g −1 2 , g 2 g −1 14 g −1 7 g −1 11 , g 3 g −1 13 g −1 11 g 7 , g 2 1 g 2 g 13 , g 2 2 g −1 6 g −1 13 , g 2 7 g −1 11 g 15 , g 1 g −1 11 g −2 15 . Presentation for the Group Γ 2 The 12 side-pairing transformations g 1 , g 2 , g 3 , g 4 , g 5 , g 8 , g 9 , g 12 , g 13 , g 14 , g 15 , g 16 that generate the group Γ 2 are represented in O + (4, 1) as follows: g 1 = 1 2       3 3 1 1 −4 1 −1 1 −1 0 −3 −3 1 1 4 1 −1 −1 1 0 −4 −4 0 0 6       , g 2 = 1 2       −1 −1 −1 1 0 −3 3 1 1 −4 3 −3 1 1 4 1 1 −1 1 0 4 −4 0 0 6       , g 3 = 1 2       −1 −1 −1 1 0 −3 3 1 1 4 3 −3 1 1 −4 1 1 −1 1 0 −4 4 0 0 6       , g 4 = 1 2       3 3 1 1 4 1 −1 1 −1 0 −3 −3 1 1 −4 1 −1 −1 1 0 4 4 0 0 6       , g 5 = 1 2       1 −1 −1 −1 0 1 1 −1 1 0 3 −1 3 1 −4 3 1 3 −1 −4 −4 0 −4 0 6       , g 8 = 1 2       1 −1 −1 −1 0 1 1 −1 1 0 3 −1 3 1 4 3 1 3 −1 4 4 0 4 0 6       , g 9 = 1 2       −1 −1 1 −1 0 1 3 3 −1 −4 1 −1 1 1 0 −1 3 3 1 −4 0 −4 −4 0 6       , g 12 = 1 2       −1 −1 1 −1 0 1 3 3 −1 4 1 −1 1 1 0 −1 3 3 1 4 0 4 4 0 6       , g 13 = 1 2       1 −1 1 −1 0 1 1 −1 −1 0 −3 −1 −1 −3 4 3 −1 −1 3 −4 −4 0 0 −4 6       , g 14 = 1 2       1 1 −1 1 0 3 1 1 −3 4 −1 1 −1 −1 0 −3 1 1 3 −4 4 0 0 −4 6       , g 15 = 1 2       1 1 −1 1 0 3 1 1 −3 −4 −1 1 −1 −1 0 −3 1 1 3 4 −4 0 0 4 6       , g 16 = 1 2       1 −1 1 −1 0 1 1 −1 −1 0 −3 −1 −1 −3 −4 3 −1 −1 3 4 4 0 0 4 6       . Presentation for the Group Γ 3 The 12 side-pairing transformations g 1 , g 2 , g 3 , g 4 , g 6 , g 7 , g 9 , g 10 , g 11 , g 12 , g 14 , g 17 that generate the group Γ 3 are represented in O + (4, 1) as follows: g 1 = 1 2       −3 −3 −1 −1 4 −1 1 −1 1 0 −3 −3 1 1 4 1 −1 −1 1 0 −4 −4 0 0 6       , g 2 = 1 2       3 −3 1 1 4 −1 −1 1 −1 0 −1 −1 −1 1 0 3 −3 −1 −1 4 4 −4 0 0 6       , g 3 = 1 2       3 −3 −1 1 −4 1 1 −1 −1 0 1 1 1 1 0 3 −3 1 −1 −4 −4 4 0 0 6       , g 4 = 1 2       −3 −3 −1 1 −4 1 −1 −1 −1 0 −3 −3 1 −1 −4 −1 1 −1 −1 0 4 4 0 0 6       , g 6 = 1 2       1 −1 1 1 0 −1 −1 −1 1 0 3 1 −3 1 4 3 −1 −3 −1 4 4 0 −4 0 6       , g 7 = 1 2       1 1 1 −1 0 −1 1 −1 −1 0 −3 −1 3 −1 4 3 −1 −3 −1 −4 −4 0 4 0 6       , g 9 = 1 2       −1 −1 1 −1 0 1 3 3 −1 −4 −1 1 −1 −1 0 1 −3 −3 −1 4 0 −4 −4 0 6       , g 10 = 1 2       1 −1 −1 −1 0 1 −3 3 1 −4 −1 −1 −1 1 0 −1 −3 3 −1 −4 0 4 −4 0 6       , g 11 = 1 2       −1 1 1 −1 0 −1 −1 −1 −1 0 1 3 −3 −1 −4 −1 3 −3 1 −4 0 −4 4 0 6       , g 12 =       0 0 0 1 0 0 2 1 0 2 1 0 0 0 0 0 −1 −2 0 −2 0 2 2 0 3       , g 14 =       2 0 0 −1 2 0 0 1 0 0 0 1 0 0 0 −1 0 0 2 −2 2 0 0 −2 3       , g 17 = 1 2       −1 1 −1 −1 0 1 1 1 −1 0 1 3 −1 3 −4 1 −3 −1 −3 4 0 −4 0 −4 6       . Presentation for the Group Γ 4 The 12 side-pairing transformations g 1 , g 2 , g 3 , g 4 , g 5 , g 6 , g 8 , g 10 , g 11 , g 13 , g 14 , g 17 that generate the group Γ 4 are represented in O + (4, 1) as follows: Table 2. Homology groups of the orientable double covers g 1 = 1 2       1 −1 −1 1 0 −3 −3 1 1 4 −3 −3 −1 −1 4 1 −1 1 −1 0 −4 −4 0 0 6       , g 2 =       0 0 −1 0 0 1 −2 0 0 2 0 0 0 1 0 −2 1 0 0 −2 2 −2 0 0 3       , g 3 =       0 0 −1 0 0 1 −2 0 0 −2 0 0 0 1 0 −2 1 0 0 2 −2 2 0 0 3       , g 4 = 1 2       1 −1 −1 1 0 −3 −3 1 1 −4 −3 −3 −1 −1 −4 1 −1 1 −1 0 4 4 0 0 6       , g 5 =       0 1 0 0 0 0 0 0 1 0 −1 0 −2 0 2 2 0 1 0 −2 −2 0 −2 0 3       , g 6 = 1 2       3 −1 −3 1 4 −1 1 −1 1 0 −3 −1 3 1 −4 1 1 1 1 0 4 0 −4 0 6       , g 8 =       0 1 0 0 0 0 0 0 1 0 −1 0 −2 0 −2 2 0 1 0 2 2 0 2 0 3       , g 10 = 1 2       −1 −1 −1 −1 0 1 −1 −1 1 0 1 3 −3 −1 4 −1 3 −3 1 4 0 4 −4 0 6       , g 11 = 1 2       −1 −1 −1 −1 0 1 −1 −1 1 0 1 3 −3 −1 −4 −1 3 −3 1 −4 0 −4 4 0 6       , g 13 =       2 0 0 1 −2 0 1 0 0 0 0 0 −1 0 0 1 0 0 2 −2 −2 0 0 −2 3       , g 14 =       2 0 0 −1 2 0 −1 0 0 0 0 0 1 0 0 −1 0 0 2 −2 2 0 0 −2 3       , g 17 = 1 2       1 1 −1 −1 0 1 3 1 3 −4 −1 1 1 −1 0 −1 3 −1 3 −4 0 −4 0 −4 6       . Name H 0 H 1 H 2 H 3 H 4 24cdc1.1 Z Z ⊕ Z 3 ⊕ Z 13 Z 2 ⊕ Z 3 ⊕ Z 13 0 0 24cdc1.2 Z Z ⊕ Z 3 ⊕ Z 13 Z 2 ⊕ Z 3 ⊕ Z 13 0 0 24cdc1.3 Z Z ⊕ Z 2 2 ⊕ Z 5 Z 2 ⊕ Z 2 ⊕ Z 5 0 0 24cdc1.4 Z Z ⊕ Z 3 3 Z 2 ⊕ Z 3 3 0 0 The face F radially projects from the origin onto Q. As Q is a fundamental polytope for Γ * , we deduce that F is a fundamental polytope for the action of Γ * on the boundary of the convex hull of Γ * v. Therefore Q is the Epstein-Penner canonical tessellation of H 4 determined by M . Hence, every isometry of M lifts to a symmetry of Q. Therefore, every isometry of M is induced by a symmetry of Q. Orientable Double Covers The orientable double cover of the flat 3-manifold N 3 2 is the 3-torus. Therefore, the orientable double cover of each of the one-cusped hyperbolic 24-cell manifolds is a hyperbolic 4-manifold with one cusp of link type the 3-torus. The homology groups of the orientable double covers of the one-cusped 24-cell manifolds are given in Table 2. Observe that manifolds 24cdc1.1 and 24cdc1.2 have the same homology groups. This suggests that 24cdc1.1 and 24cdc1.2 are isometric manifolds. In fact, these manifolds are isometric, since the side-pairings of two 24-cells that glue up the manifolds are equivalent up to symmetries of each of the two 24-cells. Table 1 . 1Homology groups of the one-cusped hyperbolic 24-cell manifolds2. The Closed Flat 3-Manifold N 3 2 Defining relators for the above set of 12 generators for Γ 2 are as follows: g 2 g 12 g 4 g −18, g 4 g −1 12 g −1 13 g −1 16 , g 5 g 8 g −1 12 g 16 , g 4 g −1 16 g 15 g −1 5 , g 3 g −1 12 g −1 9 g −1 4 , g 1 g −1 9 g −1 16 g −1 13 , g 8 g −1 16 g −1 14 g −1 15 , g 5 g −1 9 g 13 g 8 , g 1 g 12 g 9 g −1 2 , g 3 g −1 13 g −1 5 g −1 12 , g 2 g −1 14 g −1 9 g 5 , g 3 g 4 g −1 14 g −1 16 , g 4 g 15 g 9 g 13 , g 5 g −1 13 g −1 15 g −1 14 , g 1 g −1 13 g 14 g −1 8 , g 2 g −1 16 g −1 8 g −1 9 , g 3 g −1 15 g −1 12 g 8 , g 1 g 14 g 12 g 16 , g 1 g 3 g 15 g 4 , g 1 g 4 g 2 g 14 ,Defining relators for the above set of 12 generators for Γ 3 are as follows:, g 4 g −1 12 g 2 11 , g 4 g −1 14 g 11 g 12 , g 3 g −1 10 g 14 g 4 , g 2 g 4 g 10 g −1 6 , g 3 g 9 g −1 6 g −1 10 , g 3 g 2 10 g −1 4 , g 4 g 9 g −1 17 g −1 11 , g 3 g −1 12 g 7 g −1 11 , g 3 g −1 7 g −1 10 g −1 12 , g 2 g 12 g 11 g −1 17 , g 2 1 g −1 11 g 10 , g 1 g −1 9 g −1 14 g −1 7 , g 2 g 3 g 14 g 17 , g 6 g −1 14 g 12 g 9 , g 1 g 12 g 17 g −1 14 , g 1 g −2 17 g 9 , g 1 g 14 g 6 g 3 ,Defining relators for the above set of 12 generators for Γ 4 are as follows:, g 5 g 14 g 2 10 , g 1 g −2 5 g 17 , g 1 g 6 g −1 14 g 5 , g 1 g 2 g −1 8 g 6 , g 2 2 g −1 6 g −1 11 , g 2 1 g −1 13 g 2 , g 1 g −1 17 g 14 g −1 2 .Volumes of Maximum CuspsIn this section, we determine the volume of the maximum cusp of each of the one-cusped hyperbolic 24-cell manifolds. Proof. We pass to the conformal ball model of H 4 and center the 24-cell Q at the origin. Two horoballs based at adjacent vertices of Q that project to the maximum cusp are tangent at the Euclidean midpoint of the edge of Q joining the vertices. From this observation, it is easy to work out the volume of the maximum cusp.Orders of Isometry GroupsIn this section, we determine the order of the isometry group of each of the one-cusped hyperbolic 24-cell manifolds. Proof. The number of side-pairings of the 24-cell Q for the manifold 24c1.k, for k = 1, . . . , 4, that are equivalent up to a symmetry of Q is 12, 12, 2, 8 respectively. Hence, the order of the group of isometries of the hyperbolic manifold 24c1.k, for k = 1, . . . , 4, that are induced by a symmetry of the 24-cell Q is 12, 12, 2, 8 respectively. That these are the orders of the full groups of isometries of the manifolds will follow once we prove that Q is the Epstein-Penner canonical tessellation[2]of H 4 determined by the manifolds.Let M = H 4 /Γ * be one of the hyperbolic manifolds 24c1.k, for k = 1, . . . , 4. Let C be a cusp of M . Choose an ideal vertex u of Q. Then C is covered by a horoball B based at u. Let v be the vector on the positive light cone L + such that the horosphere ∂B has the equation x • v = −1. The vector v lies on the ray from the origin in L + corresponding to u.Each ideal vertex of Q is equivalent to u by the composition of a finite sequence of side-pairing transformations, since M has a single cusp. Suppose that u is equivalent to the ideal vertex u ′ be a side-pairing transformation g. Then g is the composition rf where f is a symmetry of Q that maps u to u ′ and r is a reflection that fixes u ′ . Therefore, the orbit Γ * v contains the vectors v = v 1 , . . . , v 24 , of the same height on L + , on the rays in L + corresponding to the ideal vertices of Q. Therefore, the convex hull of the vectors v = v 1 , . . . , v 24 is a horizontal Euclidean regular 24-cell F . The remaining vectors in the orbit Γ * v are higher up on L + than v, since a nonidentity element of Γ * moves Q higher up on H 4 . Therefore F is a face of the convex hull of the orbit Γ * v. Crystallographic groups of four-dimensional space. H Brown, R Bülow, J Neubüser, H Wondratschek, H Zassenhaus, John Wiley & SonsNew YorkH. Brown, R. Bülow, J. Neubüser, H. Wondratschek, H. Zassenhaus, Crystallographic groups of four-dimensional space, John Wiley & Sons, New York, 1978. Euclidean decompositions of noncompact hyperbolic manifolds. D B A Epstein, R C Penner, J. Differ. Geom. 27D. B. A. Epstein and R. C. Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differ. Geom. 27 (1988) 67-80. Dreidimensionale euklidische Raumformen. W Hantzsche, H Wendt, Math. Ann. 110W. Hantzsche and H. Wendt, Dreidimensionale euklidische Raumformen, Math. Ann. 110 (1935), 593-611. Hyperbolic four-manifolds with one cusp. A Kolpakov, B Martelli, Geom. Funct. Anal. 23A. Kolpakov and B. Martelli, Hyperbolic four-manifolds with one cusp, Geom. Funct. Anal. 23 (2013), 1903-1933. Symmetries of Hyperbolic 4-manifolds. A Kolpakov, L Slavich, Int. Math. Res. Not. IMRN. 2016A. Kolpakov and L. Slavich, Symmetries of Hyperbolic 4-manifolds, Int. Math. Res. Not. IMRN 2016 (2016), 2677-2716. Hyperbolic 4-manifolds, colourings and mutations. A Kolpakov, L Slavich, Proc. London. Math. Soc. 113A. Kolpakov and L. Slavich, Hyperbolic 4-manifolds, colourings and mutations, Proc. London. Math. Soc. 113 (2016), 163-184. J G Ratcliffe, Foundations of Hyperbolic Manifolds. Springer Nature Switzerland AG149Third EditionJ. G. Ratcliffe, Foundations of Hyperbolic Manifolds, Third Edition, Graduate Texts in Math., vol. 149, Springer Nature Switzerland AG, 2019. The volume spectrum of hyperbolic 4-manifolds, Experimental Math. J G Ratcliffe, S T Tschantz, 9J. G. Ratcliffe and S. T. Tschantz, The volume spectrum of hyperbolic 4-manifolds, Experi- mental Math. 9 (2000), 101-125. J G Ratcliffe, S T Tschantz, Fibered orbifolds and crystallographic groups. 10J. G. Ratcliffe and S. T. Tschantz, Fibered orbifolds and crystallographic groups, Algebr. Geom. Topol. 10 (2010), 1627-1664. Some hyperbolic 4-manifolds with low volume and number of cusps. L Slavich, Topology Appl. 191L. Slavich, Some hyperbolic 4-manifolds with low volume and number of cusps, Topology Appl. 191 (2015), 1-9. E-mail address: j.g. [email protected] address: [email protected]
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[ "Quantum Language Model with Entanglement Embedding for Question Answering", "Quantum Language Model with Entanglement Embedding for Question Answering" ]
[ "Yiwei Chen ", "Senior Member, IEEEYu Pan ", "Senior Member, IEEEDaoyi Dong " ]
[]
[]
Quantum Language Models (QLMs) in which words are modelled as a quantum superposition of sememes have demonstrated a high level of model transparency and good posthoc interpretability. Nevertheless, in the current literature word sequences are basically modelled as a classical mixture of word states, which cannot fully exploit the potential of a quantum probabilistic description. A quantum-inspired neural network module is yet to be developed to explicitly capture the nonclassical correlations within the word sequences. We propose a neural network model with a novel Entanglement Embedding (EE) module, whose function is to transform the word sequence into an entangled pure state representation. Strong quantum entanglement, which is the central concept of quantum information and an indication of parallelized correlations among the words, is observed within the word sequences. The proposed QLM with EE (QLM-EE) is proposed to implement on classical computing devices with a quantum-inspired neural network structure, and numerical experiments show that QLM-EE achieves superior performance compared with the classical deep neural network models and other QLMs on Question Answering (QA) datasets. In addition, the post-hoc interpretability of the model can be improved by quantifying the degree of entanglement among the word states.Index Terms-quantum language model, complex-valued neural network, interpretability, entanglement embedding.
10.1109/tcyb.2021.3131252
[ "https://arxiv.org/pdf/2008.09943v3.pdf" ]
221,266,294
2008.09943
ac4c55586293cf35e39f4040277c52a542a80478
Quantum Language Model with Entanglement Embedding for Question Answering Yiwei Chen Senior Member, IEEEYu Pan Senior Member, IEEEDaoyi Dong Quantum Language Model with Entanglement Embedding for Question Answering 1 Quantum Language Models (QLMs) in which words are modelled as a quantum superposition of sememes have demonstrated a high level of model transparency and good posthoc interpretability. Nevertheless, in the current literature word sequences are basically modelled as a classical mixture of word states, which cannot fully exploit the potential of a quantum probabilistic description. A quantum-inspired neural network module is yet to be developed to explicitly capture the nonclassical correlations within the word sequences. We propose a neural network model with a novel Entanglement Embedding (EE) module, whose function is to transform the word sequence into an entangled pure state representation. Strong quantum entanglement, which is the central concept of quantum information and an indication of parallelized correlations among the words, is observed within the word sequences. The proposed QLM with EE (QLM-EE) is proposed to implement on classical computing devices with a quantum-inspired neural network structure, and numerical experiments show that QLM-EE achieves superior performance compared with the classical deep neural network models and other QLMs on Question Answering (QA) datasets. In addition, the post-hoc interpretability of the model can be improved by quantifying the degree of entanglement among the word states.Index Terms-quantum language model, complex-valued neural network, interpretability, entanglement embedding. I. INTRODUCTION N EURAL Network Language Model (NNLM) [1] is widely used in Natural Language Processing (NLP) and information retrieval [2]. With the rapid development of deep learning models, NNLMs have achieved unparalleled success on a wide range of tasks [3]- [10]. It becomes a common practice for the NNLMs to use word embedding [4], [11] to obtain the representations of words in a feature space. While NNLM has been very successful at knowledge representation and reasoning [7], its interpretability is often in question, making it inapplicable to critical areas such as the credit scoring system [12]. Two important factors have been summarized in [13] for evaluating the interpretability of a machine learning model, namely, Transparency and Post-hoc Interpretability. The model transparency relates to the forward modelling process, while the post-hoc interpretability is the ability to unearth useful and explainable knowledge from a learned model. Another emerging area is quantum information and quantum computation where quantum theory can be utilized to develop more powerful computers and more secure quantum communication systems than their classical counterparts [14]. The interaction between quantum theory and machine learning has also been extensively explored in recent years. On one hand, many advanced machine learning algorithms have been applied to quantum control, quantum error correction and quantum experiment design [15]- [18]. On the other hand, many novel quantum machine learning algorithms such as quantum neural networks and quantum reinforcement learning have been developed by taking advantage of the unique characteristics of quantum theory [19]- [26]. Recently, Quantum Language Models (QLMs) inspired by quantum theory (especially quantum probability theory) have been proposed [27]- [34] and demonstrated considerable performance improvement in model accuracy and interpretability on information retrieval and NLP tasks. QLM is a quantum heuristic Neural Network (NN) defined on a Hilbert space which models language units, e.g., words and phrases, as quantum states. By embedding the words as quantum states, QLM tries to provide a quantum probabilistic interpretation of the multiple meanings of words within the context of a sentence. Compared with the classical NNLM, the word states in QLM are defined on a Hilbert space which is different from the classical probability space. In addition, modelling the process of feature extraction as quantum measurement which collapses the superposed state to a definite meaning within the context of a sentence could increase the transparency of the model. Despite the fact that previous QLMs have achieved good performance and transparency, the state-of-the-art designs still have limitations. For example, the mixed-state representation of the word sequence in [32], [34] is just a classical ensemble of the word states. As shown in the left of Fig. 1, a classical probabilistic mixture of quantum states is not able to fully capture the complex interaction among subsystems. Although Quantum Many-body Wave Function (QMWF) [33] method has been applied to model the entire word sequence as the combination of subsystems, it is still based on a strong premise that the states of the word sequences are separable, as shown in the middle of Fig. 1. A general quantum state has the ability to describe distributions that cannot be split into independent states of subsystems like in the right of Fig. 1. In quantum physics, the state for a group of particles can be generated as an inseparable whole, leading to a non-classical phenomenon called quantum entanglement [35]. Quantum entanglement Fig. 1. Comparison between mixed state, product state and entangled state with a bipartite example. |ψ is the system state. |ψ 1 and |ψ 2 are the states of subsystems A and B, respectively. Parallelized correlations (denoted as arrows) exist between the two subsystems in the entangled state, while for mixed and product states the superposition only exists within the subsystem itself. Separable state is defined as the classical probabilistic mixture of product states. The product state and entangled state are defined on the tensor product (⊗) of two Hilbert spaces. can be understood as correlations (between subsystems) in superposition, and this type of parallelized correlations can be observed in human language system as well. A word can have different meanings when combined with other words. For example, the verb turn has four meanings {move, change, start doing, shape on a lathe}. If we combine it with on to get the phrase turn on, the meaning of turn will be in the superposition of change and start doing. However, this kind of correlation has not been explicitly generated in the present NN-based QLMs. Besides, a statistical method has been proposed in [36] to characterize the entanglement within the text in a post-measurement configuration, which still lacks the transparency in the forward modelling process. In this paper, we propose a novel quantum-inspired Entanglement Embedding (EE) module which can be conveniently incorporated into the present NN-based QLMs to learn the inseparable association among the word states. To be more specific, each word is firstly embedded as a quantum pure state and described by a unit complex-valued vector corresponding to the superposition of sememes. The Word Embedding neural network module is adopted from [34]. Word sequences (phrases, N -grams, etc.) are initially given as the tensor product of the individual word states, and then transformed to a general entangled state as the output of the EE module. The EE module is realized by a complex-valued neural network, which is essentially approximating the unitary operation that converts the initial product state to an entangled state. After the entanglement embedding, high-level features of the word sequences are extracted by inner products between the entangled state vector and virtual quantum measurement vectors [14]. All the parameters of the complex-valued neural network are trainable with respect to a cost function defined on the extracted features. Entanglement measures for quantifying and visualizing the entanglement among the word states can be directly applied on the output of the learned model. We conduct experiments on two benchmark Question Answering (QA) datasets to show the superior performance and posthoc interpretability of the proposed QLM with EE (QLM-EE). In addition, the word embedding dimension can be greatly reduced when compared with previous QLMs, due to the composition of word embedding and EE modules in the hierarchical structure of the neural network. Note that the current QLM-EE is proposed to implement on classical computing devices with a quantum-inspired neural network structure. The main contributions of this paper are summarized as follows. • A novel EE neural network module is proposed. The output of the EE module represents the correlations among the word states with a quantum probabilistic model which explores the entire Hilbert space of quantum pure states. The embedded states can reveal the possible entanglement between the words, which is an indication of parallelized correlations. The entanglement can be quantified to promote the transparency and post-hoc interpretability of QLMs to an unprecedented level. • A QLM-EE framework is presented by cascading the word embedding and EE modules. The word embedding module captures the superposed meanings of individual words, while the EE modules encode the correlations between the words at a higher level. The resulting cascaded deep neural network is more expressive and efficient than the shallow networks used by previous QLMs. • The superior performance of QLM-EE is demonstrated over the state-of-the-art classical neural network models and other QLMs on QA datasets. In addition, the word embedding dimension in QLM-EE is greatly reduced and the semantic similarity of the embedded states of word sequences can be studied using the tools from quantum information theory. The entanglement between the words can be quantified and visualized using analytical methods under the QLM-EE framework. This paper is organized as follows. Section II provides a brief introduction to preliminaries and related work. Entanglement embedding is presented in Section III. Section IV proposes the QLM-EE model. Experimental results are presented in Section V and the results show that QLM-EE achieves superior performance over five classical models and five quantum-inspired models on two datasets. Post-hoc interpretability is discussed in Section VI and concluding remarks are given in Section VII. II. PRELIMINARIES AND RELATED WORK A. Quantum State Mathematically, an n-level quantum system can be described by an n-dimensional Hilbert space H = C n . A pure state of the quantum system is described by Dirac notation where a ket represents a state vector, written as |ψ (equivalent to a complex-valued column vector). The conjugate transpose, denoted by †, of a state vector is called bra, denoted as ψ|, i.e., ψ| = (|ψ ) † . Denote a chosen orthonormal basis of H as {|0 , |1 , . . . , |n − 1 }. Any quantum pure state can be described by a unit vector in H, which may be expanded on the basis states as |ψ = n−1 i=0 α i |i ,(1) with complex-valued probability amplitudes {α i } satisfying n−1 i=0 |α i | 2 = 1.(2) Note that the set {|α i | 2 } defines a classical discrete probability distribution. A quantum system can be in the superposition of distinct states at the same time, with the probability of being |i given by |α i | 2 . For example, we consider a quantum bit (qubit) that is the basic information unit in quantum computation and quantum information, which can be physically realized using e.g., a photon, an electron spin or a two-level atom [14]. The state of a qubit can be described as |ψ = α 0 |0 + α 1 |1 ,(3) where α 0 , α 1 ∈ C and |0 = 1 0 , |1 = 0 1 .(4) The state of a composite system |ψ AB consisting of two subsystems A and B can be described by the tensor product (⊗) of the states of these two subsystems |ψ A and |ψ B as |ψ AB = |ψ A ⊗ |ψ B .(5) For example, if two qubits are in |ψ 1 = α 0 |0 + α 1 |1 and |ψ 2 = α 3 |0 + α 4 |1 , respectively, the state of the two-qubit system can be described by |ψ 12 = α 0 α 3 |00 + α 0 α 4 |01 + α 1 α 3 |10 + α 1 α 4 |11 ,(6) where we have denoted |0 ⊗ |0 = |00 =     1 0 0 0     , |0 ⊗ |1 = |01 =     0 1 0 0     , |1 ⊗ |0 = |10 =     0 0 1 0     , |1 ⊗ |1 = |11 =     0 0 0 1     . For an open quantum system or a quantum ensemble, its state needs to be described by a density matrix ρ satisfying tr(ρ) = 1, ρ † = ρ and ρ ≥ 0. In this paper, we mainly focus on quantum pure states, and thus the inputs and outputs of the neural network modules are complex-valued vectors that stand for the pure states. If a quantum system is in the state in (1), then the system is physically in the superposition state of {|i }. Similar superposition, although not physically, may also exist in the human language systems, which is expressed as the superposition of multiple meanings of a semantic unit. B. Quantum Entanglement Quantum entanglement is one of the most fundamental concepts in quantum theory. Entanglement describes the nonclassical correlation between quantum systems. To be more specific, a many-body quantum system is in an entangled state if the state of one subsystem is determined by the measurement result of the other subsystem. Mathematically speaking, the joint state of an entangled quantum system cannot be decomposed into the states of subsystems by tensor product. For example, we consider two quantum systems A and B defined in Hilbert spaces H A and H B , respectively. Assume the basis state vectors of the two subsystems are {|i } and {|j }. The joint state is then defined on the tensor product space H A ⊗H B , whose basis state vectors are given by the set {|i ⊗|j }. A general pure state |ψ of the composite quantum system can be written as follows |ψ = n i,j α ij |i ⊗ |j ,(7) where {α ij } are complex-valued probability amplitudes. The pure state is separable if it can be decomposed as |ψ = |ψ 1 ⊗ |ψ 2 ,(8) where |ψ 1 = i α 1i |i and |ψ 2 = j α 2j |j are pure states of the subsystems. Otherwise, the pure state is entangled. According to (7) and (8), separable pure states only constitute a small potion of the quantum states that can be defined on H A ⊗ H B , which means that a significant amount of correlations between the subsystems cannot be characterized by the separable states. For example, one of the entangled Bell States or EPR pairs [14] is defined by |ψ = 1 √ 2 (|00 + |11 ),(9) which can not be written as a tensor product of two pure states of the subsystems. The composite system is in the superposition of two basis states |00 and |11 . If we measure the state of the first system and the measurement result is |1 , then the state of the second system is |1 . However, since the first system is a superposition of two states |0 and |1 , the measurement result can be |0 with equal probability. In that case, the state of the second system is |0 . This kind of non-classical correlation cannot be modelled by classical probability. Quantum entanglement can be used to model the superposition of correlations between the subsystems, or in our case, the superposition of multiple meanings between the word states. C. Quantum Measurement Quantum measurement is used to extract information from a quantum system. A widely-used measurement is the projective measurement (von Neumann measurement). For example, when measuring a pure state |ψ = n−1 i=0 α i |i by projecting onto the measurement basis states {|i }, the quantum state will collapse to one of the basis states with the probability of p i (|ψ ) = |α i | 2 = | i|ψ | 2 ,(10) and the inner product i|ψ of |i and |ψ is calculated as i|ψ = (|i ) † |ψ .(11) In a more general setting, projective measurements can be performed using any state vector |x (i.e., not just the computational basis), with the probability of obtaining |x given by p x (|ψ ) = | x|ψ | 2 .(12) D. Quantum Fidelity In quantum information theory, fidelity is a real-valued measure of the similarity between two quantum pure states, which is defined as F(|ψ a , |ψ b ) = | ψ a |ψ b | 2 .(13) According to (10), fidelity is just the probability of collapsing |ψ a to |ψ b if |ψ a is measured by |ψ b , or the probability of collapsing |ψ b to |ψ a if |ψ b is measured by |ψ a . In other words, fidelity is the probability that one quantum state will pass the test to be identified as the other. E. Complex-valued Word Embedding Complex-valued word embedding module aims to model words as quantum pure states in the semantic Hilbert space. In this paper, the complex-valued word embedding module is adopted from [34]. As shown in Fig. 2, each word is firstly encoded to a one-hot vector with a fixed length. Then, the amplitude embedding and phase embedding modules map the one-hot vector into a pair of real-valued amplitude and phase vectors {[r 1 · · · r n ] T , [φ 1 · · · φ n ] T }. After that, the amplitude vector is normalized to a unit vector and the polar form representation of the word state is given by |s = n j=1 r j e iφj |j ,(14) where i is the imaginary number with i 2 = −1. Note that {|j } n j=1 are the basis sememes of the semantic Hilbert space, which represent the minimum semantic units of the word meaning. Finally, the pair of vectors is transformed into a complex-valued vector as [α 1 · · · α n ] T , and the word state can be written as |s = n j=1 α j |j(15) with n j=1 |α j | 2 = 1. The complex-valued word embedding in [37] defines the real-valued amplitude as the semantic meaning of the word, and the complex phases as the positional information of word in the sequence. In contrast, the complexvalued word embedding in this paper aims to model words using quantum state representation, and no specific meaning is given to the phase or amplitude. Instead, the semantic meanings and their quantum-like superposition are jointly determined by the amplitude and phase. F. Related Work In [38], van Rijsbergen argued that quantum theory can provide a unified framework for the geometrical, logical and probabilistic models for information retrieval. Coecke et al. [27] introduced DisCo formalism based on tensor product composition and Zeng et al. [28] presented a quantum algorithm to categorize sentences in DisCo model. Kartsaklis [39] used the traditional machine learning method to quantify entanglement between verbs in DisCo model. Sordoni et al. [29] proposed a quantum language modelling approach for information retrieval and the density matrix formulation was used as a general representation for texts. The single and compound terms were mapped into the same quantum space, and term dependencies are modelled as projectors. In [30], Quantum entropy minimization method has been proposed in learning concept embeddings for query expansion, where concepts from the vocabulary were embedded in rank-one matrices, and documents and queries were described by the mixtures of rank-one matrices. In [31], a quantum language model was presented where a "proof-of-concept" study was implemented to demonstrate its potential. Two NN-based Quantum-like Language Models (NNQLMs) have been proposed, namely NNQLM-I and NNQLM-II [32]. Words were modelled as quantum pure states in the semantic Hilbert space and a word sequence was also modelled in the same space by mixing the word states in a classical way as ρ = k c k |s k s k | ,(16) where |s k was the word state representing the k-th word in the sentence and c k is the weight of k-th word state satisfying k c k = 1. By (16), the semantic meaning of the word sequence is mainly determined by the word states with larger weights. In NNQLM-I, the representation of a single sentence was obtained by the first three layers as a density matrix corresponding to a mixed state, and then the joint representation of a question/answer pair was generated in the fourth layer by matrix multiplication. The last softmax layer was invoked to match the question/answer pair. NNQLM-II adopted the same first 4-layer network structure to obtain the joint representation of a question/answer pair as NNQLM-I, and employed a 2-dimensional convolutional layer instead to extract the features of the joint representation for comparing the question/answer pairs. In [33], a Quantum Many-body Wave Function (QMWF) method was presented in which the representation of a single sentence was given by the tensor product of word vectors. Operation that mimics the quantum measurement was applied on the product state to extract the correlation patterns between the word vectors. To be more specific, a three-layer Convolutional Neural Network (CNN) was used, in which the first layer generated the product state representation of a word sequence and the projective measurement on the product state was simulated by a 1D convolutional layer with product pooling. A complex-valued network called CNM was presented in [34]. Similar to NNQLMs, CNM embedded the word sequence as a mixed state but with a complex-valued neural network. Then a number of trainable measurement operations were applied on the complex-valued density matrix representations to obtain the feature vectors of question/answer for comparison. CNM achieved comparable performance over the state-of-the-art models based on CNN and recurrent NN. More importantly, CNM has shown its advantage in interpretability, since the model has simulated the generation of a quantum probabilistic description for the individual words with complex-valued word states, and the projection of the superposed sememes onto a fixed meaning by quantum-like measurement within a particular context. The work [40] presented a survey of the quantum-inspired information retrieval and drew a road-map of future directions. Ref. [41] proposed a decision-level fusion strategy for predicting sentiment judgments inspired by quantum theory. In the existing works, some potential of quantum-inspired NN modules have been explored for language modelling. However, most of the existing works assume that the word states are placed in the same Hilbert space, while in QLM-EE the word states are placed in the tensor product of Hilbert spaces. This enables an explicit entanglement analysis between the word states, which is not possible if the word states are mixed in the same space. Although [33] has modelled the words in independent Hilbert spaces, it was assumed that the interaction among the words are described by product states, which limits the expressive power of quantum state space. This paper continues these efforts and proposes a novel entanglement embedding module for question answering task. III. ENTANGLEMENT EMBEDDING Quantum probabilistic superposition of states models the polysemy of words and sequences. Quantum states span the entire complex Hilbert space, while the amplitudes and complex phases of the states give the probability distribution of the multiple meanings of words. An N -gram is the composition of N words whose joint state is defined on the tensor product of N Hilbert spaces. However, the joint state itself is not necessarily the tensor product of N states of the Hilbert spaces. The set of entangled state is significantly larger than that of the tensor-product states, and compound semantic meaning lying within the N -gram is mainly captured by the entanglement representation. Then the parameters of the entangled states is trained by the EE module, which characterizes the features of entanglement between the word states within the sequence. The EE module is trained to capture high-level features between the words in the N -gram, while the word embedding module can focus on encoding the basic semantic meanings of       Fig. 3. Pipeline of the EE module for a 3-word sequence. ⊗ denotes the tensor product of input vectors. individual words with reduced dimension. The clear separation of duties in the hierarchical structure of the neural network increases the transparency of the model to an unprecedented level, which allows an accurate interpretation of the intermediate states using the tools borrowed from quantum information theory. In line with the previous works, the complex-valued word embedding module is used to transform the one-hot representation of the word into a quantum pure state in the Hilbert space of word states H w , expressed as a state vector |ψ = [α 0 · · · α i · · · ] T , with i |α i | 2 = 1. The general quantum pure state for describing a word sequence is defined on the tensor-product Hilbert space H s := ⊗ i (H w ) i , and can be formulated as |ψ s = i1,...,i N β i1...i N |i 1 ⊗ · · · ⊗ |i N ,(17) where i1,...,i N |β i1...i N | 2 = 1. |ψ s is used to characterize the probability distribution of the sememes. If the word embedding dimension is D, then a general pure state vector defined on the tensor product of N Hilbert spaces contains D N elements. The EE module starts with composing the contiguous sequences of words as N -grams [42]; see Fig. 3 for an example. For each N -gram, a separable pure state is generated as the tensor product of its word states taking the following form |ψ ss = |ψ 1 ⊗ · · · ⊗ |ψ N . The word states are defined on Hilbert spaces which are isomorphic to each other, with the same semantic basis states. For this reason, N -grams are defined on the same tensorproduct space before entering the EE module. The complexvalued EE module is then connected to transform the initial separable state to an unnormalized vector |ψ s = [β i1...i N ] which will then be normalized to determine a general pure state |ψ s = [β i1...i N ] in the form of (17). The transformation induced by the NN can be formally written as |ψ s = W|ψ ss ,(19) where W is the weight matrix. Then the output vector must be normalized by β i1...i N =β i1...i N i1,...,i N |β i1...i N | 2(20) to be consistent with quantum theory. Note that the operations (19)- (20) induce an endomorphism of this Hilbert space and the EE module transfers one pure state to another. The pure states are unit vectors in the tensor-product Hilbert space. Any unit vectors in the same Hilbert space can be connected by a unitary matrix which induces a rotation. It is known that for any two quantum pure states |a and |b of an N -qubit register, there exists a gate sequence to physically realize the unitary matrix U such that |b = U |a [43]. Since the model proposed in this paper is only quantum-inspired, we are using a linear matrix multiplication and a normalization layer to approximate the effect of the transformation matrix by the classical way of training. In principle, if one layer of linear matrix multiplication is not sufficient for learning the transformation, it is always possible to stack several layers of linear operations to accurately approximate any transformation, which is consistent with the traditional neural network theory. We take Fig. 3 as an example to illustrate the working mechanism of the EE module. We denote the state vectors for the first two words as [α 1 α 2 ] T and [α 3 α 4 ] T . The two word states form the separable state as the input to the NN layer by the following tensor product α 1 α 2 ⊗ α 3 α 4 =     α 1 α 3 α 1 α 4 α 2 α 3 α 2 α 4     .(21) The output of the NN layer is an unnormalized vector [β 00β01β10β11 ] T . After normalization, we obtain     β 00 β 01 β 10 β 11     ,(22) which is in the most general form of the state vector on the joint Hilbert space. If [β 00 β 01 β 10 β 11 ] T cannot be written in a decomposable form just like the RHS of (21), then the output vector is a representation for an entangled state which captures the non-classical correlations between the word states. IV. MODEL The structure of QLM-EE for QA is shown in Fig. 4. It is a complex-valued, end-to-end neural network optimized by back-propagation on classical computing devices. The QLM-EE can be divided into three major steps. • Word embedding and entanglement embedding. The word embedding module generates the word states, and the entanglement embedding module generates the complete quantum probabilistic description of the sequence of word states. The complete probabilistic description is given by a generic quantum pure state vector, which may be entangled to encode the information on the nonclassical correlations between the word states. Entanglement embedding and word embedding modules encode the states as feature vectors at different levels, which improves the transparency of the quantum probabilistic modelling process. In particular, the distance between the feature vectors in the embedding space can be calculated based on the well-established measures from the quantum information theory for comparing quantum states, which could be used to reveal the relations between the words and phrases on a deeper semantic level. For example, in the classical Word2Vec, the cosine similarity is defined on real-valued embedded vectors as C( a, b) = a · b || a|| · || b|| .(23) In our case, the cosine similarity is defined on the complex-valued unit vectors as C( a, b) = a † b || a|| · || b|| = a † b.(24) In general, C( a, b) is a complex number, which means that the state representations could differ by a complex phase factor. Since complex phases are hard to visualize, 5. The forward process for the phrase solar system. The distance between state vectors is defined by the quantum fidelity. The fidelities of (solar system, united nations), (solar system, the comet), (solar system, the sun) are 0.523, 0.365, 0.335, respectively, while the fidelities of randomly selected combinations (solar system, in fact), (solar system, such as) are 0.026, 0.015, respectively. we employ the quantum fidelity measure to compare the words and word sequences. In Fig. 5, we visualize the process how the state of the phrase solar system evolves in our model using the quantum fidelity measure. After the word embedding layer, the state vector that represents the word solar is close to {sun, crazy, dominated}, which reflects how the model comprehends the meaning of the input words. The word state vector of system is embedded close to {party, committee, ordered}. After the EE module, the phrase solar system can be linked to other high-level phrases, e.g., united nations, while its state vector is still very close to the phrase the sun and the comet which share a similar meaning. • Measurement operation. Projective measurement operation is equivalent to the calculation of fidelity between quantum states. That is, the output of the measurement can be seen as a distance between the measured state and a measuring state, corresponding to the distance between the semantic entangled state and a measuring sememe which is used for comparison. Therefore, performing the same set of measurement operations provides a way to compare the semantic similarity of the question and answering word sequences. After entanglement embedding, a series of parameterized measurements {|m i } are performed on the state |ψ s via the formula p i (|ψ s ) = | m i |ψ s | 2 .(25) Here p i is defined as the measurement output, which indicates the probability of the state |ψ s possessing the semantic meaning represented by the measurement vector |m i . Note that the unit vectors {|m i } are optimized in a data-driven way. By using pure states as measurement basis, the computation cost for measuring a quantum state virtually is reduced from O(n 3 ) to O(n) compared to CNM [34], in which density matrices were used as the measurement basis. be more specific, the down-sampling takes the maximum value of each row of the matrix to form a reduced feature vector. Then a vector-based similarity metric can be employed in the matching layer for evaluating the distance between the pair of feature vectors for question and answer. The answer with the highest matching score is chosen as the predicted result among all candidate answers. V. EXPERIMENT A. Experiment Details 1) Dataset: We conduct the experiments on two benchmark datasets for QA, namely TREC-QA [44] and W IKI QA [45]. TREC-QA is used in the Text REtrieval Conference. W IKI QA is an open-domain QA dataset released by Microsoft Research. The statistics of the datasets are given in TABLE I. 2) Evaluation metrics: The metrics called Mean Average Precision (MAP) and Mean Reciprocal Rank (MRR) [46] are utilized to evaluate the performance of the models. MAP for a set of queries Q is the mean of the Average Precision scores AveP(q) for each query, formulated as MAP = |Q| q=1 AveP(q)/|Q|. MRR is the average of the Reciprocal Ranks of results for a sample of queries Q, calculated as MRR = 1 |Q| |Q| i=1 1/rank i , where rank i refers to the rank position of the first relevant document for the i-th query. 3) Baselines: a) Classical models for TREC-QA including • Unigram-CNN [47]: Unigram-CNN is a CNN-based model that utilizes 1-gram as the input to obtain the representations of questions and answers for comparison. It is composed of one convolutional layer and one average pooling layer. • Bigram-CNN [47]: Bigram-CNN has the same network structure as Unigram-CNN but it extracts the representations from bi-gram inputs. • ConvNets [48]: ConvNets is built upon two distributional sentence models based on CNN. These underlying sentence models work in parallel to map questions and answers to their distributional vectors, which are then used to learn the semantic similarity between them. • QA-LSTM-avg [49]: QA-LSTM-avg generates distributed representations for both the question and answer independently by bidirectional LSTM outputs with average pooling, and then utilizes cosine similarity to measure their distance. • aNMM-1 [50]: aNMM-1 employs a deep neural network with value-shared weighting scheme in the first layer, which is followed by fully-connected layers to learn the sentence representation. A question attention network is used to learn question term importance and produce the final ranking score. b) Classical models for W IKI QA including LSTM and a max pooling layer to obtain the representation of questions and answers, and then computes the cosine similarity between the two representations. • LSTM-attn [52]: LSTM-attn firstly obtains the representations for the question and answer from independent LSTM models, and then adds an attention model to learn the pair-specific representation for prediction on the basis of the vanilla LSTM. c) Quantum models for TREC-QA and W IKI QA including QLM-MLE [29], NNQLM-I, NNQLM-II [32], QMWF-LM [33], and CNM [34]. These models have been briefly introduced in Section II.E. 4) Hyper-parameters: Cosine similarity defined by (23) is used as the distance metric between the real-valued feature vectors after pooling. The hinge loss [53] for training the model is given by L = max{0, 0.1 − C + + C − },(26) where C + is the cosine similarity of a ground truth answer, C − is the cosine similarity of an incorrect answer randomly In line with CNM, the concatenation of the feature vectors for N = 1, 2, 3 has also been tested. Larger N has been tried, but N ∈ {1, 2, 3} shows better performance than N ≥ 4. All the parameters are initialized from standard normal distributions except the measurement vectors, which are initialized by orthogonal vectors. B. Performance The experimental results on TREC-QA and W IKI QA are shown in TABLE II. We compare the performances of the classical models, including CNNs, Recurrent NNs and attention models with the performances of QLMs. QLM-EE is consistently better than all the QLMs and classical models on both datasets if MAP is used as the metric. If we use MRR as the metric, QLM-EE performs slightly worse than CNM on TREC-QA while better than the other models, and slightly worse than QMWF-LM while better than all the other models on W IKI QA. Our word embedding dimension is selected from D ∈ {6, 8, 10, 12, 14, 16}, which is much smaller than the word embedding dimension of previous QLMs selected from {50, 100, 200}. As a consequence, the amount of parameters for the word embedding layer has seen dramatic reduction while the performance of model has been improved. Fig. 6 illustrates how the word embedding dimension affects the performances of different models in terms of MAP and MRR on TREC-QA. It is clear that the model with the concatenation of the feature vectors for N = 1, 2, 3 performs best and the best word embedding dimension is 8, which is significantly smaller than {50, 100, 200} for the previous QLMs. As D increases, the dimension of the state vector after the entanglement embedding is increased according to the formula D N . As pointed out in the literature [54], learning word embedding has the risk of underfitting or overfitting. Embedding dimension that is too small (less than 50) or too large (more than 300) will degrade the performance. In QLM-EE, the N -gram lies in the tensor-product space, which tends to result in a high-dimensional representation and poor performance. For example, if the word embedding dimension is 8, then 3-grams are described by complex-valued 512dimensional vectors, or real-valued 1024-dimensional vectors. In addition, the dimension of EE module and measurement vectors will increase accordingly, leading to a model that easily overfits the data. In our case, the performance of 3-gram model is worse than that of the 2-gram model, which is a sign of overfitting in this relatively small-sized dataset. However, it is possible that high-dimensional representation will improve the performance if a much larger dataset is considered. Model k M AP p M AP k M RR p M RR QLM-ME (D = 8) 1 0.17% 1 0.02% QLM-ME (D = 64) 1 3.18% 1 6.75% QLM-SE (D = 8) 1 0.01% 1 0.02% QLM-EE-Real (D = 8) 1 1.12% 1 1.18% C. Ablation Test Ablation test studies the contribution of certain components to the overall performance by removing these components from the model. We conduct ablation tests to evaluate the effectiveness of entanglement embedding on 2-gram QLM. QLM-SE has removed thxe entanglement embedding module and measurements are directly performed on the separable joint states. QLM-ME generates the mixed-state embedding by (16), and measurements are performed on the density matrix ρ. In the setting of complex-valued QLM-EE, we let D = 8 and M = 3000. For the complex-valued QLM-ME whose structure is the same as CNM, we set D = 8 and D = 64 with M = 3000 for comparison. Note that when D = 64, the complex-valued QLM-ME has the same dimension for 2-grams as QLM-EE. However, since the word embedding dimension has been increased, the number of parameters for the word embedding module has been increased accordingly. We have doubled the word embedding dimension to D = 16 in the real-valued QLM-EE (QLM-EE-Real), and thus the dimension of 2-grams is 256 which is four times larger than the dimension of 2-grams in the complex-valued QLM-EE. Besides, the dimension of the measurement vectors has been increased four times accordingly. To make the number of parameters compatible, we set M = 1500 for this case. We have run the experiment 10 times. The mean and standard deviation of metrics, the number of parameters and Floating Point Operations (FLOPs) are reported in TABLE III. We can see that QLM-EE achieves the best performance for 2-gram model when the word embedding dimension is 8. The performances QLM-ME with D = 64 and QLM-EE with D = 8 are close, but the number of parameters and FLOPs used by the former is significantly greater than the latter. The result of QLM-EE-Real confirms that the complex-valued neural network indeed promotes the performance of QLMs. The two-sample Kolmogorov-Smirnov (K-S) test result on the performance metrics is shown in TABLE IV, which verifies the performance differences of these models. VI. POST-HOC INTERPRETABILITY von Neumann entanglement entropy S [55] is an accurate measure of the degree of quantum entanglement for a bipartite quantum pure state. The entanglement entropy is calculated as follows S = − K i=1 |λ i | 2 log(|λ i | 2 ),(27) where λ i is the Schmidt coefficient of the composite pure state and K is the minimal dimension of the subsystems, i.e., K = min(dim(H A ), dim(H B )). Apart from the analytical entanglement measure (27) for bipartite states, it is also worth mentioning that efficient numerical methods are available for quantifying quantum entanglement for multipartite states [56]. TABLE V shows the selected most and least 2-grams of words, ranked by the von Neumann entanglement entropy. The most entangled pairs are mostly set phrase or some wellknown combinations of words, e.g., how long. However, is cataracts is clearly not a set phrase or well-known combination of words. We found that is cataracts appears many times, and cataracts is always next to is in this particular training dataset. This may be the reason why the learned model takes is cataracts as a fixed combination of words. The least entangled pairs consist of words with fixed semantic meaning such as names {Quarry, China} and interrogatives {what, who}, some of which appear only once in the dataset. In other words, there is not so much semantic ambiguity or superposition with these combinations that demands a quantum probabilistic interpretation. TABLE VI shows the selected most and least entangled 3-grams, whose von Neumann entanglement entropies are calculated to indicate the entanglement between the first two words, and the remaining word (the third word). Similar to 2-grams, the most entangled 3-grams are fixed collocations and combinations that often appear together in the training set, such as a set phrase. Interestingly, punctuation marks also appear in the most entangled 3-grams. For example, entanglement in at times . is large, which implies that the phrase at times is often at the end of the answers. However, there worldwide ? is among the least entangled combinations, since there worldwide is not in any of the question sentences for training. The third word in the least entangled 3-grams are mainly names and numbers, which cannot combine with the first two words to form a fixed phrase. In Fig. 7, we also visualize the entanglement entropy in some selected sentences, where the degree of darkness indicates the level of entanglement. It can be seen that words with multiple meanings in different contexts, e.g., {is, does, the, film}, have greater capabilities to entangle with their neighboring words. VII. CONCLUSION In this paper, we proposed an interpretable quantuminspired language model with a novel EE module in the neural network architecture. The EE enables the modelling of the word sequence by a general quantum pure state, which is capable of capturing all the classical and non-classical correlations between the word states. The expressivity of the neural network is greatly enhanced by cascading the word embedding and EE modules. The complex-valued model has demonstrated superior performance on the QA datasets, with much smaller word embedding dimensions compared to previous QLMs. In addition, the non-classical correlations between the word states can be quantified and visualized by appropriate entanglement measures, which improves the post-hoc interpretability of the learned model. The future plan is to apply the adaptive methods [57], [58] for optimizing the virtual measurement operations to increase the efficiency in feature extraction. The QLM-EE model is expected to be more powerful on huge datasets in which the semantic meanings of words and their correlations are far more complex. With a larger dataset and richer semantic superpositions between the words, several entanglement embedding modules can be cascaded to form a deeper neural network, which could encode the multipartite correlations within the text at different scales. This work was supported by the National Natural Science Foundation of China (No. 61703364) and the Australian Research Council's Discovery Projects funding scheme under Project DP190101566. Y. Chen is with the Institute of Cyber-Systems and Control, Zhejiang University, Hangzhou, 310027, China. (email: [email protected]). Y. Pan is with the Institute of Cyber-Systems and Control, College of Control Science and Engineering, Zhejiang University, Hangzhou, 310027, China. (email: [email protected]). D. Dong is with the School of Engineering and Information Technology, University of New South Wales, Canberra, ACT 2600, Australia. (email: [email protected]). Fig. 2 . 2Pipeline of the word embedding module for a 3-word sequence. Fig. 4 . 4The structure of QLM-EE. Fig. 5. The forward process for the phrase solar system. The distance between state vectors is defined by the quantum fidelity. The fidelities of (solar system, united nations), (solar system, the comet), (solar system, the sun) are 0.523, 0.365, 0.335, respectively, while the fidelities of randomly selected combinations (solar system, in fact), (solar system, such as) are 0.026, 0.015, respectively. For a sentence described by the concatenation of L word sequences, the feature matrix which stores all the measurement results has L × M entries if M measurement vectors are used. • Similarity Measure. A max-pooling layer is applied on each row of the feature matrix for down-sampling. To • Bigram-CNN [47]; • PV-Cnt [45]: PV-Cnt is the Paragraph Vector (PV) model combined with Word Count. The model score of PV is the cosine similarity score between the question vector and the sentence vector. • CNN-Cnt [45]: CNN-Cnt employs a Bigram-CNN model with average pooling and combines it with Word Count. • QA-BILSTM [51]: QA-BILSTM uses a bidirectional Fig. 6 . 6Experimental results of models with different N and different word embedding dimensions on TREC-QA. N = 1, 2, 3 refers to the model in which 1-gram, 2-gram and 3-gram entanglement embeddings are applied in parallel on the questions and answers, and the feature vectors are concatenated as a single vector for comparison. chosen from the entire answer space. The parameters in the QLM-EE are determined by the set of hyper-parameters Θ := {N, D, M }, where N is the number of words in a sequence, D is the word embedding dimension and M is the number of measurement vectors. N = 1 means embedding the words without composition, and in this case no entanglement can be generated. We test single-layer and twolayer fully connected neural networks with {128, 256, 512} neurons for entanglement embedding. A grid search is conducted using N ∈ {1, 2, 3}, D ∈ {6, 8, 10, 12, 14, 16}, M ∈ {500, 1000, 2500, 5000}, batch size in {16, 32, 64} and learning rate in {0.01, 0.1, 0.5}. Fig. 7 . 7Entanglement entropy of the 2-grams in the sentences. Darker color is an indication of larger entanglement between adjacent words. TABLE I ISTATISTICS OF THE DATASETSDataset Train(Q/A) Dev(Q/A) Test(Q/A) TREC-QA 1,229/53,417 65/1,134 68/1,478 W IKI QA 873/8,627 126/1,130 243/2,351 TABLE II EXPERIMENTAL IIRESULTS ON TREC-QA AND W IKI QATREC-QA W IKI QA Model MAP MRR Model MAP MRR Unigram-CNN 0.5470 0.6329 Bigram-CNN 0.6190 0.6281 Bigram-CNN 0.5693 0.6613 PV-Cnt 0.5976 0.6058 ConvNets 0.6709 0.7280 CNN-Cnt 0.6520 0.6652 QA-LSTM-avg 0.6819 0.7652 QA-BILSTM 0.6557 0.6695 aNMM-1 0.7385 0.7995 LSTM-attn 0.6639 0.6828 QLM-MLE 0.6780 0.7260 QLM-MLE 0.5120 0.5150 NNQLM-I 0.6791 0.7529 NNQLM-I 0.5462 0.5574 NNQLM-II 0.7589 0.8254 NNQLM-II 0.6496 0.6594 QMWF-LM 0.7520 0.8140 QMWF-LM 0.6950 0.7100 CNM 0.7701 0.8591 CNM 0.6748 0.6864 QLM-EE 0.7713 0.8542 QLM-EE 0.6956 0.7003 TABLE III ABLATION IIIRESULTS ON TREC-QA. QLM WITH SEPARABLE-STATE EMBEDDING (QLM-SE) AND QLM WITH MIXED-STATE EMBEDDING (QLM-ME) ARE COMPARED WITH QLM-EE. QLM-EE-REAL IS THE REAL-VALUED QLM-EE.Model D N MAP MRR Params. FLOPs QLM-EE 8 2 0.7302±0.0101 0.8071±0.0156 1.45M 4.20M QLM-ME 8 2 0.6905±0.0711 0.7584±0.0059 0.99M 4.81M QLM-ME 64 2 0.7267±0.0194 0.8171±0.0151 7.88M 124.56M QLM-EE-Real 16 2 0.6778±0.0139 0.7505±0.0187 1.97M 6.01M QLM-SE 8 2 0.6934±0.0149 0.7804±0.0156 1.32M 3.28M TABLE IV THE IVTWO-SAMPLE K-S TEST STATISTICS k AT p SIGNIFICANCE LEVEL BETWEEN THE PERFORMANCE METRICS OF QLM-EE (D = 8) AND OTHER MODELS IN ABLATION TEST. k = 1 INDICATES THAT THE TWO SETS OF METRIC DATA ARE SAMPLED FROM DIFFERENT DISTRIBUTIONS. TABLE V SELECTED VENTANGLED 2-GRAMS IN TREC-QA Type Word combinations Most entangled in questions annual revenue; as a; is cataracts; how long; tale of; first movie; ethnic background Least entangled in questions introduced Jar; the name; what year; the main; who is; whom were ; is a; in what Most entangled in answers ends up; never met; plane assigned; in kindergarten; academy of; going to; agricultural farming; secure defendants Least entangled in answers skinks LRB; while some; grounded in; of Quarry; of seven; he said; in China; responsibility and TABLE VI SELECTED VIENTANGLED 3-GRAMS IN TREC-QA Type Word combinations Most entangled in questions the company Rohm; Hale Bopp comet; how often does; Insane Clown Posse; Capriati 's coach Least entangled in questions there worldwide ?; What is Crips; Criminal Court try; When did James Most entangled in answers after a song; comet 's nucleus; Out of obligation; at times .; Black Panther Party Least entangled in answers Queen , Tirado; came from Britain; Nobel Prize last; appears in the; Americans over 50 A neural probabilistic language model. 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[ "Phonon superradiance and phonon laser effect in nanomagnets", "Phonon superradiance and phonon laser effect in nanomagnets" ]
[ "E M Chudnovsky \nDepartment of Physics and Astronomy\nLehman College\nCity University of New York\n250 Bedford Park Boulevard West10468-1589BronxNew YorkU.S.A\n", "D A Garanin \nDepartment of Physics and Astronomy\nLehman College\nCity University of New York\n250 Bedford Park Boulevard West10468-1589BronxNew YorkU.S.A\n\nInstitut für Physik\nJohannes-Gutenberg-Universität\nD-55099MainzGermany\n" ]
[ "Department of Physics and Astronomy\nLehman College\nCity University of New York\n250 Bedford Park Boulevard West10468-1589BronxNew YorkU.S.A", "Department of Physics and Astronomy\nLehman College\nCity University of New York\n250 Bedford Park Boulevard West10468-1589BronxNew YorkU.S.A", "Institut für Physik\nJohannes-Gutenberg-Universität\nD-55099MainzGermany" ]
[]
We show that the theory of spin-phonon processes in paramagnetic solids must take into account the coherent generation of phonons by the magnetic centers. This effect should drastically enhance spin-phonon rates in nanoscale paramagnets and in crystals of molecular nanomagnets. PACS numbers: 75.50.Xx, 43.35.+d The problem of spin-lattice interactions is almost as old as quantum theory of solids[1]. It is about computing the amplitude of the transition between spin states of a magnetic atom due to the spontaneous emission of a phonon or due to, e.g., the Raman scattering of an existing thermal phonon. These processes are responsible for the width of the paramagnetic resonance and for other spin-lattice relaxation effects. In this Letter we shall demonstrate that all previous works on the subject have missed the collective nature of the relaxation when the wavelength of the phonon is comparable to the distance between the magnetic atoms. General theoretical treatment of this effect has become possible due to the recently established mechanism of the dynamic spinphonon coupling [2],where S is the spin of the atom, Ω is the angular velocity of the local rotation of the crystal due to a transverse phonon, and u is the phonon displacement field. Eq. (1) has the following origin. The crystal field (i.e., the magnetic anisotropy) that determines spin states of an atom in a solid is defined in a local coordinate frame coupled to the crystal axes. In that coordinate frame the local rotation of the lattice is equivalent to the magnetic field, B = Ω/gµ B (with g being the gyromagnetic factor for the spin), that has Zeeman interaction with S. Since the local rotations of the crystal lattice by phonons are small, all matrix elements computed in the rotating frame and in the laboratory frame are practically identical. The beauty of this approach is that it has no unknown parameters. For, e.g., spontaneous emission of a phonon, all parameters of the crystal field enter the interaction through the phonon frequency only. The latter is determined by the distance between the spin levels and can be directly measured, leaving no fitting parameters for comparison between theory and experiment. Another way to look at this interaction is through the conservation of the total angular momentum, spin + lattice. In order to conserve the angular momentum, the transition between the spin states of an atom produces a local elastic twist that propagates out as a transverse phonon. This Letter is based upon observation that an assembly of the magnetic atoms in the excited state must interact via transverse phonons very much the same as atoms inter-act via photons in the laser physics. The deep analogy between the two problems can be seen by noticing from Eq. (1) thatu/2gµ B is equivalent to the vector potential of the electromagnetic field. The huge (c/v t ) 3 ratio of the phonon density of states to the photon density of states (with v t being the speed of the transverse sound) makes such a phonon laser much more powerful than the photon laser based upon magnetic dipoles. We consider two nearly degenerate states of spin S, described by the Hamiltonian for pseudospin 1/2,where σ is the Pauli matrix, ∆ is the splitting of the levels at the avoided crossing, and W = E −1 − E 1 is the energy difference for the two degenerate states at ∆ = 0 (±1 denoting spin up and down). We label these states as ψ −1 and |ψ 1 , then E ±1 = ψ ±1 H S ψ ±1 and ∆ = 2 ψ −1 H S |ψ 1 , where H S is the crystal-field Hamiltonian for the spin S. Choosing the new z axis for pseudospins in the direction of ω 0 , one obtains the eigenvalues and the eigenfunctions of H 0 : E ± = ±(1/2) √ W 2 + ∆ 2 and ψ ± = C ∓ |ψ 1 ± C ± ψ −1 √ 2 with C ± = 1 ± W/ √ ∆ 2 + W 2 , whereas ω 0 = √ W 2 + ∆ 2 e z . The projection of the spin-phonon interaction, Eq. (1), onto the two-state basis isFor tunneling between up and down states of spin S, ψ ±1 = |±S and g x = e z SC + C − = e z S ∆ √ W 2 + ∆ 2 , g y = 0For many two-state particles coupled to phonons the effective many-body Hamiltonian is H = − 2 i ω 0 ·σ i − 2 i σ i · ← → g · [∇ ×u(r i )]+ H ph .(5)
10.1103/physrevlett.93.257205
[ "https://export.arxiv.org/pdf/cond-mat/0404239v1.pdf" ]
45,016,027
cond-mat/0404239
820be38c7585a778c90e1d285daeea2305da1cd9
Phonon superradiance and phonon laser effect in nanomagnets 9 Apr 2004 E M Chudnovsky Department of Physics and Astronomy Lehman College City University of New York 250 Bedford Park Boulevard West10468-1589BronxNew YorkU.S.A D A Garanin Department of Physics and Astronomy Lehman College City University of New York 250 Bedford Park Boulevard West10468-1589BronxNew YorkU.S.A Institut für Physik Johannes-Gutenberg-Universität D-55099MainzGermany Phonon superradiance and phonon laser effect in nanomagnets 9 Apr 2004(Dated: March 22, 2022) We show that the theory of spin-phonon processes in paramagnetic solids must take into account the coherent generation of phonons by the magnetic centers. This effect should drastically enhance spin-phonon rates in nanoscale paramagnets and in crystals of molecular nanomagnets. PACS numbers: 75.50.Xx, 43.35.+d The problem of spin-lattice interactions is almost as old as quantum theory of solids[1]. It is about computing the amplitude of the transition between spin states of a magnetic atom due to the spontaneous emission of a phonon or due to, e.g., the Raman scattering of an existing thermal phonon. These processes are responsible for the width of the paramagnetic resonance and for other spin-lattice relaxation effects. In this Letter we shall demonstrate that all previous works on the subject have missed the collective nature of the relaxation when the wavelength of the phonon is comparable to the distance between the magnetic atoms. General theoretical treatment of this effect has become possible due to the recently established mechanism of the dynamic spinphonon coupling [2],where S is the spin of the atom, Ω is the angular velocity of the local rotation of the crystal due to a transverse phonon, and u is the phonon displacement field. Eq. (1) has the following origin. The crystal field (i.e., the magnetic anisotropy) that determines spin states of an atom in a solid is defined in a local coordinate frame coupled to the crystal axes. In that coordinate frame the local rotation of the lattice is equivalent to the magnetic field, B = Ω/gµ B (with g being the gyromagnetic factor for the spin), that has Zeeman interaction with S. Since the local rotations of the crystal lattice by phonons are small, all matrix elements computed in the rotating frame and in the laboratory frame are practically identical. The beauty of this approach is that it has no unknown parameters. For, e.g., spontaneous emission of a phonon, all parameters of the crystal field enter the interaction through the phonon frequency only. The latter is determined by the distance between the spin levels and can be directly measured, leaving no fitting parameters for comparison between theory and experiment. Another way to look at this interaction is through the conservation of the total angular momentum, spin + lattice. In order to conserve the angular momentum, the transition between the spin states of an atom produces a local elastic twist that propagates out as a transverse phonon. This Letter is based upon observation that an assembly of the magnetic atoms in the excited state must interact via transverse phonons very much the same as atoms inter-act via photons in the laser physics. The deep analogy between the two problems can be seen by noticing from Eq. (1) thatu/2gµ B is equivalent to the vector potential of the electromagnetic field. The huge (c/v t ) 3 ratio of the phonon density of states to the photon density of states (with v t being the speed of the transverse sound) makes such a phonon laser much more powerful than the photon laser based upon magnetic dipoles. We consider two nearly degenerate states of spin S, described by the Hamiltonian for pseudospin 1/2,where σ is the Pauli matrix, ∆ is the splitting of the levels at the avoided crossing, and W = E −1 − E 1 is the energy difference for the two degenerate states at ∆ = 0 (±1 denoting spin up and down). We label these states as ψ −1 and |ψ 1 , then E ±1 = ψ ±1 H S ψ ±1 and ∆ = 2 ψ −1 H S |ψ 1 , where H S is the crystal-field Hamiltonian for the spin S. Choosing the new z axis for pseudospins in the direction of ω 0 , one obtains the eigenvalues and the eigenfunctions of H 0 : E ± = ±(1/2) √ W 2 + ∆ 2 and ψ ± = C ∓ |ψ 1 ± C ± ψ −1 √ 2 with C ± = 1 ± W/ √ ∆ 2 + W 2 , whereas ω 0 = √ W 2 + ∆ 2 e z . The projection of the spin-phonon interaction, Eq. (1), onto the two-state basis isFor tunneling between up and down states of spin S, ψ ±1 = |±S and g x = e z SC + C − = e z S ∆ √ W 2 + ∆ 2 , g y = 0For many two-state particles coupled to phonons the effective many-body Hamiltonian is H = − 2 i ω 0 ·σ i − 2 i σ i · ← → g · [∇ ×u(r i )]+ H ph .(5) We show that the theory of spin-phonon processes in paramagnetic solids must take into account the coherent generation of phonons by the magnetic centers. This effect should drastically enhance spin-phonon rates in nanoscale paramagnets and in crystals of molecular nanomagnets. The problem of spin-lattice interactions is almost as old as quantum theory of solids [1]. It is about computing the amplitude of the transition between spin states of a magnetic atom due to the spontaneous emission of a phonon or due to, e.g., the Raman scattering of an existing thermal phonon. These processes are responsible for the width of the paramagnetic resonance and for other spin-lattice relaxation effects. In this Letter we shall demonstrate that all previous works on the subject have missed the collective nature of the relaxation when the wavelength of the phonon is comparable to the distance between the magnetic atoms. General theoretical treatment of this effect has become possible due to the recently established mechanism of the dynamic spinphonon coupling [2], H s−ph = − S · Ω, Ω(r) = 1 2 ∇ ×u(r),(1) where S is the spin of the atom, Ω is the angular velocity of the local rotation of the crystal due to a transverse phonon, and u is the phonon displacement field. Eq. (1) has the following origin. The crystal field (i.e., the magnetic anisotropy) that determines spin states of an atom in a solid is defined in a local coordinate frame coupled to the crystal axes. In that coordinate frame the local rotation of the lattice is equivalent to the magnetic field, B = Ω/gµ B (with g being the gyromagnetic factor for the spin), that has Zeeman interaction with S. Since the local rotations of the crystal lattice by phonons are small, all matrix elements computed in the rotating frame and in the laboratory frame are practically identical. The beauty of this approach is that it has no unknown parameters. For, e.g., spontaneous emission of a phonon, all parameters of the crystal field enter the interaction through the phonon frequency only. The latter is determined by the distance between the spin levels and can be directly measured, leaving no fitting parameters for comparison between theory and experiment. Another way to look at this interaction is through the conservation of the total angular momentum, spin + lattice. In order to conserve the angular momentum, the transition between the spin states of an atom produces a local elastic twist that propagates out as a transverse phonon. This Letter is based upon observation that an assembly of the magnetic atoms in the excited state must interact via transverse phonons very much the same as atoms inter-act via photons in the laser physics. The deep analogy between the two problems can be seen by noticing from Eq. (1) thatu/2gµ B is equivalent to the vector potential of the electromagnetic field. The huge (c/v t ) 3 ratio of the phonon density of states to the photon density of states (with v t being the speed of the transverse sound) makes such a phonon laser much more powerful than the photon laser based upon magnetic dipoles. We consider two nearly degenerate states of spin S, described by the Hamiltonian for pseudospin 1/2, H 0 = − 1 2 ω 0 · σ, ω 0 = W e z + ∆e x ,(2) where σ is the Pauli matrix, ∆ is the splitting of the levels at the avoided crossing, and W = E −1 − E 1 is the energy difference for the two degenerate states at ∆ = 0 (±1 denoting spin up and down). We label these states as ψ −1 and |ψ 1 , then E ±1 = ψ ±1 H S ψ ±1 and ∆ = 2 ψ −1 H S |ψ 1 , where H S is the crystal-field Hamiltonian for the spin S. Choosing the new z axis for pseudospins in the direction of ω 0 , one obtains the eigenvalues and the eigenfunctions of H 0 : E ± = ±(1/2) √ W 2 + ∆ 2 and ψ ± = C ∓ |ψ 1 ± C ± ψ −1 √ 2 with C ± = 1 ± W/ √ ∆ 2 + W 2 , whereas ω 0 = √ W 2 + ∆ 2 e z . The projection of the spin-phonon interaction, Eq. (1), onto the two-state basis is H s−ph = − 2 σ· ← → g · (∇ ×u) ≡ − 2 σ α g αβ (∇ ×u) β . (3) For tunneling between up and down states of spin S, ψ ±1 = |±S and g x = e z SC + C − = e z S ∆ √ W 2 + ∆ 2 , g y = 0 g z = −e z S 1 2 C 2 + − C 2 − = −e z S W √ W 2 + ∆ 2 .(4) For many two-state particles coupled to phonons the effective many-body Hamiltonian is H = − 2 i ω 0 ·σ i − 2 i σ i · ← → g · [∇ ×u(r i )]+ H ph . (5) where H ph is the Hamiltonian of non-interacting phonons. We use canonical quantization of phonons, u = 2M N kλ e kλ e ik·r √ ω kλ a kλ + a † −kλ ,(6) where M is the mass of the unit cell, N is the number of cells, e kλ are unit polarization vectors, λ = t, t, l denotes polarization, and ω kλ = v λ k is the phonon frequency. The spin-phonon Hamiltonian then becomes H s−ph = − 2 kλ i e ik·ri σ i ·G kλ a kλ − a † −kλ ,(7) where G kλ is due to transverse phonons only, λ = t, t, G kλ ≡ ω kλ 2M N ← → g · [k × e kλ ] , G kλ = −G −kλ . (8) In the Heisenberg representation for time-dependent operators, i [d O(t)/dt] = [ O(t), H], one obtains, with account of the commutation relations, [σ iα , σ jβ ] = 2iǫ αβγ σ iγ δ ij and [a kλ , a † k ′ λ ′ ] = δ kk ′ δ λλ ′ , the following coupled equations: σ i = σ i × [ω 0 + kλ e ik·ri G kλ (a kλ − a † −kλ )] (9) a kλ = − (iω kλ + γ kλ ) a kλ + i 2 i e −ik·ri σ i ·G kλ . (10) Due to the linear interaction between σ i and u, Eq. (5) conserves ( i σ i ) 2 for uniform rotations. Consequently, σ i in Eqs. (9) and (10) can be treated as a semiclassical average over distances large compared to the spacing between magnetic atoms but small compared to the phonon wavelength. We are interested in the interaction of such local polarization with a semiclassical phonon field. One can write the Heisenberg operators as σ i+ (t) =σ i+ (t)e −iω0t , a kλ (t) =ã kλ (t)e −iω kλ t ,(11) whereσ i+ (t) andã kλ (t) are slow variables. Keeping only resonant terms in Eqs. (9) and (10), and neglecting the hybridization of phonon and pseudospin modes, one obtains after integration on time a kλ = 1 4 i e −ik·ri σ i+ G kλ,− ω kλ − ω 0 − iγ kλ ,(12) where G kλ,± = G kλ,x ± iG kλ,y . The insertion of a kλ and a † −kλ into Eq. (9) then gives the system of equations: σ i± = ∓iω 0 σ i± − 1 2 σ iz j Γ ij σ j± σ iz = 1 4 j Γ ij (σ j+ σ i− + σ j− σ i+ )(13) with Γ ij = 1 N k e ik·(ri−rj ) V 2 k 2γ k (ω k − ω 0 ) 2 + γ 2 k(14) and V 2 k ≡ N 4 λ=t,t G 2 kλ,x + G 2 kλ,y .(15) For a small sample of size, L λ 0 ∼ v t /ω 0 , containing N ≫ 1 magnetic atoms or molecules (1 ≪ N ≪ N ) that are initially polarized in one direction, one can use e ik·(ri−rj ) ∼ = 1 in Eq. (14). Eqs. (13) then preserve the length of the total polarization, j σ j . The unit vector σ = (1/N )( j σ j ) satisfies the Landau-Lifhitz equation, σ = [σ × ω 0 ] − α [σ× [σ × ω 0 ]] ,(16) where α = N Γ 1 /(2ω 0 ) is the dimensionless damping constant and Γ 1 = N −1 k V 2 k 2πδ (ω k − ω 0 ) is the one-atom spin-phonon rate. Replacing N −1 k . . . by v 0 d 3 k/(2π) 3 . . . (v 0 being the unit cell volume) and writing V 2 k with the help of Eqs. (4), (8), and (15) as V 2 k = S 2 8 ∆ 2 M v 2 t ω 3 k ω 2 0 1 − k 2 z k 2 ,(17) one obtains for the one-atom spin-phonon rate [2]: Γ 1 = S 2 12π ∆ 2 M v 2 t ω 3 0 ω 3 D = S 2 12π ∆ 2 ω 3 0 ρv 5 t ,(18) where ω D ≡ v t /v 1/3 0 is the Debye frequency for the tranverse phonons and ρ = M/v 0 is the mass density of the crystal. It is easy to check that Γ SR = N Γ 1 is the rate of the relaxation of σ z in Eq. (16), that is, the longitudinal relaxation rate. The transverse relaxation rate for σ ± is Γ SR /2. This is Dicke superradiance [3] with the rate exceeding the one-atom spin-phonon rate by a large factor N . Eq. (16), among other situations, describes the case when all pseudospins are initially prepared in the excited state, σ z = −1, and then relax collectively towards the ground state, σ z = 1, by emitting coherent phonons of frequency ω 0 . Because V 2 k of Eq. (17) is proportional to sin 2 θ = 1 − k 2 z /k 2 , the angular distrubution of the emitted phonons is maximal in the direction perpendicular to the quantization axis z. For large samples, λ L, the phase of the emitted phonons is no longer constant throughout the sample, and one should also study inhomogeneous solutions of Eqs. (13). For that purpose it is convenient to switch to the Fourier harmonics σ i± = N −1 q e ±i q·r i σ q± and σ iz = N −1 q e i q·r i σ qz . When the paramagnetic body is a small part of a large solid matrix, the phonon modes are practically continuous in the k-space while q is quantized. The first of Eqs. (13) then becomeṡ σ q± = ∓iω 0 σ q± − 1 2 q ′′ Γ(q ′′ )σ [±(q−q ′′ )],z σ q ′′ ± ,(19) where Γ(q) = 1 N k V 2 k |F k−q | 2 γ k (ω k − ω 0 ) 2 + γ 2 k ,(20) and F k = N −1 i e ik·ri ∼ = V −1 dre ik·r . For a rectangular body of dimensions L α (α = x, y, z) it gives F k = α sin(k α L α /2) k α L α /2 (21) whereas for a sphere of radius R one obtains F k = f (x) ≡ 3x −3 (sin x − x cos x) with x = kR. Superradiance corresponds to the q = 0 mode in Eq. (19). For γ k → 0 it occurs at a rate Γ SR = N v 0 d 3 k (2π) 3 V 2 k |F k | 2 2πδ (ω k − ω 0 ) = N Γ 1 V 2 k0 |F k0 | 2 V 2 k0 ,(22) where . . . denotes the average over the directions of k 0 that satisfy ω k0 = ω 0 . For k α L α ≪ 1 one can replace F k0 by 1, and the second of Eqs. (22) reduces to the Dicke rate, Γ SR = N Γ 1 . In the general case, the enhancement of the relaxation rate due to superradiance is given by Γ SR Γ 1 = n M k 3 0 Υ, n M = N V .(23) where n M is concentration of magnetic atoms and Υ = k 3 0 V V 2 k0 |F k0 | 2 V 2 k0(24) depends on the wavelength and on the shape of the magnetic body. Since the maximal value of Υ is of order 1, the maximal enhancement is always given by the dimensionless ratio n M /k 3 0 . For a sphere one obtains Υ = (4π/3) (k 0 R) 3 f 2 (k 0 R). The destructive interference, Υ = 0, occurs at k 0 R ∼ = πn, where n is an integer. At k 0 R ≫ 1, Υ(k 0 R) ∼ 1/(k 0 R). That is, Γ SR goes down to Γ 1 only at k 0 R ∼ n M /k 3 0 , that can be very large. For a box of dimensions L α = b α L, the geometrical factor Υ must be computed numerically. For a cube of size L, Υ(k 0 L) is qualitatively similar to that of the sphere. It almost vanishes at k 0 L = 2πn, which corresponds to the destructive interference of phonons emitted perpendicular to the cube faces. The choice of L x = L y causes a more complicated pattern shown in Fig. 1a. An interesting observation is that the increase of the z dimension of the body beyond the wavelength of the radiation does not lead to the decrease of the superradiant rate. The latter is the consequence of the suppression of phonons emitted in that direction, Eq. (17). For a sample of a prism shape with k 0 L x ∼ k 0 L y ≪ 1 and arbitrary k 0 L z , one obtains Υ = (k 0 L x ) (k 0 L y )Υ (k 0 L z ), whereΥ (u) ∼ = u at u ≡ k 0 L z ≪ 1 andΥ (u) ∼ = 3π/2 at u ≫ 1, see Fig. 1b. Another interesting case is a flat sample with k 0 L x ≪ 1, b y ∼ b z ∼ 1 and arbitrary u = k 0 L. In this case one can write Υ = (k 0 L x )Υ (u, b y , b z ), wherẽ Υ ∼ = u 2 b y b z at u ≪ 1 andΥ ∼ = 3π at u ≫ 1, Fig. 1b. The latter is a thin-film limit for which Υ depends neither on b y,z nor even on the shape of the film area. The dependence of Υ on the thickness of such yz-film is 10 1: a -Υ for rectangular geometry; b -Υ for the film and prism geometries. Lα = bαL, α = x, y, z. Υ = 6π 1 − cos (k 0 L x ) k 0 L x .(25)∼ k 0 L x , k 0 L y <<1; b z =1 (Prism) k 0 L z <<1; b x =b y =1 b k 0 L x <<1; b y =b z =1 U k 0 L FIG. In this geometry the phonons are emitted in the direction x perpendicular to the film. The emission rate can be as large as Γ SR,max ≈ 13.7 n M /k 3 0 Γ 1 at k 0 L x ≈ 2.33, while the intensity of sound is proportional to the film surface. One should notice the difference from the case of the xy-film, Fig. 1b, for which the superradiant rate decreases with increasing the dimensions of the film surface beyond the phonon wavelength. Let us now consider the relaxation of the non-zero σ q harmonics in macroscopic samples for a particular case of the maximal inversion, σ z ∼ = −1. At k 0 L ≫ 1, because of the factor |F k−q | 2 in Eq. (20), the phonon momenta k that contribute to Eq. (20) must be very close to q. Under the above conditions, Eq. (19) simplifies tȯ σ q± = ∓iω 0 σ q± + 1 2 Γ(q)σ q± ,(26) where Γ(q) is now the increment rate given by Γ(q) = V 2 q 2γ q (ω q − ω 0 ) 2 + γ 2 q .(27) The highest rate corresponds not to q = 0, as for the superradiance, but to q that provides the minimal value of ω q −ω 0 determined by the previously neglected hybridization of the spin and phonon modes, (∆ω) min ∼ V k0 . This is the laser mode that grows at a rate Γ L = V 2 k0, max γ k0 V 2 k0, max + γ 2 k0 , V 2 k0, max = S 2 8 ∆ 2 ω 0 M v 2 t ,(28) where V 2 k0, max is the maximal value of V 2 k0 achieved for k perpendicular to the z axis. The maximal value of Γ L on γ k0 is Γ L,max = V k0, max , and the enhancement of the spin-phonon relaxation is very large: Γ L,max Γ 1 = 3 √ 2 S ω D ω 0 3 ω 0 ∆ 1/2 M v 2 t ∆ 1/2 . (29) At V k0, max ≪ γ k0 , Eq. (28) reduces to the well-known laser formula, Γ L = V 2 k0, max /γ k0 . As in optical lasers, the increment rate Γ L must compete with the dephasing rate Γ 2 due to, e.g., inhomogeneous broadening of ∆. The laser generation requires V 2 k0,max > γ k0 Γ 2 . It will be the preferred decay mode of the inverted population of spin states if ω 0 = √ W 2 + ∆ 2 / coincides with one of the frequencies of the quantized phonon modes of the crystal. For a small crystal, this condition can be satisfied only at certain values of W and ∆. This can be achieved by continuously changing longitudinal or transverse magnetic field. Since the distance between these resonant field values is inversely proportional to the size of the body, the resonance condition, at a finite γ k0 , must be satisfied at any field for a body of size L v t /γ k0 . Our results on phonon superradiance alter the existing theory of EPR in nanomagnets at low temperature. They show that one cannot properly account for the width of the EPR by simply computing one-phonon processes for isolated magnetic atoms. When the wavelength of the phonons, λ 0 = 2π/k 0 , is greater than the distance between magnetic atoms, the collective processes come into play, greatly enhancing the rates. Observation of EPR lines requires k B T ω 0 , which at T 1 K translates into λ 0 50 nm. Thus, in the kelvin temperature range, our results on superradiance are applicable to nanomagnets of size below 50 nm and to thin films of thickness less than 50 nm. Larger samples should exhibit superradiance at lower temperatures. The direct study of phonon superradiance requires inverted population of spin states. In conventional paramagnets this can be achieved in a pulsed magnetic field. In microscrystals of molecular nanomagnets the inverted population of spin states is easy to achieve even in a slowly varying field [4]. Note that in crystals of small size the phonon superradiance always wins over the electromagnetic superradiance [5]. The existence of the laser phonon mode may be consistent with the experimental evidence [6,7,8,9] that the low-temperature spin-phonon rates in molecular nanomagnets are few orders of magnitude higher than the computed one-molecule spin-phonon rates [10,11]. In molecular magnets the two-level system is formed every time the external magnetic field, B, produces a resonance between the states described by the magnetic quantum numbers m and m ′ . For the (m, m ′ ) resonance, S in all our formulas must be replaced by |m − m ′ |/2, while W = |m − m ′ |gµ B B. Consider, e.g., a typical field-sweep experiment in Fe 8 , when the field crosses the (10, −8) resonance and is 0.1T past the resonance. Taking ∆ ≃ 10 −4 K [12], M ≃ 5 × 10 −21 g, and v t ≃ 10 5 cm/s, one obtains: ω 0 ≃ 10 11 s −1 , Γ 1 ≃ 10 −5 s −1 , but Γ L,max ≃ 10 4 s −1 , which would provide the m = 10 ⇒ m ′ = −8 spinphonon relaxation in less than 1-ms time. This may explain spin-phonon avalanches observed in sufficiently large crystals of molecular magnets [13,14]. The laser rate Γ L = V 2 k0, max /γ k0 has a distinctive ω 0 dependence that should be seen in experiment. At low temperature γ k0 ∝ ω 4 0 due to the scattering of phonons by the inhomogeneities of the crystal [15], while V 2 k0, max ∝ ω 0 . Consequently, Γ L ∝ ω −3 0 , which is opposite to the ω 0 dependence of the one-molecule spin-phonon rate, Γ 1 ∝ ω 3 0 . This work has been supported by the NSF Grant No. EIA-0310517. PACS numbers: 75.50.Xx, 76.30.-v, 43.35.+d E G See, A Abragam, B Bleaney, Electron Paramagnetic Resonance of Transition Ions. OxfordClarendon Pressand references thereinSee, e.g., A. Abragam and B. Bleaney, Electron Param- agnetic Resonance of Transition Ions (Clarendon Press, Oxford, 1970) and references therein. . E M Chudnovsky, Phys. Rev. Lett. 723433E. M. Chudnovsky, Phys. Rev. Lett. 72, 3433 (1994); . E M Chudnovsky, X Martinez-Hidalgo, Phys. Rev. 6654412E. M. Chudnovsky and X. Martinez-Hidalgo, Phys. Rev. B66, 054412 (2002); . E M Chudnovsky, Phys. Rev. Lett. 92120405E. M. Chudnovsky, Phys. Rev. Lett. 92, 120405 (2004). . R H Dicke, Phys. Rev. 9399R. H. Dicke, Phys. Rev. 93 99 (1954). . R Sessoli, D Gatteschi, A Caneschi, M A Novak, Nature. 365141R. Sessoli, D. Gatteschi, A. Caneschi, and M. A. Novak, Nature (London) 365, 141 (1993). . E M Chudnovsky, D A Garanin, Phys. Rev. Lett. 89157201E. M. Chudnovsky and D. A. Garanin Phys. Rev. Lett. 89, 157201 (2002). . J R Friedman, M P Sarachik, J Tejada, R Ziolo, Phys. Rev. Lett. 763830J. R. Friedman, M. P. Sarachik, J. Tejada, and R. Ziolo, Phys. Rev. Lett. 76, 3830 (1996). . J R Friedman, M P Sarachik, R Ziolo, Phys. Rev. 5814729J. R. Friedman, M. P. Sarachik, and R. Ziolo, Phys. Rev. B58, R14729 (1998). . Kyungwha Park, M A Novotny, N S Dalal, S Hill, P A Rikvold, J. Appl. Phys. 917167Kyungwha Park, M. A. Novotny, N. S. Dalal, S. Hill, and P. A. Rikvold, J. Appl. Phys. 91, 7167 (2002). . M Dressel, B Gorshunov, K Rajagopal, S Vontragool, A A Mukhin, Phys. Rev. 6760405M. Dressel, B. Gorshunov, K. Rajagopal, S. Vontragool, and A. A. Mukhin, Phys. Rev. B67, 060405 (2003). . J Villain, F Hartmann-Boutron, R Sessoli, A Rettori, Europhys. Lett. 27159J. Villain, F. Hartmann-Boutron, R. Sessoli, and A. Ret- tori, Europhys. Lett. 27, 159 (1994). . D A Garanin, E M Chudnovsky, Phys. Rev. 5611102D. A. Garanin and E. M. Chudnovsky, Phys. Rev. B56, 11102 (1997). . W Wernsdorfer, R Sessoli, Science. 284133W. Wernsdorfer and R. Sessoli, Science 284, 133 (1999). C Paulsen, J.-G Park, in Quantum Tunneling of Magnetization -QTM'94. L. Gunther and B. BarbaraDordrecht, NetherlandsKluwerC. Paulsen and J.-G. Park, pp. 189-207 in Quantum Tun- neling of Magnetization -QTM'94, Edited by L. Gunther and B. Barbara (Kluwer, Dordrecht, Netherlands, 1995). . E Barco, J M Hernandez, M Sales, J Tejada, H Rakoto, J M Broto, E M Chudnovsky, Phys. Rev. 6011898E. del Barco, J. M. Hernandez, M. Sales, J. Tejada, H. Rakoto, J. M. Broto, and E. M. Chudnovsky, Phys. Rev. B60, 11898 (1999). . E G See, D A Garanin, Ann. Phys. (N.Y.). 218293and references thereinSee, e.g., D. A. Garanin, Ann. Phys. (N.Y.) 218, 293 (1992), and references therein.
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[ "Can the clocks tick together despite the noise? Stochastic simulations and analysis *", "Can the clocks tick together despite the noise? Stochastic simulations and analysis *" ]
[ "Stéphanie M C Abo ", "José A Carrillo ", "Anita T Layton " ]
[]
[]
The suprachiasmatic nucleus (SCN), also known as the circadian master clock, consists of a large population of oscillator neurons. Together, these neurons produce a coherent signal that drives the body's circadian rhythms. What properties of the cell-to-cell communication allow the synchronization of these neurons, despite a wide range of environmental challenges such as fluctuations in photoperiods? To answer that question, we present a mean-field description of globally coupled neurons modeled as Goodwin oscillators with standard Gaussian noise. Provided that the initial conditions of all neurons are independent and identically distributed, any finite number of neurons becomes independent and has the same probability distribution in the mean-field limit, a phenomenon called propagation of chaos. This probability distribution is a solution to a Vlasov-Fokker-Planck type equation, which can be obtained from the stochastic particle model. We study, using the macroscopic description, how the interaction between external noise and intercellular coupling affects the dynamics of the collective rhythm, and we provide a numerical description of the bifurcations resulting from the noise-induced transitions. Our numerical simulations show a noise-induced rhythm generation at low noise intensities, while the SCN clock is arrhythmic in the high noise setting. Notably, coupling induces resonance-like behavior at low noise intensities, and varying coupling strength can cause period locking and variance dissipation even in the presence of noise.
null
[ "https://export.arxiv.org/pdf/2202.12300v2.pdf" ]
247,154,875
2202.12300
5acf38bf8852e648f46ac8b2977e93c1f2947f66
Can the clocks tick together despite the noise? Stochastic simulations and analysis * Stéphanie M C Abo José A Carrillo Anita T Layton Can the clocks tick together despite the noise? Stochastic simulations and analysis * Mean-field limitDiffusionSCNSynchronisation MSC codes 92B2582C3160H10 The suprachiasmatic nucleus (SCN), also known as the circadian master clock, consists of a large population of oscillator neurons. Together, these neurons produce a coherent signal that drives the body's circadian rhythms. What properties of the cell-to-cell communication allow the synchronization of these neurons, despite a wide range of environmental challenges such as fluctuations in photoperiods? To answer that question, we present a mean-field description of globally coupled neurons modeled as Goodwin oscillators with standard Gaussian noise. Provided that the initial conditions of all neurons are independent and identically distributed, any finite number of neurons becomes independent and has the same probability distribution in the mean-field limit, a phenomenon called propagation of chaos. This probability distribution is a solution to a Vlasov-Fokker-Planck type equation, which can be obtained from the stochastic particle model. We study, using the macroscopic description, how the interaction between external noise and intercellular coupling affects the dynamics of the collective rhythm, and we provide a numerical description of the bifurcations resulting from the noise-induced transitions. Our numerical simulations show a noise-induced rhythm generation at low noise intensities, while the SCN clock is arrhythmic in the high noise setting. Notably, coupling induces resonance-like behavior at low noise intensities, and varying coupling strength can cause period locking and variance dissipation even in the presence of noise. 1. Introduction. The suprachiasmatic nucleus (SCN) in the brain serves as the central clock in mammals and regulates most circadian rhythms in the body [38,73]. The SCN is remarkable -it not only synchronizes the biological rhythms to the external light-dark cycle [1], but also generates robust rhythmic outputs with an endogenous period of around 24 h in constant darkness [18,55].The specific mechanism responsible for this behaviour continues to be the subject of numerous experimental and theoretical studies. The rhythmic output emanates from a regulatory circuit with a negative feedback loop. We refer to the reviews [70,41] for a description of the architecture of the SCN clock. Although single neurons produce autonomous oscillations, the emergence of global and robust oscillations of the SCN activity requires the synchronization of neural cells [27]. Oscillations at the global level arise from the interaction, also called coupling, between SCN neurons. In this work, we study how a population of SCN neurons manages to synchronize and remain synchronized despite external perturbations. Our focus is on the effect of coupling strength and external noise on synchronization dynamics. Experimental studies have shown that cell-to-cell coupling in the SCN is carried out in part by neurotransmitters [48,41]. Vasoactive intestinal polypeptide (VIP), arginine vasopressin (AVP) and gamma-aminobutyric acid (GABA) are examples of neurotransmitters which play a role in the coupling [31]. The SCN is divided into two hemispheres, each of which contains two groups of neurons: a dorsomedial shell (DM) and a ventrolateral (VL) core. These two sets of neurons differ by their light sensitivity, the neurotransmitters they produce, hence their coupling properties. DM cells mainly express AVP, whereas VL neurons express VIP [2,36]. Yet, all SCN neurons express the neurotransmitter GABA [64]. In addition to coupling, SCN function is influenced by stochastic noise, which includes exogenous and endogenous cellular noise [79]. Exogenous noise results from changes in the environment [55,79], such as fluctuations in light signals, and has been shown to play an important role in the amplitudes of neural oscillators and the entrainment to a new environmental cycle [34,22]. The endogenous noise is caused by low molecular counts of the mRNA and protein species involved [79]. A number of mathematical models of coupled oscillators have been developed to study the SCN properties such synchrony, the ensemble period and the entrainment ability of the SCN [48,26,68,72,9,32,58,33]. Most of these models are in the form of coupled ordinary differential equations, and are therefore deterministic. Some recent modeling and experimental studies, however, investigate the influence of noise [34,55,47] on the circadian clock by means of stochastic differential equations or experimental analysis of stochastic rhythms. All of these models based on the particle-like description of a set of interacting neurons are called individual-based models (IBM), and often used in animal swarming [62,16]. The topologies often considered are all-to-all coupling between the neurons [48,26,68] and small world networks [88,78]. For a large number of interacting agents, the collective motion in the system can be studied through macroscopic descriptions based on the evolution of a density of individuals. These models are known as continuum models, and the scaling limit is called the mean-field limit [16,10,42]. These continuum models are useful in reducing IBMs into an effective one-body problem: the particle probability density [10]. Naturally, noise at the level of the IBMs which represent the SCN network is essential since the neuronal activity is not totally deterministic. The randomness should be reflected in the macroscopic description. As pointed out in [15], stochastic IBMs lead to Fokker-Planck type equations in the mean-field limit for second order models. The proof of this stochastic mean-field limit relies on standard hypotheses: global Lipschitz continuity and linear growth condition of the drift and diffusion coefficients, and the Lipschitz continuity of the interaction function [60,61,83]. In the present article, we consider a stochastic system of interacting SCN neurons in a diffusive scaling and study the effects of external noise, as the network size approaches infinity, on SCN properties: robust oscillation amplitude and period. We also investigate the effect of noise on bifurcation boundaries. SCN neurons are characterized by small size and high density [9], and all express GABA [64]. We assume, based on this information, that intercellular coupling is carried out by chemical signals released by each cell and that spatial transmission is fast in comparison to the time scale of the oscillations (24h). We derive the mean-field equation for a system of globally-coupled Goodwin-type neurons with noise. The Goodwin model is commonly employed for circadian oscillators because it describes a biological process with a negative feedback loop -one of the key circadian clock regulation mechanisms [68]. Many studies have considered the SCN as a network where neurons are globally connected [46,48,26,58,32], but other network topologies for coupling oscillators have also been studied: Newman-Watts (NW) small-world networks [36,87], regular networks [9,50], random networks [35] and scale-free networks [35,36]. To the best of our knowledge, no study has discussed the influence of external noise on the circadian clock through mean-field equations. We present numerical results on the relation between amplitude of circadian oscillations and coupling strength. We also discuss the effect of noise on bifurcation boundaries. The question arises as to whether fluctuations in the noise level can influence bifurcation boundaries and therefore influence the robustness of circadian oscillations with respect to external noise. Moreover, synchronization will be understood to mean the dissipation of the empirical variance. This approach does not rely on the stability properties of individual neurons nor on the existence of limiting oscillatory behaviors. [11]. The work is organized as follows: in Section 2, the mean-field model is introduced to describe a network of coupled SCN neurons with noise. Simulation results about the dependence of the rhythms on the coupling strength and noise intensity are discussed in Section 3. Then, we assess the accuracy of our numerical scheme in Section 4. The conclusions and discussion are presented in Section 5. A complete description of the numerical scheme is available in Appendix C. A minimal SCN model and its mean-field description. 2.1. Model description. In this paper, we propose a mathematical model for describing the collective activity of SCN neurons. The core architecture in mammals of the circadian clock consists of two feedback loops that interact to generate biochemical oscillations with a period of nearly 24 h [74]. The primary feedback loop is driven by clock proteins CLOCK and BMAL1. The proteins dimerize to create the CLOCK-BMAL1 complex which initiates the transcription of the target period (PER) and cryptochrome (CRY) genes. Negative feedback is achieved through PER-CRY heterodimers that repress their own transcription after delays due to cellular processes, such as transcription, translation, and nuclear transport [3]. In a secondary loop, CLOCK-BMAL1 proteins activate the transcription of Rev-Erbα. After being translated into proteins, Rev-Erbα downregulates Bmal1 transcription, thus completing the loop [74,46]. The Goodwin model, a negative feedback oscillator with variables X, Y and Z, can be used to represent these two regulatory loops; see (2.1-2.2) and Fig 1. In general, the mechanism behind biological oscillators consists of delayed negative feedback loops. [84]. The Goodwin model has been widely studied theoretically [90,3,25] and applied to various biological systems, such as circadian clocks [48,26,1,68] or enzymatic regulation [29]. The temporal evolution of a single Goodwin-type neuron is governed by the following equations: (2.1)Ẋ = f (Z) − k 2 X,Ẏ = k 3 X − k 4 Y,Ż = k 5 Y − k 6 Z, where (2.2) f (Z) = k 1 K n i K n i + Z n . In this model, X denotes the mRNA concentration of a certain clock gene, Y is the matching protein, and Z is a transcriptional inhibitor in the nuclear form or the phosphorylated form of the protein. The inhibition term is described by a nonlinear and hyperbolic function (i.e. Hill function), f (Z). All other terms are linear. The Hill function is parametrized by a Hill coefficient n characterizing the response steepness, and an inhibition threshold K i that describes the concentration of inhibitor that halves the production rate, i.e., half-maximal repression occurs when Z = K i . In many organisms, the Hill function has been employed to characterize transcriptional repression: Neurospora [76,54], Drosophila [52,76,85] and mammals [53,74,26,48]. Hill functions are often employed to describe cooperative binding of repressors to the gene promotor in transcription [24] or repression based on multisite phosphorylation [25]. The latter is a more realistic mechanism, especially because a large Hill exponent is required for oscillations in the Goodwin model (n > 8). The recent work of Cao et al. [12], which builds upon [75,65], provides some evidence for repression based on multisite phosphorylation in mammals. The authors show that removal of CLOCK-BMAL1 involves phosphorylation (hyperphosphorylation) of CLOCK, which is accomplished by CK1δ when CRY and PER deliver CK1δ to the CLOCK-BMAL1 complex in the nucleus of cells. A different transcriptional repression mechanism based on protein sequestration has been proposed to describe the negative feedback underlying circadian oscillators. See [44,45,43,17] for details. The various rate constants parametrize transcription (k 1 , k 3 ), degradation (k 2 , k 4 , k 6 ), and nuclear import (k 5 ). Note that all reaction rates are positive. Concentrations X, Y, Z and K i have units nM. Rate constants have units h −1 , except for k 1 which has units nM h −1 . Depending on parameter values, the model can produce limit cycle oscillations. Following the approach in [90], we reformulate the equations in (2.1) in dimensionless form. Assuming equal degradation rates (k 2 = k 4 = k 6 ), we introduce the new variables: x = k 3 k 5 k 2 2 K i X, y = k 5 k 2 K i Y, z = Z K i , t = k 2 T τ , with τ chosen to make the intrinsic period of the oscillator 23.5. We obtain, (2.3) dx dt = α 1 + z n − x, dy dt = x − y, dz dt = y − z, where (2.4) α ≡ k 1 k 3 k 5 k 3 2 K i is the only parameter for a given n. Fig 2 represents the bifurcation and stability diagrams for system (2.3). At the critical value α H , the system transitions from a stable steady state to an unstable steady state (via a Hopf bifurcation), and a periodic solution arises (Fig 2a). The steady state is stable if α < α H and unstable otherwise. To obtain limit-cycle oscillations, the Hill coefficient must be larger than 8 (Fig 2b). We now consider a population of N identical neurons. Let x i (t), y i (t), z i (t) ∈ R denote the concentration of mRNA, protein and inhibitor protein of neuron i at time t, respectively. To produce a functional SCN network, there must be reciprocal signalling between neurons. We k 1 = 1nM· h −1 , k 3 = k 5 = 1h −1 , k 2 = k 4 = k 6 = 0.1h −1 , K i = 1nM , and n = 10 in (2.1). The oscillation period is about 40h. examine our network under all-to-all coupling conditions. Each neuron in the group adjusts its production of mRNA (x) by averaging with all the others. We obtain, (2.5)      dx i dt = α 1+z n i − x i + K(x − x i ) dy i dt = x i − y i dz i dt = y i − z i Here, and throughout this paper, the coupling parameter K is assumed to be the same for all oscillators.x(t) is the average value of all individual variables x i (t) at time t: (2.6)x(t) = 1 N N i=1 x i (t) 2.2. Stochastic extension and mean-field limit. Our goal is to analyze a system of the type (2.5) with Gaussian noise. Biological clocks, although noisy at the microscopic level due to both external noise (e.g., fluctuations in photoperiods) and inherent stochasticity (e.g., coupling between cells), can be relatively precise at the macroscopic level [4,39]. We represent stochasticity via additive white noise acting through the first variable x. Specifically, we will consider a large network of N interacting R 3 -valued processes (x i (t), y i (t), z i (t)) t≥0 with Hill exponent n > 8 is required for oscillations. If n < 8, the steady state is a stable focus for all values of α. 1 ≤ i ≤ N solution of (2.7)          dx i (t) = α 1+z i (t) n − x i (t) + K N N j=1 H(x i (t) − x j (t)) dt + √ 2D dW i (t) dy i (t) = x i (t) − y i (t) dt dz i (t) = y i (t) − z i (t) dt where H(ω) := −ω and with independent and identically distributed initial data (x 0 i , y 0 i , z 0 i ), 1 ≤ i ≤ N . The processes (W i (t)) t≥0 with 1 ≤ i ≤ N are independent Brownian motions in R and the noise intensity is √ 2D, with D > 0. To our knowledge, the stochastic mean-field limit description of a system of coupled Goodwin-type neurons has not yet been considered. All neurons have the same distribution on R 3 at time t due to the symmetry of the initial configuration and of the evolution [10]. For any t > 0 the neurons become correlated due to the coupling term K N N j=1 H(x j − x i ) in the evolution, though they are independent at t = 0. However, given the order 1/N of the interaction term, it seems reasonable that any fixed number k of these interacting neurons become less correlated as N gets large. This property is called propagation of chaos [10]. The following assumptions hold for the stochastic model: 1) global Lipschitz continuity of the drift and diffusion terms; 2) linear growth condition of the drift and diffusion terms; 3) Lipschitz continuity of the coupling function. These assumptions imply that the system of stochastic differential equations (2.7) is well-posed. It follows from the general theory of Sznitman [83] (see also the more recent [59,10]) that our N interacting processes (x i (t), y i (t), z i (t)) t≥0 respectively behave as N → ∞ like the processes (x i (t),ỹ i (t),z i (t)) t≥0 , solutions of the kinetic McKean-Vlasov type processes on R 3 : (2.8)                dx i (t) = ψ(ρ) dt + √ 2D dW i (t) dỹ i (t) = (x i (t) −ỹ i (t)) dt dz i (t) = (ỹ i (t) −z i (t)) dt x 0 i ,ỹ 0 i ,z 0 i = x 0 i , y 0 i , z 0 i , ρ = law x i (t),ỹ i (t),z i (t) ψ ρ x i ,ỹ i ,z i , t = α 1+z i (t) n −x i (t) + K H ρ(x i ,ỹ i ,z i , t) The Brownian motions (W i (t)) t≥0 in (2.8) are those governing the evolution of (x i (t), y i (t), z i (t)) t≥0 . The processes (x i (t),ỹ i (t),z i (t)) t≥0 with i ≥ 1 are independent since the initial conditions and governing Brownian motions are independent. Notice that they are identically distributed and, by the Itô formula, their common law ρ at time t should evolve according to the kinetic McKean-Vlasov equation (2.9) ∂ρ ∂t = D∂ 2 x ρ − ∂ x ξ(ρ)ρ − ∂ y (x − y)ρ − ∂ z (y − z)ρ where (2.10) ξ ρ x, y, z, t = α 1 + z n − x + K H ρ with (2.11) H ρ(x, y, z, t) = R 3d H(x − w)ρ(w, y, z, t) dw dy dz . Since (2.8) models the evolution of concentrations (x i (t),ỹ i (t),z i (t)) t≥0 we constrained the solutions of the network when performing numerical simulations to ensure that they remained in a smooth positive domain in R 3 + . In particular, we used the compact support [0, 2] × [0, 2] × [0, 2]. The boundaries of the domain are instantaneously reflecting in an oblique direction. See [23,37] and the references therein for a detailed discussion of Euler schemes for reflected stochastic differential equations. In this case, the general theory of Snitzman [82,57,83] still applies, but we recover (2.9) with no-flux boundary conditions. 3. Numerical results. We focus on the dynamic evolution of solutions in the mean field scaling of the SCN network with noise. Our model (2.9) is a nonlocal nonlinear transport equation with no-flux boundary conditions. Such characteristics of the mean-field equation make it difficult to conduct a theoretical study using center manifold or bifurcation theory [19,20]. Instead, we investigate the nature of the bifurcations numerically and compare the solution to the mean-field equation with that of the finite SCN network. We consider two main parameters in our analysis, namely the strength of the coupling K and the noise level D. In all our simulations, we refer to the marginal probability densities for x, y and z as ρ 1 , ρ 2 , and ρ 3 , respectively. A noise-free network analysis is outside the scope of this article, but we refer the reader to [90,17] for details on the stability of the noise-free system and associated bifurcation diagrams. In the case of the noise free network with identical Goodwin cells (2.5), the linear stability of steady states is tractable and bifurcation diagrams can be determined by solving for the steady states of the amplitude equations [91]. Moreover, for oscillators with weak coupling the phase-locked states could be studied using weakly coupled oscillator theory, where the network can be reduced to its phase model description [21,51,67]. The stability of the synchronous state for strong coupling can be studied using the master stability function, which allows to calculate the stability as determined by a particular choice of stability measure, like Lyapunov or Floquet exponents [71,80]. 3.1. Coupling strength and synchronization. The SCN coordinates physiological cycles throughout the body with incredible precision [40]. Nevertheless, local oscillations at the level of individual neurons can be substantially different from global network-wide oscillations [40,77]. In fact, defining characteristics of circadian rhythms such as period, large amplitude, accuracy and synchrony arise due to coupling [77,89,63]. In this section, we investigate the effect of coupling in the overall dynamics of the SCN. We perform a numerical bifurcation analysis by varying the parameter K, which quantifies the strength of the chemical signal that is transmitted to oscillator cells. A value of K < 1 signifies a decay of the chemical signal before it reaches the target cell, while K = 1 represents the perfect transduction of the chemical signal to the target cell. In particular, K = 0 implies an uncoupled network where oscillations are localized in individual neuronal cells. Following the approach done in [19,6], a stationary solution of (2.9) characterises asynchronous activity, a state in which neurons exhibit out-of-phase oscillations. Synchronized activity refers to a state where the mean-field is periodic in time. In Fig 3 we show a bifurcation diagram of the spatial averages in x, y, and z as a function of the coupling strength K. E[x], E[y] and E[z] are calculated from the solution to the mean-field equation (2.9). We call E[·] the average over the values of a given variable across the space domain. We assume a low level of noise with D = 0.01. Fig 3a shows the transition from a stable regime without oscillations to a regime of sustained oscillations, which correspond to the progression of solutions toward a periodic orbit [28]. For values of K less than 0.3, the mean-field solution shows damped oscillations that eventually result in an invariant distribution. This indicates that the network of SCN neurons is out of sync. However, past the critical value K H ≈ 0.3, synchronized activity emerges within the network as shown by a periodic solution to (2.9). Figs 3a and 3b suggest the existence of a bifurcation of the asynchronous state. At this bifurcation emerges a limit cycle corresponding to a stable and robust sinusoidal oscillation. Mathematically, the expansion of amplitude as the periodic orbit moves away from the bifurcation boundary (Fig 3a) and the relatively constant period of the oscillations (Fig 3b) are characteristic of a supercritical Hopf bifurcation. Moreover, the amplitude grows as the coupling strength increases, and this phenomena is especially prominent near the bifurcation boundary (Fig 3a). Previous studies have shown that coupling can induce amplitude expansion near the Hopf bifurcation [1,77,5]. This so-called "resonance" may be enhanced by synchronizing factors, here represented by the coupling strength K. Examining the temporal evolution of the empirical variance offers a complementary perspective on the role of coupling. Synchronization can be understood as the dissipation of the empirical variance for the spatial averages of x, y and z computed from the solution of (2.9) as time goes by. In Fig 4 and . The solution to the mean-field equation approaches a steady state when K = 0 as shown by damped oscillations in E[x], but looses its stability to rapidly settle into a periodic orbit when K = 1. The other variables y and z, which are not shown here, are qualitatively similar with x. It should be noted that regardless of initial conditions, the system follows the same trajectories (data not presented). For each scenario, the time evolution of the variance for all three variables is recorded in Fig 5. Assuming a low level of noise (D = 0.01), we observe when K = 0 an asynchronous state represented by an empirical variance of constant order in time and which is greater than the input noise level. However, for perfect coupling K = 1, the empirical variance dissipates to a minimum equal to the magnitude of the external noise, and the global output is rhythmic (see Fig 4B and Fig 5B). In the ideal case of perfect coupling, the variance in y and z decreases to about zero, whereas the variance in x decreases to about 0.01 (the level of input noise D). Our numerical experiment suggests the following about the qualitative behavior of the coupled SCN network in terms of D: for D > 0 the neuron trajectories are enclosed in a band whose width rises with D, whereas a noise-free network will ultimately reach perfect synchronisation where the empirical variance decreases to zero. The direct consequence of these observations is that the threshold K H is itself a function of D. To see this, consider Fig 6 which shows the joint probability distribution between x and z at t = 600h, ρ(t = 600, x, z), for different values of K. We choose a long integration time in order to not be influenced by the initial conditions. As the coupling parameter increases, variance decreases and the distribution ρ(t = 600, x, z) tends to concentrate on a delta function in the z dimension and to be distributed only along the spatial dimension x due to noise. Overall, studying the effects of coupling on the solution to the mean-field limit gave the following numerical predictions: increasing coupling strength can lead to period locking, variance dissipation, and larger amplitudes due to resonance effects. 3.2. Effect of noise on bifurcation boundaries. Using our stochastic mean-field model, we examine how the robustness of the SCN clock is affected by noise. This refers to how the noise intensity in the system affects the distance to a bifurcation point. We proceed by varying the parameter α in (2.9) for different noise levels, and we look for synchronized activity within the network. Robustness is used here to denote the persistence of a certain type of dynamic behavior over a significant range of parameter values. The term "robustness" refers to the persistence of a specific dynamic behavior over a wide range of parameter values. Consider D = 0.01 as an example of low noise setting. As shown in Fig 7, there is in the low noise setting a critical value α H at which the system's stability appears to change from a stable stationary distribution to an oscillatory solution. This result shows that the development and maintenance of a global rhythmic output requires the synchronization of single-cell rhythms [27]. The existence of such invariant distributions characterizes a state We investigate further how increasing the level of noise affects bifurcation boundaries. Fig 9 shows a high-low plot for the peaks and troughs of oscillations of the means in x, y and z as a function of the parameter α for different noise levels. Since it is difficult to numerically estimate exact bifurcation values, we use dotted lines to represent intervals containing exact bifurcation points, which we call α H . We estimate α H ∈ (1.72, 1.73], (1.8, 1.9] or (2.0, 2.1] when the noise intensity is low, moderate or high, respectively. As the noise level increases, the value of α H necessary to obtain sustained circadian oscillations also increases. From Fig 9d, the amplitude of these oscillations gradually decreases as noise intensity increases, These findings indicate that higher noise levels render synchronisation more difficult to achieve. That is because a new and larger value α H becomes the sine qua non condition for oscillations. Given the biological meaning of α in (2.3), its value could increase to recover oscillations if: 1) activation rate k 1 , k 3 , or k 5 increases for x, y or z, respectively; 2) degradation rates decrease for all three clock components (k 2 , k 4 , k 6 where k 2 = k 4 = k 6 ); 3) inhibition of x by z is attenuated (i.e., lower K i ). We argue that noise can affect synchrony-dependent rhythmicity. The noise can affect ensemble properties of oscillators including their coupling and their period of oscillations. In Fig 10, we present solutions to (2.9) which are qualitatively different from the solutions shown in Fig 6: as the intensity of the noise increases, the variance increases and the distribution tends to widen and shorten in all directions, indicating that a wider spread of values is possible and that external noise impairs the synchrony of the system. Although we showed that uniformly coupled networks can robustly synchronize (Fig 6), it can also be concluded that noise weakens synchronization degree and affects the robustness of the system. Fig 11 illustrates that the SCN is able to withstand higher noise levels with increasing coupling strength. However, noise eventually abrogate the oscillation in the SCN (Fig 11b). For instance, when K = 0.6, a noise intensity superior to 0.08 is sufficient to desynchronize the neurons as indicated by the null period in Fig 11a. When D < 0.08, the system remains in synchrony with an ensemble period between 24.5 and 22. 4. Convergence studies. In this section, we report numerical results relating to the spatial accuracy of the numerical method. Our discussion focuses on a mesh convergence study to validate the order of convergence of the scheme in space. If the solution ρ is sufficiently smooth, then the spatial discretization is expected to be second-order accurate (see Appendix C). In default of an analytical solution to our problem, we instead compute relative errors using different grid sizes. More precisely, we compute deviations from the estimated solution on a 3D fine mesh made of 384 3 cells. Fig 12 illustrates the results in L 1 and L ∞ norms. The time step ∆t is determined by the CFL condition derived in equation (C.11) and the spatial step size is uniform in all directions (∆x = ∆y = ∆z). We used zero-flux boundary conditions. The initial probability density function ρ(x, y, z, 0) is Gaussian, Fig 12 that the numerical scheme is at least second-order accurate, as anticipated. Fig 13 shows a qualitative similarity between solutions to the network equations and those obtained by solving the mean-field equation. In particular, we expect that the network and mean-field equation solutions will condense their mass around the periodic orbit [7]. This is shown in Fig 13a where the dynamics evolve to sustained oscillations. Table 1 summarizes all of the parameters involved in the aforementioned simulations. (4.1) ρ(x, y, z, 0) = 1 (2π) 3/2 σ x 0 σ y 0 σ z 0 e −(x−µx 0 ) 2 /(2σ 2 x 0 )−(y−µy 0 ) 2 /(2σ 2 y 0 )−(z−µz 0 ) 2 /(2σ 2 z 0 ) . It appears from Discussion and conclusions. To summarize, we have conducted an analysis of the Goodwin model using the mean-field limit approach from kinetic theory. We developed a minimal yet effective macroscopic model of the SCN circuit level dynamics and investigated the impact of noise on the emerging properties of the SCN -rhythmicity (i.e., synchronisation), amplitude expansion, and ensemble period. We applied a positivity-preserving finite volume scheme developed in [13] for our numerical simulations. Initial conditions Domain Goodwin neuron µ x 0 = 1 x min = 0 α = 1.8 µ y 0 = 0.9 x max = 2 n = 20 µ z 0 = 0.9 y min = 0 K = 0.5 σ 2 x 0 = 0.05 y max = 2 D = 0.005 σ 2 y 0 = 0.02 z min = 0 τ = 6.4885 σ z 0 = 0.01 z max = 2 ∆x = 0.002 ∆y = 0.002 ∆z = 0.002 Table 1: Model parameters. These parameters apply to the validation results presented in Section 4. We presented simulation results indicating that coupling is important in maintaining the synchronization and amplitude expansion characteristics of the SCN, at least in the meanfield limit. Notably, increasing the coupling strength leads to phase transitions. We provided numerical evidences for the existence of Hopf bifurcations, with respect to the coupling parameter, which is synonymous with synchronized activity (Figs 3 and 6). On the one hand, low coupling strengths result in a decrease of the amplitude of the SCN rhythm. In particular, if the coupling strength is less than a certain threshold (K H ), the oscillation amplitude becomes null, meaning that the circadian rhythm is lost due to neuronal oscillators being out of phase with each other. Our findings, on the other hand, show that significant coupling causes resonance effects. This leads to amplitude expansion and the rapid establishment of a coherent evolution implying the dissipation of variance in the system. Moreover, we provided a numerical description of the bifurcations that govern the instabilities caused by noise-induced transitions, i.e., Hopf bifurcations. Our approach allows us to identify where the system of coupled SCN neurons exhibit a stable stationary state (incoherence within the SCN network) or limit cycle oscillations (synchronized activity). We suggest that noise weakens synchrony-dependent rhythmicity and affects the robustness of the system (Fig 10). Robustness to external noise decreases in proportion to the noise level: bifurcation boundaries are pushed forward as the noise level increases, making it more difficult to reach the oscillatory regime (Fig 9). However, in a biological context, rhythmicity could be recovered with higher activation rates (k 1 , k 3 ), lower degradation rates (k 2 , k 4 , k 6 ) or slower inhibition of the mRNA by its inhibitor protein (K i ) in equation (2.4). The repression mechanism used in modelling the negative feedback loop in circadian clocks can affect significantly properties of models, including robustness to perturbations. We use a Hill-type repression function to explain how transcriptional activity decreases as repressor concentration rises (see equation 2.2). Recently, a new mechanism of transcriptional repression based on protein sequestration has been proposed: repressors tightly bind activators to form an inactive 1:1 stoichiometric complex (see [44] for details). In Hill-type (HT) and proteinsequestration (PS) models, Kim and Forger investigated the qualitative differences based on the repression mechanisms [43,44]. According to their analyses, the HT and PS models have different prerequisites for generating rhythms: a large Hill exponent and a 1:1 molar ratio between repressor and activator, respectively. Kim and colleagues [45] also showed that the coupled periods are near the mean period of the SCN when transcriptional repression occurs via protein sequestration, whereas the collective period is farther from the mean if modeled with Hill-type regulation. Apart from the repression mechanism, the models mentioned above are IBMs and differ from ours in that the coupling function is different, cells are heterogeneous in terms of period, and noise is not taken into account. In our mean-field model that uses Hill-type repression, we observe that the collective period is close to the intrinsic period of the cells in the presence of low to moderate noise. This could be explained by our use of a homogeneous network (see Fig 3b and Fig 11b). Moreover, according to Chen et al. [17], there exist coupling strengths c in both HT and PS models such that the collective frequency equals the average frequency of individual cells. For the HT model, such strength c is larger. Despite these differences between HT and PS models, many intercellular coupling properties are shared between the two and some general trends are similar. For instance, the logarithmic sensitivity of the repression function should be greater than 8 at steady state for both models to generate oscillations [43], and increasing coupling strength causes amplitude expansion in both models [43,17]. Our results can be extended when the protein sequestration function is used instead of the Hill function up to a constant in the bifurcation values for homogeneous networks of cells. Further study is needed when heterogeneous oscillators with different periods are coupled. In addition to nonlinearity in the repression function, oscillations require sufficiently long delays in feedback loops. This can be achieved by adding intermediate steps in the ODE formulation or by introducing explicit delays representing the durations of post-translational regulations. Our numerical scheme, first developed in [13], applies when time-independent delays are modeled with noise, white or colored additive and multiplicative. However, numerical challenges may arise from adding explicit time delays. First, a delay may further constrain the stability condition on the time step so that the solver's time step is smaller than its value. Second, delays require storing a history of the function, which can be memory-prohibitive. Not the least is the noise effect, as noise causes the stochastic solution to disperse around the deterministic solution and the empirical variance stabilizes but for large time, with a limit value which increases as spatial regularity decreases and noise intensity increases. Importantly, the major numerical challenges would be due neither to noise nor to delay, but rather to the nature of the equation whose type degenerates at certain points of the domain of definition or at the boundary of this domain. The proposed scheme is able to cope with non-smooth stationary states, different time scales including metastability, as well as concentrations and self-similar behavior induced by singular nonlocal kernels [13]. Lastly, our model has limitations, which should be acknowledged. It has been shown that the SCN is a heterogeneous network, consisting of two groups of neurons that are structurally and functionally different. Namely, the ventralateral part (VL) which receives light information and transmits it to the dorsalmedial part (DM). This second group is only indirectly sensitive to light [69]. Within these regions different neurotransmitters are used for communication between the cells [34]. Network topology, in addition to network heterogeneity, has a substantial influence on the SCN's collective behaviour. In this article, we tested an all-toall linear coupling between neurons, which may not be a realistic architecture for the SCN network. Extensions of our work could include a dual-network representation of the VL-DM architecture, as well as an emphasis on nonlinear cross-regional coupling. Future research could also look at the molecular details of the repression pathway, which could include both phosphorylation and protein sequestration. Appendix A. Convergence of stochastic and mean-field solutions. We present, in this section, supplementary convergence results: the convergence of error between solutions to the stochastic system (2.8) and the mean-field equation (2.9) when the number of neurons tends to infinity. We have used a population of 10,000 Goodwin-type neurons and ran 10,000 Monte Carlo simulations of the network model using the Euler-Maruyama method [8]. Solutions of the network are constrained to remain in a smooth positive domain D. Namely, we simulate the case where the boundary ∂D is instantaneously reflecting in an oblique direction. See [23,37] and the references therein. The most classical way to show convergence is to reason in terms of trajectories and to show that, when the number of agents tends to infinity, the behavior of the stochastic system converges to the mean-field approximation almost surely or in probability. Thus, we present the Kullback-Leibler divergence D KL ρ N etwork . An estimation of the relative error in L ∞ norm at time T is given by: (A.1) e ∞ ∆x = µ M F ∆x (T ) − µ M C ∆x (T ) L∞(Ω) µ M C ∆x (T ) L∞(Ω) , where µ M F ∆x represents the average in x of the probability density computed on a uniform mesh of size ∆x, and µ M C ∆x represents the average in x from the Monte Carlo simulations of the network equations using a similar mesh size. Relative errors e ∞ ∆y and e ∞ ∆z are computed similarly. Results are shown in Fig 14b. For reference, a dashed black line of slope two is added. We see that the slope of the dashed line appears to match well that of the error curves, suggesting that the mean-field equation accurately describes the network for large N . Appendix B. Derivation of the continuum model (2.9). The equations in (2.9-2.11) can be derived via the mean-field limit. Here we present a simple formal description of this procedure. Starting from the deterministic model, define the empirical distribution density associated to a solution (x(t), y(t), z(t)) of (2.5) and given by ρ N (x, y, z, t) = 1 N N i=1 δ(x − x i (t))δ(y − y i (t))δ(v − z i (t)), t > 0, where δ is the Dirac delta probability measure. Let us denote by P R k the space of probability measures on R k . Let us assume that the particles remain in a fixed compact domain (x i (t), y i (t), z i (t)) ∈Ω ⊂ R × R × R for all N in the time interval t ∈ [0, T ]. Our model (2.5) satisfies this assumption if for instance the initial configuration is obtained as an approximation of an initial compactly supported probability measure ρ 0 [16]. Since for each t the measure ρ N (t) := ρ N (·, ·, ·, t) is a probability measure in P R 3 with the uniform support in N, then Prohorov's theorem implies that the sequence is weakly- * -relatively compact. Assume there exists a subsequence ρ N k k and ρ : [0, T ] → P R 3 such that ρ N k → ρ (k → ∞) in the w * -convergence sense in P R 3 , pointwise in time. Following the approach in [16], let us consider the test function ϕ ∈ C 1 0 R 3 . To simplify the notation we will write ϕ for ϕ x i (t), y i (t), z i (t) , and x i , y i , and z i for x i (t), y i (t), and z i (t), respectively. We compute d dt ρ N (t), ϕ = 1 N N i=1 d dt ϕ x i (t), y i (t), z i (t) = 1 N N i=1 ∂ x ϕ α 1 + z n i − x i + K N N j=1 H(x i − x j ) + 1 N N i=1 ∂ y ϕ (x i − y i ) + 1 N N i=1 ∂ z ϕ (y i − z i ) = ρ N (t), ∂ y ϕ (x − y) + ρ N (t), ∂ z ϕ (y − z) + 1 N N i=1 ∂ x ϕ α 1 + z n i − x i + 1 N N i=1 K N N j=1 H(x i − x j ) ∂ x ϕ = ρ N (t), ∂ y ϕ (x − y) + ρ N (t), ∂ z ϕ (y − z) + ρ N (t), α 1 + z n − x ∂ x ϕ + ρ N (t), K N N j=1 H(x − x j ) ∂ x ϕ . We can rewrite 1 N N j=1 H(x − x j ) = 1 N N j=1 H(x − ω), δ(ω − x j ) x = H m ρ N (y, z, t), where m ρ N (y, z, t) = R ρ N (x, y, z, t)dx = 1, 1 N N j=1 δ(ω − x j )δ(y − y j )δ(z − z j ) x ; Collecting all the terms we obtain d dt ρ N (t), ϕ = ρ N (t), ∂ x ϕ α 1 + z n − x + K H m ρ N + ∂ y ϕ (x − y) + ∂ z ϕ (y − z) . After integration by part in x, y, and z, we obtain ∂f ∂t + ∂ x ξ ρ N ρ N + ∂ y (x − y)ρ N + ∂ z (y − z)ρ N , ϕ = 0 or, in the strong form, ∂f ∂t + ∂ x ξ ρ N ρ N + ∂ y (x − y)ρ N + ∂ z (y − z)ρ N = 0, where ξ is defined by ξ ρ x, y, z, t = α 1 + z n − x + K H ρ , with H ρ(x, y, z, t) = R 3d H(x − w)ρ(w, y, z, t) dw dy dz . Letting k → ∞ in the subsequence ρ N k leads formally to ∂ρ ∂t − ∂ x ξ(ρ)ρ − ∂ y (x − y)ρ − ∂ z (y − z)ρ = 0. The case with noise in (2.9) follows a similar approach using the so-called coupling method introduced by Sznitman [83] together with [57,82] to deal with boundary conditions. Defining a system of uncoupled copies of McKean-Vlasov particles and comparing the error with respect to the coupled system of particles is a common approach in many areas of applications of interacting particle systems in mathematical biology, see [16] for instance. By taking the difference between the two particle systems, one can develop direct Gronwall inequalities for the 2-Wasserstein distance among the marginals of the joint probability distributions. We refer the reader for the details to [10] for instance. Appendix C. Presentation of the numerical scheme. In this section, we present our finite volume scheme for (2.9) preserving the structure of the gradient flow in the case of identical oscillators. We also prove the positivity preserving property for this scheme. Inspired by [13,14,49], we construct a discrete numerical scheme in the variables x, y and z in (2.9) as follows. We introduce a Cartesian mesh consisting of the cells C i,j,k := x i− 1 2 , x i+ 1 2 × y j− 1 2 , y j+ 1 2 × z k− 1 2 , z k+ 1 2 , which for the sake of simplicity are assumed to be of uniform size ∆x∆y∆z, that is, x i+ 1 2 − x i− 1 2 ≡ ∆x, ∀ i, y j+ 1 2 − y j− 1 2 ≡ ∆y, ∀ j, and z k+ 1 2 − z k− 1 2 ≡ ∆z, ∀ k. Here, we denote by (C.1)ρ i,j,k (t) = 1 ∆x∆y∆z C i,j,k ρ(x, y, z, t) dx dy dz the computed cell averages of the solution ρ, which we assume to be known or approximated at time t ≥ 0. A discrete finite volume scheme is obtained by integrating (2.9) over each cell C i,j,k and is given by the following system of ODEs forρ i,j,k : dρ i,j,k (t) dt = − F x i+ 1 2 ,j,k (t) − F x i− 1 2 ,j,k (t) ∆x − F y i,j+ 1 2 ,k (t) − F y i,j− 1 2 ,k (t) ∆y − F z i,j,k+ 1 2 (t) − F z i,j,k− 1 2 (t) ∆z , (C.2) where F x i+ 1 2 ,j,k , F y i,j+ 1 2 ,k and F z i,j,k+ 1 2 are upwind numerical fluxes and approximate the continuous fluxes in the x, y and z directions, respectively. For simplicity, we will omit the dependence of the computed quantities on t ≥ 0. In order to construct the upwind fluxes, we first construct piecewise linear polynomials in each cell C i,j,k , (C.3)ρ i,j,k (x, y, z) =ρ i,j,k + (ρ x ) i,j,k (x − x i ) + (ρ y ) i,j,k (y − y j ) + (ρ z ) i,j,k (z − z k ), (x, y, z) ∈ C i,j,k and compute the right ("east"), ρ E i,j,k , and left ("west"), ρ W i,j,k , point values at the corresponding cell interfaces ( x i+ 1 2 , y j , z k ), (x i− 1 2 , y j , z k ), (x i , y j+ 1 2 , z k ), (x i , y j− 1 2 , z k ), (x i , y j , z k+ 1 2 ) and (x i , y j , z k+ 1 2 ). Namely, ρ Ex i,j,k =ρ i,j,k (x i+ 1 2 − 0, y j , z k ) =ρ i,j,k + ∆x 2 (ρ x ) i,j,k , ρ Wx i,j,k =ρ i,j,k (x i− 1 2 + 0, y j , z k ) =ρ i,j,k − ∆x 2 (ρ x ) i,j,k . (C.4) and analogously for the other two variables. These values will be second-order accurate provided the numerical derivatives (ρ x ) i,j,k , (ρ y ) i,j,k and (ρ z ) i,j,k are at least first-order accurate approximations. To ensure the point values in (C.4) are both second-order and nonnegative, the slopes (ρ x ) i,j,k , (ρ y ) i,j,k , (ρ z ) i,j,k are calculated according to the following adaptive procedure. First, the centered-difference approximations (ρ x ) i,j,k = ρ i+1,j,k − ρ i−1,j,k 2∆x , (ρ y ) i,j,k = ρ i,j+1,k − ρ i,j−1,k 2∆y and (ρ z ) i,j,k = ρ i,j,k+1 − ρ i,j,k−1 2∆z (C. 5) are used for all i, j, k. Then, if the reconstructed point values in some cell C i,j,k become negative (i.e., either ρ E i,j,k < 0 or ρ W i,j,k < 0), we recalculate the corresponding slopes (ρ x ) i,j,k , (ρ y ) i,j,k or (ρ z ) i,j,k using a monotone nonlinear slope limiter, which guarantees that the reconstructed point values are nonnegative as long as the cell averagesρ i,j,k are nonnegative for all i, j, k. In our numerical experiments, we have used the one-parameter family of the generalized minmod limiter [13,56,66,81,86]: (ρ x ) i,j,k = minmod θρ i+1,j,k −ρ i,j,k ∆x ,ρ i+1,j,k −ρ i−1,j,k 2∆x , θρ i,j,k −ρ i−1,j,k ∆x (C. 6) and analogously for the other two variables, where the minmod function and its parameters are chosen as in [13]. Given the polynomial reconstruction (C.3) and its point values (C.4), the upwind numerical fluxes in (C.2) are defined as where the discrete values ξ i+ 1 2 ,j,k , u i,j+ 1 2 ,k and v i,j,k+ 1 2 of the velocities at midpoints are obtained as follows, Finally, the semi-discrete scheme (C.2) is integrated using a stable and accurate ODE solver. In all our numerical examples, the third-order strong preserving Runge-Kutta (SSP-RK) ODE solver [30] is used. (C.8) ξ i+ 1 2 ,j,k = − D ∆x logρ i+1,j,k ρ i,j,k − f x i,j,k + f x i+1,j,k 2 − K ∆x∆y∆z i,j,k xρ i,j,k − x i+ 1 2 u i,j+ 1 2 ,k = f y i,j,k + f y i,j+1,k 2 , v i,j,k+ 1 2 = f z i,j,k + f z i,j, Remark C.1. The second-order finite volume scheme (C.2),(C.7)-(C.9), reduces to the first-order scheme if the piecewise constant reconstruction is used instead of (C.3), in which case we haveρ i,j,k (x, y, z) =ρ i,j,k and therefore ρ Ex i,j,k = ρ Wx i,j,k = ρ Ey i,j,k = ρ Wy i,j,k = ρ Ez i,j,k = ρ Wz i,j,k =ρ i,j,k , ∀i, j, k. Remark C.2. Given initial data ρ 0 (x) ≥ 0 for system (2.9), the semi-discrete finite-volume scheme (C.2),(C.7)-(C.9) preserves positivity for all t > 0. A CFL condition can be computed explicitly using equation (C.2) which is discretized by the forward Euler method. Specifically, the computed cell averagesρ i,j,k ≥ 0, ∀ i, j, k provided that the following CFL condition is Remark C.3. Numerical simulations with GPUs.-The finite volume algorithm for solving the mean-field equation described in Appendix C is computationally very expensive. In fact, when the discretization steps ∆x, ∆y, and ∆z are small, we must also maintain ∆t small enough to ensure the algorithm's stability (see C.2). The simulations will undoubtedly slow down as a result of this. We were able to mitigate this issue by employing more powerful hardware, specifically graphical processing units (GPUs). Through GPU computing we were able to adopt a more accurate and stable ODE solver, namely the strong stability-preserving Runge-Kutta (SSP-RK) solver of order three [30], thus allowing for three calls per time step at a lower computational cost. Figure 1 : 1Limit cycle oscillations for the following parameter values: Figure 2 : 2(a) Bifurcation diagram of the stable periodic solutions near the Hopf bifurcation point (α H = 1.633) when n = 20. At α H , the system's stability switches from stable (solid black line) to unstable (dashed black line) and a periodic solution arises. The bifurcation is supercritical (solid blue curves). (b) Two-parameter bifurcation diagram in terms of α and n. Figure 3 : 3(a) Bifurcation diagram associated with the coupling strength K in the mean-field model. E[x], E[y] and E[z] refer to spatial averages. Numerical simulations are run until a steady state is reached or until the oscillation amplitude becomes stable. For the latter, the graph shows the peaks and troughs of oscillations as a function of K. (b) Period of oscillations of E[z]. Noise level is constant at D = 0.01 in both (a) and (b). A Hopf bifurcation appears around the critical value K H ≈ 0.3. Fig 5 we show two extreme scenarios in this regard: absence of coupling where K = 0 and perfect coupling with K = 1. Fig 4 gives the time evolution of E[x] Figure 4 : 4Evolution of the average in x for two limiting scenarios: (a) no coupling with K = 0 and (b) perfect coupling with K = 1. E[x] is computed from the solution to the mean-field equation (2.9). Figure 5 : 5Time evolution of the empirical variance in two limiting scenarios: (a) no coupling with K = 0 and (b) perfect coupling with K = 1. D = 0.01 in all simulations.of incoherence within the SCN network (seeFig 7a). Noting that α = k 1 k 3 k 5 /k 3 2 K i , Fig 7 suggests that the circadian system is more favorable to lower degradation rates in the presence of noise.To formally investigate the bifurcation inFig 7,we use the same parameters and initial conditions as inFig 7,with the exception of the parameter α which now varies from 1.5 to 3. Results are displayed inFig 8a.When α < 1.73, the network is not synchronised and evolves towards a steady state. Above the critical value α H ≈ 1.73, the network is synchronised and the solution to the mean-field equation (2.9) is periodic in time. As characteristic of a local Hopf bifurcation, the cycle that is born is nearly elliptical with a small amplitude; seeFig 8b. Figure 6 :Figure 7 :Figure 8 : 678Joint probability distribution between x and z at t = 600. (A) K=0.1, (B) K=0.2, (C) K=0.4, (D) K=0.6, (E) K=0.8 and (F) K=1. Noise level D = 0.01. Evolution of the marginal density of x in the presence of noise. (a) steady state regime with α = 1.5, (b) oscillatory regime with α = 2. Other parameters: n = 20, (a) Bifurcation diagram associated with the parameter α. Details of the diagram are the same as those of Fig 3. (b) Example of a periodic orbit when α = 2. Other parameters: n = 20, K = 0.6, D = 0.01. and the oscillations disappear through a supercritical Hopf bifurcation, thus resulting in trivial behavior with a single fixed point. Figure 9 : 9Bifurcation diagrams associated with the parameter α for the spatial averages (a) E[x], (b) E[y] and (c) E[z]. (d) Stable limit cycles in the E[x]-E[z] plane when α = 3. We model low noise (D = 0.01), medium noise(D = 0.025) and high noise (D = 0.05). Dotted lines represent intervals containing exact bifurcation values α H . Details of the figures are the same as those of Fig 3. Figure 10 : 10Joint probability distribution between x and z at t = 600. (a) D = 0.01, (b) D = 0.02, (c) D = 0.04, (d) D = 0.08, (e) D = 0.12 and (f) D = 0.15. Coupling strength K = 0.6. Figure 11 : 11(a) Bifurcation diagram associated with the noise intensity (D). A Hopf bifurcation appears around the critical value D H ≈ 0.8 for K fixed at 0.6. Details of the figure are the same as those of Fig 3. (b) Period of oscillations of E[z] as a function of noise for different coupling strengths. Figure 12 :Figure 13 : 1213Convergence of error for the solution to the mean-field equation (2.9) in L 1 and L ∞ norms. Grid cells are uniform in size across all three variables, h = ∆x = ∆y = ∆z. The final time is t f inal = 1. (a) Time evolution of the averages in x, y, and z obtained by simulating the network equations (solid curves) and the mean-field equation (dashed curves). We ran 100 Monte Carlo simulations of the network with network size N=100 up to time t f inal = 400. (b)Comparison between marginal probability densities ρ 1 (t, x), ρ 2 (t, y), ρ 3 (t, z) derived from the network and mean-field equation solutions. We conducted 10,000 Monte Carlo simulations with a network size N=10,000 up to time t f inal = 1. Figure 14 : 14distributions.For increasing values of network size N, we ran 10,000 Monte Carlo simulations of the network equations until t f inal = 1. As seen inFig 14a,the Kullback-Leibler divergence decreases as N increases, validating the efficiency of the mean-field model even for relatively small values of N. We conclude that the solution to the mean-field equation (2.9) accurately represents the network's average behaviour. These results highlight the accuracy of the numerical method in preserving long time behavior of the solutions. Solution remain strictly positive for all t > 0, (a) Kullback-Leibler divergence between the marginal probability densities ρ 1 (t, x), ρ 2 (t, y), ρ 3 (t, z) calculated from the network and mean-field equation solutions as network size N increases. (b) Convergence of error between solutions to the stochastic system (2.8) and the mean-field equation (2.9) for the averages in x, y, and z in L 1 and L ∞ norms. Grid cells are assumed to be of uniform size in all three variables, ∆x = ∆y = ∆z. We conducted 10,000 Monte Carlo simulations with a network size N=10,000 up to time t f inal = 1.thus problem is not degenerate (see C.2).Next, consider the spatial averages of x, y, and z separately, i.e. E[x], E[y] and E[z] for the other two variables. We note that x = [row vector of (inter)face values of the cells in the x-direction, and the values f x i,j,k , f y i,j,k , f z i,j,k are calculated by discretizing (C.10): f x (x, y, z) := α 1 + z n − x, f y (x, y, z) := x − y, f z (x, y, z) := y − z. (C.10) F x i+ 1 2 x1,j,k = ξ + i+ 1 2 ,j,k ρ Ex i,j,k + ξ −i+ 1 2 ,j,k ρ Wx i+1,j,k F y i,j+ 1 2 ,k = u + i,j+ 1 2 ,k ρ Ey i,j,k + u − i,j+ 1 2 ,k ρ Wy i,j+1,k F z i,j,k+ 1 2 = v + i,j,k+ 1 2 ρ Ez i,j,k + v − i,j,k+ 1 2 ρ Wz i,j,k+1 , (C.7) Acknowledgments. The authors would like to thank the anonymous referees for valuable suggestions and remarks. Coupling governs entrainment range of circadian clocks. U Abraham, A Granada, P Westermark, M Heine, A Kramer, H Herzel, 10.1038/msb.2010.92Mol. Syst. Biol. 6U. Abraham, A. Granada, P. Westermark, M. Heine, A. Kramer, and H. Herzel, Coupling governs entrainment range of circadian clocks, Mol. Syst. Biol., 6 (2010), https://doi.org/10.1038/ msb.2010.92. 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[ "Wetting at Curved Substrates: Non-Analytic Behavior of Interfacial Properties", "Wetting at Curved Substrates: Non-Analytic Behavior of Interfacial Properties" ]
[ "R Evans \nMax-Planck-Institut für Metallforschung\nHeisenbergstr. 370569StuttgartGermany\n\nITAP\nUniversität Stuttgart\nPfaffenwaldring 5770569StuttgartGermany\n", "R Roth \nMax-Planck-Institut für Metallforschung\nHeisenbergstr. 370569StuttgartGermany\n\nITAP\nUniversität Stuttgart\nPfaffenwaldring 5770569StuttgartGermany\n", "P Bryk \nMax-Planck-Institut für Metallforschung\nHeisenbergstr. 370569StuttgartGermany\n\nITAP\nUniversität Stuttgart\nPfaffenwaldring 5770569StuttgartGermany\n\nDep. of Modeling of Physico-Chemical Processes\nMCS University\n20-031LublinPoland\n", "H H Wills ", "\nPhysics Laboratory\nUniversity of Bristol\nBS8 1TLBristolUK\n" ]
[ "Max-Planck-Institut für Metallforschung\nHeisenbergstr. 370569StuttgartGermany", "ITAP\nUniversität Stuttgart\nPfaffenwaldring 5770569StuttgartGermany", "Max-Planck-Institut für Metallforschung\nHeisenbergstr. 370569StuttgartGermany", "ITAP\nUniversität Stuttgart\nPfaffenwaldring 5770569StuttgartGermany", "Max-Planck-Institut für Metallforschung\nHeisenbergstr. 370569StuttgartGermany", "ITAP\nUniversität Stuttgart\nPfaffenwaldring 5770569StuttgartGermany", "Dep. of Modeling of Physico-Chemical Processes\nMCS University\n20-031LublinPoland", "Physics Laboratory\nUniversity of Bristol\nBS8 1TLBristolUK" ]
[]
PACS. 05.70.Np -Interface and surface thermodynamics. PACS. 68.08.Bc -Wetting.Abstract. -We argue that for complete wetting at a curved substrate (wall) the wall-fluid surface tension is non-analytic in R −1 i , the curvature of the wall and that the density profile of the fluid near the wall acquires a contribution proportional to the gas-liquid surface tension ×R −1 i plus higher-order contributions which are non-analytic in R −1 i . These predictions are confirmed by results of density functional calculations for the square-well model of a liquid adsorbed on a hard sphere and on a hard cylinder where complete wetting by gas (drying) occurs. The implications of our results for the solvation of big solvophobic particles are discussed.c EDP Sciences
10.1209/epl/i2003-00445-5
[ "https://export.arxiv.org/pdf/cond-mat/0305109v1.pdf" ]
119,354,370
cond-mat/0305109
d13945d50ad62165744c190b8a7833114b98bbb7
Wetting at Curved Substrates: Non-Analytic Behavior of Interfacial Properties 6 May 2003 R Evans Max-Planck-Institut für Metallforschung Heisenbergstr. 370569StuttgartGermany ITAP Universität Stuttgart Pfaffenwaldring 5770569StuttgartGermany R Roth Max-Planck-Institut für Metallforschung Heisenbergstr. 370569StuttgartGermany ITAP Universität Stuttgart Pfaffenwaldring 5770569StuttgartGermany P Bryk Max-Planck-Institut für Metallforschung Heisenbergstr. 370569StuttgartGermany ITAP Universität Stuttgart Pfaffenwaldring 5770569StuttgartGermany Dep. of Modeling of Physico-Chemical Processes MCS University 20-031LublinPoland H H Wills Physics Laboratory University of Bristol BS8 1TLBristolUK Wetting at Curved Substrates: Non-Analytic Behavior of Interfacial Properties 6 May 2003Europhysics Letters PREPRINT PACS. 05.70.Np -Interface and surface thermodynamics. PACS. 68.08.Bc -Wetting.Abstract. -We argue that for complete wetting at a curved substrate (wall) the wall-fluid surface tension is non-analytic in R −1 i , the curvature of the wall and that the density profile of the fluid near the wall acquires a contribution proportional to the gas-liquid surface tension ×R −1 i plus higher-order contributions which are non-analytic in R −1 i . These predictions are confirmed by results of density functional calculations for the square-well model of a liquid adsorbed on a hard sphere and on a hard cylinder where complete wetting by gas (drying) occurs. The implications of our results for the solvation of big solvophobic particles are discussed.c EDP Sciences Understanding the adsorption of fluids at solid substrates has taken on new importance with recent advances in the controlled fabrication of tailored surfaces for applications in microfluidics and other areas [1]. Much experimental [2] and theoretical effort [3] is concerned with wetting and associated interfacial transitions in wedge geometry and recently attention has turned to wetting at an apex [4]. It is becoming increasingly clear that substrate geometry can have a profound influence on the nature of fluid adsorption and, in particular, on wetting characteristics making these quite different from those at a planar substrate. Here we consider complete wetting at two substrates that have simple geometries, namely a single sphere of radius R s and an infinitely long cylinder of radius R c . Using an effective interfacial Hamiltonian approach [5] combined with exact microscopic sum-rules for the density profile of the fluid near a hard wall, we show that in the limit R i → ∞, with i = s, c, the surface tension of the substrate (wall)-fluid interface is non-analytic in the curvature R −1 i and that the density of the fluid in contact with the hard wall acquires a contribution proportional to γ gl (∞)/R i , where γ gl (∞) is the surface tension of the planar interface between coexisting gas and liquid, as well as higher-order non-analytic terms in R −1 i . Our results, which are confirmed fully by the results of microscopic density functional (DFT) [6] calculations, show that non-zero curvature leads to unexpected and subtle effects on interfacial properties when complete wetting occurs, even for the simplest of substrate geometries. In previous theoretical studies of wetting on spheres and cylinders [7,8,9,10] the thrust was on understanding how a finite radius limits the thickness of a wetting film and modifies the wetting transition that can occur at a planar substrate. Little attention was paid to the effect of curvature on the surface tension and on the form of the density profile near the substrate which are the main concerns of this Letter. Our results have repercussions for the general theory of solvation of big solvophobic solute particles, for wetting of colloidal particles and fibers [1,10] and, possibly, for the surface tension of drops and bubbles [11]. We consider first the general case of a bulk fluid phase a, at chemical potential µ, in contact with a wall w and implement a standard, coarse-grained, effective interfacial Hamiltonian approach [5,8] in which the wetting film of fluid phase b is characterized only by its thickness l. For short-ranged (finite ranged, exponentially or faster decaying) wall-fluid and fluid-fluid potentials the binding potential, i.e. the excess (over bulk) grand potential per unit area, takes the form [5,8]: ω i wa (l) = γ i wb (R i ) + γ i ba (R i + l) +ã(T )e −l/ξ + (ρ a − ρ b )δµ l + α i γ i ba (R i + l)l R i + O(l/R i ) 2 ,(1) with i = s, c and α s = 2, α c = 1 for the spheres and cylinders, respectively. γ i wb is the surface tension of the wall-phase b interface, γ i ba is the tension of the ba fluid-fluid interface located near R i + l,ã(T ) > 0 is a coefficient that we need not specify further for complete wetting, and ξ is the true bulk correlation length of the wetting phase b. The fourth term in eq. (1) accounts for the increase in grand potential per unit area associated with the volume of a film of phase b; δµ ≡ µ − µ co (T ) is the chemical potential deviation and ρ a , ρ b are the number densities of bulk phases a and b at coexistence µ co (T ). For R i = ∞ the binding potential (1) reduces to that appropriate for complete wetting at a planar interface in models where the range of the wall-fluid potential is shorter than ξ and, henceforward, we shall assume this is the case. Then minimizing eq. (1) w.r.t. l yields the well-known logarithmic divergence of the equilibrium film thickness: l eq (∞) = −ξ ln (δµ ξ(ρ a − ρ b )/ã(T )), as δµ → 0. When R i = ∞ two important modifications arise: i) γ i wb and γ i ba now depend on curvature and ii) the surface area of the ba interface now depends on the film thickness l. Equation (1) assumes very large radii R i and that l/R i ≪ 1. Note that the fifth term in (1) is proportional to the Laplace pressure across the fluid-fluid ba interface. Since γ ba is always non-zero, away from the bulk critical point, this term ensures that the film thickness remains finite even at µ co (T ), i.e. minimizing eq. (1) yields l i eq (R i ) = −ξ ln[α i ξγ i ba (R i )/(R iã (T ))] for δµ = 0, where we have ignored terms O(l/R i ) 2 and the l dependence in γ i ba -this can be justified a posteriori. Several authors [7,8,9,10] have argued that for large R i the quantity α i γ i ba (R i )/R i should play the same role in complete wetting at a curved wall, with δµ = 0, as the effective bulk field (ρ a − ρ b )δµ plays for the planar wall. We shall pursue this argument further in the present Letter. The wall-fluid surface tension is given by γ i wa (R i ) ≡ω i wa (l eq ) = γ i wb (R i ) + γ i ba (R i ) + (ξ + l eq ) α i γ i ba (R i ) R i + (ρ a − ρ b )δµ . (2) For the planar interface the final term vanishes at δµ = 0 and the wall-fluid tension reduces to the sum of the planar tensions: γ wa (∞) = γ wb (∞) + γ ba (∞), appropriate to complete wetting by a macroscopic film of phase b. When δµ = 0, γ wa (∞) acquires a non-analytic contribution proportional to −|δµ| ln |δµ| [5]. For finite R i we set δµ = 0 and obtain γ i wa (R i ) = γ i wb (R i ) + γ i ba (R i ) 1 + α i ξ R i + α i ξγ i ba (R i ) R i ln(R i × const) + H.O.T..(3) That the surface tension contains a contribution which is non-analytic in the curvature is clearly a direct manifestation of complete wetting. However, the necessity for such a contribution does not seem to have been widely recognized. An exception is Ref. [7] but in that paper the consequences were not discussed. In order to elucidate these we must examine the other terms in eq. (3). At the non-wet wb interface we do not expect non-analyticities in γ i wb (R i ), provided the system exhibits short-ranged forces [12]. For a spherical fluid-fluid interface it is usually assumed [11] (for short-ranged forces) that γ s ba (R s ) = γ ba (∞)(1 − 2δ s T /R s + H.O.T.),(4) where δ s T is the Tolman length [13] familiar in studies of liquid drops, whereas for cylindrical interfaces it has been argued [14] that γ c ba (R c ) = γ ba (∞)(1 + b H ln R c /R c + H.O.T.).(5) If the latter form were correct eq. (3) would imply that the wall-fluid tension γ c wa (R c ) should have a contribution γ ba (∞)(b H + ξ) ln(R c )/R c . Provided the length b H is comparable with the bulk correlation length ξ, as is expected on physical grounds, the term in b H should be easily identified in numerical work. The result (3) is valid for any complete wetting situation, with the proviso that the forces are short-ranged. We specialize now to the case of a hard wall exerting a purely repulsive potential on the fluid: V i (r) = ∞ for radius r < R i and is zero for r > R i . It is well-known that the phenomenon of complete drying occurs at a planar hard wall: the interface between the bulk liquid (l) and the wall is wet by a macroscopic film of gas (g) as µ → µ + co (T ) for all temperatures T at which gas and liquid coexist [15,16]. At µ = µ + co (T ) the density profile of the fluid is a composite of the planar wall-gas and the (free) gas-liquid interfacial profiles and γ wl (∞) = γ wg (∞) + γ gl (∞); phase a ≡ l and phase b ≡ g. Whilst drying at a planar hard wall has been investigated extensively using computer simulations [15] and DFT [16], we are not aware of studies on curved substrates. We shall see below that curvature leads to some striking new results. When R i = ∞ thick films of gas will still develop at the wall but the thickness will remain finite at µ + co (T ) and the wall-liquid tension γ i wl (R i ) should exhibit the non-analyticities described above. Moreover the density profile ρ(r) is no longer a perfect composite of the two separate interfacial profiles and exhibits interesting features near the wall. A particular advantage of hard walls is that there is an exact statistical mechanical sumrule [17] which relates the fluid density at contact, ρ i (R + i ), to the pressure p of the bulk (reservoir) fluid and the wall-fluid tension: k B T ρ i (R + i ) = p + α i γ i wf (R i ) R i + ∂γ i wf (R i ) ∂R i T,µ .(6) In the limit R i = ∞ the contact density reduces to p/k B T , the well-known planar contact theorem. Equation (6) is valid for any one-component fluid at a hard wall. We now insert our result (3) for the tension of the 'dry' interface, γ i wl (R i ), into (6) to obtain the contact density ρ i liq (R + i ) at µ = µ + co (T ). By subtracting ρ i gas (R + i ), the contact density when the bulk phase is gas at µ = µ − co (T ), we eliminate both the pressure p(µ co ) and the (analytic) wall-gas tension obtaining for a hard spherical wall: and for a hard cylindrical wall: k B T (ρ s liq (R + s ) − ρ s gas (R + s )) = 2γ gl (∞) R s + 2ξγ gl (∞) R 2 s ln R s + H.O.T.(7)k B T (ρ c liq (R + c ) − ρ c gas (R + c )) = γ gl (∞) R c + (ξ + b H )γ gl (∞) R 2 c + H.O.T.(8) where H.O.T. refers to terms of higher order in R −1 i . For a planar wall the contact density is p(µ co )/k B T in both phases so the difference vanishes identically. There are two remarkable features in these results. First the contact density in the liquid phase ('dry' interface) depends on γ gl (∞), the tension of the planar gas-liquid interface, which can be very far from the wall as R i → ∞: the first term on the r.h.s. of (7) and (8) is simply the Laplace pressure across the gas-liquid interface. Second the contact density difference for the sphere contains a next to leading order R −2 s ln R s non-analyticity, whereas for the cylinder the next to leading order term is analytic, i.e. O(R −2 c ), despite the fact that both surface tensions are non-analytic at order R −1 i ln R i . In order to test the predictions of the coarse-grained theory we have adopted a fully microscopic DFT approach and performed calculations for a square-well fluid at hard curved walls. The fluid-fluid pair potential φ(r) is infinite for r < σ HS , the hard sphere diameter, the width of the well is 0.5σ HS and the well-depth is ε. We treat the hard-sphere part of the free-energy functional by means of Rosenfeld's fundamental measure theory [18] and the attractive part using a simple mean-field approximation [6,16], taking φ att (r) = −ε for r < 1.5σ HS and zero otherwise. By minimizing the grand potential functional we determine the equilibrium density profile ρ(r) and the wall-fluid tension for any thermodynamic state; details of the numerical treatment will be given elsewhere. Our DFT approach has the following advantageous features: i) the coexisting densities of gas and liquid, ρ g and ρ l , can be calculated precisely, ii) the DFT obeys the sum-rule (6) and the Gibbs adsorption theorem and iii) the key quantities ξ and γ gl (∞) which enter eqs. (3,7,8) can be obtained from independent calculations (for the bulk and for the planar interface) using the same functional. fig. 2 but now for the difference between the surface tensions. For both geometries the leading order curvature dependence is proportional to ln Ri/Ri -see eq. (3). In both cases the coefficient is confirmed to be αiξγ gl (∞)/kBT , with ξ the correlation length in the bulk gas. We focus first on the density profile. In fig. 1 we display results for ρ(r), at a reduced temperature k B T /ε = 1 and chemical potential µ + co (T ), for the square-well fluid adsorbed on a hard sphere with radius R s = 2500σ HS . In the same figure we show the profiles for the wall-gas interface at µ − co (T ) and for the planar interface with δµ ≡ µ− µ co (T ) = 2γ gl (∞)/(R s (ρ l − ρ g )). One can observe that this translation between bulk field for a planar wall and curvature yields wall-liquid profiles which lie on top of each other. This is a non-trivial result. Although it is evident that within the coarse-grained treatment of eq. (1) the film thickness l i eq is identical for planar and curved walls, there is no reason a priori why the complete microscopic density profiles should be especially close. In fig. 2 we seek to test the validity of eqs. (7,8) by examining the difference in the contact densities for the two phases as a function of R −1 i . Our DFT results are in excellent agreement with the prediction of the coarse grained theory for both spheres and cylinders. In particular we find the data is fit best with a R −2 s ln R s contribution for spheres and without such a contribution for cylinders. It is straightforward to extract the coefficient of the R −1 i contribution and we find close agreement, to 1 part in 10 4 , with our independent (planar) results for the dimensionless quantity α i γ gl (∞)σ 2 HS /k B T . In fig. 3 we plot the difference between the wall-liquid and wall-gas surface tensions, evaluated at µ ± co (T ), respectively versus R −1 i for the square-well fluid adsorbed at spheres and cylinders, again for k B T /ε = 1. In accordance with eqs. (3) and (4) we expect that for the sphere the leading order curvature correction to the difference is proportional to R −1 s ln R s with a coefficient given by 2ξγ gl (∞) where ξ is the correlation length in the bulk gas. For cylinders we employ eqs. (3) and (5) which imply the same type of leading order correction but now with a coefficient (ξ + b H )γ gl (∞). Our numerical DFT results are completely consistent with these predictions with the relevant coefficients agreeing with the planar result to better than 1%. In our calculations we were able to obtain reliable results for R s up to 30000σ HS and R c up to 5000σ HS . For the cylinder our best fit yields a coefficient that has b H = 0 (or a very small fraction of ξ) and we conclude that there is no evidence for the ln R c /R c term conjectured [14] for the cylindrical fluid-fluid interface -see eq. (5). We have confirmed that the same level of agreement between the coarse-grained theory and DFT holds at other temperatures. Since there is nothing special about the square-well model our predictions for hard spheres and cylinders should be valid for any fluid with short-ranged fluid-fluid potentials that exhibits gas-liquid coexistence and this has important repercussions for the solvation of big solvophobic solute particles. The excess chemical potential of a single hard sphere of radius R s in a solvent is given generally byμ HS (R s ) = p4πR 3 s /3 + γ s wf (R s )4πR 2 s , where γ s wf is the wall-fluid surface tension. Using eq. (3) it follows that the difference betweenμ HS evaluated in the liquid, where drying occurs, and in the gas phase, at µ ± co (T ), is (μ HS liq −μ HS gas ) 4πR 2 s = γ gl (∞)(1 + 2ξR −1 s ln R s ) + H.O.T.,(9) for R s → ∞. Whilst it is known from recent studies [19] of hard-sphere solutes thatμ HS liq should contain a gas-liquid surface tension contribution γ gl (∞)4πR 2 s , the presence of the nonanalytic correction is striking -especially when we recall that the excess chemical potential is the derivative of the excess free energy of the bulk mixture w.r.t. the solute density in the limit of vanishing solute. The coarse grained treatment that leads to eq. (3) is valid for wetting by either fluid phase. Thus, provided the attractive part of the wall-fluid potential is sufficiently strong and has finite range or decays on a length scale that is shorter than the bulk correlation length ξ, now referring to the wetting liquid, adsorption from the saturated gas, at µ − co (T ), will give rise to a wall-gas tension which has a term α i ξγ gl (∞)R −1 i ln R i . Although the contact theorem (6) is modified when the wall-fluid potential is no longer purely hard the contact density in the presence of a wetting liquid film will still acquire terms which depend on the gas-liquid surface tension, i.e. we expect results equivalent to (7) and (8). Both the coarse-grained and DFT approaches presented here are mean-field like in that they omit effects of capillary-wave fluctuations in the wetting film [5,6]. For complete wetting in three dimensions at a planar interface, renormalization group studies based on effective interfacial Hamiltonians with the binding potential (1) find that critical exponents are not altered from their mean-field values but the amplitude of the equilibrium film thickness l eq (∞) is changed from ξ to ξ(1 + ω/2), for ω < 2, when fluctuations are included [5]. Here ω = k B T /(4πγ gl (∞)ξ 2 ) is the usual parameter which measures the strength of capillary-wave fluctuations: ω = 0 corresponds to mean-field. We conjecture that our present results for the leading non-analytic term in the surface tension are modified in a similar fashion, i.e. the third term in eq. (3) simply has ξ replaced by ξ(1 + ω/2). As regards the results (7) and (8) for the difference in the contact densities, the leading-order (Laplace pressure) terms will be unchanged by fluctuations whereas the next to leading order term in eq. (7) will require the same replacement for ξ. For real fluids dispersion forces are always present giving rise to r −6 power-law decay of the fluid-fluid pair potential. This leads, in turn, to a wetting film thickness which diverges at coexistence as l eq (R i ) ∼ R 1/3 i , for R i → ∞ [10,20]. The present coarse-grained analysis suggests that the non-analyticities in curvature which arise for wetting with such potentials will be power laws rather than terms involving logarithms and we are presently investigating these using DFT [21]. We have shown that the isomorphism between bulk field δµ(ρ l − ρ g ) for a planar substrate and Laplace pressure at a curved substrate implied by the binding potential (1) leads to striking consequences for interfacial properties: first one cannot obtain the surface excess free energy (surface tension) of a fluid that wets completely a non-planar substrate by expanding only in powers of the curvature(s). Second the true microscopic density profile near the curved substrate depends on the surface tension of the gas-liquid interface which, for large R i , can be far from the substrate. A detailed explanation of this curious behavior will be given elsewhere. Here we merely state that it is associated with the exponential decay (for short-ranged potentials) of the tails of the density profile of the 'free' gas-liquid interface. This is not the first time that analysis of complete wetting or drying at a hard wall has caused surprises and lead to new insight into the fundamental physics of fluid interfaces [15,22]. We have benefited from conversations with J.R. Henderson and M. Thomas. R.E. is grateful to S. Dietrich for kind hospitality and to the Humboldt Foundation for support under GRO/1072637 during his stay in Stuttgart. Fig. 1 - 1The density profile of a square-well fluid adsorbed at a hard wall for kBT /ε = 1. Solid curve refers to a liquid at bulk coexistence µ + co (T ) near a spherical wall of radius Rs = 2500σHS and the dotted curve to the gas at µ − co (T ) at the same wall. Symbols refer to the liquid near a planar wall at chemical potential µ − µco(T ) = 2γ gl (∞)/(Rs(ρ l − ρg)) (see text). One observes that the two wall-liquid profiles are indistinguishable in the region of the gas-liquid interface and close to the wall -see inset. Contact occurs at distance h = σHS/2. Fig. 2 Fig. 2 - 2The difference between contact densities of a square-well fluid in the liquid and in the gas phases, at µ ± co (T ), adsorbed at a hard spherical (full line) and hard cylindrical (dotted line) wall of radius Ri for kBT /ε = 1. From the linear portion of each curve near the origin, we can extract the gas-liquid surface tension γ gl (∞). The next to leading order term is proportional to R −2 s ln Rs for the sphere and to R −2 c for the cylinder -see eqs.(7)and(8). Fig. 3 - 3As in . G Dietrich S. ; C, Caccamo, New Approaches to Problems in Liquid State Theory. Kluwer, Dordrecht529197e.g. Dietrich S., in New Approaches to Problems in Liquid State Theory, edited by C. Caccamo et al. NATO ASI Ser. C, Vol. 529 (Kluwer, Dordrecht) 1999, p. 197. . L Bruschi, A Carlin, G Mistura, Phys. Rev. Lett. 89166101Bruschi L., Carlin A., and Mistura G., Phys. Rev. Lett., 89 (2002) 166101. . C E.G. Rascón, A O Parry, Nature. 986e.g. Rascón C. and Parry A.O., Nature (London), 407 (2000) 986. . A O Parry, M J Greenall, J M Romero-Enrique, Phys. Rev. Lett. 9046101Parry A.O., Greenall M.J. and Romero-Enrique J.M., Phys. Rev. Lett., 90 (2003) 046101. For a general review of wetting see: Dietrich S. Phase Transitions and Critical Phenomena. C. Domb and J.L. LebowitzLondonAcademic121For a general review of wetting see: Dietrich S., in Phase Transitions and Critical Phenomena, edited by C. Domb and J.L. Lebowitz (Academic, London), Vol. 12 1988, p. 1. For, R Evans, Fundamentals of Inhomogeneous Fluids. D. HendersonNew YorkDekker85For a review of DFT see Evans R.,: in Fundamentals of Inhomogeneous Fluids, edited by D. Henderson (Dekker, New York) 1992, p. 85. . R Holyst, A Poniewierski, Phys. Rev. B. 361Holyst R. and Poniewierski A., Phys. Rev. B, 36 (1987) 5628. These authors obtained a R −1 s ln Rs contribution to the surface tension in a Cahn-like treatment of wetting on a sphere. s ln Rs contribution to the surface tension in a Cahn-like treatment of wetting on a sphere. . M P Gelfand, R Lipowsky, Phys. Rev. B. 368725Gelfand M.P. and Lipowsky R., Phys. Rev. B, 36 (1987) 8725. . P J Upton, J O Indekeu, J M Yeomans, Phys. Rev. B. 40666Upton P.J., Indekeu J.O. and Yeomans J.M., Phys. Rev. B, 40 (1989) 666. . T Bieker, S Dietrich, Physica A. 25285and references thereinBieker T. and Dietrich S., Physica A, 252 (1998) 85 and references therein. . . G Rowlinson, J S , J. Phys.: Condens. Matt. 61e.g. Rowlinson J.S., J. Phys.: Condens. Matt., 6 (1994) A1. DFT studies for hard spheres adsorbed on a hard sphere or cylinder find no contributions to the surface tension of the form R −1 i ln Ri. Rather they find γ i wb (Ri) = γ wb (∞). DFT studies for hard spheres adsorbed on a hard sphere or cylinder find no contributions to the surface tension of the form R −1 i ln Ri. Rather they find γ i wb (Ri) = γ wb (∞)( . H O T ) , See Bryk, R Roth, K R Mecke, Dietrich S , to be publishedH.O.T.), see Bryk, Roth R., Mecke K.R., and Dietrich S., to be published. . R C Tolman, J , Chem. Phys. 17333Tolman R.C, J. Chem. Phys., 17 (1949) 333. . J R Henderson, J S Rowlinson, J. Phys. Chem. 886484Henderson J.R. and Rowlinson J.S., J. Phys. Chem., 88 (1984) 6484. . . G Henderson, J R Van Swol, F , Mol. Phys. 561313and references thereine.g. Henderson J.R. and van Swol F., Mol. Phys., 56 (1985) 1313 and references therein. . P Tarazona, R Evans, Mol. Phys. 52847Tarazona P. and Evans R., Mol. Phys., 52 (1984) 847; . A O Parry, R Evans, Mol. Phys. 65455Parry A.O. and Evans R., Mol. Phys., 65 (1988) 455. -see eq. (12.84a). Our eq. (6) follows by defining γ i wf (Ri) as the excess grand potential per unit area. J R Henderson, Fluid Interfacial Phenomena. C. A. CroxtonNew YorkWiley555Henderson J.R., in Fluid Interfacial Phenomena, edited by C. A. Croxton (Wiley, New York) 1986, p. 555 -see eq. (12.84a). Our eq. (6) follows by defining γ i wf (Ri) as the excess grand potential per unit area. . Y Rosenfeld, Phys. Rev. Lett. 63980Rosenfeld Y., Phys. Rev. Lett., 63 (1989) 980. . K Lum, D Chandler, J D Weeks, J. Phys. Chem. B. 1034570e.g. Lum K., Chandler D. and Weeks J.D., J. Phys. Chem. B, 103 (1999) 4570; . D M Huang, D Chandler, Phys. Rev. E. 1501Huang D.M. and Chandler D., Phys. Rev. E, 61 (2000) 1501; . J R Henderson, J. Chem. Phys. 1165039Henderson J.R., J. Chem. Phys., 116 (2002) 5039. . F Brochard, J. Chem. Phys. 844664Brochard F., J. Chem. Phys., 84 (1986) 4664; . T Gil, L V Mikeev, Phys. Rev. E. 52772Gil T. and Mikeev L.V., Phys. Rev. E, 52 (1995) 772. Since the upper critical dimension for complete wetting with power law potentials is < 3, fluctuations are not important in this case. Since the upper critical dimension for complete wetting with power law potentials is < 3, fluctuations are not important in this case. . A O Parry, R Evans, Mol. Phys. 781527and references thereinParry A.O. and Evans R., Mol. Phys., 78 (1993) 1527 and references therein.
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[ "Autonomous Driving From the Sky: Design and End-to-End Performance Evaluation", "Autonomous Driving From the Sky: Design and End-to-End Performance Evaluation" ]
[ "Matteo Bordin ", "Marco Giordani \nDepartment of Information Engineering\nUniversity of Padova\nItaly\n", "Michele Polese ", "Tommaso Melodia ", "Michele Zorzi \nDepartment of Information Engineering\nUniversity of Padova\nItaly\n", "\nInstitute for the Wireless Internet of Things\nNortheastern University\nBostonMAUSA\n" ]
[ "Department of Information Engineering\nUniversity of Padova\nItaly", "Department of Information Engineering\nUniversity of Padova\nItaly", "Institute for the Wireless Internet of Things\nNortheastern University\nBostonMAUSA" ]
[]
For autonomous vehicles to operate without human intervention, information sharing from local sensors plays a fundamental role. This can be challenging to handle with bandwidthconstrained communication systems, which calls for the adoption of new wireless technologies, like in the millimeter wave (mmWave) bands, to solve capacity issues. Another approach is to exploit Unmanned Aerial Vehicles (UAVs), able to provide human users and their cars with an aerial bird's-eye view of the scene otherwise unavailable, thus offering broader and more centralized observations. In this article we combine both aspects and design a novel framework in which UAVs, operating at mmWaves, broadcast sensory information to the ground as a means to extend the (local) perception range of vehicles. To do so, we conduct a full-stack end-to-end simulation campaign with ns-3 considering real UAV data from the Stanford Drone Dataset, and study four scenarios representing different UAV-toground communication strategies. Our results focus on the tradeoff between centralized data processing in the sky vs. distributed local processing on the ground, with considerations related to the throughput, latency and reliability of the communication process.
10.1109/gcwkshps56602.2022.10008495
[ "https://arxiv.org/pdf/2205.12203v1.pdf" ]
249,017,546
2205.12203
2a39a36dc598d77ecad005b80c962a5cd17539a4
Autonomous Driving From the Sky: Design and End-to-End Performance Evaluation Matteo Bordin Marco Giordani Department of Information Engineering University of Padova Italy Michele Polese Tommaso Melodia Michele Zorzi Department of Information Engineering University of Padova Italy Institute for the Wireless Internet of Things Northeastern University BostonMAUSA Autonomous Driving From the Sky: Design and End-to-End Performance Evaluation This paper has been submitted to IEEE for publication. Copyright may change without notice.Index Terms-UAVsvehicular networksmillimeter wavesoffloadingend-to-end performancens-3 For autonomous vehicles to operate without human intervention, information sharing from local sensors plays a fundamental role. This can be challenging to handle with bandwidthconstrained communication systems, which calls for the adoption of new wireless technologies, like in the millimeter wave (mmWave) bands, to solve capacity issues. Another approach is to exploit Unmanned Aerial Vehicles (UAVs), able to provide human users and their cars with an aerial bird's-eye view of the scene otherwise unavailable, thus offering broader and more centralized observations. In this article we combine both aspects and design a novel framework in which UAVs, operating at mmWaves, broadcast sensory information to the ground as a means to extend the (local) perception range of vehicles. To do so, we conduct a full-stack end-to-end simulation campaign with ns-3 considering real UAV data from the Stanford Drone Dataset, and study four scenarios representing different UAV-toground communication strategies. Our results focus on the tradeoff between centralized data processing in the sky vs. distributed local processing on the ground, with considerations related to the throughput, latency and reliability of the communication process. I. INTRODUCTION The scientific community is witnessing an increasing interest in research and experimentation on autonomous driving vehicles, powered by the several benefits they provide (from improved safety to more efficient traffic management) and the market potential they generate [1]. For future vehicles to be fully autonomous, they will be equipped with diverse and heterogeneous sensors, from optical cameras to Light Detection and Ranging (LiDAR) sensors, able to perceive the environment and identify road entities in the surroundings [2]. In this scenario, more robust scene understanding can be achieved if vehicles share sensory data with other vehicles, which however imposes strict demands in terms of data rates, that may be difficult to support with legacy bandwidth-constrained communication systems [3]. One way to solve this issue is to compress and process the data before transmission [4], as well as to operate at high frequencies, e.g., in the millimeter wave (mmWave) bands, where the large spectrum available, in combination with Multiple Input Multiple Output (MIMO) technologies, can support ultra-high transmission rates [5]. At the same time, Unmanned Aerial Vehicles (UAVs), mainly known as drones, have rapidly became popular thanks to the ease of deployment, low maintenance and operating costs, and native support for ubiquitous broadband coverage. When equipped with sensors, UAVs can enable several services, from crowd monitoring [6] to airspace surveillance and border patrol [7]. Drones have been further studied as a solution to provide connectivity to ground users and first responders in emergency situations [8], e.g., when cellular infrastructures are unavailable or no longer operational [9]. In recent years, UAVs have been also considered to support autonomous driving applications, especially for vehicular edge computing [10] and traffic management [11]. In fact, UAVs operating from the sky can guarantee a birds'-eye wide perception of the scene than it would be possible from vehicles' (local) sensor acquisitions, thus achieving more centralized and precise observations. Despite these benefits, however, the limited battery power and computational capacity available at the UAVs raise the questions of where to process and how to disseminate sensory data on the ground, in view of latency constraints. Today, UAV communication is typically enabled by legacy wireless technologies such as Long Term Evolution (LTE) [12] which, however, may not satisfy the boldest latency and throughput requirements of future vehicular networks. In this respect, several prior works have demonstrated the feasibility of operating UAVs at mmWaves [13], and characterized the optimal beamforming and deployment options for aerial nodes [14]. Based on the above introduction, in this paper we evaluate the feasibility of implementing an autonomous driving framework by relying on UAV's observations, and whether sensory information from the sky can be efficiently delivered to ground vehicles, possibly operating at mmWaves. To do so, we investigate several communication options between the UAV and the vehicles, each of which involves three main components: a UAV where sensory data are generated, a base station (BS) acting as a relay, and multiple autonomous vehicles. Notably, we study whether autonomous driving tasks based on these data (e.g., object detection) should be processed on board of the UAV, or delegated to on-the-ground nodes. The performance of the different schemes will be evaluated in ns-3 using the mmwave module [15] and real-world UAV data collected in the Stanford Drone Dataset [16], which promotes extreme levels of realism and permits to analyze the network considering full-stack end-to-end metrics and a real-world dataset as input. Our preliminary results demonstrate that data processing at the BS guarantees more efficient communication, in view of the limited power and computational capabilities of the UAV. The rest of this paper is organized as follows. In Sec. II we review the most recent works on UAV-based autonomous driving research, in Sec. III we present our system model and communication scenarios, in Sec. IV we describe how we extended Network Simulator 3 (ns-3) to simulate UAVto-ground communication, in Sec. V we present our main numerical results, whereas conclusions are summarized in Sec. VI. II. STATE OF THE ART In this section, we discuss the recent research in the area of UAV-based autonomous driving, specifically computational offloading and data dissemination. Hayat et al., in [17], evaluate the burden of data (image) processing for UAV autonomous navigation, that can be done on board, or fully/partially offloaded to an edge server. Similarly, in [18] the authors study the performance of UAV edge computing using Hydra, an architecture for the establishment of flexible sensing-analysiscontrol pipelines over autonomous airborne systems. If cellular infrastructures are unavailable (e.g., damaged by natural disasters), data offloading can be also between ground vehicles, whose limited computing and energy resources make it difficult to execute computationally sensitive mobile applications on board, and aerial platforms. In [19], the author suggest to offload computing tasks from the ground to UAVs that carry edge serves, and propose an algorithm to minimize the total energy and time required for the UAVs to complete the offloaded tasks, while optimizing their 3D flying height and horizontal positions. Computational offloading may be also assisted by high altitude platforms (HAPs), as proposed in [20], where the authors designed a framework to offload communication and computational resources to aerial nodes to maximize the total number of user device requests with satisfied delay requirements while minimizing the total energy consumption. Similarly, in our prior work [10], we formalized an optimization problem in which tasks are modeled as a Poisson arrival process, and applied queuing theory to identify how ground vehicles should offload resource-hungry tasks to UAVs, HAPs, or a combination of the two. With respect to the state of the art, in this paper we do not focus on computational offloading, but rather on how UAV data are disseminated to ground vehicles, and where to process them. Moreover, while most literature focuses on UAV-toground communication in the legacy bands, and/or considers link-level evaluations, we perform end-to-end simulations in ns-3 considering mmWave frequencies, as well as both onboard and fully-offloaded computation. III. UAV-TO-GROUND COMMUNICATION SCENARIOS In this section we present four possible strategies for UAVto-ground communication, as illustrated in Fig. 1. Notably, each scenario consists of the following elements: a UAV that is recording videos from the sky, N autonomous cars on the ground, and a BS (or gNB, in 5G NR parlance) forwarding data from the UAV to the cars. The UAV is placed at the center of the scene (e.g., at a road intersection) at height h, the BS is placed on the ground, perpendicular to the UAV in order to maintain a stable connection, and vehicles are allocated randomly within a rectangle. The four models differ in the way the data are broadcast, and the location of the computing platform (the chip icon in Fig. 1) where UAV sensory data is processed to detect critical road entities in the scene. a) Scenario 1 -Multiple full frames (MFF): In the first scenario ( Fig. 1a) the drone is sending frames via the BS at a rate F frame to each ground vehicle, that will eventually perform object detection using its own on-board computational capacity. The frame rate is not optimized so, in a situation where all packets are delivered without errors (best-case scenario), the total frame rate in the first link (UAV-BS) would be equal to N F frame , i.e., the sum of the frames sent in each second link (BS-vehicle). On the downside, computation on board of vehicles may incur non-negligible delays given the limited capacity of budget vehicles, and the data rate in the first link would be N times larger than the total data rate of each second link. In turn, this approach does not require coordination with the BS. b) Scenario 2 -Broadcast full frames (BFF): In the second scenario (Fig. 1b), frames are sent at an optimized rate. While in the MFF scenario sensory data in the first link were replicated N times, with optimized settings the UAV sends only one frame to the BS, that will create N copies of the received packets and eventually forward them to each of the N cars. Finally, each vehicle will perform object detection on the received data, which may still incur long delays due to computational limitations. Ideally, the throughput in the first link would be equal to the throughput of each second link. In other words, the ideal throughput of the first link is N times smaller than the sum of the throughput of all the second links. c) Scenario 3 -Broadcast frames and annotations (BFA): In the third scenario (Fig. 1c) the UAV sends one copy of all the frames to the BS, which then performs object detection on the received data. This approach promotes faster processing than in the previous scenarios, as BSs are typically connected to continuous power sources and do not pose strict limitations in terms of computational capacity, space and storage. Eventually, the processed output (i.e., the bounding boxes of the detected objects, also referred to as annotations) is returned to the ground vehicles at a frame rate F anno , in a packet of a much smaller size than the original frame, which allows to reduce the communication latency on the second links. In particular, the size of an annotation α is calculated as α = N β, where β is the memory size of a bounding box and N is the number of detected objects. To find the value of β, we made offline simulations to generate bounding boxes from real-world UAV video recordings collected in the Stanford Drone Dataset [16]. To do so, we used the YoloV5 algorithm [21], a common benchmark in this field. d) Scenario 4 -Broadcast annotations only (BAO): In the fourth scenario (Fig. 1d), object detection is performed as soon as the frame is generated, i.e., on board of the UAV. While this allows low-size annotations to be sent already on the first link, as well as on the second link, thereby reducing the overall communication latency, the computational capacity of aerial nodes is generally lower than that available at the BSs, which may increase the processing delay compared to the BFA scenario. Ideally, the per-user throughput in the first link is equal to the per-user throughput in each second link. IV. NS-3 IMPLEMENTATION In this section we describe how we extended the ns-3 simulator to implement the four communication scenarios presented in Sec. III. While most simulators focus on Physical (PHY) and Medium Access Control (MAC) layer designs, and sacrifice the accuracy of the higher layers to reduce the computational complexity, ns-3 incorporates accurate models of the whole protocol stack, thus enabling scalable end-to-end simulations. In particular, in our work communication nodes operate at mmWaves. As such, we use the ns3-mmwave module, described in [15], which enables the simulation of 5G-NR-compliant end-to-end cellular networks at mmWave frequencies. It features a complete stack for User Equipments (UEs) and gNBs, with custom PHY (described in [22]) and MAC layers with an Orthogonal Frequency Division Multiplexing (OFDM) frame structure, dynamic Time Division Duplexing (TDD), Adaptive Modulation and Coding (AMC), and several scheduler implementations. Thanks to the integration with ns-3, it also features a complete implementation of the User Datagram Protocol (UDP) stack. The simulator also features a 3GPP-compliant channel model, as well as antenna and beamforming models for mmWave communications [23]. In terms of the implementation, in the MFF scenario UDP is installed at the end vehicles: the UAV is set up as a client while vehicles as servers. With this configuration, the UAV sends the same amount of packets N F frame to every car. On the other hand, to implement broadcast communications in the other scenarios, some changes were applied to the ns3-mmwave module that manages the forwarding of the packets at the BS. With these changes, the UAV is sending only one copy of each packet to the BS, that in turn produces N copies of the received packet, which are transmitted to the vehicles on the ground. Finally, the packet size and the packet sending rate are set. For the simulations where the size of a UAV frame is larger than the maximum size of a UDP packet (UDP pck ), the data must be split and sent in smaller packets of size UDP pck . In MFF and BFF and in the first link of BFA, the packets sending rate is equal to 1/((i · F frame )/UDP pck ) where i is the total size in Byte of the sensory frame to be sent, and F frame is the source frame rate. For the second link in BFA, and in BAO, the annotation rate is 1/(F anno /(β · N )). V. PERFORMANCE EVALUATION In this section we introduce our performance evaluation setup, and discuss the simulation performance of the different UAV-to-ground communication scenarios. A. Simulation Parameters Our simulator implements the communication scenarios described in Sec. III. The main parameters for the simulations are described in Table I. We also provide the source code and simulation scripts as a reference. 1 We run the simulations with the ns-3 Simulation Execution Manager (SEM) library [24], which takes care of running multiple statistically independent instances of the same scenario and collecting relevant metrics. The ns-3 simulation time is set to 15 s, and we consider the following parameters: • Application frame rate. According to the Stanford Drone Dataset, the camera of the UAV records at F frame ∈ {15, 30} frames per second (FPS). Therefore, we set the frame rate of the ns-3 application generating UAV data to 1/F frame (independently on weather annotations or full frames are transmitted). Drone Dataset videos to be sent from the UAV, as shown in Fig. 2. Notice that the frame size does not follow an increasing trend with the number of vehicles in the scene, i.e., the size of each full frame does not necessarily correlate with the number of vehicles. • Annotation size α. It is the size of an annotation produced after object detection. It is modeled as α = βN , where β = 39.7 bytes is the average size of a single bounding box detected by the YoloV5 detection algorithm [21], and N is the number of vehicles in the scene. We consider 90 random UAV frames from the Stanford Drone Dataset, and run multiple statistically independent simulations to capture the following end-to-end metrics: • Per-user throughput. It corresponds to the total number of received bytes per user divided by the total simulation time, averaged over all connected vehicles. • Per-user reliability. It is measured as the ratio between the number of packets delivered to the cars without errors and the total number of packets transmitted by the UAV. • Per-user latency. It is modeled as L1 + L2, where L1 represents the latency between the UAV and the BS (uplink) and L2 represents the latency between the BS and the vehicles (downlink), averaged over all connected vehicles. Both L1 and L2 account for the transmission time as well as the queuing time resulting from NRspecific scheduling and buffering, as modeled in the ns3-mmwave module. B. Performance Evaluation In the following paragraphs we compare the performance of the different communication scenarios described in Sec. III. Fig. 3 reports the end-to-end throughput, latency and reliability for all configurations. MFF scenario. The results in Fig. 3 clearly highlight that the wireless network (despite a bandwidth of 1 GHz) cannot support MFF with more than 11 vehicles and 30 FPS. Notably, the throughput (Fig. 3a) and reliability (Fig. 3c) decrease, while the latency (Fig. 3b) increases to more than 200 ms. This is due to the a bottleneck in the first link (UAV-BS), which transmits N times more data compared to each second link (BS-car). Notice that the latency of the second link is not particularly representative, as it is relative to only the correctly received packets. Given that most packet losses happen on the first link, which makes the system less congested, the (few) packets that make it to the second link are then transmitted with very low latency. This also explains the latency plateau at around 200 ms, due to the fact that the UAV transmit buffer (e.g., at the Radio Link Control (RLC) layer) overflows for more than 11 cars. On the other hand, the MFF configuration can better support an application generating data at 15 FPS, as a consequence of the 50% less traffic on the UAV-BS link, and the resulting less populated RLC queues at the UAV. The system performance is stable for up to 19 vehicles (Fig. 3a). After this threshold the UAV buffer saturates causing a degradation in latency (which reaches the 200 ms plateau) and reliability. BFF scenario. The BFF strategy is more efficient than MFF, as it does not saturate the UAV buffer and the capacity of the first link by avoiding unnecessary duplication of the frames (c) Per-user reliability. Fig. 3: Performance evaluation for the 4 communication scenarios, with two different frame rates. MFF stands for multiple full frames, with the UAV sending one frame for each vehicle, then relayed by the BS. BFF stands for broadcast full frames, with the UAV sending a common reference frame for all vehicles, then relayed by the BS. With broadcast frames and annotations, or BFA, the UAV sends the common reference frame, and the BS forwards only the annotations. Finally, with broadcast annotations only, or BAO, the UAV sends annotations which are then relayed by the BS. in the uplink. The performance of BFF with an application rate of 30 FPS only degrades for more than 20 connected vehicles. Unlike MFF, this is due to a saturation of resources in the downlink, i.e., in the second links from the BS to the end vehicles. In fact, Fig. 2 shows that the file size of a frame with few users (e.g., 4) is comparable with the file size of a frame with 21 users. In the first case, however, the resources on the wireless link are split only among the UAV (uplink) and 4 more users (downlink), while in the latter more than 20 users contend for the same downlink resources, which may saturate the available capacity. BFF also easily sustains the performance with 15 FPS at the application. This strategy, while being more efficient than MFF, requires the support for multicasting support at the RAN, a feature that has been only recently standardized in 5G networks [25], [26]. BFA scenario. BFA, which assumes data processing at the BS and the transmission of annotations in the downlink, manages to easily support the traffic for all the users in the tested environment. Fig. 3c and Fig. 3b show that this scheme provides ultra-high reliability, with an average of 98.472% correctly delivered annotations, and an end-to-end latency as low as 3 ms, respectively. In this case, the performance with 30 and 15 FPS is comparable. Notice that the BFA throughput is reasonably lower than that of MFF and BFF. This is due to the much lower size of annotations compared to full frames (on average up to 4 orders of magnitude), which limits the source rate on the second link. In Fig. 4 we plot the average per-user latency in the uplink L 1 , i.e., the UAV-BS link, and in the downlink L 2 , i.e., the BS-car links. We can see that in BFA the throughput in the uplink, where full frames are transmitted, is higher than the throughput in the downlink. Nonetheless, the latency is lower in the first link. This can be explained by considering the resource scheduling process implemented in the simulated 5G BS. As discussed in Sec. IV, this follows a TDD scheme, where the resources are first split between the uplink and the downlink, and then assigned to each uplink or downlink user. In this case, the UAV is the only uplink user -thus it does not have to contend for resources with other users. Additionally, to accommodate for the analog beamforming implemented at the BS to improve the link budget with the vehicles at mmWaves, each symbol at the PHY layer is allocated to a single user at a time. This further deteriorates the contention performance in the downlink, and results into lower efficiency (in the resource allocation) with higher latency. Future developments can try to address this by using different scheduler implementations, frequency range (which does not require beamforming), or hybrid beamforming schemes [27]. BAO scenario. In this case we assume data processing at the UAV, and the transmission of only annotations (with a size proportional to the number of vehicles) in the two links. While the throughput is as low as 0.208 Mbit/s (0.104 Mbit/s) for 30 (15) Overall end-to-end comparison. This analysis clearly highlights that throughput should not be the only metric used to profile the performance of data dissemination systems in the context of vehicular networks. A more important metric, which is independent of the application source rate, is indeed the end-to-end reliability, which indicates (in this case) how many frames or annotations are received correctly. The throughput analysis then can provide inputs on what kind of dissemination strategy a certain wireless network (in this case, a 5G mmWave deployment) can support. In Fig. 3a the highest throughput is obtained in the BFF scenario with 30 FPS, where the UAV is sending only one copy of each frame in the first link and then the BS is broadcasting the received information to each connected vehicle. However, with more than 20 users the latency drastically increases (Fig. 3b) and the reliability decreases (Fig. 3c). The scenarios that are transmitting only annotations (BFA only in the downlink, BAO in both uplink and downlink) have the highest reliability and the lowest latency overall. Moreover, in both scenarios, the information that each car is receiving is the same, but in one case the object detection has to be performed by the BS while in the other it has to be performed by the UAV. On one side, the UAV has generally more energy and computational constraints than the BS, which makes BOA more desirable. On the other side, BFA may be the only available choice in those environments lacking coverage from terrestrial infrastructures [28]. Further studies on power consumption will help understand what is the final choice for UAV-to-ground communication, depending on whether it is feasible to perform object detection at the UAV. VI. CONCLUSIONS In this paper we presented and evaluated different communication techniques between UAVs and ground vehicles for the dissemination of UAV sensory observations to augment vehicles' autonomous driving capabilities, based on high-capacity mmWave links. We assessed the performance of four different communication scenarios, with applications transmitting data at 15 and 30 FPS with UDP at the transport layer. An extensive performance evaluation based on real-world UAV data and considering ns-3 simulations showed that those configurations that transmit annotations (rather than full frames) achieve the best performance in terms of latency and reliability. For up to 21 connected users, they guarantee a latency of around 2 ms, and a reliability above 99%. Our results provide a first, quantitative evaluation of the feasibility of complementing vehicle's on-board sensors with UAV data from the sky. As part of our future work, we will combine the communication network model with an energy model that profiles the computational complexity of the object detection task on the UAV, BS, and ground vehicles hardware. Fig. 1 : 1An illustration of the four UAV-to-ground communication scenarios. A chip icon is placed adjacent to the node that is performing the object detection. Fig. 2 : 2Size of each frame vs. the number of detected objects/vehicles.• Number of vehicles (N ). It corresponds to the number of objects detected in each frame of the Stanford Drone Dataset videos, and sets the number of vehicles in each simulation. Based on offline simulations, we obtained that each processed frame featured from 4 to 21 vehicles, as shown in Fig. 2. • Full frame size. It is the size of a frame of the Stanford ) Per-user latency. 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[ "Limitations of the Lipschitz constant as a defense against adversarial examples", "Limitations of the Lipschitz constant as a defense against adversarial examples" ]
[ "Todd Huster [email protected] \nPerspecta Labs\n07920Basking RidgeNJUSA\n", "Cho-Yu Jason Chiang \nPerspecta Labs\n07920Basking RidgeNJUSA\n", "Ritu Chadha \nPerspecta Labs\n07920Basking RidgeNJUSA\n" ]
[ "Perspecta Labs\n07920Basking RidgeNJUSA", "Perspecta Labs\n07920Basking RidgeNJUSA", "Perspecta Labs\n07920Basking RidgeNJUSA" ]
[]
Several recent papers have discussed utilizing Lipschitz constants to limit the susceptibility of neural networks to adversarial examples. We analyze recently proposed methods for computing the Lipschitz constant. We show that the Lipschitz constant may indeed enable adversarially robust neural networks. However, the methods currently employed for computing it suffer from theoretical and practical limitations. We argue that addressing this shortcoming is a promising direction for future research into certified adversarial defenses.
10.1007/978-3-030-13453-2_2
[ "https://arxiv.org/pdf/1807.09705v1.pdf" ]
51,721,243
1807.09705
3cc8a615164a040774ae68bc25ce8342b68e5c05
Limitations of the Lipschitz constant as a defense against adversarial examples 25 Jul 2018 Todd Huster [email protected] Perspecta Labs 07920Basking RidgeNJUSA Cho-Yu Jason Chiang Perspecta Labs 07920Basking RidgeNJUSA Ritu Chadha Perspecta Labs 07920Basking RidgeNJUSA Limitations of the Lipschitz constant as a defense against adversarial examples 25 Jul 2018adversarial examples, Lipschitz constant Several recent papers have discussed utilizing Lipschitz constants to limit the susceptibility of neural networks to adversarial examples. We analyze recently proposed methods for computing the Lipschitz constant. We show that the Lipschitz constant may indeed enable adversarially robust neural networks. However, the methods currently employed for computing it suffer from theoretical and practical limitations. We argue that addressing this shortcoming is a promising direction for future research into certified adversarial defenses. Introduction Machine learning models, such as deep neural networks (DNNs), have been remarkably successful in performing many tasks [5] [7] [9]. However, it has been shown that they fail catastrophically when very small distortions are added to normal data examples [6] [14]. These adversarial examples are easy to produce [6], transfer from one model to another [11] [15], and are very hard to detect [2]. Many methods have been proposed to address this problem, but most have been quickly overcome by new attacks [1] [3]. This cycle has happened regularly enough that the burden of proof is on the defender that her or his defense will hold up against future attacks. One promising approach to meet this burden is to compute and optimize a certificate: a guarantee that no attack of a certain magnitude can change the classifier's decision for a large majority of examples. In order to provide such a guarantee, one must be able to bound the possible outputs for a region of input space. This can be done for the region around a specific input [8] or by globally bounding the sensitivity of the function to shifts on the input, i.e., the function's Lipschitz constant [13] [16]. Once the output is bounded for a given input region, one can check whether the class changes. If not, there is no adversarial example in the region. If the class does change, the model can alert the user or safety mechanisms to the possibility of manipulation. We argue in this paper that despite the achievements reported in [13], Lipschitzbased approaches suffer from some representational limitations that may prevent them from achieving higher levels of performance and being applicable to more complicated problems. We suggest that directly addressing these limitations may lead to further gains in robustness. This paper is organized as follows: Section 2 defines the Lipschitz constant and shows that classifiers with strong Lipschitz-based guarantees exist. Section 3 describes a simple method for computing a Lipschitz constant for deep neural networks, while Section 4 presents experimental and theoretical limitations for this method. Section 5 describes an alternative method for computing a Lipschitz constant and presents some of its limitations. Finally, Section 6 presents conclusions and a long term goal for future research. Lipschitz Bounds We now define the Lipschitz constant referenced throughout this paper. Definition 1. Let a function f be called k-Lipschitz continuous if ∀x 1 , x 2 ∈ X : d Y (f (x 1 ), f (x 2 )) ≤ kd X (x 1 , x 2 ) (1) where d X and d Y are the metrics associated with vector spaces X and Y , respectively. Loosely speaking, a Lipschitz constant k is a bound on the slope of f : if the input changes by ǫ, the output changes by at most kǫ. If there is no valuê k where f isk-Lipschitz continuous andk < k, then we say k is the minimal Lipschitz constant. In this paper, we restrict our analysis to Minkowski L p spaces with distance metric · p . We now show that global Lipschitz constants can in principle be used to provide certificates far exceeding the current state-of-the-art, and thus are worthy of further development. Proposition 1. Let D be a dataset D = (x i , y i ) | i = 1, ..., m, x i ∈ R d , y i ∈ {−1, 1} where x i = x j for y i = y j . Let c be a positive scalar such that ∀i, j : y i = y j → ||x i − x j || p > c (2) for p ≥ 1. There exists a 2 c -Lipschitz function f : X → R where ∀i : sign(f (x i + δ)) = y i for ||δ|| p < c 2 . Proof. We relegate the full proof to appendix A.1, but we define a function meeting the criteria of the proposition that can be constructed for any dataset: f (x) =      1 − 2 c ||x − x + || p if ||x − x + || p < c 2 −1 + 2 c ||x − x − || p if ||x − x − || p < c 2 0 otherwise(3) where x + and x − are the closest vectors to x in D with y = 1 and y = −1, respectively. The function f described above shows that the Lipschitz method can be used to provide a robustness guarantee against any perturbation of magnitude less than c 2 . This can be extended to a multi-class setting in a straightforward manner by using a set of one vs. all classifiers. Table 1 shows the distance to the closest out-of-class example for the 95th percentile of samples; i.e., 95% of samples are at least c away from the nearest neighbor of a different class. Proposition 1 implies the existence of a classifier that is provably robust for 95% of samples against perturbations of magnitude c 2 . This bound would far exceed the certifications offered by current methods, i.e., [8] [13] [16], and even the (non-certified) adversarial performance of [10]. It is important to note that the existence of a c 2 -Lipschitz function in Proposition 1 does not say anything about how easy it is to learn such a function from examples that generalizes to new ones. Indeed, the function described in the proof is likely to generalize poorly. However, we argue that current methods for optimizing the Lipschitz constant of a neural network suffer much more from underfitting than overfitting: training and validation certificates tend to be similar, and adding model capacity and training iterations do not appear to materially improve the training certificates. This suggests that we need more powerful models. The remainder of this paper is focused on how one might go about developing more powerful models. Atomic Lipschitz Constants The simplest method for constructing a Lipschitz constant for a neural network composes the Lipschitz constants of atomic components. If f 1 and f 2 are k 1and k 2 -Lipschitz continuous functions, respectively, and f (x) = f 2 (f 1 (x)), then f is k-Lipschitz continuous where k = k 1 k 2 . Applying this recursively provides a bound for an arbitrary neural network. For many components, we can compute the minimal Lipschitz constant exactly. For linear operators, l W,b (x) = W x + b, the minimal Lipschitz constant is given by the matrix norm of W induced by L p : W p = sup x =0 W x p x p(4) For p = ∞, this is equivalent to the largest magnitude row of W : W ∞ = max wi∈W w i 1(5) The L 2 norm of W is known as its spectral norm and is equivalent to its largest singular value. The element-wise ReLU function ReLU (x) = max(x, 0) has a Lipschitz constant of 1 regardless of the choice of p. Therefore, for a neural network f composed of n linear operators l W1,b2 , ..., l Wn,bn , and ReLUs, a Lipschitz constant k is provided by k = n i=1 W i p(6) Several recent papers have utilized this concept or an extension of it to additional layer types. [14] uses it to analyze the theoretical sensitivity of deep neural networks. [4] and [12] enforce constraints on the singular values of matrices as a way of increasing robustness to existing attacks. Finally, [16] penalizes the spectral norms of matrices and uses equation 6 to compute a Lipschitz constant for the network. Limitations of Atomic Lipschitz Constants One might surmise that this approach can solve the problem of adversarial examples: compose enough layers together with the right balance of objectives, overcoming whatever optimization difficulties arise, and one can train classifiers with high accuracy, guaranteed low variability, and improved robustness to attacks. Unfortunately, this does not turn out to be the case, as we will show first experimentally and then theoretically. Experimental Limitations First, we can observe the limits of this technique in a shallow setting. We train a two layer fully connected neural network with 500 hidden units f = l W2,b2 • ReLU • l W1,b1 on the MNIST dataset. We penalize W 1 p W 2 p with weight λ p . We denote the score for class i as f i (x) and the computed Lipschitz constant of the difference between f i (x) and f j (x) as k ij . We certify the network for example x with correct class i against a perturbation of magnitude ǫ by verifying that show results for L ∞ and L 2 , respectively. In both cases, adding a penalty provides a larger region of certified robustness, but increasing the penalty hurts performance on unperturbed data and eventually ceases to improve the certified region. This was true for both test and training (not shown) data. This level of certification is considerably weaker than our theoretical limit from Proposition 1. f i (x) − f j (x) − k ij ǫ > 0 for i = j. There also does not appear to be much certification benefit to adding more layers. We extended the methodology to multi-layer networks and show the results in figures 1 (c) and (d). Using the λ ∞ penalty proved difficult to optimize for deeper networks. The λ 2 penalty was more successful, but only saw a mild improvement over the shallow model. The results in (d) also compare favorably to those of [16], which uses a 4 layer convolutional network. Theoretical Limitations We now consider the set of neural networks with a given atomic Lipschitz bound and the functions it can compute. This set of functions is important because it limits how well a neural network can split a dataset with particular margins, and thus how strong the certificate can be. Definition 2. Let A p k be the set of neural networks with an atomic Lipschitz bound of k in L p space: A p k l Wn,bn • · · · • ReLU • l W1,b1 | i W i p ≤ k, n ≥ 2(7) We focus our analysis here on L ∞ space. To show the limitations of A ∞ k , consider the simple 1-Lipschitz function f (x) = |x|. Expressing f with ReLU's and linear units is simple exercise, shown in figure 2. However, since 1 −1 ∞ 1 1 ∞ = 2,(8) the neural network in figure 2 is a member of A ∞ 2 , but not A ∞ 1 . This is only one possible implementation of |x|, but as we will show, the atomic component method cannot express this function with a Lipschitz bound lower than 2, and the situation gets worse as more non-linear variations are added. We now provide two definitions that will help delineate the functions that the neural networks in A ∞ k can compute. For a function f : R → R, let the total variation be defined as V b a (f ) sup T ∈T ti∈T |f (t i ) − f (t i−1 )| (9) where T is the set of partitions of the interval [a, b]. The total variation captures how much a function changes over its entire domain, which we will use on the gradients of neural networks. V ∞ −∞ is finite for neural network gradients, as the gradient only changes when a ReLU switches states, and this can only happen a finite number of times for finite networks. Clearly, for the slope of the absolute value function, this quantity is 2: the slope changes from -1 to 1 at x = 0. I(f ) V ∞ −∞ (f ) + |f (∞)| + |f (−∞)|(10) and call it the intrinsic variability of f . As we will show, the intrinsic variability is a quantity that is nonexpansive under the ReLU operation. The intrinsic variability the slope of the absolute value function is 4: we add the magnitude of the slopes at the extreme points, 1 in each case, to the total variation of 2. We now begin a set of proofs to show that A ∞ k is limited in the functions it can approximate. This limit does not come from the Lipschitz constant of a function f , but by the intrinsic variability of its derivative, f ′ . Lemma 1. For a linear combination of functions f (x) = i w i f i (x), I(f ′ ) ≤ i |w i |I(f ′ i ).(11) Proof. Proof is relegated to appendix A.2 Definition 5. Let a function f : R → R be called eventually constant if ∃t − ∈ R, f ′ (t) = f ′ (t − ), t ≤ t − (12) ∃t + ∈ R, f ′ (t) = f ′ (t + ), t ≥ t +(13) Lemma 2. Let f (t) be a function where f ′ (t) is eventually constant. For the ReLU activation function g(t) = max(f (t), 0), I(g ′ ) ≤ I(f ′ )(14) Proof. Proof is relegated to appendix A.3 Theorem 1. Let f ∈ A ∞ k be a scalar-valued function f : R d → R. Let h W0,b0 = f • l W0,b0 where W 0 ∈ R d×1 , b 0 ∈ R d and W 0 ∞ = 1.I(h ′ W0,b0 ) ≤ 2k.(15) Proof. Proof is relegated to appendix A.4 A function in A ∞ k has a hard limit on the intrinsic variability of its slope along a line through its input space. If we try to learn the absolute value function while penalizing the bound k, we will inevitably end up with training objectives that are in direct competition with one another. One can imagine more difficult cases where there is some oscillation in the data manifold and the bounds deteriorate further: for instance sin(x) is also 1-Lipschitz, but can only be approximated with arbitrarily small error by a member of A ∞ ∞ . While this limit is specific to A ∞ k , since W 2 ≤ W ∞ , it also provides a limit to A 2 k . Paired-layer Lipschitz Constants and Their Limitations We have shown the limitations of the atomic bounding method both experimentally and theoretically, so naturally we look for other approaches to bounding the Lipschitz constant of neural network layers. A fairly successful approach was given by [13]. [13] presents a method for bounding a fully connected neural network with one hidden layer and ReLU activations, which yielded impressive performance on the MNIST dataset. This approach optimizes the weights of the two layers in concert, so we call it the paired-layer approach. The paper does not attempt to extend the method to deeper neural networks, but it can be done in a relatively straightforward fashion. Certifying a Two-layer Neural Network Ignoring biases for notational convenience, a two-layer neural network with weights W 1 and W 2 can be expressed f (x) = W 2 diag(s)W 1 x(16) where s = W 1 x > 0. We consider a single output, although extending to a multiclass setting is straightforward. If s were fixed, such a network would be linear with Lipschitz constant W 2 diag(s)W 1 p . [13] accounts for a changeable s by finding the assignment of s that maximizes the L ∞ Lipschitz constant and using this as a bound for the real Lipschitz constant: k ≤ max s∈{0,1} d W 2 diag(s)W 1 ∞(17) They convert this problem to a mixed integer quadratic program and bound it in a tractable and differential manner using semi-definite programming, the details of which are explained in [13]. We can add a penalty on this quantity to the objective function to find a model with relatively high accuracy and low Lipschitz constant. We did not have access to the training procedure developed by [13], but we were able to closely replicate their results on MNIST and compare them to the atomic bounding approach, shown in figure 3 (a). Figure 3 shows that there are practical benefits to the paired-layer approach, and we can also show a corresponding increase in expressive power. Similar to A p k , we define a set of neural networks M k , although we will restrict the definition to 2 layer networks in L ∞ space: Definition 6. Let M k be the set of two-layer neural networks with a pairedlayer Lipschitz bound of k in L ∞ space: Theoretical Benefits and Limitations of Paired-layer Approach M k l W2,a2 • ReLU • l W1,a1 | max s∈{0,1} d W 2 diag(s)W 1 ∞ ≤ k(18) M k can express functions that A ∞ k cannot. For example, we can apply the paired-layer method to the neural network in figure 2 by enumerating the different cases. In this case the bound is tight, meaning that the neural network is in M 1 . From Theorem 1, we know that this function cannot be expressed by any member of A ∞ 1 . It is easy to see that any two layer neural network in A ∞ k is also in M k , so we can say confidently that the paired-layer bounds are tighter than atomic bounds. This additional expressiveness is not merely academic. Figure 3 (b) shows the output of the networks from (a) along a particular line in input space, scaled by the given Lipschitz bound. The function learned by the paired-layer method does in fact exhibit an intrinsic variability larger than 2k, meaning that function cannot be represented by a network in A ∞ k . This suggests that the gains in performance may be coming from the increased expressiveness of the model family. It is still easy to construct functions for which the paired-layer bounds are loose, however. Figure 4 shows a 1-Lipschitz function and a corresponding neural network that is only in M 2 . The problem arises from the fact that the two hidden units cannot both be on, but the quadratic programming problem in equation 17 implies that they can. For a 1-D problem, the bound essentially adds up the magnitudes of the paths with positive weights and the paths with negative weights and takes the maximum. A higher dimensional problem can be reduced to a 1-D problem by considering arbitrary lines through the input space. The expressive limitations of M k are apparent when we consider its components. Any neural network in M k is a sum of combinations of the four basic forms in figure 5, with various biases and slopes. The sum of the slope magnitudes from the positive paths can be no greater than k, and likewise for the negative paths. Each form has a characteristic way of affecting the slope at the extremes and changing the slope. For instance form (a) adds a positive slope at +∞ as well as a positive change in f ′ . From here we can see that there is still a connection between the total variation and extreme values of f ′ and the bound k. While the paired-layer bounds are better than the atomic ones, they still become arbitrarily bad for e.g., oscillating functions. Conclusions We have presented a case that existing methods for computing a Lipschitz constant of a neural network suffer from representational limitations that may be preventing them from considerably stronger robustness guarantees against adversarial examples. Addressing these limitations should enable models that can, at a minimum, exhibit strong guarantees for training data and hopefully extend these to out-of-sample data. Ideally, we envision universal Lipschitz networks: a family of neural networks that can represent an arbitrary k-Lipschitz function with a tight bound. The development of such a family of models and methods for optimizing them carries the potential of extensive gains in adversarial robustness. that each continuously differentiable piece is 2 c -Lipschitz. Using Definition 1, we must show |f (x) − f (x + δ)| ≤ 2 c ||δ|| p .(21) For the first condition of f with a fixed x + , we get 1 − 2 c ||x − x + || p − 1 − 2 c ||x + δ − x + || p ≤ 2 c ||δ|| p (22) 2 c ||x − x + || p − ||x + δ − x + || p ≤ 2 c ||δ|| p ,(23) which holds for p ≥ 1 due to the Minkowski inequality. The same holds for the second condition. Since the third condition is constant, f (x) must be 2 c -Lipschitz and the proof is complete. ⊓ ⊔ A.2 Proof of Lemma 1 Proof. Using the chain rule, we get f ′ (t) = i w i f ′ i (t).(24) The triangle inequality gives us the following two inequalities |f ′ (t)| = i w i f ′ i (t) ≤ i |w i f ′ i (t)| = i |w i ||f ′ i (t)| (25) |f ′ (t i ) − f ′ (t i−1 )| ≤ i |w i f ′ i (t i ) − w i f ′ i (t i−1 )| = i |w i ||f ′ i (t i ) − f ′ i (t i−1 )| (26) Let T f ′ be a maximal partition for V ∞ ∞ (f ′ ), giving us I(f ′ ) = ti∈T f ′ |f ′ (t i ) − f ′ (t i−1 )| + |f ′ (∞)| + |f ′ (−∞)|(27) We complete the proof by substituting with (25) and (26) and reordering the terms : I(f ′ ) ≤ i |w i | ti∈T f |f ′ i (t i ) − f ′ i (t i−1 )| + |f ′ i (∞)| + |f ′ i (−∞)| = i |w i |I(f ′ i ). (28) ⊓ ⊔ where w i u,v is element (u, v) of W i . Therefore V ∞ −∞ (σ ′ 0,u ) = 0(39) We also have ∀t, |σ ′ 0,u | = |w 0 u,1 |, so by Definition 4 I(σ ′ 0,u ) = 2|w 0 u,1 | ≤ 2.(40) We recursively define functions for each unit in layers 1 to n: g i,v (t) = u w i u,v σ i−1,u (t)(41) σ i,u (t) = max(g i,u (t), 0) Applying Lemma 2 and noting that a function composed of ReLU and linear operators is eventually constant, we get I(σ ′ i,u ) ≤ I(g ′ i,u )(43) Applying Lemma 1, we get I(g ′ i,v ) ≤ u |w i u,v |I(σ i−1,u )(44) Furthermore, we can say max v I(g ′ i,v ) ≤ W i ∞ max u I(g ′ i−1,u )(45) Finally, we conclude the proof by recursively applying (45) on the base case in (40) to yield I(h ′ W0,b0 ) = I(g ′ n,1 ) ≤ 2 n i=1 W i ∞ ≤ 2k(46) ⊓ ⊔ Fig. 1 . 1Experimental results from atomic Lipschitz penalties. On the left, the L∞ norm is used for both the perturbation and the penalty, while on the right, L2 is used Fig. 2 . 2The absolute value function (left) and a neural network that implements it (right) Definition 3. Definition 4 . 4For a function f : R → R, define a quantity For any selection of W 0 and b 0 , Fig. 3 . 3(a) Results comparing penalizing the atomic Lipschitz bound and the pairedlayer bound (b) Neural network outputs along the line w0t and their intrinsic varibilities. Values are scaled by the given Lipschitz constant Fig. 4 . 4A 1-Lipchitz function (left) and a neural network that implements it (right) Fig. 5 . 5The four forms of components of a two layer neural network, and their distinguishing characteristics Table 1 . 1Distances to closest out-of-class example, 95th percentile.Metric MNIST CIFAR-10 L1 29.4 170.8 L2 4.06 4.58 L∞ 0.980 0.392 Acknowledgement: This research was partially sponsored by the U.S. Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-13-2-0045 (ARL Cyber Security CRA). The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation here on.A ProofsA.1 Proof of Proposition 1Proof. Consider the functionwhere x + and x − are the closest vectors to x in D with y = 1 and y = −1, respectively. Since ||x + − x − || p > c, the conditions are mutually exclusive. When y i = 1 and ||δ|| p < c 2 ,The inverse is true for y i = −1, therefore sign(f (x i + δ)) = y i holds for all i. f is continuous at the non-differentiable boundaries between the piecewise conditions of f and the selections of x + and x − . Therefore, it suffices to show). In this case,Similarly,Putting the different intervals together, we getSo the statement holds when our assumption about f is met. To address cases where f has negative values in [t − , t + ], consider an interval (t 1 ,We note that f ′ (t 1 ) < 0 and f ′ (t 2 ) > 0. Since f ′ must transition from f ′ (t 1 ) to f ′ (t 2 ), over (t 1 , t 2 ),Since g ′ transitions from f ′ (t 1 ) to 0 to f ′ (t 2 ) over (t 1 , t 2 ) so,Applying this to all such intervals gives usand therefore I(g ′ ) ≤ I(f ′ ) ⊓ ⊔A.4 Proof of Theorem 1Proof. Combining the definition of h W0,b0 with Definition 2, we can see that h W0,b0 = l Wn,bn •· · ·•ReLU •l W1,b1 •l W0,b0 and n i=0 W i ∞ ≤ k. We consider the additional linear transform as the zeroth layer of a modified network. Consider unit u in the zeroth layer as a function σ 0,u (t). σ ′ 0,j (t) is constant, with σ ′ 0,u (t) = |w 0 u,1 | ≤ 1 (38) Obfuscated gradients give a false sense of security: Circumventing defenses to adversarial examples. A Athalye, N Carlini, D A Wagner, CoRR abs/1802.00420Athalye, A., Carlini, N., Wagner, D.A.: Obfuscated gradients give a false sense of security: Circumventing defenses to adversarial examples. CoRR abs/1802.00420 (2018) Adversarial examples are not easily detected: Bypassing ten detection methods. N Carlini, D A Wagner, AISec@CCSCarlini, N., Wagner, D.A.: Adversarial examples are not easily detected: Bypassing ten detection methods. In: AISec@CCS (2017) Towards evaluating the robustness of neural networks. N Carlini, D A Wagner, IEEE Symposium on Security and Privacy (SP. Carlini, N., Wagner, D.A.: Towards evaluating the robustness of neural networks. 2017 IEEE Symposium on Security and Privacy (SP) pp. 39-57 (2017) M Cissé, P Bojanowski, E Grave, Y Dauphin, N Usunier, Parseval networks: Improving robustness to adversarial examples. Cissé, M., Bojanowski, P., Grave, E., Dauphin, Y., Usunier, N.: Parseval networks: Improving robustness to adversarial examples. 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[]
[ "Causal Calculus in the Presence of Cycles, Latent Confounders and Selection Bias", "Causal Calculus in the Presence of Cycles, Latent Confounders and Selection Bias" ]
[ "Patrick Forré \nInformatics Institute University of Amsterdam\nThe Netherlands\n", "Joris M Mooij [email protected] \nInformatics Institute University of Amsterdam\nThe Netherlands\n" ]
[ "Informatics Institute University of Amsterdam\nThe Netherlands", "Informatics Institute University of Amsterdam\nThe Netherlands" ]
[]
We prove the main rules of causal calculus (also called do-calculus) for i/o structural causal models (ioSCMs), a generalization of a recently proposed general class of non-/linear structural causal models that allow for cycles, latent confounders and arbitrary probability distributions. We also generalize adjustment criteria and formulas from the acyclic setting to the general one (i.e. ioSCMs). Such criteria then allow to estimate (conditional) causal effects from observational data that was (partially) gathered under selection bias and cycles. This generalizes the backdoor criterion, the selection-backdoor criterion and extensions of these to arbitrary ioSCMs. Together, our results thus enable causal reasoning in the presence of cycles, latent confounders and selection bias. Finally, we extend the ID algorithm for the identification of causal effects to ioSCMs.The ioSCM will be denoted by M = (G + , X , P U , g). Definition 2.4 (Modular structural causal model, see[10,11]). A modular structural causal model (mSCM) is an ioSCM without input nodes, i.e. J = ∅. Remark 2.5 (Composition of ioSCMs). Consider two ioSCMs M 1 , M 2 and an identification of subsets I 1 ⊆ V + 1 with I 2 ⊆ J 2 and maps g i2 : X i1 → X i2 , for i 1 corresponding to i 2 , e.g. g i2 = id if possible. We can now "glue" them together to get a new ioSCM M 3 given bywhere we add the the edges i 1 i 2 , and the mechanisms g i2 and P U3 := P U1 ⊗ P U2 . Example 2.6 (Constructing mSCMs from ioSCMs). Given an ioSCM M = (G + , X , P U , g) with graph G + = (V∪U∪J, E + ) we can construct a well-defined mSCM by specifying a product distribution P J := j∈J P j on X J := j∈J X j and following 2.5 with M 1 with only U 1 := J 2 without any edges and gluing maps g i := id.The actual joint distributions on the observed space X V and thus the random variables attached to any ioSCM will be defined in the following. Definition 2.7. Let M = (G + , X , P U , g) be an ioSCM with G + = (V∪U∪J, E + ). The following constructions will depend on the choice of a fixed value x J ∈ X J .
null
[ "https://export.arxiv.org/pdf/1901.00433v2.pdf" ]
57,373,914
1901.00433
914dccfe293baec09c3144370f94616dc7a89e85
Causal Calculus in the Presence of Cycles, Latent Confounders and Selection Bias 3 Jul 2019 Patrick Forré Informatics Institute University of Amsterdam The Netherlands Joris M Mooij [email protected] Informatics Institute University of Amsterdam The Netherlands Causal Calculus in the Presence of Cycles, Latent Confounders and Selection Bias 3 Jul 2019 We prove the main rules of causal calculus (also called do-calculus) for i/o structural causal models (ioSCMs), a generalization of a recently proposed general class of non-/linear structural causal models that allow for cycles, latent confounders and arbitrary probability distributions. We also generalize adjustment criteria and formulas from the acyclic setting to the general one (i.e. ioSCMs). Such criteria then allow to estimate (conditional) causal effects from observational data that was (partially) gathered under selection bias and cycles. This generalizes the backdoor criterion, the selection-backdoor criterion and extensions of these to arbitrary ioSCMs. Together, our results thus enable causal reasoning in the presence of cycles, latent confounders and selection bias. Finally, we extend the ID algorithm for the identification of causal effects to ioSCMs.The ioSCM will be denoted by M = (G + , X , P U , g). Definition 2.4 (Modular structural causal model, see[10,11]). A modular structural causal model (mSCM) is an ioSCM without input nodes, i.e. J = ∅. Remark 2.5 (Composition of ioSCMs). Consider two ioSCMs M 1 , M 2 and an identification of subsets I 1 ⊆ V + 1 with I 2 ⊆ J 2 and maps g i2 : X i1 → X i2 , for i 1 corresponding to i 2 , e.g. g i2 = id if possible. We can now "glue" them together to get a new ioSCM M 3 given bywhere we add the the edges i 1 i 2 , and the mechanisms g i2 and P U3 := P U1 ⊗ P U2 . Example 2.6 (Constructing mSCMs from ioSCMs). Given an ioSCM M = (G + , X , P U , g) with graph G + = (V∪U∪J, E + ) we can construct a well-defined mSCM by specifying a product distribution P J := j∈J P j on X J := j∈J X j and following 2.5 with M 1 with only U 1 := J 2 without any edges and gluing maps g i := id.The actual joint distributions on the observed space X V and thus the random variables attached to any ioSCM will be defined in the following. Definition 2.7. Let M = (G + , X , P U , g) be an ioSCM with G + = (V∪U∪J, E + ). The following constructions will depend on the choice of a fixed value x J ∈ X J . INTRODUCTION Statistical models are governed by the rules of probability (e.g. sum and product rule), which link joint distributions with the corresponding (conditional) marginal ones. Causal models follow additonal rules, which relate the observational distributions with the interventional ones. In contrast to the rules of probability theory, which directly follow from their axioms, the rules of causal calculus need to be proven, when based on the definition of structural causal models (SCMs). As SCMs will among other things depend on the underlying graphical structure (e.g. with or without cycles or bidirected edges, etc.), the used function classes (e.g. linear or non-linear, etc.) and the allowed probability distributions (e.g. discrete, con-tinuous, singular or mixtures, etc.) the respective endeavour is not immediate. Such a framework of causal calculus contains rules about when one can 1.) insert/delete observations, 2.) exchange action/observation, 3.) insert/delete actions; and about when and how to recover from interventions and/or selection bias (backdoor and selection-backdoor criterion), etc. (see [1, 4, 5, 14, 21-24, 26, 27, 32-35]). While these rules have been extensively studied for acyclic causal models, e.g. (semi-)Markovian models, which are attached to directed acyclic graphs (DAGs) or acyclic directed mixed graphs (ADMGs) (see [1,4,5,14,[21][22][23][24]26,27,[32][33][34][35]), the case of causal models with cycles stayed in the dark. To deal with cycles and latent confounders at the same time in this paper we will introduce the class of input/output structural causal models (ioSCMs), a "conditional" version of the recently proposed class of modular structural causal models (mSCMs) (see [10,11]) to also include "input" nodes that can play the role of parameter/context/action/intervention nodes. ioSCMs have several desirable properties: They allow for arbitrary probability distributions, non-/linear functional relations, latent confounders and cycles. They can also model non-/probabilistic external and probabilistic internal nodes in one framework. The cycles are modelled in a least restrictive way such that the class of ioSCMs still becomes closed under arbitrary marginalizations and interventions. All causal models that are based on acyclic graphs like DAGs, ADMGs or mDAGs (see [9,28]) can be interpreted as special acyclic ioSCMs. Besides feedback over time ioSCMs can also express instantaneous and equilibrated feedback under the made model assumptions (e.g. the ODEs in [2,18]). All models where the non-trivial cycles are "contractive" (negative feedback loops, see [11]) are ioSCMs without further assumptions. Thus ioSCMs generalize all these classes of causal models in one framework, which goes beyond the acyclic setting and also allows for conditional versions of those (e.g. CADMGs), expressed via external non-/probabilistic "input" nodes. Also the generalized directed global Markov property for mSCMs (see [10,11]) generalizes to ioSCMs, i.e. ioSCMs entail the conditional independence relations that follow from the σ-separation criterion in the underlying graph, where σ-separation generalizes the usual d-separation (also called m-or m * -separation, see [9,20,24,28,38]) from acyclic graphs to directed mixed graphs (DMGs) (and even HEDGes [10] and σ-CGs [11]) with or without cycles in a non-naive way. This paper now aims at proving the mentioned main rules of causal calculus for ioSCMs and derive adjustment criteria with corresponding adjustment formulas like generalized (selection-)backdoor adjustments. We also provide an extension of the ID algorithm for the identification of causal effects to the ioSCM setting, which reduces to the usual one in the acyclic case. The paper is structured as follows: We will first give the precise definition of ioSCMs closely mirroring mSCMs from [10,11]. We will then review σ-separation and generalize its criterion from mSCMs (see [10,11]) to ioSCMs. As a preparation for the causal calculus, which relates observational and interventional distributions, we will then show how one can extend a given ioSCM to one that also incorporates additional interventional variables indicating the regime of interventions on the observed nodes. We will then show how the rules of causal calculus directly follow from applying the σ-separation criterion to such an extended ioSCM. We then derive the mentioned general adjustment criteria with corresponding adjustment formulas. Finally, we introduce the right definitions for ioSCMs to extend the ID algorithm for the identification of causal effects to the general setting. INPUT/OUTPUT STRUCTURAL CAUSAL MODELS In this section we will define input/output structural causal models (ioSCMs), which can be seen as a "conditional" version of modular structural causal models (mSCMs) defined in [10,11]. We will then construct marginalized ioSCMs and intervened ioSCMs. To allow for cycles we first need to introduce the notion of loop of a graph and its strongly connected components. Sc G (v) := Anc G (v) ∩ Desc G (v). The set of strongly connected components is S(G). Remark 2.2. Let G = (V, E) be a directed graph. 1. We always have v ∈ Sc G (v) and Sc G (v) ∈ L(G). 2. If G is acyclic then: L(G) = {{v} | v ∈ V }. In the following all spaces are meant to be equipped with σ-algebras and all maps to be measurable. Whenever (regular) conditional distributions occur we implicitly assume standard measurable spaces (to ensure existence). X v for every v ∈ V + , X := v∈V + X v , 3. a product probability measure P U = u∈U P u on the latent space X U := u∈U X u , 4. a directed graph structure G + = (V + , E + ) with the properties: (a) V = Ch G + (U ∪ J), (b) Pa G + (U ∪ J) = ∅, where Ch G + and Pa G + stand for children and parents in G + , resp., 1 5. a system of causal mechanisms g = (g S ) S∈L(G + ) S⊆V : g S : v∈Pa G + (S)\S X v → v∈S X v , 2 that satisfy the following global compatibility conditions: For every nested pair of loops S ′ ⊆ S ⊆ V of G + and every element x Pa G + (S)∪S ∈ v∈Pa G + (S)∪S X v we have the implication: g S (x Pa G + (S)\S ) = x S =⇒ g S ′ (x Pa G + (S ′ )\S ′ ) = x S ′ , where x Pa G + (S ′ )\S ′ and x S ′ denote the corresponding components of x Pa G + (S)∪S . 1 To have a "reduced" form of the latent space one can in addition impose the condition: Ch G + (u1) Ch G + (u2) for every two distinct u1, u2 ∈ U . This can always be achieved by gathering latent nodes together if Ch G + (u1) ⊆ Ch G + (u2). 2 Note that the index set runs over all "observable loops" S ⊆ V , S ∈ L(G + ), not just the sets {v} for v ∈ V . The latent variables are given by (X u ) u∈U ∼ P U , i.e. by the canonical projections X u : X U → X u , which are jointly P U -independent. We put X do(xJ ) u := X u , i.e., independent of x J . 2. For j ∈ J we put X do(xJ ) j := x j , the constant variable given by the j-component of x J . 3. The observed variables (X do(xJ ) v ) v∈V are inductively defined by: X do(xJ ) v := g S,v (X do(xJ ) w ) w∈Pa G + (S)\S , where S := Sc G + (v) := Anc G + (v) ∩ Desc G + (v) and where the second index v refers to the vcomponent of g S . The induction is taken over any topological order of the strongly connected components of G + , which always exists (see [10]). 4. By the compatibility condition for g we then have that for every S ∈ L(G + ) with S ⊆ V the following equality holds: X do(xJ ) S = g S (X do(xJ ) Pa G + (S)\S ), where we put X A := v∈A X v and X A := (X v ) v∈A for subsets A. 5. We define the family of conditional distributions: P U (X A |X B , X J = x J ) := P U (X A |X B , do(X J = x J )) := P U (X do(xJ ) A |X do(xJ ) B ), for A, B ⊆ V and x J ∈ X J . Note that in the following we will use the do and the do-free notation (only) for the J-variables interchangeably. 6. If we, furthermore, specify a product distribution P J = j∈J P j on X J , then we get a joint distribution P on X V ∪J by setting: P(X V , X J ) := P U (X V | do(X J )) ⊗ P J (X J ). Remark 2.8. Let M = (G + , X , P U , g) be an ioSCM with G + = (V∪U∪J, E + ). For every subset A ⊆ V we get a well-defined map g A : X Pa G + (A)\A → X A , by recursively plugging in the g S into each other for the biggest occuring loops S ⊆ A by the same arguments as before. These then are all globally compatible by construction and satisfy: X do(xJ ) A = g A (X do(xJ ) Pa G + (A)\A ). Similar to mSCMs (see [10,11]) we can define the marginalization of an ioSCM. g S ′ ,v (x Pa (G + ) \W (S ′ )\S ′ ) := g S,v x Pa G + (S)\(S∪{w}) , g {w} (x Pa G + (w)\{w} ) , where (G + ) \W is the marginalized graph of G + (see Supplementary Material B), S ′ ⊆ V \W := V \ W is any loop of (G + ) \W and S the corresponding induced loop in G + . Similar to mSCMs (see [10,11]) we now define what it means to intervene on observed nodes in an ioSCM. The remaining functions then are clearly globally compatible and we get a well-defined ioSCM M do(W ) . Here we generalize conditional independence for structured families of distributions. The main application will be the distributions (P U (X V | do(X J = x J ))) xJ ∈XJ coming from ioSCMs, but the following definition might be of more general importance. Definition 3.1 (Conditional independence). Let X V := v∈V X v and X J := j∈J X j be product spaces and P := (P V (X V |x J )) xJ ∈XJ a family of distributions on X V (measurably 3 ) parametrized by X J . For subsets A, B, C ⊆ V∪J we write: X A ⊥ ⊥ P X B | X C if and only if for every product distribution P J = j∈J P j on X J we have: X A ⊥ ⊥ PV ∪J X B | X C , i.e.: P V ∪J (X A |X B , X C ) = P V ∪J (X A |X C ) P V ∪J -a.s., where P V ∪J (X V ∪J ) := P V (X V |X J ) ⊗ P J (X J ) is the distribution given by X J ∼ P J and then X V ∼ P V (_|X J ). Remark 3.2. 1. The definition 3.1 assumes that the input variables J are considered independent, in contrast to [3,29], where all J are implicitely assumed to be jointly confounded. We discuss this further in Supplementary Material C. 2. In contrast with [3,6,29] definition 3.1 can accommodate any variable from V or J at any spot of the conditional independence statement. 3. ⊥ ⊥ P satisfies the separoid axioms (see [6,7,13,25] or see rules [1][2][3][4][5] for ⊥ ⊥ P ) as these rules are preserved under conjunction. σ-SEPARATION In this section we will define σ-separation on directed mixed graphs (DMG) and present the generalized directed global Markov property stating that every ioSCM will entail the conditional independencies that come from σ-separation in its induced DMG. We will again closely follow the work in [11]. Definition 4.1 (Directed mixed graph (DMG)). A directed mixed graph (DMG) G consists of a set of nodes V together with a set of directed edges ( ) and bidirected edges ( ). In case G contains no directed cycles it is called an acyclic directed mixed graph (ADMG). 3 We require that for every measurable F ⊆ XV the map XJ → [0, 1] given by xJ → PV (XV ∈ F |xJ ) is measurable. Such families of distributions are also called channels or (stochastic) Markov (transition) kernels (see [16]). Definition 4.2 (σ-Open walk in a DMG). Let G be a DMG with set of nodes V and C ⊆ V a subset. Consider a walk π in G with n ≥ 1 nodes: v 1 · · · v n . 4 The walk will be called C-σ-open if: 1. the endnodes v 1 , v n / ∈ C, and 2. every triple of adjacent nodes in π that is of the form: (a) collider: v i−1 v i v i+1 , satisfies v i ∈ C, (b) left chain: v i−1 v i v i+1 , satisfies v i / ∈ C or v i ∈ C ∩ Sc G (v i−1 ), (c) right chain: v i−1 v i v i+1 , satisfies v i / ∈ C or v i ∈ C ∩ Sc G (v i+1 ), (d) fork: v i−1 v i v i+1 , satisfies v i / ∈ C or v i ∈ C ∩ Sc G (v i−1 ) ∩ Sc G (v i+1 ). Similar to d-separation we define σ-separation in a DMG. Definition 4.3 (σ-Separation in a DMG). Let G be a DMG with set of nodes V . Let A, B, C ⊆ V be subsets. 1. We say that A and B are σ-connected by C or not σseparated by C if there exists a walk π (with n ≥ 1 nodes) in G with one endnode in A and one endnode in B that is C-σ-open. In symbols this statement will be written as follows: A σ ⊥ ⊥ G B | C. 2. Otherwise, we will say that A and B are σseparated by C and write: A σ ⊥ ⊥ G B | C. Remark 4.4. 1. In any DMG we will always have that σ-separation implies d-separation, since every C-d-open walk is also C-σ-open because {v} ⊆ Sc G (v). If a DMG G is acyclic, i.e. an ADMG, then σseparation coincides with d-separation (also called m-or m * -separation in this context). It was shown in [10] that σ-separation satisfies the graphoid/separoid axioms (see [6,7,13,25]): Lemma 4.5 (Graphoid and separoid axioms). Let G be a DMG with set of nodes V and A, B, C, D ⊆ V subsets. Then we have the following rules for σ-separation in G (with ⊥ ⊥ standing for ⊥ ⊥ σ G ): 1. Redundancy: A ⊥ ⊥ B | A always holds. 2. Symmetry: A ⊥ ⊥ B | D =⇒ B ⊥ ⊥ A | D. 3. Decomposition: A ⊥ ⊥ B ∪ C | D =⇒ A ⊥ ⊥ B | D. 4. Weak Union: A ⊥ ⊥ B∪C | D =⇒ A ⊥ ⊥ B | C ∪D. 5. Contraction: (A ⊥ ⊥ B | C ∪ D) ∧ (A ⊥ ⊥ C | D) =⇒ A ⊥ ⊥ B ∪ C | D. 6. Intersection: (A ⊥ ⊥ B | C ∪ D) ∧ (A ⊥ ⊥ C | B ∪ D) =⇒ A ⊥ ⊥ B ∪ C | D, whenever A, B, C, D are pairwise disjoint. 7. Composition: (A ⊥ ⊥ B | D) ∧ (A ⊥ ⊥ C | D) =⇒ A ⊥ ⊥ B ∪ C | D. It was also shown that σ-separation is stable under marginalization (see [10,11]): Theorem 4.6 (σ-Separation under marginalization, see [10,11]). Let G be a DMG with set of nodes V . Then for any sets A, B, C ⊆ V and L ⊆ V \ (A ∪ B ∪ C) we have the equivalence: A σ ⊥ ⊥ G B | C ⇐⇒ A σ ⊥ ⊥ G \L B | C, where G \L is the DMG that arises from G by marginalizing out the variables from L. A GLOBAL MARKOV PROPERTY The most important ingredient for our results is a generalized directed global Markov property that relates the graphical structure of any ioSCM M to the conditional independencies of the observed random variables via a σ-separation criterion. Since we have no access to the latent nodes u ∈ U of an ioSCM with graph G + we need to marginalize them out (see Supplementary Material B). This will give us an induced directed mixed graph (DMG) G. Definition 5.1 (Induced DMG of an ioSCM). Let M = (G + , X , P U , g) be an ioSCM with G + = (V∪U∪J, E + ). The induced directed mixed graph (DMG) G of M is defined as follows: 1. G contains all nodes from V ∪ J. 2. G contains all the directed edges of G + whose endnodes are both in V ∪ J. 3. G contains the bidirected edge v w with v, w ∈ V if and only if v = w and there exists a u ∈ U with v, w ∈ Ch G + (u), i.e. v and w have a common latent confounder. The following generalized directed global Markov property directly generalizes from mSCMs (see [10,11]) to ioSCMs. An alternative version with confounded input is given in C.5. Theorem 5.2 (σ-Separation criterion). Let M be an ioSCM with induced DMG G. Then for all subsets A, B, C ⊆ V ∪ J we have the implication: A σ ⊥ ⊥ G B | C =⇒ X A ⊥ ⊥ P X B | X C . In words, if A and B are σ-separated by C in G then the corresponding variables X A and X B are conditionally independent given X C under P, i.e. under the joint dis- tribution P U (X V | do(X J )) ⊗ P J (X J ) for any product distribution P J = j∈J P j . Proof. As mentioned, after specifying the product distribution P J the ioSCM M constitutes a well-defined mSCM with the same induced DMG G. So the σseparation criterion for ioSCMs directly follows from the mSCM-version proven in [10,11]. Remark 5.3. Note that, since σ-separation is stable under marginalization (see [10,11]), also the σ-separation criterion is stable under marginalization. Remark 5.4 (Causal calculus for mechanism change). The σ-separation criterion 5.2 can be viewed as the causal calculus for mechanism change (also sometimes called "soft" interventions, see [8,17,19,24]). As an ex- ample consider A, B ⊆ V , I ⊆ J. Then the graphical separation A ⊥ ⊥ σ G I | B ∪ (J \ I) implies that the condi- tional probability P U (X A |X B , do(X J )) is independent of the actual input variables in I. THE EXTENDED IOSCM In this section we want to consider (perfect) interventions onto the observed nodes and improve upon the general rules mentioned in 5.4. For an elegant treatment of this we need to gather for a given ioSCM M all interventional ioSCMs M do(W ) , where W runs through all subsets of observed variables, and glue them all together into one big extended ioSCMM . To consider all interventions at once we will need to introduce additional intervention variables I v to the graph G + , v ∈ V , which indicate which interventional mechanisms to use. Such techniques were already used in the acyclic case in [21,22,24]. The definition will be made in such a way thatM will still be a well-defined ioSCM. So all the results for ioSCMs will apply toM , most importantly the σ-separation criterion (Thm. 5.2). Definition 6.1. Let M = (G + , X , P U , g) be an ioSCM with G + = (V∪U∪J, E + ). The extended ioSCMM = (Ĝ + ,X , P U ,ĝ) will be defined as follows: 1. For every v ∈ V define the interventional domain I v := X v∪ { v }, where v is a new symbol corre- sponding to the observational (non-interventional) regime. For a set A ⊆ V we put I A := v∈A I v and A := ( v ) v∈A . 2. LetĜ + be the graph G + with the additional intervention nodes I v and directed edges I v v for every v ∈ V . For a uniform notation we sometimes write I j instead of j for j ∈ J. So we have: J := J ∪ {I v | v ∈ V } = {I w | w ∈ V∪J}. 3. For every A ⊆ V we will define the mechanism: g A :X PaĜ + (A)\A = I A ×X Pa G + (A)\A → X A =X A . First, for x A ∈ I A we put I(x A ) := {v ∈ A|x v = v }. Consider the subgraph of G + : H(x A ) := (Pa G + (A) ∪ A) do(I(xA)) . Then define recursively for v ∈ A: g A,v (x A , x Pa G + (A)\A ) := x v if v ∈ I(x A ), g S,v (x Pa H(x A ) (S)\S ) if v / ∈ I(x A ), where S := Sc H(xA) (v) is also a loop in G + . 4. These functions then are again globally compatible andM constitutes a well-defined ioSCM. 5. All the distributions inM then are given by the general procedure of ioSCMs (see Def. 2.7). We introduce the notation for C ⊆ V and (x C , x J ) ∈ I C × X J : P U (X V |I C = x C , X J = x J ) := P U (X V | do((I C , I V \C , X J ) = (x C , V \C , x J )). 6. The extended DMGĜ of G + is then the induced DMG ofĜ + , i.e. the induced DMG G with the ad- ditional edges I v v for every v ∈ V . The following result now relates the interventional distributions of the ioSCM M with the ones from the extended ioSCMM . These relations will be used in the following. Proposition 6.2. Let M = (G + , X , P U , g) be an ioSCM with G + = (V∪U∪J, E + ) andM the extended ioSCM. Let A, B, C ⊆ V be pairwise disjoint set of nodes and x C∪J ∈ X C∪J . Then we have the equations: P U (X A |X B , do(X C∪J = x C∪J )) = P U (X A |X B , I C = x C , X J = x J ) = P U (X A |X B , I C = x C , X C = x C , X J = x J ). Proof. This follows from I(x C , V \C ) = C. See Sup- plementary Material D.1. THE THREE MAIN RULES OF CAUSAL CALCULUS Notation 7.1. Since everything has been defined in detail in the last section we now want to make use of a simplified and more suggestive notation for better readability. 1. We identify variables X A with the set of nodes A. We omit values x V and the subscript in P U . E.g. we write P(Y |I T , T, Z, do(W )) instead of P U X Y | I T = x T , X T = x T , X Z = x Z , do(X W = x W ) , where the latter comes from the extended ioSCM of the intervened ioSCM M do(W ) := M do(W \J) of M . 3. We abbreviate X Y ⊥ ⊥ PU (_| do(XW =xW )) X T |X Z as Y ⊥ ⊥ P T |Z, do(W ), etc.. 4. We write Y ⊥ ⊥ σ G I X | X, Z, do(W ) to mean Y ⊥ ⊥ σ G do(W ) I X | X, Z, whereĜ do(W ) is the extended DMG of the intervened graph G + do(W ) . Insertion/deletion of observation: If Y σ ⊥ ⊥ G X|Z, do(W ) then: P(Y |X, Z, do(W )) = P(Y |Z, do(W )). Action/observation exchange: If Y σ ⊥ ⊥ G I X |X, Z, do(W ) then: P(Y | do(X), Z, do(W )) = P(Y |X, Z, do(W )). Insertion/deletion of actions: If Y σ ⊥ ⊥ G I X |Z, do(W ) then: P(Y | do(X), Z, do(W )) = P(Y |Z, do(W )). The proofs follow directly from the σ-separation criterion 5.2 and Prp. 6.2 applied to the extended ioSCM and can be found in Supplementary Material E.1. ADJUSTMENT CRITERIA Notation 8.1. Let M = (G + , X , P, g) be an ioSCM with G + = (V∪U∪J, E + ). The following set of nodes/variables will play the described roles: Y : the outcome variables, X: the treatment or intervention variables, Z 0 : the core set of adjustment variables, Z + : additional adjustment variables, Z := Z 0 ∪ Z + : all actual adjustment variables, L: "marginalizable" adjustment variables, C: context variables, W : default intervention variables containing J, S: variables inducing selection bias given S = s. We are interested in finding a "do(X)-free" expression for the (conditional) causal effect P(Y |C, do(X), do(W )) only using data for C, X, Y, Z that was gathered under selection bias S = s and intervention do(W ) and additional unbiased observational data for C, Z given do(W ). The task can be achieved via the following criterion, which is a generalization of the acyclic case of the selection-backdoor criterion (see [1]), the backdoor criterion (see [21,22,24]) and its extensions (also see [4,26,27,32]) to general ioSCMs. The proof again follows directly from the σ-separation criterion 5.2 and Prp. 6.2 applied to the extended ioSCM and can be found in the Supplementary Material F.1. Figure 1: An induced DMG G with input node I X (the others are left out for readability). The variables satisfy the general adjustment criterion for P(Y |C, do(X)) with L = {L 1 , L 2 } and Z + = {Z 1 , Z 2 }. Note that L 2 could also have been a latent variable. Different colours for different node and/or edge types. Remark 8.3. Note that the adjustment formula in theorem 8.2 does not depend on L. This thus allows us to even choose variables for L that come from an ioSCM M ′ that marginalizes to M , e.g. L ⊆ U or by extending directed edges v w by v ℓ w with ℓ ∈ L. This technique was used in [32] to find all adjustment sets in the acyclic case with C = S = ∅. Corollary 8.4. Let the notations be like in 8.1 and 8.2 and W = J = ∅. We have the following special cases, which in the acyclic case will reduce to the ones given by the indicated references: I X X W Y Z 0 L 1 C Z 1 Z 2 L 2 S 1. General selection-backdoor (see [4]): C = ∅. 2. Selection-backdoor (see [1]): C = L = ∅. 3. Extended backdoor (see [26,32]): C = S = ∅. 4. Backdoor (see [21,22,24]): C = S = L = Z + = ∅: (a) Z σ ⊥ ⊥ G I X , and (b) Y σ ⊥ ⊥ G I X | X, Z, implies: P(Y | do(X)) = P(Y |X, Z) dP(Z). More details can be found in the Supplementary Material F.2. Also a generalization of the criterion for selection without/partial external data of [4,5] is given there. [21,22,24]): For rule 3 in Thm. 7.2 one usually requires Y ⊥ ⊥ d X|Z, W in the graph G do(W ) that is further mutilated on the set X(Z), the set of all X-nodes that are not ancestors of any Z-node in G do(W ) . 2. For the backdoor criterion instead of L ⊥ ⊥ d G I X we could have written that L does not contain any descendent of X; and for Y ⊥ ⊥ d G I X | X, Z that Z blocks all "backdoor paths" from X to Y . We presented the results in the formulaic terms of σseparation because the relations to their use is directly indicated (e.g. in the proofs), it makes the generalization to ioSCMs possible and when reduced to the acyclic case it will be equivalent to the usual description. IDENTIFYING CAUSAL EFFECTS Here we extend the ID algorithm for the identification of causal effects to ioSCMs. The main references are [12,14,15,24,29,[34][35][36][37]. The task is to decide if a causal effect P(Y | do(W )) in an ioSCM can be identified from (i.e., expressed in terms of) the observational distributions P(V | do(J)) and the induced graph G. Note that having more dependence structure (like latent confounders, feedback cycles, etc.) will leave us with less identifiable causal effects in general. Due to space limitations, we can only provide here the bare necessities to state the generalized ID algorithm. We assume that the reader is already familiar with the ID algorithm formulated for ADMGs (for example, the treatment in [36]). We generalize the notion of districts / C-components: Definition 9.1 (Consolidated districts). Let G be a di- rected mixed graph (DMG) with set of nodes V . Let v ∈ V . The consolidated district Cd G (v) of v in G is given by all nodes w ∈ V for which there exist k ≥ 1 nodes (v 1 , . . . , v k ) in G such that v 1 = v, v k = w and for i = 2, . . . , k we have that the bidirected edge v i−1 v i is in G or that v i ∈ Sc G (v i−1 ). For B ⊆ V we write Cd G (B) := v∈B Cd G (v). Let CD(G) be the set of consolidated districts of G. We also generalize the notion of topological order: Definition 9.2 (Apt-order, see [10]). Let G be a DMG with set of nodes V . An assembling pseudo-topological order (apt-order) of G is a total order < on V with the following two properties: 1. For every v, w ∈ V we have: w ∈ Anc G (v) \ Sc G (v) =⇒ w < v. 2. For every v 1 , v 2 , w ∈ V we have: v 2 ∈ Sc G (v 1 )∧(v 1 ≤ w ≤ v 2 ) =⇒ w ∈ Sc G (v 1 ). Remark 9.3. Let G be a DMG. 1. If G is acyclic then an apt-order < is the same as a topological order (i.e. w ∈ Pa G (v) =⇒ w < v). If G has a topological order then G is acyclic. 3. For any DMG G there always exists an apt-order < (in contrast to topological orders). Notation 9.4. Let G be a DMG with set of nodes V and < a apt-order on G. For elements v ∈ V and subsets B ⊆ V we put: 1. Pred G < (v) := {w ∈ V | w < v}, 2. Pred G ≤ (v) := {w ∈ V | w = v or w < v}, 3. Pred G ≤ (B) := v∈B Pred G ≤ (v), 4. Pred G < (B) := Pred G ≤ (B) \ B. Remark 9.5. If B is strongly-connected, then Pred G ≤ (B) is ancestral in G, i.e., Anc G (Pred G ≤ (B)) = Pred G ≤ (B). The notion of input variables enables the following convenient and intuitive construction: Keep all functions g S with S ⊆ C. 4. P U [C] := u∈U [C] P u , i.e. the marginal of P U and we will use the notation P U (or just P) for both. For C ⊆ V ∪ J with C ∩ V = ∅ put M [C] := M [C∩V ] . By the definition of the random variables induced by an ioSCM we immediately get the following basic result: Lemma 9.7. Let M = (G + , X , P U , g) be an ioSCM with G + = (V∪U∪J, E + ). For C ⊆ V , we have (indices for emphasis): P M [C] (C| do(Pa G (C) \ C)) = P M (C| do(J ∪ W )), for any W ⊆ V \C that contains (Pa G (C)∩V )\C. As a special case: if A ⊆ G is ancestral, i.e., Anc G (A) = A, P M [A] (A ∩ V | do(A ∩ J)) = P M (A ∩ V | do(J ∪ W )) for any W ⊆ V \ A ∩ V . The ID algorithm works by repeatedly applying the previous lemma and the following rules: Proposition 9.8. Let M = (G + , X , P U , g) be an ioSCM with G + = (V∪U∪J, E + ) and < an apt-order for G + . 1. P(V | do(J)) = S∈S(G) S⊆V P(S|Pred G < (S)∩V, do(J)). For For D ⊆ V a consolidated district of G: P(D| do(J ∪ V \ D)) = S∈S(G) S⊆D P(S|Pred G < (S) ∩ V, do(J)). Proof. 1. uses the chain rule; 2. is proved in Supplementary Material G.2; 3. is shown by applying 1. and Remark 9.7 to G [D] and then making use of 2.. Note that the product might not be well-defined as the consolidated districts i.g. are not totally ordered by < (in contrast to strongly connected components), even in the acyclic case. For example, consider the graph: v 1 v 2 v 3 v 4 This problem is usually not addressed in the literature. The problem disappears if every strongly connected component S ⊆ V comes with a measure µ S such that P(V | do(J)) has a density w.r.t. the product measure S∈S(G) S⊆V µ S . Then the densities p(D| do(J ∪ V \ D)) can be multiplied in any order and the integration can be separately done via the µ S in reverse order of <. We now have all the prerequisites to state the generalized ID algorithm (Algorithm 1) and prove its correctness: Remark 9.11. 1. We make no claim about the completeness of the algorithm here. 2. The algorithm reduces to the usual version in the acyclic case (see [29,[35][36][37] CONCLUSION We proved the three main rules of causal calculus and general adjustment criteria with corresponding formulas to recover from interventions and selection bias Algorithm 1 ID: Generalized ID algorithm for the identification of causal effects in general ioSCMs. 1: function ID(G, Y, W, P(V | do(J))) 2: require: Y ⊆ V , W ⊆ V , Y ∩ W = ∅ 3: H ← Anc G V \W (Y ) 4: for C ∈ CD(H) do 5: Q[C] ← IDCD(G, C, Cd G (C), Q[Cd G (C)]) 6: if Q[C] = FAIL then for S ∈ S(G [A] ) s.t. S ⊆ Cd G [A] (C) do 23: R A [S] ← P(S|Pred G < (S)∩A, do(J ∪V \A)) 24: end for 25: end if 28: end function for general ioSCMs, which allow for arbitrary probability distributions, non-/linear functional relations, latent confounders, external non-/probabilistic parameter/action/intervention/context/input nodes and cycles. This generalizes all the corresponding results of acyclic causal models (see [1,4,21,22,24,26,27,32]) to general ioSCMs. We also showed how to extend the ID algorithm for the identification of causal effects from the acyclic setting to general ioSCMs. In supplementary material A we also show how to do counterfactual reasoning in ioSCMs. Future work might address completeness questions of the ID algorithm (see [14,24,33,34]). Q[Cd G [A] (C)] ← S∈S(G [A] ) S⊆Cd G [A] (C) R A [S] SUPPLEMENTARY MATERIAL A TWIN NETWORKS AND COUNTERFACTUALS In addition to probabilistic and causal reasoning about interventions, ioSCMs allow for counterfactual reasoning. Given an ioSCM M with graph G + = (V∪U∪J, E + ), a set W ⊆ V ∪ J and the corresponding intervened ioSCM M do(W ) with graph G + do(W ) one can construct a (merged) twin ioSCM M twin similarly to the acyclic case (see [24]), or a single world intervention graph (SWIG, see [30]). This is done by identifying/merging the corresponding nodes, mechanisms and variables from the non-descendants of W , i.e., NonDesc G + (W ) and NonDesc G + do(W ) (W ), which are unchanged by the action do(W ). Then one has the two different branches Desc G + (W ) and Desc G + do(W ) (W ) in the network. This construction then allows one to formulate counterfactual statements like in the acyclic case (see [24]), but now for general ioSCMs. E.g., one could state the assumption of strong ignorability (see [24,31]) as: Y do( ) , Y do(X) σ ⊥ ⊥ Gtwin X | Z, or the conditional ignorability (see [31,32]) as: Y do(X) σ ⊥ ⊥ Gtwin X | Z. All the causal reasoning rules derived in this paper can thus also be applied to reason about counterfactuals. B MARGINALIZATION OF DIRECTED MIXED GRAPHS For completeness, we provide here the definition of marginalization of directed mixed graph. For more details and the relationship with the marginalization of an mSCM (or as a straightforward generalization, an ioSCM), we refer the reader to [10]. 1. v 1 v 2 ∈ E ′ iff there exist k ≥ 0 nodes w 1 , . . . , w k ∈ W such that the directed walk: v 1 w 1 · · · w k v 2 lies in G (the corner case v 1 v 2 ∈ E also ap- plies). 2. v 1 v 2 ∈ B ′ iff there exist k ≥ 0 nodes w 1 , . . . , w k ∈ W and an index 0 ≤ m ≤ k such that a walk of the form: v 1 w 1 · · · w m · · · w k v 2 lies in G with m ≥ 1 or a walk of the form: v 1 w 1 · · · w m m≥0 w m+1 · · · w k k−m≥0 v 2 lies in G (including the corner cases v 1 v 2 ∈ B and v 1 w v 2 in G with w ∈ W ). C CONDITIONAL INDEPENDENCE AND ITS ALTERNATIVE WITH CONFOUNDED INPUTS Here we want to give a generalization of [3,29] in the flavor of definition 3.1. The main point is that the approaches of conditional independence for families of distributions/Markov kernels in [3,29] implicitely assume that the input variables J are jointly confounded. The definition 3.1 of conditional independence, in contrast, assumes (via the product distributions) that the variables J are jointly independent. The approach in definition 3.1 can be easily adapted to the confounded input setting as follows. C.1 INPUT CONFOUNDED CONDITIONAL INDEPENDENCE Definition C.1 (Input confounded conditional independence). Let X V := v∈V X v and X J := j∈J X j be the product spaces of any measurable spaces and P V (X V |X J ) a Markov kernel (i.e. a family of distributions on X V measurably 5 parametrized by X J ). For subsets A, B, C ⊆ V∪J we write: X A ⊥ ⊥ PV (XV |XJ ),• X B | X C if and only if for every joint distribution P J on X J we have: X A ⊥ ⊥ PV ∪J X B | X C , which means that for all measurable F ⊆ X A we have: P V ∪J (X A ∈ F |X B , X C ) = P V ∪J (X A ∈ F |X C ) P V ∪J -a. s., 5 We require that for every measurable F ⊆ XV the map XJ → [0, 1] given by xJ → PV (XV ∈ F |XJ = xJ ) is measurable. where P V ∪J (X V ∪J ) := P V (X V |X J ) ⊗ P J (X J ), the distribution given by X J ∼ P J and then X V ∼ P V (_|X J ). Lemma C.2. Let the situation be like in C.1 and assume all spaces X v , v ∈ V , to be standard measurable spaces. Let A, B, C be pairwise disjoint, A ∩ J = ∅ and J ⊆ B ∪ C. Then every statement implies the one below: 1. There is a version of P V (X A |X B , X C ) such that for all x B , x ′ B ∈ X B , x C ∈ X C : P V (X A |X B = x B , X C = x C ) = P V (X A |X B = x ′ B , X C = x C ). 2. X A ⊥ ⊥ PV (XV |XJ ),• X B | X C . 3. X A ⊥ ⊥ PV (XV |XJ ) X B | X C (using definition 3.1). 4. X A ⊥ ⊥ PV (XV |XJ )⊗δx J (XJ ) X B | X C for every x J ∈ X J . If there is a Markov kernel P(X A |X C ) that is a version of P V ∪J (X A |X C ) for every Dirac delta distribution P J = δ xJ (e.g. if J ⊆ C) then the last point also implies the first. Proof. 1. =⇒ 2.: Functional dependence only on x C . 2. =⇒ 3. =⇒ 4.: Every product distribution is a joint distribution and every Dirac delta distribution is a product distribution. 1. ⇐= 4.: Let N ⊆ X B∪C be the measurable set on which the Markov kernels P V (X A |X B , X C ) and P(X A |X C ) (considered as functions of (x B , x C )) differ. For every x J ∈ X J we have by assumption: X A ⊥ ⊥ PV (XV |XJ )⊗δx J (XJ ) X B | X C . This shows that: P V (X A |X B = x B , X C = x C ) = P(X A |X C = x C ) for (x B , x C ) outside of a P V (X (B∪C)\J |X J = x J )zero set, for which we can take the section N xJ of N . This implies that N is a P V (X (B∪C)\J |X J )-zero set. So P(X A |X C ) is a version of P V (X A |X B , X C ) and satisfies 1.. Remark C.3. 1. The existence of the Markov kernel P(X A |X C ) under the assumption 4. in lemma C.2 always/only holds up to measurability questions, because for every fixed P J the regular conditional probability distribution P V ∪J (X A |X B , X C ) always exists in standard measurable spaces and agrees with P V ∪J (X A |X C ) (by the assumption 4.). The existence of the Markov kernel P(X A |X C ) follows for standard measurable spaces X v , v ∈ V , if either: (a) J ⊆ C and assumption 4. holds, or: (b) X J is discrete and assumption 2. holds, or: (c) P V (X V |X J ) comes as P U (X V |X J ) from an ioSCM and assumptions 2.-4. even hold in form of the corresponding σ-separation statement in the induced DMG G. We plan in future work to address all these subtleties in more detail. 2. Lemma C.2 shows that definition C.1 (and also already definition 3.1) generalizes the one from [29] (when applied symmetrized). The clear correspondence/generalization is that for any (not necessarily disjoint) A, B, C ⊆ V ∪ J: [3] shows that definitions 3.1, C.1 also generalize the one from [3] in the same sense. 4. In contrast with [3,6,29], definition C.1 can accommodate any variable from V or J at any position of the conditional independence statement. 5. Also note that ⊥ ⊥ PV (XV |XJ ),• is well-defined for any measurable spaces and is not restricted to discrete variables or distributions/Markov kernels that come with densities. 6. Furthermore, ⊥ ⊥ PV (XV |XJ ),• satisfies the separoid axioms (see [6,13,25] or see rules 1-5 in Lem. 4.5 for ⊥ ⊥ PV (XV |XJ ),• ). Indeed, every single ⊥ ⊥ PV ∪J satisfies the separoid axioms (see [3,6]) and an arbitrary intersection of separoids is again a separoid (see [7]): X A ⊥ ⊥ [29] X B | X C : ⇐⇒ X A ⊥ ⊥ PV (XV |XJ ),• X B∪J | X C ∨ X B ⊥ ⊥ PV (XV |XJ ),• X A∪J | X C . Thm. 4.4 in ⊥ ⊥ PV (XV |XJ ),• = PJ ⊥ ⊥ PV ∪J . C.2 INPUT CONFOUNDED GLOBAL MARKOV PROPERTY We can also prove a global Markov property for the input confounded version of conditional independence. For this we need to modify the graphical structures a bit and introduce a few more notations. Note that all spaces are assumed to measurable (but not necessarily standard). 3. E + • := E + ∪ {• j | j ∈ J}, 4. add g {j} , the canonical projection from X • onto X j , to g for j ∈ J. With this setting M • is a well-defined ioSCM. Furthermore, let G • be the input confounded induced DMG, i.e. the induced DMG of G + • where • is marginalized out. In other words, G • arises from the induced DMG G of G + by just adding j 1 j 2 for all j 1 , j 2 ∈ J, j 1 = j 2 , to G. A σ ⊥ ⊥ G• B | C =⇒ X A ⊥ ⊥ PU (XV | do(XJ )),• X B | X C . In words, if A and B are σ-separated by C in G • then the corresponding variables X A and X B are conditionally independent given X C for any distribution P U (X V | do(X J )) ⊗ P J (X J ) for any joint distribution P J on X J . Proof. This directly follows from the σ-separation criterion/global Markov property 5.2 applied to the input confounded ioSCM M • and G + • , or, alternatively, again from the mSCM-version proven in [10,11] for each fixed joint distribution P J on X J = X • . Note that G • is a marginalization of G + • and σ-separation is stable under marginalization. D THE EXTENDED IOSCM -PROOFS Proposition D.1. Let M = (G + , X , P U , g) be an ioSCM with G + = (V∪U∪J, E + ) andM the extended ioSCM. Let A, B, C ⊆ V be pairwise disjoint set of nodes and x C∪J ∈ X C∪J . Then we have the equations: P U (X A |X B , do(X C∪J = x C∪J )) = P U (X A |X B , I C = x C , X J = x J ) = P U (X A |X B , I C = x C , X C = x C , X J = x J ). Proof. Consider the first equality. For any subset D ⊆ V the variable X do(XC∪J =xC∪J ) D was recursively defined in M do(C) via g using G + do(C) , whereas the variable X do((IC ,I V \C ,XJ )=(xC , V \C ,xJ )) D was recursively defined inM via the same g but using I(x C , V \C ) and G + do(I(xC, V \C )) . Since x C ∈ X C we have that I(x C , V \C ) = C and thus G + do(I(xC , V \C )) = G + do(C) . It directly follows that: X do(XC∪J =xC∪J ) D = X do((IC ,I V \C ,XJ )=(xC, V \C ,xJ )) D . This shows the equality of top and middle line. For the equality between the middle and bottom line note that: I C = x C xC∈XC =⇒ X C = x C . E Action/observation exchange: If Y σ ⊥ ⊥ G I X |X, Z, do(W ) then: P(Y | do(X), Z, do(W )) = P(Y |X, Z, do(W )). Insertion/deletion of actions: If Y σ ⊥ ⊥ G I X |Z, do(W ) then: P(Y | do(X), Z, do(W )) = P(Y |Z, do(W )). Proof. 1. Thm. 5.2 applied to G do(W ) gives: Y σ ⊥ ⊥ G X|Z, do(W ) 5.2 =⇒ Y ⊥ ⊥ P X|Z, do(W ). The latter directly gives the claim: P(Y |X, Z, do(W )) = P(Y |Z, do(W )). 2. The σ-separation criterion 5.2 w.r.t. toĜ do(W ) gives: Y σ ⊥ ⊥ G I X |X, Z, do(W ) 5.2 =⇒ Y ⊥ ⊥ P I X |X, Z, do(W ). Together with Prp. 6.2 (applied to M do(W ) ) we have: P(Y | do(X), Z, do(W )) 6.2 = P(Y |I X , X, Z, do(W )) Y ⊥ ⊥ IX |X,Z,do(W ) = P(Y |X, Z, do(W )). 3. As before we have: Then one can estimate the conditional causal effect P(Y |C, do(X), do(W )) via the adjustment formula: Y σ ⊥ ⊥ G I X |Z, do(W ) 5.2 =⇒ Y ⊥ ⊥ P I X |Z, do(W ).P(Y |C, do(X), do(W )) = P(Y |X, Z, C, S = s, do(W )) dP(Z|C, do(W )). Proof. Since C, do(W ) occur everywhere as a conditioning set, we will suppress C, do(W ) in the following everywhere. Then note that the σ-separation criterion 5.2 implies the corresponding conditional independencies in the following when indicated. The adjustment formula then derives from the following computations: Proof. First note that the σ-separation criterion Theorem 5.2 implies the corresponding conditional independencies in the following when indicated. We implicitly make use of Proposition 6.2 when needed. The adjustment formula then derives from the following computations: P(Y | do(X)) = P(Y |Z 0 , L, do(X)) dP(Z 0 , L| do(X)) 6.2 = P(Y |I X , X, Z 0 , L) dP(Z 0 , L|I X ) Y ⊥ ⊥ IX |X,P(Y | do(X)) Y ⊥ ⊥ S | do(X) = P(Y |S, do(X)) chain rule = P(Y |Z 0 , S, do(X)) dP(Z 0 |S, do(X)) Z0 ⊥ ⊥ IX | S = 6.2 P(Y |Z 0 , S, do(X)) dP(Z 0 |S) dP(Z+|Z0,S)=1 = P(Y |Z 0 , S, do(X)) dP(Z + , Z 0 |S) Y ⊥ ⊥ Z+ | Z0,S,do(X) = P(Y |Z + , Z 0 , S, do(X)) dP(Z + , Z 0 |S) Z=Z+∪Z0 = P(Y |Z, S, do(X)) dP(Z|S) Y ⊥ ⊥ IX | X,Z,S = 6.2 P(Y |Z, S, X) dP(Z|S). The following theorem generalizes the adjustment criterion of type III in [5]. For this we have to introduce even more adjustment sets: Z A 0 , Z B 0 , Z A 1 , Z B 1 , Z 2 , Z 3 and L 0 , L 1 . We write Z 0 = (Z A 0 , Z B 0 ), Z A ≤1 = (Z A 0 , Z A 1 ) , etc.. Theorem F.4 (General adjustment with partial external data). Assume that data was collected under selection bias, P(V |S = s), but we have unbiased data from P(Z B ≤1 ). Further assume that the variables satisfy: 1. (L 0 , Z 0 ) ⊥ ⊥ I X , 2. Y ⊥ ⊥ Z 1 | L 0 , Z 0 , do(X), 3. Z A ≤1 ⊥ ⊥ S | Z B ≤1 , 4. L 0 ⊥ ⊥ I X | Z ≤1 , 5. Y ⊥ ⊥ S | Z ≤1 , do(X), 6. (L 1 , Z 2 ) ⊥ ⊥ I X | S, Z ≤1 , 7. Y ⊥ ⊥ Z 3 | L 1 , S, Z ≤2 , do(X), 8. L 1 ⊥ ⊥ I X | S, Z, 9. Y ⊥ ⊥ I X | X, S, Z. Then we have the adjustment formula: P(Y | do(X)) = P(Y |S = s, Z, X) dP(Z\Z B ≤1 |S = s, Z B ≤1 ) dP(Z B ≤1 ). Note that this formula does not depend on L 0 and L 1 . So L 0 and L 1 can be chosen in a graph G ′ that marginalizes to G. Proof. P(Y | do(X)) chain rule = P(Y |L 0 , Z 0 , do(X)) dP(L 0 , Z 0 | do(X)) (L0,Z0) ⊥ ⊥ IX = 6.2 P(Y |L 0 , Z 0 , do(X)) dP(L 0 , Z 0 ) dP(Z1|L0,Z0)=1 = Z ≤1 =Z0∪Z1 P(Y |L 0 , Z 0 , do(X)) dP(L 0 , Z ≤1 ) Y ⊥ ⊥ Z1 | L0,Z0,do(X) = P(Y |L 0 , Z ≤1 , do(X)) dP(L 0 , Z ≤1 ) chain rule = Z ≤1 =Z A ≤1 ∪Z B ≤1 P(Y |L 0 , Z ≤1 , do(X)) dP(L 0 |Z ≤1 ) dP(Z A ≤1 |Z B ≤1 ) dP(Z B ≤1 ) Z A ≤1 ⊥ ⊥ S | Z B ≤1 = P(Y |L 0 , Z ≤1 , do(X)) dP(L 0 |Z ≤1 ) dP(Z A ≤1 |S, Z B ≤1 ) dP(Z B ≤1 ) L0 ⊥ ⊥ IX | Z ≤1 = 6.2 P(Y |L 0 , Z ≤1 , do(X)) dP(L 0 |Z ≤1 , do(X)) dP(Z A ≤1 |S, Z B ≤1 ) dP(Z B ≤1 ) chain rule = P(Y |Z ≤1 , do(X)) dP(Z A ≤1 |S, Z B ≤1 ) dP(Z B ≤1 ) Y ⊥ ⊥ S | Z ≤1 ,do(X) = P(Y |S, Z ≤1 , do(X)) dP(Z A ≤1 |S, Z B ≤1 ) dP(Z B ≤1 ) chain rule = P(Y |L 1 , Z 2 , S, Z ≤1 , do(X)) dP(L 1 , Z 2 |S, Z ≤1 , do(X)) dP(Z A ≤1 |S, Z B ≤1 ) dP(Z B ≤1 ) Z ≤2 =Z ≤1 ∪Z2 = P(Y |L 1 , S, Z ≤2 , do(X)) dP(L 1 , Z 2 |S, Z ≤1 , do(X)) dP(Z A ≤1 |S, Z B ≤1 ) dP(Z B ≤1 ) (L1,Z2) ⊥ ⊥ IX | S,Z ≤1 = 6.2 P(Y |L 1 , S, Z ≤2 , do(X)) dP(L 1 , Z 2 |S, Z ≤1 ) dP(Z A ≤1 |S, Z B ≤1 ) dP(Z B ≤1 ) Y ⊥ ⊥ Z3 | L1,S,Z ≤2 ,do(X) = P(Z3|L1,S,Z ≤2 )=1 P(Y |L 1 , S, Z ≤2 , Z 3 , do(X)) dP(L 1 , Z 2 , Z 3 |S, Z ≤1 ) dP(Z A ≤1 |S, Z B ≤1 ) dP(Z B ≤1 ) chain rule = Z=Z ≤2 ∪Z3 P(Y |L 1 , S, Z, do(X)) dP(L 1 |S, Z) dP(Z \ Z B ≤1 |S, Z B ≤1 ) dP(Z B ≤1 ) L1 ⊥ ⊥ IX | S,Z = 6.2 P(Y |L 1 , S, Z, do(X)) dP(L 1 |S, Z, do(X)) dP(Z \ Z B ≤1 |S, Z B ≤1 ) dP(Z B ≤1 ) chain rule = P(Y |S, Z, do(X)) dP(Z \ Z B ≤1 |S, Z B ≤1 ) dP(Z B ≤1 ) Y ⊥ ⊥ IX | X,S,Z = P(Y |S, Z, X) dP(Z \ Z B ≤1 |S, Z B ≤1 ) dP(Z B ≤1 ). G IDENTIFYING CAUSAL EFFECTS Remark G.1 (More remarks about the ID-algorithm). 1. The extended version of the ID algorithm is equivalent to applying the ID algorithm to the acyclification G +,acy of G + , which here is meant to be the conditional ADMG that arises by adding edges v w ′ if v / ∈ Sc G (w) ∋ w ′ and v w ∈ G + , and erasing all edges inside Sc G (w), w ∈ V (see [10]). A consolidated district in G then is the same as a district in G acy . 3. Every apt-order of G is a topological order of G acy . 4. So identifiability in G acy implies identifiability in G. This leads to the rule of thumb that causal effects where both cause and effect nodes are inside one strongly connected component of G are not identifiable from observational data alone, and, that the causal effects of sets of nodes between strongly connected components follow rules similar to the acyclic case. 6. Similarly, the corner cases for the identification of conditional causal effects P(Y |R, do(W )) in G that are not covered by the identification of P(Y, R| do(W )) in G follow from the (acyclic) conditional ID-algorithm from [36] applied to G acy and then translated back to G by the above correspondences. Lemma G.2. Let M = (G + , X , P U , g) be an ioSCM with G + = (V∪U∪J, E + ) and < an apt-order for G + and G its induced DMG (with nodes V∪J). Proof. First note that since D is a union of strongly connected components and all other variables in G [D] have no parents the total order < is also an apt-order for G [D] . It follows that we have the equality of sets of nodes: Then we get the relations between the sets of nodes: V = R <∪ D∪ R >∪ P <∪ P > D = D <∪ S∪ D > , P = P <∪ P >∪ P J , Pred G < (S) ∩ V = D <∪ R <∪ P < , J = J <∪ J > . Since Pred G ≤ (S) is ancestral in G and Pred (S|D < , do(P < , P > , P J )) 9.7 = P M [D] (S|D < , do(P < , P > , P J , D > )) 9.7 = P M (S|D < , do(P < , P > , J, D > , R < , R > )). So the equality between those expressions and thus the claim follows by the 2nd rule of causal calculus in Theorem 7.2 with the σ-separation statement: S σ ⊥ ⊥ G I R<,P< | D < , R < , P < , do(J, R > , P > , D > ). To prove the latter note that the intervention do(R > , P > , D > ) allows us to restrict to the ancestral subgraph Pred G ≤ (S) ∪ J. Now let π be a path from an indicator variable from I R<,P< to S (in Pred G ≤ (S) ∪ J). Then the path can only be of the form: v i · · · v p v d · · · v s , with v i ∈ I R<,P< , v p ∈ P < , v d ∈ D, v s ∈ S, as there cannot be any bidirected edge or directed edge in the other direction between R < ∪ P < and D by the definition of consolidated districts and P = Pa G (D) \ D. Since we condition on P < the path π is σ-blocked. Remark G.3. Another way to deal with the problem that consolidated districts are not topologically ordered in the extended ID-algorithm (see Algorithm 1 and theorem 9.10) as discussed in remark 9.9 is to work with unions of consolidated districts directly instead of working with each single consolidated district at a time (and then having problems multiplying them in a ordered way). The corresponding ID-algorithm then iterates taking the ancestral closure and taking (the unions of) consolidated districts of the queried set until convergence. If the sets agree the causal effect is identifiable and the occuring products can be computed like in proposition 9.8 point 3, with D now a union of consolidated districts. The soundness then follows again with proposition 9.8 and lemma G.2, which also work in this case, but the algorithm might more often respond with "FAIL". Definition 2 . 9 ( 29Marginalization of ioSCMs). Let M = (G + , X , P, g) be an ioSCM with G + = (V∪U∪J, E + ) and W ⊆ V a subset. The marginalized ioSCM M \W w.r.t. W can be defined by plugging the functions g S related to W into each other. For example, when marginalizing out W = {w} we can define (for the non-trivial case w ∈ Pa G + (S) \ S): Definition 2. 10 ( 10Perfect interventions on ioSCMs). Let M = (G + , X , P, g) be an ioSCM with G + = (V∪U∪J, E + ). Let W ⊆ V ∪ J be a subset. We then define the post-interventional ioSCM M do(W ) w.r.t. W : 1. Define the graph G + do(W ) by removing all the edges v w for all nodes w ∈ W and v ∈ Pa G + (w). 2. Put V do(W ) := V \ W and J do(W ) := J ∪ W . 3. Remove the functions g S for loops S with S ∩ W = ∅. Theorem 7.2 (The three main rules of causal calculus). Let M be an ioSCM with set of observed nodes V and input nodes J and induced DMG G. Let X, Y, Z ⊆ V and J ⊆ W ⊆ V ∪ J be subsets. Theorem 8. 2 (X 2General adjustment criterion and formula). Let the setting be like in 8.1. Assume that data was collected under selection bias, P(V |S = s, do(W )) (or under P(V | do(W )) and S = ∅), and there are unbiased samples from P(Z|C, do(W )). Further assume that the variables satisfy: X , Z + ) | C, X, Z 0 , L, do(W ), | C, Z, do(W ).Then one can estimate the conditional causal effect P(Y |C, do(X), do(W )) via the adjustment formula: P(Y |C, do(X), do(W )) = P(Y |X, Z, C, S = s, do(W )) dP(Z|C, do(W )). Remark 8. 5 . 5The conditions in theorems 7.2, 8.2 and corollary 8.4 are in the acyclic setting usually phrased in terms of sub-structures of the graph G (see Definition 9. 6 ( 6Sub-ioSCMs). Let M = (G + , X , P U , g) be an ioSCM with G + = (V∪U∪J, E + ). For C ⊆ V non-empty define the ioSCM M [C] as follows: 1. Put G + [C] to be the subgraph of G + do(Pa G (C)\C) induced by C ∪ Pa G + (C). 2. V [C] := C, J [C] := Pa G + (C) \ (C ∪ U ), U [C] := U ∩ Pa G + (C). S ⊆ V a strongly connected component of G, D := Cd G (S) its consolidated district in G and P := Pa G (D) \ D: P M (S|Pred G < (S) ∩ V, do(J)) = P M [D] (S|Pred G [D] < (S) ∩ D, do(P )). Remark 9. 9 . 9Naively putting the equations of Prp. 9.8 into each other would give us the equation: P(V | do(J)) = D∈CD(G) D⊆V P(D| do(J ∪ V \ D)). Theorem 9 . 10 ( 910Consequence of 9.8, 9.9). Let M = (G + , X , P U , g) be an ioSCM with G + = (V∪U∪J, E + ) with set of observed nodes V and input nodes J and distributions P(V | do(J)). Let < be an apt-order for G + . Assume that for every strongly connected componentS ⊆ V we have a measure µ S such that P(V | do(J)) has a density w.r.t. the product measure S∈S(G) S⊆V µ S .Let Y ⊆ V and W ⊆ J ∪ V be subsets. If the extended ID algorithm (see Algorithm 1) does not "FAIL" then the causal effect P(Y | do(W )) is identifiable, i.e. it can be computed from P(V | do(J)) alone, and the expression is obtained by postprocessing the output of the algorithm. P(Y | do(J, W )) = Q[H]dx H\Y 12: end function 13: function IDCD(G, C, D, Q[D]) 14: require: C ⊆ D ⊆ V , CD(G D ) = {D} 15: A ← Anc G [D] (C) return IDCD(G, C, Cd G [A] (C), Q[Cd G [A] (C)])27: Definition B. 1 ( 1Marginalization of DMGs). Let G = (V, E, B) be a directed mixed graph (DMG) with set of nodes V , directed edges E and bidirected edges B. Let W ⊆ V be a subset of nodes. We define the marginalized DMG G \W := G ′ = (V ′ , E ′ , B ′ ) ("marginalizing out W "), also called latent projection of G onto V \ W , with set of nodes V ′ := V \ W via the following rules (for v 1 , v 2 ∈ V \ W = V ′ ): Definition C. 4 ( 4Input confounded ioSCM). Let M = (G + , X , P U , g) be an ioSCM with graph G + = (V∪U∪J, E + ). The corresponding input confounded ioSCM M • is then constructed from M by the following changes: 1. V • := V ∪ J and U • := U , 2. J • := {•} with a new node • with space X • := X J , Theorem C. 5 ( 5Input confounded directed global Markov property). Let M be an ioSCM with input confounded induced DMG G • . Then for all subsets A, B, C ⊆ V ∪ J we have the implication: P THE THREE MAIN RULES OF CAUSAL CALCULUS -PROOFS Theorem E.1 (The three main rules of causal calculus). Let M be an ioSCM with set of observed nodes V and intervention nodes J and induced DMG G. Let X, Y, Z ⊆ V and J ⊆ W ⊆ V ∪ J be subsets. 1. Insertion/deletion of observation: (Y |X, Z, do(W )) = P(Y |Z, do(W )). And again:P(Y | do(X), Z, do(W )) 6.2 = P(Y |I X , Z, do(W )) Y ⊥ ⊥ IX |Z,do(W )= P(Y |Z, do(W )). (X General adjustment criterion and formula). Let the setting be like in 8.1. Assume that data was collected under selection bias, P(V |S = s, do(W )) (or under P(V | do(W )) and S = ∅), and there are unbiased samples from P(Z|C, do(W )). Further assume that the variables satisfy: X , Z + ) | C, X, Z 0 , L, do(W ), | C, Z, do(W ). (. 2 . 2Z0,L) ⊥ ⊥ IX P(Y |X, Z 0 , L) dP(Z 0 , L) dP(Z+|Z0,L)=1 = P(Y |X, Z 0 , L) dP(Z + |Z 0 , L) dP(Z 0 , L) Y ⊥ ⊥ Z+|X,Z0,L = P(Y |X, Z 0 , Z + , L) dP(Z + , Z 0 , L) Z=Z+∪Z0 = P(Y |X, Z, L) dP(Z, L) = P(Y |X, Z, L) dP(L|Z) dP(Z) L ⊥ ⊥ X|Z = P(Y |L, X, Z)dP(L|X, Z) dP(Z) Let the notations be like in 8.1 and 8.2 and W = J = ∅. We have the following special cases, which in the acyclic case will reduce to the ones given by the indicated references: P(Y | do(X)) = P(Y |X, Z, S = s) dP(Z|S = s). Let S ⊆ V be a strongly connected component of G and D ⊆ V be any union of consolidated districts in G with S ⊆ D (e.g. D = Cd G (S)) and P := Pa G (D) \ D. Then we have the equality (indices for emphasis): P M (S|Pred G < (S) ∩ V, do(J)) = P M [D] (S|Pred G [D] < (S) ∩ D, do(P )). ) ∩ D = Pred G < (S) ∩ D =: D < .Now we introduce the following further abbreviations:D > := D \ (S ∪ D < ), P < := Pred G < (S) ∩ (P ∩ V ), P > := (P ∩ V ) \ Pred G < (S), P J := P ∩ J, J < := Pred G < (S) ∩ J, J > := J \ Pred G < (S), R < := Pred G < (S) ∩ V \ (D ∪ P ), R > := V \ (D ∪ P ∪ Pred G < (S)). ancestral in G[D] , resp., we can by remark 9.7 arbitrarily intervene on all variables outside of these sets without changing the distributionsP M (S|Pred G < (S) ∩ V, do(J)) and P M [D] (S|Pred G [D] <(S) ∩ D, do(P )), resp.. With these remarks and our new notations we have the equalities:P M (S|Pred G < (S) ∩ V, do(J)) = P M (S|D < , R < , P < , do(J)) 9.7 = P M (S|D < , R < , P < , do(J, R > , P > , D > ));and:P M [D] (S|Pred G [D] < (S) ∩ D, do(P )) = P M [D] in G such that all the intermediate nodes are also in S (if any). The sets S = {v} are also considered as loops (independent of v v ∈ E or not). 2. The set of loops of G is written as L(G). 3. The strongly connected component of v in G is defined to be:Definition 2.1 (Loops). Let G = (V, E) be a directed graph (with or without cycles). 1. A set of nodes S ⊆ V is called a loop of G if for every two nodes v 1 , v 2 ∈ S there are two directed walks v 1 · · · v 2 and v 2 · · · v 1 Definition 2.3 (Input/Output Structural Causal Model). An input/output (i/o) structural causal model (ioSCM) by definition consists of: 1. a set of nodes V + = V∪U∪J, where elements of V correspond to output/observed variables, elements of U to probabilistic latent variables and elements of J to input/intervention variables. 2. an observation/latent/action space The stacked edges are meant to be read as an "OR" at each place independently. We also allow for repeated nodes in the walks. Some authors also use the term "path" instead, which other authors use to refer to walks without repeated nodes. In the acyclic case it was shown in[32] that when L is allowed to represent latent variables in a graph G ′ that marginalizes to G then this criterion actually characterizes all adjustment sets for G and P(Y | do(X)). AcknowledgementsThis work was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement 639466).1. General selection-backdoor (see[4]): C = ∅, and2. Selection-backdoor (see[1]3. Extended backdoor 6 (see[26,32][21,22,24]Backdoor (seeF.3 MORE ON ADJUSTMENT CRITERIAThe following generalizes the adjustment criterion of type I in[4].Theorem F.3 (General adjustment without external data). Let the setting be like in 8.1. Assume that data was collected under selection bias, P(V |S = s). 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[ "\nVINCENZO PALLOZZI LAVORANTE\n\n" ]
[ "VINCENZO PALLOZZI LAVORANTE\n" ]
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For an irreducible conic C in a Desarguesian plane of odd square order, estimating the number of points, from a Baer subplane, which are external to C is a natural problem. In this paper, a complete list of possibilities is determined for the case where C shares at least one point with the subplane.
10.1007/s10623-022-01156-7
[ "https://arxiv.org/pdf/2104.12434v2.pdf" ]
233,394,596
2104.12434
c29f83775e7b566162735d6b808ec9093d1e35bb
13 Feb 2022 VINCENZO PALLOZZI LAVORANTE 13 Feb 2022arXiv:2104.12434v2 [math.CO] EXTERNAL POINTS TO A CONIC FROM A BAER SUBPLANE 2000 MSC: Primary: 14G05, 14H50. Secondary: 14H20.Finite fieldsConicsCubic surfacesExternal points For an irreducible conic C in a Desarguesian plane of odd square order, estimating the number of points, from a Baer subplane, which are external to C is a natural problem. In this paper, a complete list of possibilities is determined for the case where C shares at least one point with the subplane. Introduction Combinatorial problems in finite projective planes often ask to count the number of points in the intersection of two, mostly geometrically defined, subsets. Such subsets are the subplanes, arcs, ovals, unitals, blocking-sets, their complements, and, in odd order planes, also the set of external, and that of internal points to an oval. Usually, the size of such an intersection heavily depends on the mutual position of the two subsets. Even the problem of finding non-trivial estimates on it may be challenging and any attempt to solve it would require sophisticated counting arguments. In a desarguesian plane, that is in a projective plane P G(2, q) over a finite field F q , the study of this and similar combinatorial problems can greatly benefit from the powerful theory of finite fields and algebraic geometry in positive characteristic. A typical problem of this kind, posed in [1], is the following. The points of a projective plane PG(2, q) fall into three classes with respect to an (absolutely) irreducible conic, namely the points lying on two tangent lines (external ), on no tangent line (internal ) and the point s on the conic. Let C and D be two distinct irreducible conics. The points of D fall into one of three subsets, namely those points E C (D) of D that are external to C, those points I C (D) that are internal, and C ∩ D. This gives rise to the functions ε C (D) = |E C (D)| and ι C (D) = |I C (D)| defined over the set of all conics D distinct from C. The combinatorial problem is to compute, or estimate the value sets of ε = ε C (D) (or equivalently of ι = ι C (D)). A solution is given in [1]: either ε = 0, q − 1, q, q + 1, or 1 1 2 (q −1) −( √ q −3) ≤ ε ≤ 1 2 (q −1) −( √ q + 3) . Let PG(2, q) be the projective plane defined over a finite field F q of odd order, canonically embedded in the projective plane PG(2, q 2 ) over the quadratic extension F q 2 of F q . Let C be an (absolutely) irreducible conic of PG(2, q 2 ) with homogeneous equation F (X 0 , X 1 , X 2 ) = 0 where F ∈ F q 2 [X 0 , X 1 , X 2 ] is an irreducible quadratic form. Then the number of points of C lying in PG(2, q) is at most 4 unless C is a conic defined over F q , that is, F ∈ F q [X 0 , X 1 , X 2 ], in which case that number equals q + 1 (this because 5 points define a unique conic and the conic would through 5 points in PG(2, q) would be defined over F q ). In this chapter we are interested in the number E q (C) of external points to C which lie in PG(2, q). Since conics defined over F q 2 are not equivalent over F q in general, E q (C) viewed as a function of C is not expected to be constant when C runs over all conics of PG(2, q 2 ). Our goal is to determine the relative value set. The main results are stated in the following theorems. • for q > 5, we have E q (C) = 1 2 (q 2 + (α − 1)q − n 0 ), where n 0 ∈ {0, 1, 2, 3} and α ∈ {1, 2, 3, 4, 5, 7}, and α − n 0 is even. When q > 5 we have 13 possibilities. External points Our notation and terminology are standard; see [5,6,7,8]. In particular, for a point (X 0 : X 1 : X 2 ) of PG(2, q 2 ) we also use the shorter notation X = (X 0 : X 1 : X 2 ). Let F (X 0 , X 1 , X 2 ) = 0≤i,j≤2 a ij X i X j . where a ij ∈ F q 2 , and det(a ij ) = 0. Then C has equation X t AX = 0, for For any two distinct points P and Q in PG(2, q 2 ), the line P Q meets C in PG(2, q 2 ) or in a quadratic extension PG(2, q 4 ) of PG(2, q 2 ), and their common points arise from the roots (ξ, ϑ) of the homogeneous Joachimsthal equation A =   ξ 2 P t AP + 2ξϑP t AQ + ϑ 2 Q t AQ = 0. More precisely, if (ξ 1 , ϑ 1 ) and (ξ 2 , ϑ 2 ) are the (non necessarily distinct) non-F q -proportional solutions of the Joachimsthal equation, then the common points are U i = ξ i P + ϑ i Q for i = 1, 2. Joachimsthal equation is useful to distinguish between external and internal points of C. Therefore, in terms of the equation X t AX = ϑ 2 with ϑ ∈ F q 2 \ {0}, the problem of determining E q (C) asks to find its homogeneous solutions X = (X 0 : X 1 : X 2 ), with X i ∈ F q . The maximal case We start the discussion with a conic C defined over F q . Write the equation of C as C : aX 2 + bXY + cY 2 + dXZ + eY Z + f Z 2 = 0, with a, b, c, d, e, f ∈ F q . Lemma 3.1. Let C be a conic defined over F q with matrix A. For every point P ∈ PG(2, q) we have P t AP ∈ q 2 . Proof. P t AP is an element of F q and so a square of F q 2 . Theorem 3.2. Let C be a conic defined over F q . The number E q (C) of external points to C in PG(2, q 2 ) which lie in PG(2, q) is q 2 . Proof. Any irreducible conic defined over F q has q + 1 points over F q . From Lemma 3.1 the remaining points of PG(2, q) are either all external or all internal to the conic C. Let t P be the F q -rational tangent to C at an F q -rational point P . Now any other point of t P defined over F q is an external point to C (note that this set is non-empty as t P is defined over F q ). In particular, this means that any point of PG(2, q) is either on the conic or is external to the conic. Since |PG(2, q)| = q 2 + q + 1 points, this implies that there are other q 2 external points. 4. Conics with at least one point in PG(2, q). Up to a change of the reference system, we may assume that C contains the point (0 : 1 : 0). Then C has equation Indeed, if −bcd+ad 2 +b 2 e = αγ 2 , with α / ∈ q 2 , we only need to multiply by α the equation of C. Equation X t AX = ϑ 2 with ϑ ∈ F q 2 can be rewritten over F q as F q 2 is a finite extension of F q , that is, the elements of F q 2 are of the form z = z 1 + ǫz 2 with z 1 , z 2 ∈ F q where ǫ ∈ F q 2 is a root of an irreducible polynomial p(X) = X 2 − ω over F q . Since the other root of p(X) is ǫ q , we have ǫ + ǫ q = 0. Thus, X t AX = ϑ 2 reads over F q : (4.2) a 1 X 2 + b 1 XY + c 1 XZ + d 1 Y Z + e 1 Z 2 = t 2 1 + ωt 2 2 a 2 X 2 + b 2 XY + c 2 XZ + d 2 Y Z + e 2 Z 2 = 2t 1 t 2 where a = a 1 + ǫa 2 , b = b 1 + ǫb 2 , c = c 1 + ǫc 2 , d = d 1 + ǫd 2 , e = e 1 + ǫe 2 , ϑ = t 1 + ǫt 2 and ω = ǫ 2 . Since we have b = 0 or d = 0, we can assume d 2 = 0 or b 2 = 0 without loss of generality. From the second equation then (4.3) Y = −e 2 Z 2 + 2t 1 t 2 − c 2 XZ − a 2 X 2 d 2 Z + b 2 X . Note that we lose the point (0 : 1 : 0). Substituting Y by the expression on the right hand side gives Proof. The determinant of the matrix associated to the polynomial (4.4) defining C is (4.4) 2t 1 t 2 (b 1 X + d 1 Z) − (t 2 1 + ωt 2 2 )(b 2 X + d 2 Z) + AX 3 + BX 2 Z + CXZ 2 + DZ 3 = 0, where A = −a 2 b 1 + a 1 b 2 , B = b 2 c 1 − b 1 c 2 − a 2 d 1 + a 1 d 2 , C = −c 2 d 1 + c 1 d 2 + b 2 e 1 − b 1 e 2 , D = d 2 e 1 − d 1 e 2 and ω = ǫ 2 is a non-square of F q . Note that Equation (4.4) is equivalent to: (2t 1 t 2 −(a 2 X 2 +c 2 XZ+e 2 Z 2 ))(b 1 X+d 1 Z)+(a 1 X 2 +c 1 XZ+e 1 Z 2 −t 2 1 −ωt 2 2 )(b 2 X+d 2 Z) = 0.a 1 b 2 = a 2 b 1 , c 1 b 2 = c 2 b 1 , d 1 b 2 = d 2 b 1 , e 1 b 2 = e 2 b 1 .1 4 (−a 1 − a 2 ǫ + (b 1 + b 2 ǫ)(c 1 + c 2 ǫ − (b 1 + b 2 ǫ)(e 1 + e 2 ǫ))), where we write every element z of F q 2 as z = z 1 + ǫz 2 with z 1 , z 2 ∈ F q where ǫ ∈ F q 2 is a root of an irreducible polynomial p(X) = X 2 + β over F q . Since d = 1 and D = 0 we have that e 2 = 0. Furthermore, A = 0, C = 0 and B = 0 imply respectively: (4.5) a 1 = a 2 b 1 b 2 , c 2 = b 2 e 1 and a 2 = b 2 c 1 − b 1 c 2 . Then det(A) = 0. if b 1 d 2 − b 2 d 1 = 0. Proof. Note that we can write the equation of S as H(t 1 , t 2 , X, Z) + G(X, Z) = 0, with F of degree 1 in X and Z. Hence the only possibility for S to be reducible is the following one: (4.6) (k 1 X + k 2 Z)(H 1 (t 1 , t 2 ) + G 1 (X, Z)) = 0, where H 1 (t 1 , t 2 ) + G 1 (X, Z) may be reducible itself. Consider now b 1 d 2 − b 2 d 1 = 0. Then the plane π : b 1 X + d 1 Z = 0 is a component of the cubic surface S. Indeed, using (4.4), we have (b 1 X +d 1 Z)(b 2 t 2 1 −2b 1 t 1 t 2 +b 2 ωt 2 2 +(b 1 a 2 −a 1 b 2 )X 2 +(b 1 c 2 −b 2 c 1 )XZ +(b 1 e 2 −b 2 e 1 )Z 2 ) = 0. Hence S is reducible. On the other hand, if S is reducible, using Equation (4.6) and the identity principle of polynomials, we have    k 1 X + k 2 Z = h(b 1 X + d 1 Z) k 1 X + k 2 Z = j(b 2 X + d 2 Z) which implies h(b 1 X + d 1 Z) = j(b 2 X + d 2 Z), for some h, j ∈ F * q and so b 1 d 2 = b 2 d 1 . Irreducible case For a survey on cubic surfaces see [9]. In this section we suppose S irreducible. In particular we know that b 1 d 2 − b 2 d 1 = 0 or equivalently d b / ∈ F q .|S(F q )| = q 2 + αq + 1, with α ∈ {−2, −1, 0, 1, 2, 3, 4, 5, 7}. The missing case is when S is singular. We start our investigation from the possible singularities of S. Proof. Let S : F (t 1 , t 2 , X, Z) = 0. The condition ∂F ∂t 1 = ∂F ∂t 2 = 0 implies t 2 1 − ǫ 2 t 2 2 = 0 or X = Z = 0. This means that t 1 = t 2 = 0, as ǫ ∈ F q 2 \ F q , or X = Z = 0. When X = Z = 0 together with ∂F ∂X = ∂F ∂Z = 0 imply (5.1) 2t 1 t 2 b 1 − b 2 (t 2 1 + ωt 2 2 ) = 0 2t 1 t 2 = 0 Hence t 1 = t 2 = 0. We need to study (5.2)      ∂F ∂X = 3AX 2 + 2BXZ + CZ 2 = 0 ∂F ∂Z = BX 2 + 2CXZ + 3DZ 2 = 0 If A = 0 then Z = 0 implies X = 0, so we can have only solutions of the form (0 : 0 : β : 1). The system becomes and then ω = 1, which is a contradiction as ω is a non-square of F q . We are going to study the tangent cone at a singular point P to investigate the number of points of S. See [3]. Remember that the tangent cone T P (S) is the set of all tangent lines at a singular point P of S. When S is a cubic surface we have four possibilities for the tangent cone T P (S): • a quadric cone; • a line (the intersection of two planes defined over F q 2 .); • a couple of distinct planes; • a repeated plane. Theorem 5.4. With the notation above, the tangent cone T P (S) at P = (0 : 0 : 1 : 0) or P = (0 : 0 : β : 1) is a quadric cone with the exception of β = −1, 0. In these cases it is a couple of planes either defined over F q 2 \ F q or over F q . In particular there are q + 1, 1, 2q + 1 tangent lines through P , respectively. Proof. The point P = (0 : 0 : 1 : 0) is singular if and only if A, B = 0 (and so C = 0 by hypothesis). In this case the associated matrix of T P (S) is T =       −b 2 b 1 0 0 b 1 −b 2 ω 0 0 0 0 0 0 0 0 0 C       It follows that T P (S) is a quadric cone. When P = (0 : 0 : β : 1) the tangent cone has the following associated matrix: T =       −b 2 β b 1 β + 1 0 0 b 1 β + 1 −b 2 βω 0 0 0 0 B + 3Aβ C + Bβ 0 0 C + Bβ 3D + Cβ       = T 1 0 0 T 2 . Note that β satisfies 3D + Cβ = −(Bβ 2 + Cβ) and C + Bβ = −(3Aβ 2 + Bβ). This implies that |T | = 0. Indeed, In both situations the rank of T equals 2. |T 2 | = −(Bβ + 3Aβ 2 )(Bβ + C) + (Bβ + 3Aβ 2 )(Bβ + C) = 0. We want to study the maximum number of lines through P = (0 : 0 : β : 1) entirely contained in S. We apply to S the invertible projectivity defined by (t 1 : t 2 : X : Z) → (t 1 : t 2 : X − βZ : Z), so that S ′ : Z(2(1+b 1 β)t 1 t 2 −(b 2 β)(t 2 1 +ωt 2 2 )+(3Aβ +B)X 2 )+X(2b1t 1 t 2 −b 2 (t 2 1 +ωt 2 2 )+AX 2 ) = 0 and P ′ = (0 : 0 : 0 : 1). This means that we need to study the system Proof. The homogeneous polynomial φ 2 cannot be a factor of φ 3 . Thus, we have at most 6 points of intersection. According whether φ 2 (t 1 , t 2 , 0) is irreducible or not over F q we lose or have two intersections and for every solution (t 1 : t 2 : 1) we have also the solution (−t 1 : −t 2 : 1). This implies that α = 1. Now note that we have at most two solutions with X = 1. They are given precisely by t 1 and t 2 satisfying (5.4) φ 2 (t 1 , t 2 , X) := 2(1 + b 1 β)t 1 t 2 − (b 2 β)(t 2 1 + ωt 2 2 ) + (3Aβ + B)X 2 = 0 φ 3 (t 1 , t 2 , X) := X(2b 1 t 1 t 2 − b 2 (t 2 1 + ωt 2 2 ) + AX 2 ) =t 1 t 2 = c 1 , where c 1 ∈ F q \ {0} depends on A, B and β. Remark 5.6. Note that when φ 2 is reducible (β = 0 or β = −1) we have α = 0 when we deal with two complex planes. In particular for β = 0 this cannot happen and hence α = 4. We are ready to state the main theorem. S q =              q 2 + αq + 1, if β = 0, −1 q 2 + 3q + 1, if β = 0 q 2 + q + 1, if β 1 / ∈ q , β = −1 q 2 + (α − 1)q + 1, if β 1 ∈ q , β = −1 where α ∈ {0, 2, 4} if β 1 / ∈ q and α ∈ {2, 4} if β 1 ∈ q . Proof. This proof relies on the above results. In particular, since P is a double point, every line passing through P , not in T P (S), meets S in exactly one point (different from P ). Thus, we need to subtract from q 2 + q + 1 the number of lines contained in T P (S) through P and add q whenever one of these lines lie on S. We will use the following notation. S q is the number of points defined over PG(3, q) of S; moreover we set n 0 and n ∞ to be the number of the ones with t 1 = t 2 = 0 and X = Z = 0 respectively. Lemma 5.8. With the above notation, let C be a conic defined by Equation (4.1). Then E q (C) = 1 2 (S q − n 0 − n ∞ ) Proof. The points of C can be obtained putting ϑ = 0 in the system (4.2) whereas E q (C) is obtained counting the points of S(F q ) with ϑ = 0. This means that every point of S(F q ) with t 1 = t 2 = 0 is an F q -rational point of C. Furthermore, we need to subtract the points with X = Z = 0, since they correspond to (0 : 1 : 0) which is on the conic. Note that for fixed X, Y, Z we have either 0 or 2 solution for (t 1 , t 2 ) defined over F q . The discriminant of the quadratic equation (4.4) in t 1 (or t 2 ) is actually different from 0. This because ω is not a square in F q . Thus, for every point (X : Y : Z) of C(F q ), we have two points (t 1 : t 2 : X : Z) of S(F q ) so, after we subtracted the values of n 0 and n ∞ , we need to divide by two. Lemma 5.9. With the above notation, we have n ∞ = q + 1 and n 0 ∈ {0, 1, 2, 3}. Proof. The points of S(F q ) with t 1 = 0 and t 2 = 0 can be obtained as follows: • A = 0. In this case we have at least the point (0 : 0 : 1 : 0) and at most other two points, (0 : 0 : β : 1), with β solution of BX 2 + CX + D = 0. • A = 0. The points are (0 : 0 : λ : 1), with λ that runs over the solution set of AX 3 + BX 2 + CX + D = 0, that are at most three. The computation of n ∞ follows easily. Indeed, the number of points (t 1 : t 2 : 0 : 0), for t 1 , t 2 ∈ F q , is q + 1. Now we are ready to establish the possible values for E q (C). The values of S q come from Theorems 5.2 and 5.7 , for S non-singular and singular respectively. Corollary 5.10. Let S q = |S(F q )|. With the notations above: E q (C) = 1 2 (S q − n 0 − q − 1). Corollary 5.11. With the notation above, the possible values for E q (C), when S is nonsingular are the following • E q (C) = 1 2 (q 2 + (α − 1)q − n 0 ), with n 0 = 0, 2 and α ∈ {−2, 0, 2, 4} • E q (C) = 1 2 (q 2 + (α − 1)q − n 0 ), with n 0 = 1, 3 and α ∈ {−1, 1, 3, 5, 7} Proof. We just need to study the parity of q 2 + (α − 1)q − n 0 to establish the possible values for n 0 and α. Reducible case Throughout this section we will suppose b = 0 and d b ∈ F q or equivalently b 1 d 2 = d 1 b 2 . The case d = 0 is analogous. Thus, from the proof of Lemma 4.7 we know that S splits as S = Π ∪ Q, where Π is the plane defined by b 1 X + d 1 Z = 0 (or b 2 X + d 2 Z = 0) and Q is a is a possibly degenerate quadric surface of PG(3, q) in t 1 , t 2 , X, Z. The equation of S is (b 1 X + d 1 Z)(b 2 t 2 1 − 2b 1 t 1 t 2 + b 2 ωt 2 2 + (b 1 a 2 − a 1 b 2 )X 2 + (b 1 c 2 − b 2 c 1 )XZ + (b 1 e 2 − b 2 e 1 )Z 2 ) We study the two factors separately. Remember that C is defined by equation (4.1). Lemma 6.1. The line of PG(2, q) defined by b 1 X + d 1 Z = 0 is the tangent line to C at the point (0 : 1 : 0). In particular, it contains exactly q external points to C. Proof. By a straightforward computation d b = d 1 b 1 − ωd 2 b 2 b 2 1 − ωb 2 2 = 1 b 2 b 1 b 2 1 d 2 b 1 − ωb 2 2 d 1 b 2 b 2 1 − ωb 2 2 = d 1 b 2 b 1 b 2 = d 1 b 1 . This means that the lines bX + dZ = 0 and b 1 X + d 1 Z = 0 are actually the same. In particular this is the tangent line to C at (0 : 1 : 0). From now on we focus on the quadric surface Q, defined by (6.1) b 2 t 2 1 − 2b 1 t 1 t 2 + b 2 ωt 2 2 + (b 1 a 2 − a 1 b 2 )X 2 + (b 1 c 2 − b 2 c 1 )XZ + (b 1 e 2 − b 2 e 1 )Z 2 = 0 First, note that if b 1 a 2 − a 1 b 2 = 0, b 1 c 2 − b 2 c 1 = 0 and b 1 e 2 − b 2 e 1 = 0 then the conic C is defined over F q , so we can skip it now (see Section 3). One associate matrix of Q is M =       b 2 −b 1 0 0 −b 1 b 2 ω 0 0 0 0 a ′ c ′ 2 0 0 c ′ 2 e ′       , where a ′ = b 1 a 2 − a 1 b 2 , c ′ = b 1 c 2 − b 2 c 1 and e ′ = b 1 e 2 − b 2 e 1 . As mentioned above, we can assume that at least one of a ′ , c ′ and e ′ is non-zero. Lemma 6.3. Let δ ′ := c ′2 − 4a ′ e ′ and δ = b 2 2 ω − b 2 1 . • if δ ′ = 0 then Q is a quadric cone with vertex v =    (0 : 0 : −c ′ : 2a ′ ), if a ′ = 0, (0 : 0 : 1 : 0), if a ′ = 0 . • if δ ′ = 0 then Q is a non-singular elliptic or hyperbolic quadric; Proof. The determinant of M is ∆ = − 1 4 δδ ′ = 1 4 (b 2 2 ω − b 2 1 )(4a ′ e ′ − c ′2 ). Since ωis a nonsquare of F q and b = 0, we have ∆ = 0 only when δ ′ = 0. The proof follows from the classification of quadric surfaces. See for example [7, pg. 14]. Lemma 6.4. We have E q (C) = 1 2 |Q| − |Q 0 | − |Π ∩ Q| + |Q 0 ∩ Π| + q, where Q 0 = {(t 1 , t 2 , X, Z) ∈ Q|t 1 = t 2 = 0}. Proof. We have already seen that Π gives its contribution of q points to E q (C). The remaining points that contribute to E q (C) are those on Q not in Π nor Q 0 (as the points of Q 0 correspond to points of C), so the equation follows easily. From the previous lemma, we need to understand better the mutual position between Π, Q, and Q 0 to achieve our goal. Proof. The plane Π meets Q 0 only at one point, that is (0 : 0 : −d 1 : b 1 ). Note that δ ′ = 0 and a ′ = 0 imply c ′ = 0. Thus, κ needs to be different from zero otherwise we have e ′ = 0 too. The claim follows from standard computation. Corollary 6.6. We have |Q 0 ∩ Π| = 0, if κ = 0 1, if κ = 0 Lemma 6.7. |Q 0 | =        0, if δ ′ / ∈ q 1, if δ ′ = 0 2, if δ ′ ∈ q Proof. This result follows from standard theory. See [7, Table 15.5] for more details. We are ready to describe the situation for every type of Q. Theorem 6.8. Let S = Π ∪ Q, with Q quadric cone (δ ′ = 0). Then E q (C) =              1 2 (q 2 − q) + q, if κ = 0, δ ∈ q 1 2 (q 2 + q) + q, if κ = 0, δ / ∈ q 1 2 (q 2 − 1) + q, if κ = 0 , δ = b 2 2 ω − b 2 1 . Proof. If δ ′ = 0, • κ = 0. This means that Π is a plane either meeting Q only at the vertex v or through two generators of Q. More precisely, it depends on whether δ = b 2 2 ω − b 2 1 is in q or not (note that δ cannot be equal to zero). Thus, we have |Π ∩ Q| = 2q + 1, if δ ∈ q 1, if δ / ∈ q • κ = 0. This implies that Π intersects Q in a non-singular conic with q + 1 points, not containing v and Q 0 = {v}, where v is the vertex of Q. Finally, the contribution of Π to E q (C) is q, as we have already seen. Lemma 6.9. Let S = Π ∪ Q. If δ ′ = 0, |Π ∩ Q| =        1, κ = 0, δ / ∈ q 2q + 1, κ = 0, δ ∈ q q + 1, κ = 0 Proof. Note that a point P = (t 1 : t 2 : X, − b 1 d 1 X) ∈ Π ∩ Q if and only if F 1 (t 1 , t 2 ) + X 2 κ = 0, where F 1 (t 1 , t 2 ) = b 2 t 2 1 + b 2 ωt 2 2 − 2t 1 t 2 b 1 . Thus, if κ =• if κ = 0 and δ ′ ∈ q then E q (C) =      1 2 (q 2 − q − 2) + q, δ / ∈ q , 1 2 (q 2 + q − 2) + q, δ ∈ q ; • if κ = 0 and δ ′ / ∈ q then E q (C) =      1 2 (q 2 − q) + q, δ ∈ q , 1 2 (q 2 + q) + q, δ / ∈ q ; • if κ = 0 (and δ ′ ∈ q ), then E q (C) = 1 2 (q 2 − 1) + q. Proof. The claims follows from Lemmas 6.4 and 6.9 and from [7, Tables 15.6 and 15.7] for the number of points of quadrics over F q . 7. Proof of Theorems 1.1 and 1.2 We are now able to prove our main theorems. More precisely Theorem 1.1 follows from Theorems 5.2, 6.8, 6.10 and the next result. Then |E ∩ B| ≥ q 2 −3 2 . Corollary 7.3 shows that when q > 3 then in Theorem 5.11 we can exclude the cases α = −2, −1, 0. Taking this into account, the proof of Theorem 1.2 is a direct consequence of Theorem 5.11, 5.7, 6.8 and 6.10. Example Let C 1 be a conic of equation ax 2 + bxy + dyz = 0, with b d / ∈ F q and a d ∈ F q . This means that we can rewrite the equation as a ′ x 2 + b ′ xy + yz = 0, where a ′ ∈ F q and b ′ ∈ F q 2 \ {F q }. Thus, we have 2(−bcd + ad 2 + b 2 e) = a ′ which is always a square in F q 2 . We conclude that the System (4.2) counts the number of external points to C 1 . Lemma 5.9 can be refined. Indeed, we have a ′ 1 b ′ 2 X 3 = 0 admits only the root (0 : 0 : 0 : 1). This implies that n 0 = 1 and C q = 2. From Theorem 5.10, since S is singular with δ ′ = 0, we have just one possibility for σ q : σ q = 1 2 (q 2 + 2q − 1). Using the Computational Algebra System Magma [4], we found the following values for (q, E q ): (3, 7), (5, 17), (7, 31), (9,49). Remark 1. 3 . 3When q = 3 all the values occur, that is 7 possibilities. When q = 5 the only values missing are {16, 22}, that is 2 out of 11. Lemma 2.1 ([5, Theorem 7.51]). If P runs over the set of all external points to C then the values P t AP are all squares or all non-squares. For an external point P , if P t AP is a square then Q t AQ is a non-square for every internal point Q to C. C : aX 2 + bXY + cXZ + dY Z + eZ 2 = 0 with a, b, c, d, e ∈ F q 2 where either b = 0 or d = 0. From now on we may assume b = 0. In case b = 0 we can apply the collineation (X : Y : Z) → (Z : Y : X) which swaps b and d. Lemma 4. 1 . 1If P runs over the set of all external points to C then the values P t AP are all squares or all non-squares according as −bcd + ad 2 + b 2 e is a square or a non-squarein F q 2 .Proof. Since the tangent line to C at Q = (0 : 1 : 0) has equation bX + dZ = 0, the point P = (−d/b : 0 : 1) of t Q is external to C. We have P T AP = (−bcd + ad 2 + b 2 e). Now, Lemma 4.1 follows from Lemma 2.1 . Remark 4. 2 . 2Without loss of generality we can always suppose −bcd + ad 2 + b 2 e ∈ q 2 . Remark 4. 3 . 3The number of solutions (X : Y : Z) of System (4.2) can be obtained (but it is not necessarily equal) by counting the points over F q lying on the cubic surface S : F (t 1 , t 2 , X, Z) = 0 of PG(3, q) with homogeneous equation(4.4). Here PG(3, q) stands for the projective space over F q with homogeneous coordinates (t 1 , t 2 , X, Z). Remark 4. 4 . 4Note that the conic C of equation (4.1) is defined over F q if and only if the following hold: Lemma 4. 5 . 5With the notation above, if (A, B, C, D) = (0, 0, 0, 0) then C is a singular conic. Remark 4. 6 . 6Since by hypothesis the conic C is non-singular, we cannot have (A, B, C, D) = (0, 0, 0, 0). Lemma 4. 7 . 7The cubic surface S, defined by the equation (4.4), is irreducible if and only Theorem 5. 3 . 3Let S be the cubic surface defined by equation(4.4). Then S has at most one singular point P . In this case P is a double point and is defined over F q . (P ) = 0 implies −2(b 2 2= 3AX 2 + 2BX + C = 0 ∂F ∂Z = BX 2 + 2CX + 3D = 0 Note that System (5.3) has either one or two solutions (counted with multiplicity). The second case is only possible if either the two equations are proportional, namely 3A = kB, B = kC and C = 3kD, or B = C = D = 0. In any cases we have a double root ( −1 k and 0). Hence we can just have one singular point, say P = (0 : 0 : β : 1). Note that this still remains true when the characteristic of the field is 3 and B = C = 0. Furthermore, β needs to be an element of F q , otherwise P ′ = (0 : 0 : β q : 1) would be another singular point different from P . Suppose now A = 0. System (5.= BX 2 + 2CXZ + 3DZ 2 = 0 If Z = 0 and B = 0 then we have no solutions. If Z = 0 and B = 0 we have only the solution (0 : 0 : 1 : 0). If Z = 0 we can suppose Z B needs to be different from 0. Indeed, if B = 0 then we have C = 0 and D = 0. Hence we can have at most one solution which is defined over F q .Finally, observe that P = (0 : 0 : X : Z) cannot be a triple point for S. Since both d 2 and b 2 cannot be zero, the condition ∂ X) = −2ω(b 2 X) Furthermore the rank of T is 3 except when 1. B = C = D = 0. In this case we have β = 0. 2. B = 0 and β = −B 3A . This case occurs when System (5.3) is reduced to the single equation X 2 + 2X + 1 = 0. Thus, β = −1. 0 0In fact, each point satisfying System (5.4) corresponds to a line through P ′ contained in S ′ . Theorem 5.5. With the notation above, if α is the number of lines through the singular point P ′ , then α ∈ {0, 2, 4}. Theorem 5 . 7 . 57Let S be the irreducible cubic surface defined by Equation (4.4) with a singular point, say P = (0 : 0 : β : 1), and let β 1 = (1 + b 1 β) 2 − b 2 2 β 2 ω. The following are the only possibilities for S q = |S(F q )|. Lemma 6. 5 . 5Π meets Q 0 if and only if κ = 0, whereκ := a ′ d 2 1 − c ′ d 1 b 1 + e ′ b 2 1 ,in which case |Π ∩ Q 0 | = 1. Furthermore, when δ ′ = 0, Π ∩ Q 0 is the vertex of the quadric cone Q, if a ′ = 0, and the empty set, if a ′ = 0. 0 there are the q + 1 points of a non-singular conic. On the other hand, if κ = 0 the number of solutions depend on whether δ ∈ q or not. The claim follows by [7, Par. 15.3]. Theorem 6.10. Let S = Π ∪ Q, with Q a non-singular quadric surface. Then 2 . 2Theorem 7.1([11]). In Theorem 5.2 the bounds are best possible, except that when q = 2, 3 or 5 the upper bound can be improved to α ≤ 5.The proof of Theorem 1.2 requires one last step. For further details about the next Theorem, see[2].Theorem 7.2. Let C be an oval of a projective plane of order q 2 , say Π q 2 , with q being odd. Let B denote a blocking set of Π q 2 and E denote the set of points lying on a tangent to C.If C ∩ B = k, then |E ∩ B| ≥ q 2 +1−k 2 .Proof. Each line of the plane, hence also the tangents to C, meets B. If t is a tangent to C at P ∈ C, then t \ {P } ⊆ E and hence t ∩ B ⊆ (E ∩ B) ∪ {P }. Thus E ∩ B has a point of each of the tangents to the points of C \ B. Since each point not in C is incident with either 0 or 2 tangents to C, then |E ∩ B| ≥ q 2 +1−k Corollary 7.3. Let C be an irreducible conic of PG(2, q 2 ), q odd, and let B denote a Baer subplane. Also, denote by E the set of external points to C. Theorem 1.1. Let C be a conic in the desarguesian plane PG(2, q 2 ) with at least one rational point in PG(2, q) and q ≥ 5. Then E q (C) = q 2 if and only if C is defined over F q . Theorem 1.2. In the desarguesian plane PG(2, q 2 ) let C be a conic not defined over F q with at least one F q -rational point. Then:• For q = 3, E q (C) ∈ {3, 4, 5, 6, 7, 8, 9}; • for q = 5, E q (C) ∈ {11, 12, 14, 15, 16, 17, 19, 20, 21, 22, 25}; Remark 5.1. In this case we can set d = 1. Indeed, we can divide the equation (4.1) by d, since we have d = 0 and b = 0. This also implies b 2 = 0.We want to find a bound for the number of rational points of S. If S is a smooth surface we have the following theorem, see [9, Theorem 27.1 andTable 1 §31].Theorem 5.2 (Weil). Let S be a smooth cubic surface defined over a finite field F q . Then AcknowledgementsThe research of V. Pallozzi L. was partially supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA -INdAM). On the mutual position of two irreducible conics in PG(2, q), q odd. V Abatangelo, J C Fisher, G Korchmáros, B Larato, Advances in Geometry. 114V. Abatangelo, J. C. Fisher, G. Korchmáros, and B. Larato, "On the mutual position of two irre- ducible conics in PG(2, q), q odd," Advances in Geometry, vol. 11, no. 4, pp. 603-614, 2011. An introduction to finite geometry. S Ball, Z Weiner, preprintS. Ball and Z. 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